De la superf?cie d'energia potencial a la reactivitat qu?mica Marc Caballero Puig Aquesta tesi doctoral est? subjecta a la llic?ncia Reconeixement- NoComercial 3.0. Espanya de Creative Commons. Esta tesis doctoral est? sujeta a la licencia Reconocimiento - NoComercial 3.0. Espa?a de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution-NonCommercial 3.0. Spain License. Programa de qu?mica te?rica i computacional De la superf?cie d'energia potencial a la reactivitat qu?mica Marc Caballero Puig1,2 Director: Josep Maria Bofill Vill?1,3 Codirector: Xavier Gim?nez Font1,2 1. Institut de qu?mica te?rica i computacional (IQTC-UB) 2. Departament de qu?mica f?sica de la Universitat de Barcelona 3. Departament de qu?mica org?nica de la Universitat de Barcelona Ars longa, vita brevis Hip?crates ? segles V - IV a.C. ?ndex 1. Introducci? 1 I. Motivaci? de la tesi doctoral 2 II. Alguns apunts sobre la hist?ria de la qu?mica te?rica 2 III. Aportacions de l'estructura electr?nica 13 IV. Aportacions de la din?mica 14 V. Preguntes i respostes: societat i m?n acad?mic 14 VI. Conclusions 21 2. Estructura electr?nica 24 I. L'equaci? de Schr?dinger 25 II. Solucions a l'equaci? de Schr?dinger basades en la funci? d'ona 29 III. Solucions a l'equaci? de Schr?dinger basades en la teoria del funcional de la densitat 41 IV. Exemple d'aplicaci?: la reacci? entre cl?sters d'aigua i el radical hidroxil 64 V. Conclusions 70 3. Teoria del cam? de reacci? 74 I. Cam? de reacci? 75 II. Coordenada de reacci? intr?nseca (IRC) 78 III. Gradients extremals 88 IV. Traject?ries de l'ascens gradual 100 V. Traject?ries newtonianes 114 VI. Conclusions 123 4. Din?mica 126 I. Din?mica qu?ntica 127 II. Hamiltoni? del cam? de reacci? 140 III. Assignaci? d'estats qu?ntics mitjan?ant traject?ries cl?ssiques 147 IV. Conclusions 159 5. Conclusions generals 163 Agra?ments 166 Ap?ndix i articles 168 Progam de qua?mictt?p Progam dendqua?mictt?pdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddl DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD qndsma?frt?pdi'd rda'v?dimtam?r M?rudor?adi'd rdCma?frt?pdi'd b'vt??oac?rdibr1c'varda'v?drC,d' do rua'2rC'uad1c'd a:df'dimurirdo'?d rdo'?o 'J?arad1c'do?mfmtrdrdcudumcf?uBcadr dCpudi'd rd?'t'?trd rd B?rudi?f?v?pdi'd rd1cgC?trda'V??trd?dtmCocart?mur d'udi?vt?o ?u'vdmdvc,i?vt?o ?u'vnd ?'vdo??ut?or vdi?vt?o ?u'vdi?uvd rd1cgC?trda'V??trd?dtmCocart?mur d1c'dvb3ruda?rtarad r d r?Bd ibr1c'varda'v?d3rud 'varad b'va?ctac?rd ' 'ta?Vu?trXd rd i?uGC?trd?t Gvv?trd?d 1cGua?trzd?d rda'm??rdi'dtrC?uvdi'd?'rtt?pnd F dCr?B'di' d.'adi'd1c'drdu?f' do'?vmur d:vdCm advra?v.rtam??domi'?dtmuI?J'?d i?f'?v'vdG?''vdi'dtmu'?J'C'uad2rd1c'do?mom?t?murdcurdB?rudmom?acu?aradib'Jorui??d ' vdtmu'?J'C'uavd?dtmCo'aIut?'vXd?'vc ard1c' tmCdvm?o?'u'uad' di'i?tr?dC'vmvdrd a?rtar?dcudo?m, 'Crdi'a'?C?uradC?a2ru(ruadcurdi?vt?o ?urd?di'vo?:vdi'vtm,???d1c'd i'vdibcudr a?'dGC,?adrtriIC?td' dCra'?Jdo?m, 'Crd'vda?rtardrC,dcurdi?vt?o ?urd i?.'?'ua dm,a'u?ua d ?'vc arav d o?mo?v d ibr1c' r d i?vt?o ?urn dQutr?r d C:v d vm?o?'u'uad ?'vc ardamado 'Brad1crudr?JVd'vdf?cdom?a'vd'ui?uvTdvmf?uad3rdvm?B?ad rdo?'Bcuardi'd 1c?urdi'd 'vdi?vt?o ?u'vd1c'dvb3ruda?rtarad:vd rdC:vd?ui?trirdo'?da?rtar?dcudo?m, 'Crd imuran -'?daruaXd b'u.mtrC'uadi'd b'vt??oac?rdibr1c'varda'v?dimtam?r da?ui?Gdcudtr?Gta'?d ?Co?'Burado'?dr1c'vad?ua'??mBruaTd1c?urdi?vt?o ?urd'Co?'Cd1crud'Co?'u'Cd b'vaci?d a'V??tdibcudv?va'Crd1cgC?td?d.?uvdrd1c?udocuadtmuu'ta'Cd' vdumva?'vd?'vc aravdrC,d rd ?'rta?f?aradi' dv?va'Crn qqndF Bcuvdrocuavdvm,?'d rd3?vaV??rdi'd rd1cgC?trda'V??tr UmCdor? rdi'd rd1cgC?trda'V??trdrC,durac?r ?aradtmCdrd,?rutrdi?vt?o ?ur?di?uvd rd 1cgC?trnd-'?daruaXdr,ruvdi'domi'?d'u?rmur?dvm,?'d1cId'ua'u'Cdo'?d1cgC?trda'V??trd tr d'var, ??d1cId'ua'u'Cdo'?d1cgC?trnd ?rd1cgC?trdumdvm?B'?Jdi' dumd?'vnd)vd b'fm ct?pdibcurdi?vt?o ?urdC:vdrua?Brdi'd rdCGdi'd brf'u(dtmu2cuad1c'df?c'udcurdvI??'di'di?vt?o ?u'vdHdtmCd rd.gv?trXd 'vd ?'d rdvco'?.gt?'dib'u'?B?rdoma'ut?r drd rd?'rta?f?arad1cgC?trddddddddddddddddddddddddddddddddddd? 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'Cd i' d i?uGC?tr d Ars lonndF1c'var d i?uGC?trXd r??' rir d .m?arC'uad r d r d .gv?trd 'varigva?tr d i' d srJ?' d ?d ? a?Cruud?d1c' d r??'utr d rC,d b'1crt?pd ibF??3'u?cvX d tmuu'tard' dtmu'?J'C'uada'V??tdibcudv?va'Crd1cgC?tdrC,d rdf'vvruadC:vdt?uIa?trdi' b'Jo'??C'uand?rd?u.m?Crt?pd1c'dub'Ja?rc?'Cdv'?rudtmuvaruavdi'df' mt?ara?do'?Vd arC,:da'Covdi'df?irdi'dCm Itc 'vXdi?va??,ct?pdib'u'?B?rdi?uvdcurdCm Itc rdmd rd ?'rta?f?ara d i' d trur vd ?'rta?cvndF1c'var d f'vvruad i' d r d i?uGC?tr dvb3rd .'a dCm a'vd f'Bri'vd?Co?'vt?ui?, 'do'?dar di'domi'?d?ua'?o?'ar?d'Jo'??C'uavn ?e?? d 9crud?ua'B?'Cd'1crt?muvdi'dCmf?C'uad?t Gvv?1c'vdmd1cGua?1c'vzd?d'udv'Bc?Cd b'fm ct?pdvm,?'dcurdvco'?.gt?' d ib'u'?B?rdoma'ut?r dv'Bc?Cd i?uvd ' d trCod i' d rd i?uGC?trdo'?Vd3mCd'vd?'.'?'?Jdrdr1c'vardf'vvruadtmCddrdoga,vAnolgint ndR'uadr?JVXd m,a'u?Cd rdf?v?pdC:vdo?m.cuirdvm,?'d rd?'rta?f?arad1c'd rd1cgC?trda'V??trdomadm.'??? imurad1c'dv'Bc?Cd 'vdCm Itc 'vdcurd rdcurd 'ud ' dv'cda?Guv?ado'?d rdvco'?.gt?'d ib'u'?B?r d oma'ut?r n d -mi'C d rtcCc r? d a?r2'taV??'v d t Gvv?1c'v d o'? d ar d i' d .'?d 'varigva?1c'vd1c'd'uvdro?mo?udrd?'vc aravd'Jo'??C'uar vdo' d1c'd.rdrdtmuvaruavdi'd f' mt?araX d a?ruv.'?Iut?r d ?ua?rCm 'tc r? d m d 'Ja?rCm 'tc r? d ib'u'?B?r d m d v'tt?muvd '.?trt'vndQ d ,r?Jd tmvadtmCocart?mur d ibr1c'va'vda?r2'taV??'vd 'uvdo'?C'adr,rvar?d arC,:d' dtrCodi'd rdi?uGC?trdi'dCrt?mCm Itc 'vXd?utm?om?ruad.?uvd?damadi?vvm f'uand ?a? ?a?rua d 'v d ro?mJ?Crt?muv d o'?a?u'uav d arC,: d omi'C d ca? ?a?r? d a?r2'taV??'vd t Gvv?1c'vdo'?dar dib'vaci?r?di?uGC?1c'vdumdri?r,Ga?1c'vndS?d 'vda?r2'taV??'vdvpud 1cGua?1c'vXdm,a'u?Cdi??'tarC'uad rdCrBu?acidib'.'ta'vdo?mo?vdibr1c'vadCmi' dtmCd omi'udv'?d b'.'ta'dawu' dmd?'vvmuGut?'vndQua?ruad'ud' dor?ri?BCrd1cGua?tXdarC,: a'u?Cd i?uGC?1c'vd ' 'ta?Vu?1c'vdo'? dar d i' d tmCo?'ui?' d ' vd ,'vtruf?vd ib'u'?B?r d 'ua?'di?.'?'uavd'varavdibcurdCm Itc rd?ufm ct?rird'udo?mt'vvmvdib'C?vv?pdmdr,vm?t?pd ib'u'?B?r?dr?JgdCra'?JXd:vdf?ar do'?dar dib'ua'ui?'d rd.'Cam1cgC?trn?ej? ?nd-?'Bcua'vd?d?'vomva'vTdvmt?'arad?dCpudrtriIC?t Quvda?m,'Cdirfruadi'dic'vdCru'?'vdib'u.mtr?d rd1cgC?trda'V??trnd8'vomu'uaddrd ?'d rdvco'?.gt?'dib'u'?B?rdoma'ut?r drd rd?'rta?f?arad1cgC?trddddddddddddddddddddddddddddddddddde? DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD r d o?'Bcuar d o rua'2rir d rC, d rua'??m??araX d r d ?u.m?Crt?p d 1c' d m.'?'?J d trird ro?mJ?Crt?pd:vdC:vdtmCo 'C'uaG??rd1c'dumdorvd'Jt m'uandQudr BcuvdtrvmvXdcurd ?u.m?Crt?pdi?uGC?trdomad?'vc ar?d?'icuiruad?'vo'ta'dr d1c'drom?ard2rd b'va?ctac?rd ' 'ta?Vu?trnd?br a?'vXdomadrom?ar?d?u.m?Crt?pdrii?t?mur ndF1c'vard?u.m?Crt?pdi'dC:vd omada'u??d?' 'fGut?rdmdumnd U'C d 'vara d or? rua d ic?rua d 'v d v'tt?muv d 1c' d o?'t'i'?J'u d r1c'var d vm,?' d ??'vomva'v?ndUmCd'ua:ud1c'd ' doro'?dibcurdi?vt?o ?urdtmCd rd1cgC?trda'V??trd?d 1cr v'fm d i' d 'v d i?vt?o ?u'v d 1c' d r1c'var d 'uB m,r d :v dm.'??? d o?'t?vrC'ua d r?JVXd ?'vomva'vnd?'vo?:vdi'd?'. 'J?mur?dcud?uvaruadvm,?'d rdC'urdi'd?'vomvard1c'domi'Cd m,a'u??d'vd.rdtrirdtmodC:vd?CC?u'uadtmC'u(r?Humvdrdo?'Bcuar?do' d1c'd?'r C'uad:vd ' d1c?idi'd rd1?'va?pTdirfruadi'daruard?'vomvarXd1c?urd'?rd rdo?'Bcuar7 QJo?'vvrC'uaXdic?ruad rdi?vtcvv?pdrua'??m?dvb3rdo rua'2rad rd1?'va?pd,gbrae,r e onbe?reb?rbAv?nevine rnool?e ??alon di'dCru'?rd1c'd'v1c?f:vv?Cdr1c'vadi? 'Cr d .murC'uar d .?uv d r???,r? d r d r1c'va d ocuan d - rua'2r? d cur d i?vtcvv?p d vm,?' d 1c?urd i?vt?o ?urd3rdibr.?muar?d b'vaci?dibcurd?'rtt?pd1cgC?trdimurirdrur ?a?ruad rdC'urdi'd ?'vomva'vd1c'domi'udimur?d 'vdi?f'?v'vdro?mJ?Crt?muvd1c'domi'Cd'Co?r?do'?drd.'?H 3md:vd?Comvv?, 'Td2rd3'Cdi?ad1c'di'vdi' docuadi'df?vard.murC'uar damavdvpudfG ?ivd? umd'vdomi'udtmua?ri???d?darC,:d3'Cdtmuvararad1c'di'vdi' docuadi'df?vardi'd 'vd ro ?trt?muvdi'dtrirvtcud?d.'uadcud,?'fgvv?Cdtmodibc drd rd,?, ?mB?r.?rd rd?u.m?Crt?pd 1c'drom?ardtrirvtcudumd:vd'Jt m'uad'uf'?vd rd?u.m?Crt?pd1c'drom?a'ud' vdr a?'vnd F?Jgdimutvd rd?'. 'J?pdvm,?'d1c?urd:vd rdC? m?di?vt?o ?urdo'?dar dib'vaci?r?dcurd ?'rtt?pd1cgC?trdtmut?'ard3rdi'd?rc?'d 'ud rd?'rtt?pd1cgC?trd'udvgndQvdtmu'?J'ud Cm a'vd?'rtt?muvd1cgC?1c'vTdr Bcu'vd3rud'varad'vaci?ri'vd'J3rcva?frC'ua?dibr a?'vd umC:vdor?t?r C'ua?dr Bcu'vdumd3rud'varad'vaci?ri'vXdomav'?do'?1cId'utr?rdumd'vd tmu'?J'und ???r?dcurd ?'rtt?pd1cgC?trd rd bra?r?d ib'ua?'d r1c'va'vda?'vd tra'Bm??'vd? ?'r ?a?r?dcud 'vaci?d imurad i?uvd ibr Bcurd i'd 'vd i?vt?o ?u'vd1c'dvb3rud 'vC'uarad?d i'vt??adrua'??m?C'uad:vd' d1c'd?Co ?trd b'ucut?rad,gbrae,r eonbe?reb?rbAv?nevin e rnool?e??alon nd?'vdi' docuadi'df?vardrtriIC?tdr?JVdumd:vd'Ja?rfrBruand9crud3mCd oc, ?trdcudr?a?t 'Xdvb?ut mcdcudror?aradib?ua?mictt?pd1c'd?ut mcdtmua'Jacr ?a?rt?pXd o?'t'i'uavd?dm,2'ta?cv?do'?Vdvmf?uadr1c'vardumd:vd rdor?ad1c'dC:vdra?rcd bra'ut?pdi' d 'tam?d1c'dtmCdrd'vo't?r ?vard,cvtrdi??'tarC'uad 'vdtmut cv?muvdmd rd.m?Cc rt?pdmd Progam dendqua?mictt?pddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddde? 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(T, +Hu)?, a(R)?a(R,r )= ? nvn? a=- ? ?, a(R) ?a(R ,r) ddd?eCEA 2udcr?rdr dDufxu,ad?uDu,:m borBu,a.dUDdcm,:u,tu,adcm,Dt?u?r?dJbudT , UDdb,d 2ud rdDbou?lgctud?pu,u?ftrdomau,ctr drd rd?urcat:taradJbgBtcrdddddddddddddddddddddddddddddddddde0 ?????????????????????????????????????????????????????????????????? mou?r?m?d?tlu?u,ctr drBId rdDufxu,adlm?Br) T ,=? o ? -e ? o ? o e=? ,e ddd?eC0A FJbgd?uBdt,c XDdrd? ,e d rd?uou,?T,ctrdrBId rdBrDDr.du dDtf,udtdu dDbBram?tC Hr?at,ad?ud pu(o?uDDtXdeCEdtdo?uDct,?t,ad?ud rd,marctXdDmI?ud rd?uou,?T,ctrdu,dR td r tdau,t,adu,dcmBoaudJbudHu d,mdrcabrdDmI?ud?, a ) ? a=- ? (? ,e+Hu)?, a?a= ? nvn? a=- ? ?, a?a ? a=- ? {? ,e(?, a?a)+Hu?, a?a}= ? nvn? a=- ? ?, a?a ? a=- ? {? , [(?, a? , ?a)+(?a? , ?, a) ]+?, aHu?a}= ? nvn? a=- ? ?, a?a ? a=- ? {?a(? ,e?, a)+e (? , ?a)(? , ?, a)+?, a(? ,e ?a)+?, a ? a?a}= ? nvn? a=- ? ?, a?a ?eCRA Fdor?at?d?pr?r.dtdou?dar d?udDtBo tltcr?d rd,marctX.dDpuBo?r?id rd,marctXd?ud2t?rcd rBId ?o?gn .dDufm,Dd rdJbr d rdt,auf?r d? ?? ? ?i? 8? ? j?r uDdom??trduDc?tb?u??i?8? ??j ?C FBI d rJbuDa d cr,:t d ?u d ,marctX dab A lgbngd Bb ato tcr?uB d pu(o?uDDtXd eCRd ou?d puDJbu??rdou?db,rdlb,ctX ? ?? Cdn d?uDb aradUD) ? ,e ?, ? + ? ? ?, ? +? a=- ? {e ?? j?? , ??i ? (? , ?, a)+?? j?? ,e??i ?? , a}= ? nvn?, ? ddd?eCjA 3rdlb,ctXd?pm,rdu uca?1,tcrd4rd,mdrlucardu Dd?mDdo?tBu?Ddau?BuDCd3rd?uDard?ud au?BuD.dJbudu,cr?rdltfb?u,d?t,Ddu dDbBram?tdou?drdamaDdu DduDaraDdu uca?1,tcD.dDX,du Dd r,mBu,raDdrcmI rBu,aDd,mdr?trIiatcDd?udo?tBu?dtdDufm,dm???u.d ?uDoucat:rBu,aCd H?u,u,d?u u:i,ctrdJbr,duDdcm,Dt?u?rd rdt,l bT,ctrd?ud rd?uDard?puDaraDdu uca?1,tcDd DmI?ud puDaradu uca?1,tcd??d Progam deCdnDa?bcab?rdu uca?1,tcrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddeRd ?????????????????????????????????????????????????????????????????? n,d pr,mBu,r?rd o?iv,alo?a?Ao?o?na?o duDd,uf tfut(u,damauDd uDdt,auf?r DdJbud t,:m bc?u,d?tlu?u,aDduDaraDCdHu?dr a?rdIr,?r.dUDdcm,ufbadJbud prcmI rBu,adr?trIiatcd ?udo?tBu?dm???ud{?a?? , ??a} dUDdDuBo?ud,b dDtd rdlb,ctXd?pm,rd,mdUDd?ufu,u?r?rdu,d rd Du:r d cmBom,u,a d uDorctr C dFBI d rJbuDauD d cm,Dt?u?rctm,D. d puJbrctX d eCj d Jbu?rd ?u?bw?rdr)dd (? ,e+ ? ? + ?? ??? ,e?? ? ? )?, ?= ? nvn?, ? dd?eC-6A Jbudom?uBd?uuDc?tb?udou?dar d?ud:tDbr ta?r?drBIdBUDdc r?u?radu dJbudtBo tcrd?ud rdDufxu,adBr,u?r) (T , + ? ? (R)+ ? (R))?, ? (R )= ? nvn?, ? (R) ddd?eC--A FJbgdror?ut(db,d,mbdau?Bu.d? ?RA.dJbudu,f mIrd prcmI rBu,adr?trIiatcd?udDufm,d m???udtdUDdcm,ufbadcmBdrd?viig??a?A?o?bosCdQDbr Bu,ada?mIuBdJbudrJbuDadau?Bu d :r?trdBm ad u,arBu,adu,dlb,ctXd?udR tdUD.du,d?u rctXdr d? ? .d?udIrt(rdBrf,tab?) d ro?m(tBr?rBu,ad rdo?mom?ctXd:ud?m,r?rdou?d rdDbBrd?ud uDdBrDDuDd?u Dd,bc tDd ?uJbrctXdeC0A?du,dcm,DuJxT,ctrduDdDm d,uf tft?dDmardu dJbudDpr,mBu,rdro?m(tBrctXd hm?,v8oou,?utBu?CdFt(1dou?Buado?u,??ud pu,u?ftrdu uca?1,tcrd? ? dcmBdDtdlmDdb,r u,u?ftrdomau,ctr ) (T , + ? ? (R ))? , j (R)=(T, + ' 4(R))?, j(R)= ? nvn?, j(R) d?eC-eA FBId prccuoarctXd?prJbuDauDdro?m(tBrctm,Dd,ut(d rdtBrafud?udJbudu Dd,bc tDduDd Bmbu,du,db,rd??i t?i???agA?gbgi?oA?ngb?aosA JbudUDdb,rdDm bctXd?ud puJbrctXd u uca?1,tcrd?udGc??z?t,fu?Cd3rdDbou?lgctud,mdUDdDu,DtI udrdb,dcr,:tdu,d uDdBrDDuDdtd ou?dar,ad ,md?td?rdulucauDdtDma1otcDCdFJbuDardDbou?lgctud?pu,u?ftrdomau,ctr dJbud DratDli d r d r d :ufr?r d pro?m(tBrctXd r?trIiatcr d t d r d ?ud hm?,v8oou,?utBu? dUDdu d cm,cuoaudlm,rBu,ar du,du dDtd?u dJbr duDd?uDu,:m bordamard rdaum?trd?udcrBt,Dd?ud ?urcctXdJbudu(rBt,r?uBdr dcrogam dSC Fd uDdDucctm,DdDufxu,aDduDd?b?idrdau?Budb,d?ucb duDJbuBiatcd?puDa?raTftuDdou?drd rd?uDm bctXd?ud puJbrctXd?udGc??z?t,fu?C 2ud rdDbou?lgctud?pu,u?ftrdomau,ctr drd rd?urcat:taradJbgBtcrddddddddddddddddddddddddddddddddddejd ?????????????????????????????????????????????????????????????????? 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Cd ?t,r Bu,a.d?uB d o r,au4rad rdDm bctXd?ud puJbrctXd?udGc??z?t,fu?du,dau?BuDd?prJbuDadDufm,dcm,4b,ad ?udlb,ctm,D d? ?t ? ?dcm,c?uarBu,adb,uDdJbud?uou,u,d,mBUDd?ud uDdcmm??u,r?uDd ,bc ur?DCd?mdmIDar,a.d,mdDp?rd?tad?uDdDmI?ud puDa?bcab?rd?prJbuDardBu,rd?udlb,ctm,DCd FJbuDauDdlb,ctm,Dd?r,d?udau,t?db,rdor?aduDorctr dJbud?uou,fbtdcu?arBu,ad?ud uDd cmm??u,r?uDd,bc ur?Ddou?1darBIUd?r,d?udau,t?db,dcmBom,u,ad?uduDog,Cd2ufbadrdJbud u d ,mDa?ud?rBt am,tidUDd,md?u rat:tDard ,mdau,tBdBUDd?uButdJbudt,c mb?udrJbuDad cmBom,u,ad?puDog,do?A??CddFdBUD.d pu uca?XdUDd?ud,rab?r uDrdlu?Bt1,tcrdt.dou?dar,a.d Progam deCdnDa?bcab?rdu uca?1,tcrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddS6 ?????????????????????????????????????????????????????????????????? rdlb,ctXd?pm,rd?rd?pUDDu?dr,atDtBTa?tcrd?uDoucaudr dIuDcr,:td?udJbr Du:b rd?ud uDd Du:uDdcmm??u,r?uDC 3rdBr,u?rdu,d rdJbudrt(1dDprcm,Dufbut(dUDdbat ta?r,ad?uau?Bt,r,aDdcmBdrdlb,ctX ?pm,rCd3uDdcm bB,uDdcm??uDom,u,drdlb,ctm,Dd?pm,rd?udcr?rdu uca?X.dr,mBu,r?uDd vi?nos Aon?a? 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(x) ? xSCCC? x ? dddd?eCSeA m, d u d lrcam? d??? d uDai d :u d ?m,ra d tfbr d Jbu d rIr,D d au,t,a d u, d cmBoau d rd t,?tDat,ftIt tarad?u Ddu uca?m,Ddtdou?dDu?du d,mBI?udBi(tBd?udor?u Dd?pu uca?m,DdJbud om?uBdr??tIr?drdlm?Br?Cd FJbuDauD d lb,ctm,D d ?u d ?u,DtaraD d om?u, d Du? d fu,u?r ta?r?uD d r d Bra?tbDCd Pm,c?uarBu,a.dor? r?guBd?ud rdBra?tbd?ud?u,Dtarad?udo?tBu?dm???udtd?ud rd?udDufm,d m???uCd n D d u uBu,aD d ?u d r d Bra?tb d ?u d ?u,Dtara d ?u d o?tBu? dm???u d au,u, d r d Dufxu,ad uDa?bcab?r)d ?- (x- . x- ')= ? ???(x- .xe .CCC.x ? )?(x - ' .xe .CCC.x ? ) ? xe CCC? x ? ddd?eCSSA FJbgdUDd ?u u:r,ad ?uDarcr? d Jbudu Ddu uBu,aDd?trfm,r Dd?prJbuDardBra?tbduDd cm??uDom,u,drBId rdlb,ctXd?ud?u,Dtarad?u?bw?rd?udo?tBu?dm???uCd2ufbadrdrJbuDadlua.d rda?r7rd?ud rdBra?tbdUDdtfbr drd? 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'?d' dtrMdDsf?d'Mo'tan'dDuMtn'a: oI ra'aC,T?cDR??ragmDq??ce dD? AGuMa'uG'?dcr?'n'Mdcm adDuS'nM'MdD'donmorprndf?dor(f'adDsm?'M:dw?rdD'd 'Md cxMdM'??u 'Mdo'ndr dtrMdcm?mDuc'?Mum?r dxMdonmbrb 'c'?ad rd(f'd'Mdtm?'uGdtmcdrd onmorprDmndPnr??d9d?utVm Mm?:dQ'n?dxMdomMMub 'd'conrndf?rdS'nMu-d'?trnrdcxM M'??u rdtm?'pfDrdtmcdrdc4amD'dDsAf 'ndstnfs:d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?2J .................................................................. Uud 'Mdon'?d s'(frtu-de:?dudmbM'nSr?ado'nd 'G'co 'd '?d rdMm ftu-de:F?d 'Mdomad mbM'nSrnd(f'dxMdomMMub 'dD'lu?undf?dmo'nrDmndDs'Sm ftu-da'comnr dtmcdrd F =? q(? D[ , ?H ? q ddHe:?z? ?(f'Madmo'nrDmndaxd rdonmou'aradD'd(f'dr dM'ndro utradMmbn'd rd lf?tu-dDsm?rd onmDfuni? dMu dMs'?ax? d s'Gom?'?a d tmc d f?r d lrM' d luGrDr d tmc d r dqJ d r d?J ? d f?d D'MlrMrc'?ada'comnr :dTMdrdDunr? ?d=xVq(? F d=? q(?J=x Vq(? / n=x (D[ , ?n=qJ? d? q( dddHe:?2? 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TMdcm adlitu donmpnrcrndf?donmorprDmndDsr(f'Ma'Mdtrnrta'ngMau(f'M?dMud'MdDuMomMrd Dsf?rdMfbnfau?rdD'danr?MlmncrDrdD'd?mfnu'ndniouDrdH??8??d' dti tf dD'd?H? ? xMd cm adM'??u do'ndrdf?doma'?tur dr?r gautb:d?d rdbub umpnrlur2d'Mdanmbrdf?dbm?dr?i uMudD'd s'Marbu uaradud 'lutiturd?fc4nutrdDsr(f'Madc4amD':dPmcd'MdomadtmconmSrn?dxMdf?d c4amD'd rdonu?tuor dSunafadD' d(fr dxMd rdMuco utuara:d ?xM d '? i? d Vu d Vr d DuMom?ub ' d f? d rco u d n'o'namnu: d ? d srtafr uara? d f? d D' Md onmorprDmnMd cxMd 'conra d xMd r d a4t?utr d ?P8q?ed HSmRq,C dG,'mIaq, d ? q,1D ? 2? cDrDdcDdq ?NaIqIDD?d (f' d xMd r dS'nMu- d cf autm?lupfnrtum?r dD' d 8q?? d '?a'?'?ad cf autm?lupfnrtum?r dD'd rdcra'uGrdcr?'nrd(f'd rdD'Mtnuardr dtrogam dzd(fr?d 'Md anrtarSrd sfMdD'd 'Mdlf?tum?MdDsm?rdcf autm?lupfnrtum?r Md'?d' dtrcodD'd s'Manftafnrd r A d 'tamnd ?mdMsVrfnur dD' dM'?aund r rncra do' d l'a d (f' d s'Gom?'?tur d tm?au?pfu d s'?'npur dmd ' d Vrcu am?uidD'o'?'?adD' dtm?a'Ga:dhrdlf?tu-dDsm?rdxMd rdcra'uGrdud rdDul'n4?turdonmSxdD'd rd a'ncu?m mpurdD' MdDul'n'?aMdtrcoM: b UsVrdD'da'?und'?dtmcoa'd(f'd rdD'nuSrDrd'Mortur dD'd rdlf?tu-dDsm?r?drd s'MorudD'dcmc'?aM?dMsmbaxd Muco 'c'?ado'ndonmDfta'dD'd rdlf?tu-dDsm?rdo' dcmc'?a z:d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?2? .................................................................. ' 'tan??utr:d hrdD'?mcu?rtu-dD'd8q?donmSxdD'd sfMdD' Mdr?mc'?raMdonmDfta'MdD'd?rnan'':dAMd tm?Manf'uGd rdlf?tu-dDsm?rdtmcdf?donmDfta'dDsmnbuar Md(f'dtmnn'Mom?'?drdpnrfMdD' ub'nara d Dul'n'?aM d u d (f' d a'?'? d D'o'?D4?tur d rcb d ' d a'coM: d ?(f'MaM d mnbuar Md Msr?mc'?'?dlf?tum?Mdcm?mornagtf rd'?d' dtm?a'GadD' dc4amD':d TMdu?a'n'MMr?adD'Martrndf?rdtnf? rdc'amDm ?putrdSuMub 'd '?d ' MdDmMdc4amD'Md tmc'?araM?dMud r d onmorprDmndauofMdAf 'n dMsVrdu?anmDf?a d f?r d ronmGucrtu- d '? d ' d ?rcu am?uido'ndar dDsrnnubrndrd s'Gon'MMu-de:?e?d '?d ' d trMdD' dc4amD'd ?P8q?d Msu?anmDf'uGd sronmGucrtu-drd rdlf?tu-dDsm?r:d I:z:dw?rdonmomMardDsro utrtu-do'ndrd rdDu?icutrd(fi?autr ?d rd?rafnrd 'Mdn'rttum?Mdlmam(fgcu(f'Mda'?'?dcm ardn' 'Si?tur:dq'pfadrdruG??d MsVr?dD'M'?Sm foradcm a'Mdcr?'n'MdDs'?trnrnd s'MafDudDsr(f'Mardc'?rdD'dn'rtauSuara:d w?rdD'd 'MdcxMdrSr?vrD'Mdtmnn'Mom?dr dtrcod D'd rdl'cam(fgcutr?dxMd' dtrMdD' Md'MafDuMdD'd n'rtauSuaradfau ua?r?adc'amDm mpurdD'dbmcbrd9d Mm?DrEH'?dr?p 4M drm1r?2?rI tD ?:dA?dr(f'Mard c'amDm mpur d 'Go'nuc'?ar ? dMsfau ua?r d f? d l'uGd (f'd'MdDuSuD'uGd'?dDmM?d' donuc'n?d rdbmcbr?d onmomntum?rdr dMuMa'crdf?rd'?'npurd(f'du?uturd r d n'rttu-5 d ' dM'pm?? d r dMm?Dr? dM'nS'uG d o'nd 'Gan'fn' du?lmncrtu- dD' dMuMa'crd rd f?da'coMd D'a'ncu?rado'ndar dD'domD'ndtm?4uG'nd(f4dVrd Mftt'?a:dUsfau ua?rdf?dMuMa'crdD'dn'arnDdo'ndar d (f' d ' dM'pm?d l'uGd '?anud '?d rttu-d (fr? dMupfud tm?S'?u'?a:dA d(f'dxMdfMfr d'?dr(f'MaMdtrMmMd xMdfau ua?rndf?rd ln'(??turdar d(f'd s'?'npurdrnnubudDun'tarc'?ado'ndmtrMum?rnd rd ?upfnrde:??dAMdomadmbM'nSrndtmcdf?d l'uGd9d(f'don4Surc'?adMsVrfnidDuSuDuad '?dDmMdcuaXr?vr?adf?dDuMomMuaufdrDu'?ad 9d'?dornadu?a'nrttum?rdrcbd rdcmManrdud '? d orna d (f'Dr d sranroras d '? d f?d DuMomMuauf d (f' d ' d o'nc'ani d u?tuDund Mmbn'd rdcmManrd'?d' dcmc'?admomna? Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?2z .................................................................. anr?Mutu-d(f'dD'M'?trD'?rnid rdn'rtauSuara: Q' d(f'd lrdrd 'Mdn'rttum?Mdu?uturD'Mda4ncutrc'?ad 'GuMa'uGdbub umpnrlurdMmbn'd s'MafDudDsu?uturtu-dD'dn'rttum?Mda4ncu(f'MdcuaXr?vr?adl'uGmMd iM'n6d(f'du?tuD'uG'? d Dun'tarc'?a dMmbn' d r d cmnManr d o'n? d ?m d rcb d c'amDm mpur d bmcbr d 9 dMm?Dr: d?d Dul'n4?turdD' dtrcodD'd rdlmam(fgcutr?d'?dr(f'MadtrMd rdnrDurtu-d?md'MaidDunupuDrdrd u?DfundDun'tarc'?ad r dn'rtauSuaradMu?-d (f' d 'Mdon'ax? dDm?rnd f?rd t'nard '?'npurd rd s'?amn?dcm 'tf rn?dpnrfMdD'd ub'narad(f'd?mdM-?don?ourc'?ad' Mdu?Sm ftnraMd'?d rd n'rttu-? d xMd rdDun? d s'?amn? d cm 'tf rn d ar? dDu?MdD' d r d on?our d cm 4tf r d tmcdD' d DuMMm S'?a:d Q'ndar?a?d s'MafDudD'd 'Mdanr?Ml'n4?tu'MdDs'?'npurd'?an'd' dMuMa'crdcm 'tf rndud' d DuMMm S'?adud'?an'd' MdonmouMdpnrfMdD'd ub'naradu?anrcm 'tf rnMd'MD'Sxdtrouar do'nd n'r ua?rndf?d'MafDuda'?nutdtmcd' d(f'd'MdonmomMrdrdtm?au?frtu-: A?dr(f'Madtrcod 'GuMa'uG'?d ' Mdr?mc'?raMdVrcu am?ur?Mdbr?1d9dMuMa'crd(f'd anrta'?dD'dcr?'nrdcm adDuS'nMrd 'MdomMMub 'Mdu?a'nrttum?Md'?an'd rdcm 4tf rdud' dM'fd '?amn?d'?d' dM'fda'nc'dDs'?'npurdoma'?tur :d?d rdcrXmnurdD'dtrMmMdr(f'Madoma'?tur d 'MdD'MtmcomMrdt rnrc'?ad'?dDf'MdornaM?d' d(f'dn'on'M'?ard' dMuMa'crdudxMdo'ndar?adf?d oma'?tur d DuMMmturauf? d f?r d brnn'nr d m d (fr M'Sf r d D' d 'M dmotum?M d (f' d 'M d Sm d rtm?M'pfundcmD' ua?rn5dud' d(f'dn'on'M'?ard' dbr?1? +N? +N ?? +N 0? +N ???? E +N?0 dddHe:?E? A d oma'?tur d (f' d n'on'M'?ar d ' d br?1 d p'?'nr c'?a d 'M d anrtar d DsmMtu ?rDmnMd Vrnc??utM: d?(f'MaM d oma'?tur M d ornrb? utM d a'?'? d ln'(??tu'M d (f' dMsrXfMa'? d D'd cr?'nr d tm?M'(??a d rcb d s'?amn? d (f' d u?a'?a'? d n'on'M'?arn5 d o'n? d r d M'Sr d D'o'?D4?tur d cxM d tmco utrDr d xM d srtmb rc'?a d rcb d r d n'Mar d D' d Vrcu am?ui:d q'a'ncu?rn d r(f'Ma d rtmb rc'?a d M'con' d xM d tnftur d Xr d (f' d r(f'Ma d ornic'an' d D'a'ncu?r d r d (fu? d nuac' dM'ni d s'?'npur d anr?Ml'nuDr d '?an' d ' MdDul'n'?aMdpnrfMdD'd ub'nara: d ?u d Vr d DuS'nM'M d cr?'n'M dD' d D'a'ncu?rn d rcb d nupmn d r(f'Mar d tm?Mar?ad Dsrtmb rc'?a:dq'MdD'd rdo'nMo'tauSrda'?nutrdM'con'dVudVrd rdomMMubu uaradD'dn'r ua?rn Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?22 .................................................................. ti tf Mdat?,d,q, do'ndtm?Manfund' Mdoma'?tur MdrdornaundD' Mdti tf MF:d A? d tmcoa'M dDsf? d Vrcu am?ui d D' d auofM dMuMa'cr d 9 d br?1 d arcbx d xMd omMMub 'd cmD' ua?rndf?dMuMa'crdD' dauofMdD'd 'Mdanr?Ml'n4?tu'MdD'donma-dud' 'tan-drtmb rD'M:d A MdcmD' MdDuMom?ub 'Mdarcbxdornrc'anua?'?d' Mdrtmb rc'?aMd'?an'd' dcmSuc'?adD' d onma- d u d s' 'tan- d rcb d ' dDuMMm S'?a dmba'?u?a d tr?SuMd ucomnar?aMd o' d (f' d lr d r d tmcomnarc'?adD' dMuMa'cr?:dUsrMM'?1r rdarcbxd(f'd srtmb rc'?adVrdDs'MarndbrMrad'?d r d u?a'nrttu- dD' d cmc'?a dDuom rndD' dMm fa dud r dom rnua?rbu uara dD' dDuMMm S'?a?d tm?Manfu?ad f?dcmD' dbrMrad 'Go gtuarc'?a d '? d rdomMutu-dD' donma-?d s' 'tan-dud ' d DuMMm S'?a0??J :d w?rdr anrdcr?'nrdDs'?trnrnd rdcmD' ua?rtu-dxMd'conrndf?dtrcgdD'dn'rttu-d(f'd u?t mpfu d uco gtuarc'?a d ' M d rtmb rc'?aM: d w? d 'G'co ' d xM d ' d c4amD' d ?RQ? d H?rcu am?uidD' d trcgdD'dn'rttu-d(fi?aut:dq'd sr?p 4M? dlmadqm1?uDaCq, d ?oaqf? Na1,Rq d,ad?d(f'd'MdD'MtnufnidcxMd'?DrSr?ad'?dr(f'Madcra'uGdtrogam :d?uG?d'?Md Dfni?do'n??drdn'r ua?rndti tf MdD'dM'pm?'MdD'nuSrD'Mdof?afr MdD' dMuMa'crdrcbd rd M'SrdtruGrdD'dMm Srartu-du?tmnomnrDrdD'damad' dtrcgdD'dn'rttu-:dq'pfadrd(f'd' d Vrcu am?uid(f'd'MdonmomMrdxMdcm?mDuc'?Mum?r ?dtr DnidfMrn,?'df?rdS'nMu-dtmcdrd cg?ucdbuDuc'?Mum?r :d?uG?dxMdD'pfadrd(f'd s'MafDud(f'd'MdonmomMrdtmcomnardtmcdrd cg?ucdDf'MdDuc'?Mum?M?d rdtmmnD'?rDrdD'dn'rttu-dud' dpnrfdD'd ub'narad'?d' d(f'd anr?Ml'nucd s'?'npurdcuaXr?vr?ad rdnrDurtu-dD'da'nrV'na?:d 8r?adbm?dof?adV'cd'Mtm uad' dMuMa'crdrdanrtarndMsVrdD'don'ornrnd' dVrcu am?uid rDu'?a:dPm?MarnidDsf?rdDuc'?Mu-drcbd rdtmmnD'?rDrdD'dn'rttu-d(f'dMsrtmb rnidrcbd r an'MdpnrfMdD'd ub'nara:dPmcdrdcg?ucdf?dDs' Mdtmnn'Mom?Dnidrd rdu?a'nrttu-d'?an'd' d MuMa'crdud' dDuMMm S'?a:dA dVrcu am?uidtm?au?Dnidf?da'nc'd(f'dtmnn'Mom?Dnidrd rd nrDurtu-dD'da'nrV'na??df?dtrcod' 4tanutdmMtu ?r?adrdf?rdln'(??turdrDu'?a?dDfnr?adf?d a'coMdD'a'ncu?radudrcbdf?rdu?a'?Muarad(f'dVrdD'dM'ndonmfdn'Df?Drdtmcdo'ndrd(f'd?md ofpfudmtrMum?rndl'??c'?Mdcf aulma??utMdrd rdcm 4tf r:d ?d tm?au?frtu- dMs'GomMrd f? d 'Mb-MdD' d (f' domDnurdM'nd f? d 'MafDud '? d r d g?urd 'GomMrDr:dA dpnrfdD'd ub'naradonu?tuor dtmnn'Mom?drdf?donmtxMdcmD' ua?radrcbdf?d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?2e .................................................................. oma'?tur dauofMd?mnM'dr d(fr dMsVudVrdrl'puadf?rdlf?tu-dprfMMur?rdo'ndar dD'd(f'dVud Vrpudf?rdbrnn'nrd'?an'dn'rtaufMdudonmDfta'Mdud?md?mcxMdf?domf:dw?rdlf?tu-dauofMd ?mnM'dD'o4?dbiMutrc'?adD'dDmMdornic'an'M?d rdonmlf?DuaradD' domfdud rdM'Srdlmncr?d (f'd'Mdomadn' rtum?rndrcbd rdln'(??turd(f'dtmnn'Mom?drd smMtu ?rDmndVrnc??utdr d cg?uc:d?dorna?d'MdomadluGrnd rdomMutu-dD' domfdudrl'pundf?da'nc'd(f'dMfcudf?rd D'a'ncu?rDr d (fr?auara dDs'?'npur? d ' d (f' dMsr?mc'?r d sD'Mo rvrc'?as dH?f,Gq?: dw?rd lf?tu- d prfMMur?r d xM d f? d cmD' d D' d brnn'nr d u d 'M d oma d trnrta'nua?rn d cuaXr?vr?ad ornic'an'M d (f' d n'Mom?'? d r d cmD' d D' d brnn'nr: d A? d r(f'Ma d trM? d f? d lrtamnd on''Gom?'?tur d(f'dtm?Mauaf'uGd rdM'Srdr vrDrdudf?dlrtamndrd s'Gom?'?ad(f'd udD-?r f?rdrco rDr:dQ'ndar dD'dtm??'tarnd' domfdD'd rdlf?tu-d ?mnM'dMs'conrnr?dlrtamnMd tmcf?Mdrdama'MdDf'Md lf?tum?M:d dhrd tm?Manfttu- dDsr(f'Mard lf?tu- dHi =g( ?dMsVrd n'r ua?radrcbd' MdM'p??aMdornic'an'M?d S =g(? U =D[ ?=g[ g S p([ ? (z? n J rcb U? E?tr 3cm V?? ??gz U V?g? ?JEJ tc [ ?VgSp ? J f:r:dudn J? J ?tr 3cm b v =g(? UD[ ?=g[ gvp (z rcb gvp ? e f:r: i =g(? S =g(? v =' ( He:?6?dd Pmcd'MdomadmbM'nSrn?d rdbrnn'nrdamar dM'nid'Grtarc'?ad rdMfcrdD'd rdonmlf?Duarad D' domfdud sr vrDrdD'd rdprfMMur?r?dzU ? ?Jd?tr 3cm :d?d rdlupfnrde:zd'MdomadmbM'nSrnd f?rdn'on'M'?artu-dD'd rdlf?tu-d i =g( :d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?2E .................................................................. PrDr d a'nc' dD' d br?1 d ax d r dM'p??a d lmncrdp'?'nr ? +N ?V, V?0 ? ?, z =g ?[ C?Q?=g,(( z dddHe:?F? Q'ndar dDsrMMm und 'Mdf?uaraMdmomnaf?'M?d tr da'?und'?dtmcoa'd' dl'adD'd(f'dMs'con'? d f?uaraMdom?D'nrD'Md '? d rdcrMMrdud '? d rd ln'(??tur: d8r dud tmcdMs'Mtnufd s'(frtu- d e:?F?drdf?dpnrfdD'd ub'naradD'dn'l'n4?turdXd M' d u d rtmb r d f? d mMtu ?rDmn d u5 d r(f'Mad mMtu ? rDmn d 'M d trnrta'nua?r d rcb d f?rd ln'(??turd ?u :dhs'(frtu-dtmnn'Mom?drdf?d a'nc'dD' dVrcu am?uidD' dbr?1d?d(f'daxd r d M'Sr d on?our d tm?Mar?a d Dsrtmb rc'?ad =t ?( d u d r d M'Sr d on?our d lf?tu-d Dsrtmb rc'?a d =Q?=g ,(( : d?(f'Mar d lf?tu- d D'o4?dD' dauofMdDsu?a'nrttu-d(f'd'MdSf pfud n'on'M'?arn: d Q'n d ar d D' d DuMtfaun d ' Md Dul'n'?aM d rtmb rc'?aM d '?an' d pnrfM d D'd ub'narad '? dDul'n'?turn'c dD' dDmMdauofM?d srtmb rc'?a d u?'r ? d m? d Q=g(? g d ud srtmb rc'?a d prfMMui d m?d Q=g(? . D=[ =g[ gJ (z(: d dqm?rad(f'dQ=g( dVrd D'da'?undf?uaraMdD'dDuMai?tur?dMsu?t mfdf?rd tm?Mar?a d.d (f'dcf auo utrd rdprfMMur?rdu (f'dSr d?dbmVn: Uud srtmb rc'?a d 'Md luGxMdtmcdr dQ=g(? J d ' d (f' d 'Mdn'on'M'?arnurdM'nurd f? mMtu ? rDmndVrnc??utdamar c'?admnampm?r dr dpnrfdD'd ub'naradonu?tuor :dhs'Manftafnrd ?upfnrde:z?d?d rdlupfnrdMfo'numndMsmbM'nSrd' d o'nlu dauofMd ?nM'dMfcradrdf?rdprfMMur?rd t'?anrDrdMmbn'd (jed f:r:: dAMdomadmbM'nSrnd (f' d r d brnn'nr d xMdD' d f?Md J:J?EdVrnan''M?d f?'Md?Jd?tr 3cm :d d?d rdlupfnrdu?l'numnd'Md cmManrd ' dcra'uGdoma'?tur do'n?dVrS'?a,Vud rtmb rad ' Mdan'MdmMtu ?rDmnMdD'd rdcr?'nr D'Mtnuar d r d a'Ga: d A? d r(f'Mar d mtrMu-?d MsmbM'nSr d f?r d brnn'nr d cm a d r ar? d J:z Vrnan''M: d?uG? d ?m d xM d on'mtfor?a d Xr d (f'd r(f'Ma d o'nlu d tmnn'Mom? d r d f? d Vuo'no rd rnbuanrnudo'ndr d(fr damaMd' MdcmD'MdD' dbr?1d Vr?d'MaradluGraMdrdf?rdtmmnD'?rDrdrnbuaninurd D'dJdf:r::d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?26 .................................................................. D'd rdlf?tu-d Q=g( dD'a'ncu?rnidlu?Mdrd(fu?dof?adMsrbr?Dm?rdr(f'Mardmnampm?r uaradud '?d(fu?'Mdn'pum?M:dA donuc'ndauofMdDsrtmb rc'?adH' d u?'r ?dn'on'M'?aradrd rdornad Mfo'numndD'd rdlupfnrde:2?dcmD' ua?rdf?dMuMa'crd'?d' d(f'dM'con'd'MdD-?rd' dcra'uG auofMdDsu?a'nrttu-dM'?M'd(f'dVuducortaud' dSr mndD'dg?u (f S'dDm?rDrdo' dornic'an'd C, : d?gMutrc'?a?d rduco utrtu-dxMd(f'dDfnr?ad' dl'?mc'?d(f'd'MdcunrdD'dcmD' ua?rnd ?mdxMdn' 'Sr?ad'?d(fu?dof?adD'd rdtmmnD'?rDrdD'dn'rttu-d'?Mdanmb'c:d?uG?domad n'Mf arnd(f' tmcdrlgMutd'?dM'pm?Md(fu?dauofMdD'donmt'MMmMdudo'ndruG?d'Mdon'M'?ard arcbxdf?drtmb rc'?adprfMMui:d?uG?d'?Mdo'nc'adtm?MuD'nrndf?dauofMdDsu?a'nrttu-d cm adr ard'?dr pf?rdp'mc'anurdDm?rDrdud(f'dD'trfd'Gom?'?tur c'?adrdc'Mfnrd(f' d rdtmmnD'?rDrdrbr?Dm?rdr(f'Mardp'mc'anur?dar dudtmcd'MdomadmbM'nSrndrd rdornad u?l'numndD'd rdlupfnrde:2:d w? d tmo d 'Marb 'naM d r(f'MaM d auofMd Dsrtmb rc'?aMdn'Mf ardtm?S'?u'?adrDroarn, mMd r d f? d onmtxM d lgMut d Dm?ra: d A? d ' d trM d D'd s'G'co 'd(f'd'MdDuMtfa'uG?dMsVrdDuMomMradDsf?d br?1 d anuDuc'?Mum?r : d w? d D' M d cmD'Md on'M'?ar d f? d rtmb rc'?a d prfMMui d rcb d rd tmmnD'?rDrdD'dn'rttu-dHbr?1d??:dh'Mdr an'Md D' d 'M dDuc'?Mum?M dHz d u d 2? d tmnn'Mom?'? d rd cmD'M d D' d bruGr d ln'(??tur5 d r(f'MaM d DmMd cmD'MdMsrtmb '?drcbd' donuc'nd u?'r c'?a:d ?uG? d 'M d Mu?a'aua?r d r d anrSxM d D' d s'M(f'crd on'M'?aradrd rdlupfnrde:e:d hrdln'(??turdD' MdmMtu ?rDmnMdxMdr ard'?d ' dtrMdD'dbr?1d?d=FFJ tc [ ?( d:dA?d' dtrMdD'd br?1 d z d u d br?1 d 2 d xM d bruGrd =?0J dudzJJ tc [ ? ?dn'Mo'tauSrc'?a:( :d?uG?dxMd D'pfa d r d (f' d r(f'MaM d DmM d Drnn'nM d cmD'Md n'on'M'?a'? d f?r du?a'nrttu- d '?an' dcm 4tf 'M d ?upfnr d e:2? d hr d tmmnD'?rDr d D' d n'rttu-d rtmb rDr d r d f? d mMtu ?rDmn: d?d r d pnilutrd Mfo'numnd'MdanrtardDsf?drtmb rc'?adD'dauofMd u?'r d u d r d r d u?l'numn d 'M d anrtar d Dsf?d rtmb rc'?adprfMMuid(f'daxd ' dM'fdciGucd XfMadrd(jedf:r:?d(f'dxMdm?d'Mdanmbrd s'MaradD'd anr?Mutu-: d TMdu?a'n'MMr?a dD'Martrn d (f' d ' d trcgdD'dn'rttu-dIRPdM'nurdamar c'?adn'ta'd '? d ' d onuc'n d trMd c'?an' d (f' dVrfnur dD'd D'Mtnufn'df?rdtmnbrd'?d' dM'pm?: Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?2F .................................................................. D' d DuMMm S'?a d u d r d cxM d n'b'? d nrDurtu- d D' d a'nrV'na? d (f' d tmnn'Mom? d r d bruG'Md ln'(??tu'M:d Qn4Surc'?adrd rdonmorprtu-?dMsVrdn'r ua?radf?rdn' rGrtu-dD' dor(f'adprfMMui:dA?d r(f'MadonmtxM?d'Mdonmorprd' dor(f'ad'?df?da'coMducrpu?rnu:dhs'l'ta'dDsruG?dxMd(f'd ' dor(f'ad 'Mdn' rGrdon'?'?ad ' dM'fd 'Marad rcbdSr mndonmoudcxMdbruG:dqsr(f'Mard cr?'nrdomD'cdrMM'pfnrnd(f'drd rdM'p??adDu?icutrd rdcrXmndornadD'd s'?'npurd tu?4autrdSu?DnidDm?rDrdo'nd rdnrDurtu-dD'da'nrV'na?: TMdD'Martrb ' d arcbxd ' d l'a d D' d (f' d ?m dMs'Mo'nr d rd tm?M'nSrtu-dD'd s'?'npurd?ud D' d r d ?mncr dD' d r d lf?tu-d Dsm?r d D'pfa d r d (f' d ' d oma'?tur d ?mnM' d on'M'?ard f?rdrMgcoamardudruG?dlmnvrdrd r d fau ua?rtu- dD'doma'?tur Md ?oautM d '? d r(f'Mard tmmnD'?rDrd (f' d rbMmnbunr?d r d lf?tu- d Dsm?r d (fr? d r d anr?McuMMu- d D' d or(f'ad Dsm?'MdVmdn'(f'n'uGu:d q'dtrnrdrd rdonmorprtu-?d MsVr d 'conra d ' d or(f'ad ?P8q?dud' dti tf daxd 'MdM'p??aMdtrnrta'ngMau(f'M:dUsVrdonmorprad' dor(f'adDsm?'Md Dfnr?adEJJdlM:dA dpnrfdD'd ub'naradD' dMuMa'crdtmcoardrcbd?Edlf?tum?MdD'dbrM'dD'd lf?tum?M d cm?mornagtf r d c'?an' d 'M d tmmnD'?rD'M dD' d br?1 dMsVr? dDmara d D' d ?J lf?tum?MdD'dbrM'dtrDrMtf?r:dhrdn'on'M'?artu-dD'dtrDrdtmmnD'?rDrdxMdD'dauofMdq?Rd HU,?CIDqD ? iaI,atRD ? uDrID?Ddqaq, d?? dMu?fMm?Dr d '? d ' d trMdD' d r d tmmnD'?rDr dD'd n'rttu-dudD'dlf?tum?MdDs?'ncua'dDsmMtu ? rDmndVrnc??utd '?d ' dtrMdD'd 'MdD'cxM:d hs'Manra4purdDsu?a'pnrtu-dxMdD'dauofMdP??dHP d?qadq?SDad?Q,DRc?:dqu?Mdr(f'Mard ?upfnr d e:e: d AM d n'on'M'?ar d 'M(f'ciautrc'?a d tmc d 'Mar?d Sutf rD'M d ama'M d 'M dDuc'?Mum?Md '?an' d ' 'M d '? d ' d cmD' d 'conra:dA dMuMa'crd'Maidn'on'M'?arado'ndf?rdtmmnD'?rDrdud' d br?1 d 'Mai d Dmara dD' d an'M dDuc'?Mum?M: d hr d tmmnD'?rDr d D'd n'rttu- d ?mcxMd 'Mai dDun'tarc'?a d rtmb rDr d r d f? d pnrf dD'd ub'nara?dr?mc'?radsbr?1d?s:dA Mdr an'MdDmMdpnrfMdD'd ub'narad D' dbr?1?dsbr?1dzsdudsbr?1d2s?d'Mar?drtmb raMd u?'r c'?adrcbd sbr?1d?sdudM-?d' Md?utMd(f'dn'b'?d'?'npur: Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?2? .................................................................. 'Manra4purd'MdomD'?dfau ua?rndDmMdu?a'pnrDmnMdDul'n'?aMdr VmnrdudMs'con'?d' dRf?p'd9d ?faar dDsmnDn' dEdHo'n d r d 'Md lf?tum?Mdcm?mornagtf r? d u d ' dUIhdHLf Iq ? xqDIaq,yD? AadC. ??: ? d r d lupfnr d e:E:r d 'M d omad mbM'nSrn d r d D'?Muara d n'Df?Dr d D'd onuc'ndmnDn'd(f'daxdtmcdrdSrnurb 'd rd tmmnD'?rDrdD'dn'rttu-: d?dcxMd D' d oma'?tur ? d 'M d n'on'M'?ar d rd D'?Muara d r d a'coM d u?utur d u d r d f?d a'coM d m? d 'M d oma d ron'turn d rd anr?McuMMu-dD'dD'?MuaradrdanrSxMdD'd rdbrnn'nr: TMdu?a'n'MMr?adr?r ua?rnd' dl fGd Ds'?'npurd '?an'dDul'n'?aMdpnrfMdD' ub'nara d r d c'Mfnr d (f' d rSr?vr d rd Du?icutr:dq'pfadrd(f'd' dcuaXid(f' d a'?uc d o'n d anr?Ml'nun d '?'npur d r d MuMa'crdxMdunnrDurndDun'tarc'?ad' Md pnrfMdD'd ub'narad(f'dtmnn'Mom?'?d r d mMtu ?rDmnM d rcb d bruGrd ln'(??tur? d xM d u?'Suarb ' d (f' d r(f'MaM d cmD'M d 'M d S'pu?d Subnrtum?r c'?a d omb raMd rd ?uS' Md lmnvrdr aM?dar?dr aMd(f'd'MD'S'?'?d rlgMutM d H?uS' M d Subnrtum?r M d D'd smnDn' d D' M d FJ,?J?: d ?(f'Mard a'?D4?tur? d ama d u dM'n d r d onumnu d f?d cmaufdo'ndDfbarndD'd rdbm?DradD' dcmD' ?drXfDrdarcbxdrd'?a'?Dn'dtmcdMsmbaxd rd n'rtauSuara?drdc'Mfnrd(f'd' MdDul'n'?aMdpnrfMdD'd ub'naradn'b'?d'?'npur?d srtfcf '?d '?dlmncrdDs'?'npurdSubnrtum?r dlu?Md(f'd'S'?afr c'?ad'Mdn' rG'?danr?Ml'nu?adr(f'Mard ?upfnr d e:E:r: d ? d r d u ? fManrtu- d Mfo'numn d MsVud n'on'M'?a'?d s'?'npurdoma'?tur d'?df?uaraMdra?cu(f'Md u rdD'?Muaradn'Df?DrdD'donuc'ndmnDn'do'ndr da'coM u?utur dudo'ndrdz?JdlM:dhrdornadanr?Mc'MrdD' dor(f'a Dsm?'MdxMdrco urDrdrd rdlupfnrdu?l'numn: Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?20 .................................................................. '?'npurdrdr an'MdpnrfMdD'd ub'naradr Md(fr Md'Mar?drtmb raM:d?fr?d rdtmmnD'?rDrdD'd n'rttu-dn'odr(f'Mard '?'npurd 'MdonmDf'uG'?dmMtu ? rtum?M5d(fr?d f?d lnrpc'?adD' d or(f'a d lu?r c'?a dMfo'nr d r d brnn'nr d 'MdD'Mo mc'? d ' Md ?uS' Md Subnrtum?r Md r Md mMtu ? rDmnMdrtmb raM:d?uG?d(f'Drdn'on'M'?aradrd rdlupfnrde:E:b:d Q' d (f' d lrd r Md?uS' MdSubnrtum?r Md rMMm uaMdrd rdDu?icutr?dlr?do'?Mrnd(f'd omadM'ndcm adDulgtu d(f'd'?df?dpnrfdD'd ub'nara d n'r d '? d f? d MuMa'cr d (fgcutd Msrtfcf udar?ard '?'npurdDsr(f'MadauofM:d ?m d mbMar?a? d tr d n'tmnDrn d (f' d '? d ' d cmD' d 'conradr(f'MadpnrfdD'd ub'narad n'on'M'?ard s'?amn?dD'd rdcm 4tf rdo'nd ar?ad?mdaxdo'nd(f4d'Marndn'on'M'?ar?adf?d cmD' d ?mncr dD' dSubnrtu- dDm?ra? dMu?-d (f' d 'M d oma d '?a'?Dn' d tmc d rd n'on'M'?artu-dDsf?dtm?Xf?adD'dcm 4tf 'Md rcb d r d tmmnD'?rDr d D' d n'rttu-: d Pr d o'?Mrndarcbxd(f'dMudr(f'Mardu?a'nrttu-d 'MdcmD' ua?rdrdanrSxMdD'dcxMdpnrfMdD'd ub'nara?d rdtmV'n4?turdD' dor(f'adDsm?'Md 'Mdcr?au?Dnidc'?1MdD' d(f'd 'Mdcr?axd '?dr(f'Mad'G'co 'dDm?ra:d ?upfnrde:E:b?dQmb rtu-dD' Md'MaraMdSubnrtum?r Md tmnn'Mom?'?aMdr MdpnrfMdD'd ub'narad?dudzdD' d br?1: d AM d oma d ron'turn d tmc d r d omb rtu-d DuMcu?f'uGdar?adbm?dof?ad'Mdtmc'?vrdrdDm?rnd rd anr?McuMMu-dD' dor(f'adDsm?'MdtrodrdonmDfta'?dar d u d tmc d 'M d oma dmbM'nSrn d tmcornr?a, r d rcb d rd lupfnrde:E:rd(f'd'Marb 'uGd su?Mar?adadjdz?JdlMdtmcd rd su?Mar?ad '?d ' d (fr dMsrtrbrdD'danr?Mc'an'd ' or(f'a:dR'tmnD'cd(f'd' dpnrfdD'd ub'naradbr?1d?d 'Mai d Dun'tarc'?a d rtmb ra d r d r d tmmnD'?rDr d D'd n'rttu-dc'?an'd(f'd' dbr?1dzd'Maid?mcxMdrtmb rad rdbr?1d?5drdcxM?d' dpnrfdD'd ub'naradbr?1d?d?md u?a'nrttum?r d Dun'tarc'?a d rcb d r d nrDurtu- d D'd a'nrV'na?:dd Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?eJ .................................................................. II:d?rcu am?uidD' dtrcgdD'dn'rttu-dHRQ?? II:?:dA dVrcu am?uidt iMMut A dtmMadtmcofartum?r dD'd(fr M'Sm dc4amD'dDu?Md rdDu?icutrd(fi?autrdxMdcm ad MmSu?adcrMMrdr a:dTMdf?dpnr?d4Guad(f'df?dc4amD'dtmcd ' d ?P8q?dVrpudompfad rlnm?arndti tf MdD'dlu?MdrdzedpnrfMdD'd ub'nara?do'n?druG?dxMdf?dl'adrbMm farc'?ad 'Gt'otum?r :d?fr?d'MdSm dn'r ua?rndf?dti tf d'?d' dtrcodD'd rdDu?icutrd(fi?autr d Mmbn'df?dMuMa'crdcm 'tf rndud?md'MdanrtardDsf?dMuMa'crd 'Mo'tur c'?adn'Df?a?d rd n'r ua?rtu-dDsronmGucrtum?MdudcmD' ua?rtum?MdxMdu?'Suarb ':dw?d'G'co 'd sV'cdSuMadrd srornaradr?a'numn?dm?dMsVrdn'r ua?radf?rdcmD' ua?rtu-d(f'dn'Df?rdpnrfMdD'd ub'naradrd c'nMdmMtu ?rDmnMdVrnc??utM:dA?dr(f'Madrornara?drcbdf?dornr ? ' uMc'dlmnvrdpnr?d tmcd'MdS'fnidrdtm?au?frtu-?dMsmoarnido'nd sronmGucrtu-: A dc4amD'do'ndar dD'dD'lu?undf?dVrcu am?uid 'conr?adf?d trcgdD'dn'rttu-dxM tm?'pfadtmcdrdVrcu am?uidD' dtrcgdD'dn'rttu-dH'?dr?p 4M?dRQ?? duDaCq, d?oaqf? Na1,Rq d,ad?:d?rdM'donmomMrado'ndonuc'nrdS'prDrdo'nd?u 'ndudtm ?rbmnrDmnM??:dhrd M'SrdonmomMardxMdD'lu?undf?dtrcgdD'dn'rttu-dcm?mDuc'?Mum?r dudonmX'tarn,VudrdtrDrd u?Mar?ad' dcmc'?ada'?u?ad'?dtmcoa'd s'?amn?d(frDniautdD' dtrcgdD'dn'rttu-:dquad Dsf?rdcr?'nrdcxMdp'?'nr ?d' dVrcu am?uidD'dtrcgdD'dn'rttu-dD'Mtnufdf?rdDu?icutrd n'Manu?puDrdm?d rdanr?M rtu-d'Maidamar c'?adonmX'tarDrdrd rdar?p'?adD' dtrcgd'Mtm uad ud' Mdn'Mar?aMd2E,FdpnrfMdD'd ub'naradM-?dcmD'MdSubnrtum?r Mdo'no'?Dutf rnMdr dtrcgd D'dn'rttu-:dhs'?'npurdoma'?tur dax?dDm?tM?df?rdlmncrdar dtmc? i =?Vl ??:::Vl 2? [ F(? i ?=?(? ?z ?,? ? 2 E [ F ?=?(,z l ,z ddddHe:??? hrdlmncrdD'd smo'nrDmndtu?4autd?mdxMdanuSur dD'dD'a'ncu?rndudSrnurda'?u?ad '?d tmcoa'd(fu?dtrMdMs'MafDur?dXrdMsVrdDs'Gon'MMrnd'?dr(f'Ma'MdtmmnD'?rD'Mdud' dM'fd cmc'?adtm?Xfpra: w?rdM'pm?rdS'nMu-dD' dVrcu am?uidD' dtrcgdD'dn'rttu-dxMd rdon'M'?arDrdo'nd Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?e? .................................................................. ?lu dudtm ?rbmnrDmnM?z:d?(f'MardonmomMardSrdcxMd'? idDsf?rdonmX'ttu-dD'd 'Md tmmnD'?rD'Mdud' Mdcmc'?aMdudromMardo'ndn'D'lu?und rd rpnr?pur?rdud' dVrcu am?uid DsrtmnDdrcbd' dtrcgdD'dn'rttu-danrtara:dq'd rdu?a'pnr dD'd rd rpnr?pur?rdrcbdlmncrd cxMdn'tfnn'?a?d ? qp q h=q ?DqDq (cq??qp q ? z =DqDq (8=DqDq ([ i =q(cq dddHe:?0? m?dq?xMd' dornic'an'd(f'dtrnrta'nua?rd rdtmnbrd'Gan'cr ua?rDr?d'Mdomadlitu c'?ad orMMrndrdf?rd rpnr?pur?rd(f'dD'o'?pfudD'd 'MdSrnurb 'MdDq?D?dudDq?D? ?dm?d??xMd rd m?puafDdDsrntd(f'dtrnrta'nua?rnid' dtrcgdD'dn'rttu-? ? qp q h=q ?DqD? ?DqD? (c???qp q ? z =DqDq (8=DqDq (=DqD? ([ ?[ i =q(=DqD? (cqdddHe:zJ? qsr(f'Mard rpnr?pur?rdomD'?dD'Dfun,M'dDf'Md'(frtum?MdDsAf 'nd9dhrpnr?p'd(f' ?mdM-?du?D'o'?D'?aM:dA dVrcu am?uid(f'dM's?dD'Df'uGdaxd rdM'p??ad'Manftafnr? N=?, r ?(? ?z r ?? =DqD? (8=DqD? ( ? i =q=?(( dddHe:z??d rcbdf?dcmc'?ad r ? dD'lu?uadtmcdr? r ?? =DqD? (8=DqD? (=D?Dq ( dddHe:zz? h'Md'(frtum?Mdr?a'numnMdomD'?dM'ndlmncf rD'Mdo'ndrdt'?1un,M'drdf?rdtmmnD'?rDrd D'dn'rttu-dDm?rDr?dtmcd' dIRP:dq'pfadrdnrm?Mdonitau(f'M?d' dD'M'?Sm forc'?ad MfbM'p??ad?md'conrdDun'tarc'?ad rd m?puafDdDsrntd? tmcdrdtmmnD'?rDrddMu?-df?rd crp?uafDdn' rtum?rDr? Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?ez .................................................................. u=s(? exp}?s0s ds'?gT g { dddHe:z2? UudMs'Mtf d' dtrcgdIRPdo'ndar dD'dtrnrta'nua?rnd' dVrcu am?uidt iMMut?dr(f'Madon'?d rdlmncr? H RP[ IRC ?u,pu ?? 12 pu2 g0u u 2 ?H 0[ I ? g0 ? 12 g0 T uH 0 H 0[ 1 uH 0 g0 +W 0 dddHe:ze? m?d? Jd rpnfordamaMd' Mda'nc'Md(f'd?mda'?'?dtrodD'o'?D4?turdrcb dm:dAMdomad ron'turndtmcd' da'nc'dtu?4autdD'o4?dDun'tarc'?adD'dmbdQ'ndr(f'Madcmauf?d'MdDufd(f'd VudVrdf?da'nc'dD'dcrMMrd(f'dSrnurdrcbd rdomMutu-:d?mdmbMar?a?d su?a'n4MdD' dc4amD'd '?d ' don'M'?ada'Gadn'trfdcxMdrSurad '?d rdS'nMu-d(fi?autrd(f'd 'MdD'M'?Sm fordrd tm?au?frtu-: II:z:dA dVrcu am?uid(fi?autdH?Q? Q'ndar dD'dD'M'?Sm fornd' dVrcu am?uid(fi?autdr dSm ar?adD' dIRP?dMsrDmoard rd tm?Dutu-dMfbMuDuinur?d r=q ,?(? q[ q=?(? 0 dddHe:zE? hrdD'lu?utu-dD'd smo'nrDmndtu?4autdD' dVrcu am?uid(fi?autdn'(f'n'uGdornaundD'd rd D'lu?utu-de:zzdud rdD'lu?utu-dD'dcmc'?ad(fi?aut?dojd ,u??( :d8rcbxdtr da'?und '? d tmcoa'd rdD'nuSrDrdD'd rdn'Manuttu-de:zE? >r=q , ?( >? ? [ Dq=?( D? dddHe:z6? A donmX'tamndo'ndr dcmc'?adMmbn'd' dtrcgdD'dn'rttu-dIRPd' domD'cd'Mtnufn'dD'd Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?e2 .................................................................. rdM'p??adcr?'nr? p= ?>r ?q,s ??>s ??>r ?q,s??>s ? T ?>r ?q,s ??>s ?T ?>r ?q,s ??>s? p dHe:zF? udrcbdr(f'Mard'Gon'MMu-dxMdM'??u drSr frnd' da'nc'do8o? r ? rW}r ? => I =gV?(?>?( =>I =gV?(?>?(?=> I =gV?(?>?(? => I =gV?(?>?( {}=> I =gV?(?>?(=>I =gV?(?>?(?=> I =gV?(?>?(? =>I =gV?(?>?( r {? ?= ?=> I =gV?(?>?(? =>I =gV?(?>?( (??e ] => I =gV?(?>?(? r}=>I =gV?(?>?(? =>I =gV?(?>?({? ?e ? ?] =>I =gV?(?>?(? r}=>I =gV?(?>?(? => I =gV?(?>?({??e ?= ?=> I =gV?(?>?(? =>I =gV?(?>?( (??e dddHe:z?? UfbMauafu?a d orntur c'?a d r d upfr ara d He:z6? d r d s'Gon'MMu- d He:z?? d mba'?uc? pT p== 1=>r =q,s (?>s(T =>r =q,s (?>s ( (1?4=dq=s(ds (T p= 1=>r =q,s (?>s(T=>r =q,s(?>s ( (1?4 ?= 1=>r =q,s (?>s (T =>r =q,s(?>s ( (1?4=dq=s(ds (T p= 1=>r =q,s (?>s (T =>r =q,s(?>s ( (1?4 dHe:z0? 8'?u?a d '? d tmcoa' d r dD'lu?utu- dD' d cmc'?a d (fi?aut d u d s'(frtu- d e:z6? dmba'?uc d s'Gon'MMu-dlu?r dD' d?mMan'dmo'nrDmndDs'?'npurdtu?4autr? Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?ee .................................................................. [ ?2 2 ? q T ? q? [ ? 2 2 =1=>r =q,s (?>s (T =>r =q,s (?>s ( (1?4>>s =1=>r =q,s(?>s (T =>r =q,s (?>s ( (1?4 ?=1=>r =q,s (?>s(T =>r =q,s (?>s( (1?4>>s =1=>r =q,s (?>s (T =>r =q,s (?>s( (1?4 He:2J? ?(f'Madmo'nrDmndxMdf?rdp'?'nr ua?rtu- dD'd smo'nrDmnd 'Mai?DrnDdD'd s'?'npurd tu?4autrdar d(f'd rdcrMMrd 'Maid 'Gon'MMrDrdD'dcr?'nrd (f'dD'o4?dD'd rdomMutu-:d ?(f'Ma d mo'nrDmn d Ds'?'npur d tu?4autr d D' d crMMr d Srnurb ' d MrauMli d ' M d M'p??aMd n'(f'nuc'?aM ?d r?d smo'nrDmndxMdV'ncgaut b? d s'Gom?'?a d D' d a'nc' d rcb d crMMr d D'o'?D'?a d D' d r d omMutu-?d ?>r ?q,s ??>s ?T ?>r ?q,s ??>s? ?dxMd,?dar dudtmcdMftt''uGdrd smo'nrDmnd'Mai?DrnDdrcbd crMMrdtm?Mar?a: t?dr(f'Mada'nc'dxMd?mcxMdlf?tu-dD'd?dud>?>s : ?uGgdDm?tM?d' dVrcu am?uid(fi?autdD' dtrcgdD'dn'rttu-dH?RQ??daxd rdM'p??ad 'Gon'MMu-? +HQRP? [ ?2 2 } 1m=s({1?4 >>s } 1m=s( {1?2 >>s } 1m=s( {1?4 +V =2 (=q=s(( dddHe:2?? m?di =?(=q=?(( dtmnn'Mom?drd s'Gor?Mu-dD' doma'?tur dlu?MdrdM'pm?dmnDn': II:2:d?o utrtu-dD' d?RQ?dMmbn'df?dMuMa'crd(fgcut Q'ndar dD'dSuMfr ua?rndrD'(frDrc'?ad 'Mdromnartum?MdD' d ?RQ?dMs'MafDurnid rd Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?eE .................................................................. M'Srdro utrtu-dMmbn'd' dMuMa'crdD'dt ?a'nMdDsrupfrdud' dnrDutr dVuDnmGu d(f'dMsVrd 'MafDuradfau ua?r?ad s'Manftafnrd' 'tan??utrdr dtrogam dz:III:E:d Pmcd(f'd'MdanrtardD'dMuMa'c'Mdm?d' dcmSuc'?adD' MdiamcMdDsVuDnmp'?don'?dcm ad D' d onmarpm?uMc'? d xMd 'Mo'nrb ' d f?r d tm?anubftu- d ucomnar?a dD' d s'l'ta' d a?' d rd (fr M'Sm d 'MafDudDu?icut: d q'pfa d r d r(f'Mad l'a? d sfMdD' dDu?icutr d (fi?autr d 'Mai XfMaulutra:d8'?u?ad'?dtmcoa'd' d?mcbn'dD'dpnrfMdD'd ub'narad9dM-?dSfuadiamcMdudo'nd ar?a d DuSfua d pnrfM dD' d ub'nara d '? d ' d trM d D' d Df'M d rup?M d u d f? d nrDutr d 9 d xMd ucon'Mtu?Dub 'd'conrndr pf?rdc'?rdDsronmGucrtu-:d ? dtrogam dz?d(fr?dMsVrdD'Mtnuad rdn'rttu-dDsf?rdMm rdcm 4tf rdDsrupfrdrcbdf?d nrDutr dVuDnmGu ?dMsVrdc'?tum?radf?rdt'nardom 4cutrd'GuMa'?adn'Mo'ta'dr dc'tr?uMc'd D'd rdn'rttu-:dP'narc'?a?dVudVrdt'nardtm?anmS4nMurdo' d(f'dlrdr doro'ndD' dtmco 'Gd on'n'rtauf:d?(f'Madtmco 'GdomDnurdM'ndar?adD'dauofMd ???:::?? d tmcd ' dcg?uc d p mbr d? z ?:::?? d 'MM'?adr(f'MardM'pm?rdmotu-d rd(f'd lu?r c'?adMsVrd'conradrd s'MafDudon'M'?ara d on4Surc'?a: dA d l'a d xMd (f' d r d s'MafDudon'M'?ara?2d 'MdD'cmManrd s'GuMa4?turdDsf?dof?adD'danr?Mutu-d'?an'dSr dudM'nnr rDr:d?(f'MadxMdf?dof?ad'Mo'tur d n' rtum?radrcbd' dof?adDsu?l 'Gu-d9dSr dM'nnr rDrdon'M'?aradr dtrogam d2:dUudf?dof?a Dsu?l 'Gu- d Sr d 9 d M'nnr rDr d 'M d Dm?rSrd 'Gt fMuSrc'?ad(fr?dtmcdrdcg?ucdf?dSr mn d onmoudD'd rdcranufdV'MMur?rd'nrd?f du?drdcxM?d mnampm?r dr dpnrDu'?a?dorn rn'cdDsf?dof?adD'd anr?Mutu- d Sr d 9 d M'nnr rDr d (fr? d r(f'Mard M'pm?r d tm?Dutu- d ?m d 'M d tmco 'uGu? d rd tfnSrafnrd?f ? rdD'd rdMfo'nlgtu'dS'dDm?rDrd o'n d f?r d tmcbu?rtu- d tm?tn'ar d D' d S'tamnMd onmouMdD'd rdV'MMur?r:dw?dof?adtmcdr(f'Mad MsVrd mtr ua?rad'conr?adf?rdlf?tu-dDsm?rdauofMd P?UUP? d 3 d 6,2?? d ? d CHzDl?zo?:d 8mom ?putrc'?a?dr(f'Madof?adrtafrdtmcdf?r d bulfntrtu- d (f' d lr d (f' d 'M d Df'M d motum?Md ?upfnrde:6?dR'on'M'?artu-dD'd rdlf?tu-dD'd anr?McuMMu-do'ndrd rdn'rttu-dD' dnrDutr d VuDnmGu drcbdf?rdcm 4tf rdDsrupfr:dAMd on'M'?a'?dDf'Mdanr?McuMMum?MdDul'n'?aM?d f?rdtmnn'Mom?drdf?rdDu?icutrdrcbdcrMMrd luGrdud sr anrdrdf?rdDu?icutrdrcbd?RQ?: Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?e6 .................................................................. tm?anromMrD'Mdo'nd rdom 4cutrdMupfu?drtt'MMub 'M:d hrdM'pm?rdmotu-?do'n??dMsVrd'MafDuradrcbd'Mo'tur dra'?tu-d'conr?ad' dc4amD'd ?RQ?:dAMdon'M'?a'?dDmMdo'nlu MdD'dn'rtauSuara:dA?dr(f'MaMdo'nlu M?d'Mdn'on'M'?ard rd onmbrbu uara d D' d anr?McuMMu- d '? d lf?tu- d D' d s'?'npur: d?(f'Mar d c'?r d D'd o'nlu Md Msmba'?'? d 'l'tafr?a d f?r d anr?MlmncrDr d D' d ?mfnu'n d Mmbn' d 'M d lf?tum?Md Dsrfamtmnn' rtu-dD'd rdonmorprtu-do'ndar dD'dorMMrndD' dDmcu?uda'comnr dr dDmcu?ud '?'np4aut:dR'Mo'ta'dr MdDmMdo'nlu Mdon'M'?araM?d'MdanrtardD' dn'Mf arad?RQ?d'au(f'arad tmcdrdscrMMrdSrnurb 'sdudD' dn'Mf aradDsf?rdonmorprtu-df?uDuc'?Mum?r d'conr?ad' d o'nlu d '?'np4autd IRPd 'au(f'arad tmcd rd scrMMr dtm?Mar?as: d?(f'Mard tmcornrtu- dxMd u?a'n'MMr?ado'ndar dD'dl'nd?marnd(fu?'Mdromnartum?Mdn'r ua?rd' dc4amD':d?d rdlupfnrde:6 'MdomadSr mnrndtmcd'MdD-?rdf?d' 'Srad'l'ta'da?' d(f'dS'dcrp?ulutrado'nd s'l'ta' d D' MdpnrfMdD'd ub'naradmnampm?r Mdr dtrcgdD'dn'rttu-dau?pfaMd'?dtmcoa'dr d?RQ?:d 8rcbxdxMdomMMub 'd?marndf?rdcrXmndtm?anubftu-drd s'l'ta'dr?aua?' df?dtmodMfo'nrDrd rdbrnn'nrdt iMMutr?d(f'dS'dM'?1r rDrdrcbdf?rd g?urd?'pnrdrdronmGucrDrc'?ad?z:Ed ?tr 3cm :dA?dtmcornrtu-drcbd' dtrMdD'dDf'Mdrup?MdHlupfnrde:F??d'MdomadmbM'nSrndf?d 'l'ta'dr?aua?' dcm adcrXmn:d ? d r dlupfnr d e:F d 'M d n'on'M'?a'? d Df'Md lf?tum?M d D' d anr?McuMMu- d rcb d 'M d M'S'Md tmnn'Mom?'?aMd r?i mpf'MdD' d crMMrd tm?Mar?a:d Pmnn'Mom?'?drcbDf'Mdrd rdn'rttu-dD' dnrDutr d VuDnmGu drcbdDf'Mdcm 4tf 'MdDsrupfrduda'?'?d tmcdrdan'adDuMau?aufd' dl'adD'dtmnn'Mom?Dn'drd IRPM d (f' d tf cu?'? d '? d 'MaraM d D' d anr?Mutu-d Dul'n'?a? d ' d 8Uz d u d ' d 8Uzr: d ?mMan'?d tmcomnarc'?aMdonitautrc'?aduD4?autMdrcbd' d ?putdD'Mo rvrc'?ad'?'np4aut:d Uud'Mdlrd rdtmcornrtu-d'?an'd' Mdo'nlu Mdo'nd r d trM dDsf?r d u dDf'M d rup?M d 'Md oma d l'n d f?rd mbM'nSrtu-? d s'l'ta' d a?' d xM d D' d crp?uafDd ?upfnr d e:F: d AM d cmManr d r d lf?tu- d D'd anr?McuMMu-do'ndrd rdn'rttu-dD' dnrDutr d VuDnmGu d rcb dDf'Mdcm 4tf 'MdDsrupfr:d AMdn'on'M'?ard rd tmnbrd tmnn'Mom?'?ad rd s'Mara d D' d anr?Mutu- d 8Uz d u d rd tmnnn'Mom?'?aM d r d s'Mara d D' d anr?Mutu-d 8Uzr:dA?drcbD-MdtrMmM?dMsu?t mfdarcbx rdtmnbrdo'ndrd rdonmorprtu-drcbdcrMMrd tm?Mar?a? d xM d r d Dun? d f?r d onmorprtu-d f?uDuc'?Mum?r d rcb d f? d Vrcu am?uid tm?S'?tum?r :d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?eF .................................................................. M'cb r?ado'n?d s'l'ta'dr?aua?' dxMdcm adcxMdonm?f?turad'?d' dtrMdDsf?rdrupfrdMud'Md tmcorn'?d 'Mdtmnb'Mdmbau?pfD'Mdrcbd?RQ??do'n?d?mdMud'Mdtmcorn'?d 'Mdtmnb'Md rcbdcrMMrdluGr:dqsr(fgd'MdomadD'Dfund(f'd rdtmmnD'?rDrdD'dn'rttu-d'MdtmcomnardD'd cr?'nr d M'cb r?a d '? d amaM d DmMd trMmM? d o'n? d ' M d cmD' M d mnampm?r M d r d r(f'Mard tmmnD'?rDrdrbMmnb'uG'?dcm ard'?'npurd'?d' dtrMdD' dMuMa'crdrcbdf?rdMm rdrupfr? n'Dfu?ad rdn'rtauSuarad'?dlmncrdDs'l'ta'dr?aua?' d'?dtm?M'(??tur:d III:d?MMup?rtu-dDs'MaraMd(fi?autMdcuaXr?vr?adanrX'ta?nu'Mdt iMMu(f'M III:?dUm ftum?Mdt iMMu(f'Mdrdonmb 'c'Md(fi?autM A?d' Mdtrogam Mdr?a'numnMdMsVrdrMM'?1r rad' dl'adD'd(f'd'?dcm a'MdmtrMum?MdxMd ucomMMub 'dn'Mm Dn'df?donmb 'crdDm?rad'conr?ad rdDu?icutrd(fi?autrdD'pfadr dtmMad tmcofartum?r dDsf?r d onmorprtu- dM'?M' d ronmGucrtum?M: dUsVr d rMM'?1r ra d tmc d r omMMubu uarad rdcmD' ua?rtu-dD' dMuMa'crdcuaXr?vr?adVrcu am?ur?MdD' dauofMdMuMa'crd9d br?1dudarcbxd rdomMMubu uaradDs'conrndn'Dfttum?MdD'dDuc'?Mum?r uaradrcbdr pf? tnua'nudtmcdomadM'nd' dD'dM' 'ttum?rndr pf?rdtmmnD'?rDrd(f'dDm?udD'do'ndMgdf?rdSuMu-d ico urdD' dl'?mc'?dtmcd'Mdlr?d'?d' d?RQ?: w?rdr anrdomMMubu uaradxMd rdD'dn'?f?turndtmco 'arc'?adrd rdDu?icutrd(fi?autrdud anrtarnd' dMuMa'crdtmcdf?dMuMa'crdt iMMutdmdn'on'M'?arnd' dMuMa'crdt iMMutrc'?adudrd ornaundDsr(fgdmba'?undDsr pf?rdcr?'nrdu?lmncrtu-d(fi?autr:dhrdn'??turdrdu?t mfn'd l'??c'?MdD'd?rafnr 'Mrd(fi?autrdtmcd s'l'ta'da?' dXrd'Maiduco gtuard'?d' donmoud o r?a'Xrc'?ado'n?darcbxdanmbrn'cdf?rdM4nu'dD'dDulutf araMdrl'puD'M:d?dcxM?darcbxd anmb'cdDulutf araMdrd sVmnrdDsrMMmturndanrX'ta?nu'Mdt iMMu(f'Mdrd'MaraMd(fi?autM:dw?rd D'd 'Md ucuartum?MdcxMd'SuD'?aMd9dar?d'SuD'?ad(f'dMsrMM'?1r rd'?d' donmoud?mcdD'd rd Du?icutr d 9 d xM d r d (fr?aua?rtu- d (f' d uco utr d sf?r d n'Mo'ta' d sr an'? d f?r d ?utrd anrX'ta?nurdt iMMutrd?md'?Mdomadromnarndf?rdDuManubftu-dSubnrtum?r d?udDm?rn,?mMdf?rd uD'r dMmbn' d r d n'rtauSuara dD' dMuMa'cr? dM'con' d tr d brMrn,M' d '? d f? d tm?Xf?a dD'd anrX'ta?nu'Mdo'n d ar dD' d l'n d 'MarDgMautr: dUud 'Mdaxd '? d tmcoa' d r(f'Mad l'a? d s'Mar Sud Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?e? .................................................................. tmcofartum?r dxMdn' raufdXrd(f'd' d?mcbn'dD'danrX'ta?nu'Mdt iMMu(f'Md?'t'MMinu'MdHD'd smnDn'dD' Mdcu um?Md'?dcm a'MdmtrMum?M?domadn'Mf arndarcbxdonmVubuauf: q'Mar(f'cdDm?tMdDmMdan'aMdD'd 'MdanrX'ta?nu'Mdt iMMu(f'M?df?dxMd' dtmMadr ad(f' ama d u d r dMuco ulutrtu- d c'amDm ?putr dM'pf'uG'? d tmcomnar?a d u d f? dM'pm? d xM d rd ?'t'MMuaradD'dcmD' Mdo'ndar dDsrMMucu rnd rdanrX'ta?nurdt iMMutrdrd rd (fi?autrdD'd cr?'nr d (f' d f? d tm?Xf?a d D' d anrX'ta?nu'M d t iMMu(f'M d ofpfu? d ml'nun d r d cra'uGrd u?lmncrtu- d (f' d f?r d onmorprtu- d (fi?autr: d q' d trnr d r d onuc'n d an'a d 'Mc'?ara? d ' d c4amD' d (f' d 'M d on'M'?arni d '? d r(f'Mar dM'ttu- d on'ax? d r 'fXrn d r(f'Mar d tinn'prd tmcofartum?r dn'Dfu?adtm?MuD'nrb 'c'?ad' d?mcbn'dD'danrX'ta?nu'Md?'t'MMinu'Mdo'nd ar dD'domD'ndtm?S'npund?fc4nutrc'?adf?rdDuManubftu-dD'dauofMdSubnrtum?r :dQ'ndar d D'dtmc'?arnd' dM'pm?drMo'ta'dxMdu?a'n'MMr?adonuc'nd'Mc'?arnd' dtm?t'oa'dD'd gcuad t iMMutdD'd rdc'ti?utrd(fi?autr:d nR?Rh1,q?CR-??,C?cD?Ra?1DC-d,Ca?gm-dq,Ca hrdlgMutrdt iMMutrdD'Mtnufdcm adMrauMlrta?nurc'?ad ' dtmcomnarc'?adDsmbX'ta'Md crtnmMt?outM:d?mdmbMar?a?dlr rd(fr?d'MdanrtardD'dD'Mtnufn'dMuMa'c'MdcutnmMt?outMd ar dudtmcdMsVrd 'Mc'?aradr?a'numnc'?a:dA d gcuadt iMMutdD'd rdc'ti?utrd(fi?autrd Mmnp'uGdD'd rd?'t'MMuaradDsfbutrnd'?dr pf?dof?adf?rdanr?Mutu-d'?an'd' MdMuMa'c'Md(f'd 'MdomD'?danrtarndrcbdc'ti?utrdt iMMutrdud' Md(f'dMsVr?dD'danrtarndrcbdc'ti?utrd (fi?autr: d hs'GuMa4?tur d Dsr(f'Ma d gcua d ?m d Vr d 'Mara d cru d onmSra d rcb d nupmn5 d 'Md tm?MuD'n'?dDuS'nM'MdSrnurb 'Md(f'domD'?dl'ndtr?Surnd' dtmcomnarc'?adDsf?MdMuMa'crd D' d (fi?aut d rd t iMMut?d r d crMMr?d srttu-? d f? d ?mcbn' d (fi?aut::: d r d tm?Dutu- dcxM rtt'oarDrdrd rdbub umpnrlurdxMd rdD'd(f'd rdtm?Mar?adD'dQ r?t?da'?D'uGudrd?'nm:z w?dom?adn'tfnn'?ad'?an'd rdc'ti?utrdt iMMutrdud rd(fi?autrdxMd rdonmorprtu-dD' or(f'aMdprfMMur?MdrdanrSxMdD'doma'?tur Mdornrb? utM:d?uG?dxMdD'pfadrd(f'dar?ad' Sr mnd'Mo'nradD'd rdomMutu-dtmcd' dD' dcmc'?ad'MdomD'?don'Dund'?dr(f'MaMdtrMmMd 'conr?a d c'ti?utr d t iMMutr: d?uG?dxMd litu dDs'?a'?Dn' d a'?u?a d '? d tmcoa' d ' d t rnd ornr ?' uMc'd'GuMa'?ad'?an'd' da'mn'crdDsAVn'?l'Madud s'(frtu-dD'dhumfSu ':d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?e0 .................................................................. A? d r(f'Ma d tm?a'Ga? d ?' 'n d D'Mtnuf d cuaXr?vr?a d '(frtum?M d r d onmorprtu-d M'cut iMMutrdDsf?dor(f'adDsm?'M?e:dUudMs'Gor?D'uGd' doma'?tur d'?df?rdM4nu'dlu?Mdrd M'pm?dmnDn'dr dSm ar?adDsf?dof?a dgJ dMsmbaxd' dM'p??adVrcu am?uido'ndrd s'?amn?d (frDniaut? N? [ ?????fe ?z 1 ? g z? i p? ' J8=g[ gJ(? ?z =g[ gJ ( 8 N J=g[ gJ( dddHe:2z? m? d' J8 dudN J d M-?d ' dS'tamndpnrDu'?a dud r d cranuf dV'MMur?r d r d of?a dgJ d HM-?d f?uDuc'?Mum?r Md'?dr(f'MadtrM?dxMdrdDun?d'Mtr rnM?:dd?dtm?au?frtu-d'Mdomad'Marb undf?d or(f'adprfMMuidn'l'n'?turadn'Mo'ta'dr(f'Madof?at : ?=g Vq(? 'Go=z?ddV?J?g[ gJ?z? z?,ddV r J8=g[ gJ (? z?,ddV?J( dddHe:22? A Mdornic'an'Md?J dud?J dM-?d?mcbn'Mdtmco 'GmMd(f'dD'a'ncu?'?d srco uafDdud rd ?mncr ua?rtu-dD' dor(f'adDsm?'M?dn'Mo'tauSrc'?a:d?dcxM?d?J dVrdD'dM'ndf?d?mcbn'd ?'praufd'?d rdM'Srdornadn'r dudxMdu?a'n'MMr?adD'Martrnd(f'd rd?mncr ua?rtu-dD'o4?dD'd ?J dD'pfadrd(f'd ucg?? ?=g?V?q(? 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'nz??zzd o'nc'adfau ua?rndanrX'ta?nu'Md'?df?dtm?a'Gadc'tr?m(fi?autd lu?Md' dof?ad(f'd ?'Maradlu?r dD'd 'MdanrX'ta?nu'MdM?au ua?rdo'ndp'?'nrndD'?MuaraMdD'd ?up?'ndrcbd 'Md(f'd'Mdtr tf '?donmbrbu uaraMdD'danr?Mutu-:dA dc4amD'dD'd?' 'n9 ?up?'n d xM d f? d on't'D'?a d lmnvr dMup?ulutrauf d D' d r d onmomMar d (f' d 'M d D'ar r d rd tm?au?frtu-?damadud(f'dr(f'Mad?mfdc4amD'du?anmDf'uGdcxMdronmGucrtum?M: III:2: dA d c4amD' dD' dt,dd,d'd brMra d '? d r d onmX'ttu- dD' d or(f'aMdDsm?rd t iMMutM UD?CI,rC,T?cDR?1?q cD ?(f'Mardc'amDm mpurd(f'dr?mc'?rn'cdrbn'fXrDrc'?ad ?Q?Pd '?Mdo'nc'ani d rMMmturndf?rdMm rdanrX'ta?nurdrdDuS'nMmMd'MaraMdSubnrtum?r Mdr Vmnrdrcbdf?d'SuD'?ad 'Mar Sudtmcofartum?r :d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?Ee .................................................................. 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D? d AM d tr tf r d ' d tm?Xf?a d D'd n'tmbnuc'?aMd'?an'd' dor(f'adDsm?'Mdud' d tm?Xf?a d D' d d lf?tum?M d on?ou'Md Subnrtum?r Md'Mtm ua:d8r dudtmcdMsVrdSuMad r?a'numnc'?a?d' dcmc'?adD'd rdornagtf r d lmncr d orna d D' d r d lrM' d D' d r d lf?tu-d Dsm?r5 d o'n d ar?a? d f? d crXmn d cmc'?ad tm?Df'uGdrdf?rdcrXmndln'(??turdudxMd on'SuMub ' d (f' d ruG? d rlrSmn'uGu d f? n'tmbnuc'?a d crXmn d rcb d f?r d lf?tu- d on?our d Dsf? d 'Mara d Subnrtum?r d cxMd 'Gtuara:d?d rd lupfnrde:?d 'MdcmManrdf?d 'G'co 'dD'd 'Mdlf?tum?Mdon?ou'MdDsf?d mMtu ? rDmn d Vrnc??ut d n'on'M'?arD'Md Xf?arc'?adrcbd' doma'?tur :dA dc?Df d r d(frDnradDsr(f'MaMdn'tmbnuc'?aMdD-?rd rdonmbrbu uaradD'd(f'd' dMuMa'crd'Mdanmbud'?d f?d'Marad'Martum?rnudDm?ra: '?dAMdMfc'?d 'Mdonmbrbu uaraMdmbau?pfD'MdrdornaundD'd 'MdDul'n'?aMdanrX'ta?nu'M:dA?d onu?tuoudMsVr?dDsrtfcf rndcxMdanrX'ta?nu'Mdlu?MdrdrMMm und rdtm?S'np4?turd?fc4nutrd D'd rdDuManubftu-:d l?dhsmba'?tu-dD'domb rtum?Mdar dudtmcdMsVrdD'Mtnuad'?d srornaradr?a'numnduco utrd (f' d 'M d omb rnr? d 'MaraM d t iMMutrc'?a d onmVubuaM? d mba'?u?a d f?r d DuManubftu-d r??cr rc'?adrco r:dPr dlu anrndr(f'MaMdn'Mf araMda'?u?ad'?dtmcoa'd s'?'npurdciGucrd D'd 'MdanrX'ta?nu'M:d ?upfnr d e:?? d A? d r(f'Mra d lupfnr d omD'c d S'fn'd n'on'M'?aradf?doma'?tur dVrnc??ut:d?aXr?vr?ad r d ln'(??tur d Dsr(f'Ma d omf d MsVr? d tr tf ra d ' Md Vrnc??utM d tmnn'Mom?'?aM d 'conr?a d lf?tum?Md Ds?'ncua'5 d MsVr? d n'on'M'?ara d r d r vrD'Md rnbuaninu'M: d h'Md f?uaraMdD' d 'Md tmmnD'?rD'Md 'Md omD'?dtm?MuD'nrndarcbxdrnbuaninu'M: Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?E6 .................................................................. uDaCC, d??D?qmc,acD? UsVr?d'MafDuradf?rdM4nu'dD'dn'rttum?Mdtm?MuD'nrD'Md'?dronmGucrtu-dtm ? u?'r : ?cbDf'MdM-?dtm?'pfD'Mdudln'(??arD'Mdrd rdbub umpnrlurdudo'ndar?adMmSu?ad'conrD'Md o'ndar dDsrSr frnd rdbm?DradD' Md?mfMdc4amD'M:dAMdomD'?dn'on'M'?arndtmcdrdf?d'Marad D' d anr?Mutu- d (f' d omnar d r dDmMd 'MaraMd rMucoa?autM5d r dMfo'nlgtu' d 'Mtm uDr d o'n d rd cmD' ua?rnd 'Mdn'rttum?MdxMdf?do rdbuDuc'?Mum?r dD'd rdhAQU: Aa?IDaCC,T?Q:N? hrdn'rttu-dxMdlmnvrd'Gm4nputr?drcbdf?rdDul'n4?turd'?an'dn'rtaufMdudonmDfta'MdD'd ?22d?tr 3cm :dQnitautrc'?ad?mdVudVrdbrnn'nr?drcbdf?d'MaradD'danr?Mutu-ded?tr 3cm d o'ndMmbn'dD'd srMgcoamardD'dn'rtaufM: ?d tm?au?frtu- d sro utrtu- dD'dD' d ?Q?Pd 'Gup'uGd ' d ti tf dD' d 'Md lf?tum?M d Subnrtum?r Mdon?ou'Mdrd srMgcoamardD'donmDfta'M:dA?dr(f'MadtrM?dVr?d'Maradmbau?pfD'Md 'conr?adf?rdnfau?rd?fc4nutrdD'dti tf dD'dSr mnMdudS'tamnMdonmouMdrdornaundD'd rd n'on'M'?artu-d'?dq?RdHn'on'M'?artu-dD'd rdSrnurb 'dDuMtn'ar:dq'd sr?p 4M?dU,?CIDqD? iaI,atRD?uDrID?Ddqaq, d? q'dtrnrdr dti tf dD'danrX'ta?nu'M?dMsVrd'Mtm uad' dtnua'nudDsrafnrn, 'Mdrdf?da'coMd Dm?ra:dA dcrXmndu?tm?S'?u'?adrMMmturadrdr(f'Madtnua'nudxMd' dD'd(f'dMudrtrb'?drdf?rd n'pu-dD'd s'MorudD'dtm?lupfnrtum?Mdcm adDul'n'?a?d 'Mdln'(??tu'MdrMMmturD'MdomD'?d M'ndDul'n'?aMdudo'ndar?ad rdtm?Manfttu-dD'd rdDuManubftu-dSubnrtum?r du?rD'(frDr:d?md mbMar?a?d' dauofMdD'dMfo'nlgtu'd'conrDrd'M(fuSrdr(f'Madu?tm?S'?u'?a?d' dtr?r dD'd onmDfta'Mdcr?axd rdM'Srd'Manftafnrd(frDniautrdDfnr?adamard srMgcoamar:d h'Mdtm?Dutum?Mdu?utur MdMsVr?d'Mtm uadD'dcr?'nrd(f'dama'Md 'MdanrX'ta?nu'Mda'?'?d f?rdcra'uGrd'?'npurdamar ?do'n?d 'MdtmmnD'?rD'Mdu?utur MdMsVr?drMMup?radM'pfu?adf?rd DuManubftu-dprfMMur?rdr dSm ar?adDsf?dof?adrd s'MorudD'dtm?lupfnrtum?Mdrd srMgcoamard D'dn'rtaufM:d?(f'Madof?adxMd' dH2:0?dJ:FE?:dd Q' d(f'dlrdr dti tf dD'dn'tmbnuc'?aM?dMsrDXf?ardrdcmD'dDs'G'co ulutrtu-d rdarf rd Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?EF .................................................................. '?d rd(fr d'Mdomadtmcornrndo'ndrdf?rdomMutu-dlu?r dudcmc'?adlu?r dDm?raMdtmcd s'Marad (f'dMsrMMup?rnurd'conr?ad' dbu??u?pdVuMampnicutdtmu?tuD'uGdrcbd s'MaradcxMdomb rad 'conr?ad' d?Q?P:d?DDutum?r c'?admba'?ucdt'nardomb rtu-dr Md'MaraMdonmo'nM:d qf pf hbin v?=0 v?=1 v?=2 v?=3 v?=4 v?=5 J:?2 ?2:2J ? J:FF ?:JJ J:E6 J:ze J:?e J:J0 J:0F ?6:2J ? J:e0 ?:JJ J:FJ J:?0 J:J? J:J? J:?F ?6:eJ ? J:Ee ?:J J:Fz J:z0 J:?J J:JE ?:Jz F:?2 J ?:JJ J:6F J:?6 J:JF J:J? J:JJ h'Mdomb rtum?Mdmbau?pfD'MdM-?d tmnn'ta'M?dar dud tmcd 'Mdomad tmconmSrndrd rd bub umpnrlurdar?da'?nutrdtmcd'Go'nuc'?ar z2: TMdn' 'Sr?ad'Mc'?arnd(f'dMsVr?dn'r ua?radonmS'MdrcbdDul'n'?aMdprcc'M:dA Md n'Mf araMdon'M'?araMdtmnn'Mom?'?drdf?r d?? 2J mbabd (f'dMsVrdtn'pfadmomnaf?rdo'nd a'?un d f?r d rco rDr d 'Mtru'?a d '? d tmcornrtu- d r d srco rDr dD' d tr?r : d ?rnurtum?M d nrm?rb 'MdDsr(f'Madornic'an'd?mdonmSm(f'?df?rdSrnurtu-dD'd rdomMutu-dD' dciGucd D'd rdDuManubftu-d?udD'd rdM'Srdrco rDr:d ?d 'Mdlupfn'Md0dud?Jdanmb'cd' Mdn'Mf araMdD' dc4amD'dXrd'?dlmncrdD'dDuManubftu-: hrdlupfnrd0dtmnn'Mom?drdf?rdDuManubftu-dm?dMsVr?d'conrad2JJdanrX'ta?nu'M:d?d rdlupfnrd ?Jdanmb'cdf?rdn'on'M'?artu-do'ndrdDf'Md'?'npu'MdcxMdr a'Md(f'd'?d' dtrMdr?a'numn:d Pm?tn'arc'?adM-?dD'dJ:ezd'?dudJ:?Fd'?:dUsVr?d'conrad2JdanrX'ta?nu'M:d Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?E? .................................................................. Aa?IDaCC,T?N 2) ? N [ )N 2?N 2 ) [ N ? )N 2 ?(f'MardxMdf?r dn'rttu-dD'db'Mtr?SudDsVuDnmp'?d (f' dn'on'M'?a'c darcbxdD'd lmncrdtm ? u?'r :dA MdDmMdpnrfMdD'd ub'naradM'nr?d 'MdDuMai?tu'MdP,?dD'd sVuDnmp'?d (f'd 'Mdb'Mtr?Sur:dqfnr?ad ' db'Mtr?Su?d ' Mdr an'MdVuDn?p'?MdM-?df?duco'Duc'?ad 'Ma4nutdrd(fr M'Sm dr anrdc'?rdD'dc'tr?uMc'dD'dcr?'nrd(f'druG?dlm?rc'?ard'?dorna d sronmGucrtu-:d R'Mo'ta'drd rdn'rttu-d'MafDurDrdrd srornaradr?a'numn?d rdonu?tuor dDul'n4?turdxMd rd ln'(??turdD' Mdtr?r MdrMucoa?autMd(f'dxMdcxMdbruGrd'?dr(f'MadtrM: hs'MafDudDsr(f'MadMuMa'crd'?Mdo'nc'adomMrndf?d'G'co 'dMmbn'd rdnrouD'MrdD'd ?upfnrde:?J?dUs'GomM'?dDf'MdDuManubftum?M:dUsVr?dmbau?pfadcuaXr?vr?ad2JdanrX'ta?nu'M:dAMd DuMau?p'uG'?do'nd s'?'npurdamar dD'd 'MdanrX'ta?nu'M:dhrdlupfnrdD'd s'M(f'nnrdtmnn'Mom?drdJ:ez'?dud rd D'd rdDn'ardrdJ:?Fd'?5d?mdmbMar?a?d 'MdDuManubftum?MdM-?dM'cb r?aMdD'pfadr dDul'n'?adn'ornauc'?adD'd s'?'npurdSubnrtum?r drd su?Mar?ad'Mtm uado'ndar dD'dornrnd rdDu?icutr:dhsrco rDrdD'd rdDuManubftu-d on'SuMardo' d?QQ?Pdamn?rdrdM'n?dD'd?mf?d 'fp'nrc'?adMfo'numn: ?upfnr d e:0? d quManubftu- d Subnrtum?r d mbau?pfDrd cuaXr?vr?ad2JJdanrX'ta?nu'M:dhrd g?urdS'nc' rdcmManrd rd DuManubftu- d Subnrtum?r d mbau?pfDr d rcb d bu??u?pd VuMampnicut:dh'Md g?u'Mdb rS'MdcmMan'?d rdMftt'MMuSrd tm?S'np4?turdD' dc4amD'd?QQ?Pdudu ?fMan'?d rdniouDrd tm?S'np4?turdD' dcra'uG:dTMdD'Martrb 'd rdM'cb r?vrd D'd rdDuManubftu-d'Gt'oa'do' d(f'dlrdrd srco rDr?df?rd cutrdMfo'numn?dD' d?QQ?P: Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?E0 .................................................................. tm?S'np4?tur dD' d c4amD' d ?Q?P: d?d r d lupfnr d e:?? d 'M d omD'? dmbM'nSrn dDf'Md DuManubftum?M?df?rd(f'dtmnn'Mom?drd2Jdrtfcf rtum?MdD'danrX'ta?nu'Mdudf?rdr anrd(f'd tmnn'Mom?drd6J:dUsVudron'turdtmcd rd(f'dtmnn'Mom?dr d?Q?Pdonitautrc'?ad?mdSrnurd c'?an'd(f'd rd(f'd tmnn'Mom?dr dt,dd,d'd VuMampnicutd '?trnrdcmManrdDul'n4?tu'Md n' 'Sr?aMdrd rdomb rtu-dD' dM'pm?d'MaradHorMMrdD'dJ:F?drdJ:6z?: ?Sr frtu-dD' Mdn'Mf araMdmbau?pfaM UsVrdo r?a'Xrad sfMdDsf?rd?mSrdc'amDm mpurdo'ndr dt,dd, ' :d?(f'Mardc'amDm mpurd ax d r dM'Sr d brM' d a'?nutr d '? d ' Md 'MafDuMd c'?tum?raMdD' d ?' 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VuMampnicutd'MdomadMu?a'aua?rnd'?d rd mbM'nSrtu-dD'd(f'dr?a'numnc'?adtrDrdanrX'ta?nurdomnarSrdrdomb rndf?d?utd'Maradu?d'?d tr?Su?d' d ?QQ?Pd' Mdomb rdamaMdrcbdtrDrdanrX'ta?nurdmba'?u?adf?rdtm?S'np4?turd cm adcxMdS' mv:ddhrdtmcornrtu-drcbdr an'Mdc4amD'MdD'dt,d ,d' dxMdf?rdo'nMo'tauSrd arcbxdu?a'n'MMr?a: ?upfnrde:???dUsmbM'nSrd srtfcf rtu-dD'dn'Mf araMdD' d?QQ?Pdo'ndrdtrDrd?mSrdanrX'ta?nurd'?db rfdu '?dS'nc' d' dn'Mf aradlu?r dD' dt,dd,d' dVuMampnicutdo'ndrd2JdH'M(f'nnr?dud6JdHDn'ar?danrX'ta?nu'M?d n'Mo'tauSrc'?a:dA Mdn'Mf araMdD' d ?QQ?Pda'?D'uG'?dniouDrc'?adtrodrdf?rdDuManubftu-duD4?autrd c'?an'd(f'dxMdomMMub 'd'?trnrdanmbrndDul'n4?tu'Md'?d' dtrMdD' dbu??u?pdVuMampnicut: Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?6J .................................................................. A Mdn'Mf araMdon' ucu?rnMdmbau?pfaMdlr?do'?Mrnd(f'df?rdrco rDrdrnbuaninurdD' d or(f'a dDsm?'Md ?mdonmSmtr d r a'n?rtum?MdMfbMar?tur Md r d ti tf dD' dDuManubftum?Md Subnrtum?r M:dA?dtr?Su?d s'Mar Sudtmcofartum?r d(f'd'Mdo r?a'Xrdrd sVmnrdD'dtm?S'npund 'M dDuManubftum?Md (f'Dr d tm?luncra: d?d br?Dr dDsruG?? d xMd ucon'Mtu?Dub ' d 'conrnd MuMa'c'MdDul'n'?aMdo'ndar dD'danmbrndomMMub 'Mdof?aMdl'b 'MdD' dc4amD':d 8rcbx d xM d tm?S'?u'?a d 'Go mnrn d tmc d 'M d tmcomnar d ' d c4amD' d lmnr d D' d rd tm ? u?'r uarad'Manutar: I?:dPm?t fMum?M A?d' MdD'M'?Sm forc'?aMd(f'dMsVr?dDfadrda'nc'd '?dr(f'Madtrogam dVudVrdf?d D'?mcu?rDmn d tmc? d prun'bx dM'con' d xM d ?'t'MMinur d f?r d ronmGucrtu-: d?(f'Mard ronmGucrtu-d'MdDffdrda'nc'drd?uS' dc'amDm ?putdDu?Md' dtrcodD'd rdDu?icutrd (fi?autr d 9 d Xr d Mupfu d 'conr?a d f? d cmD' d o'n d Muco ulutrn d ' d onmb 'cr d m d f?rd ronmGucrtu-dMmbn'd 'Md'(frtum?Md9dmdbxd'Mdomnardr da'nn'?1dD'd rdc'ti?utrdt iMMutr tr?Sur?adtmco 'arc'?ad' dornrDupcrdmdbfMtr?adf?dtrcodu?a'nc'Dud'?an'damaMdDmMd ornrDupc'MdtmcdxMd rdDu?icutrdM'cut iMMutr:d A?d ' d trcodD'd rdcmD' ua?rtu-dMsVrdompfad tm?Mararnd ' dpnr?d 'Mlmnvd(f' dxMd ?'t'MMrnudo'ndar dDsronmGucrnd' dcmD' drd rdn'r uara:dhrdbub umpnrlurd'Maido '?rdD'd Mfpp'nuc'?aMd o'n? d '? d ama'M d ' 'M dM'con' dVu dVr d f? d of?a dDsrnbuanrnu'ara dDulgtu d Ds'?lmtrn:d?uG?dlrd(f'df?d'MafDudDu?icutdn'r ua?rad'?dr(f'Ma'Mdtm?Dutum?MdMupfudD'd l'adf?d'MafDudDmb '?df?d(f'danrtardrcbdamard rdtmco 'GuaradD'd rdDu?icutrdudf?dr an'd (f'danrtard rdcr?'nrdD'dcr?a'?und' dcmD' dluMutmcra'ciautd'?dtm?arta'drcbd r d n'r uara: ?(f'MaMdDfba'MdMs'MSr'uG'?dMudMs'con'?dc4amD'MdronmGucraMd(f'd' ucu?'?dpnrfMd D' d ub'nara dmd fau ua?'? d ronmGucrtum?MdDsr pf?r d c'?r d r d r dDu?icutr d (fi?autr:d Al'tauSrc'?a? d cm a'MdS'prD'MdxMdomMMub ' d anrtarn d ' d onmb 'cr dM'?M' dVrS'n dD'd cmD' ua?rnd' dMuMa'cr:d?mdmbMar?a?d rdl'u?rd'?dr(f'MardmtrMu-dn'trfd'?d rdon?ourd Progam de:dqu?icutrdddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd?6? .................................................................. XfMaulutrtu-dD' dc4amD':dA dpnr?d?mcbn'dD'dc4amD'Mdon'M'?aMdxMda'Maucm?udMflutu'?ad D'd(f'dtrodMm ftu-d?mdxMdonmfdMrauMlrta?nurd'?damaMd' MdrMo'ta'Mdtmcdo'ndDm?rnd' d onmb 'crdo'ndar?tra:d hsromnartu-dcxMdu??mSrDmnrdn'r ua?rDrd'?dr(f'Madtrogam dxMd' dcmD' dDsrtauSrtu-d a4ncutr dD' d n'rttum?Md 'conr?a d nrDurtu- dD' d a'nrV'na? d ud ' d c4amD' d ?Q?P: dA Md n'Mf araMdmbau?pfaMdud 'GomMraMdcxMd rcf?a d lr? d o'?Mrnd (f' d rcbDf'Md g?u'MdM-?d onmc'a'Dmn'Mdo'n?dtr d'?trnrdronmlf?Dun,Vudo'ndrtrbrndD'dD'lu?und 'MdSunafaMdud' Md D'l'ta'MdD' MdM'fMdo r?a'Xrc'?aM?dn'lu?rn, mMdudlu?r c'?adDfn, mMdrdofb utrtu-: ?ub umpnrlur ?: d q: d ?: d 8r??mn: dxdqI cmCq, d ? q ? lmadqm1 ? SDCfad,C?b ? 3 ? q,1D(cDrDdcDdq? rDI?rDCq,yD:dw?uS'nMua1dUtu'?t'd?m?MdHzJJF? z: d ?:d ?'MMurV: dlmadqm1?SDCfad,C?:d ?mnaVd,d ?m r?DdQfb uMVu?pdPmcor?1d H?06F? 2: d?:d?M?rndud?:dU:dPr?cr??dfD?M mIdaR? G?)fD1,CaR?of??,C??d68?dzF0edH?0F?? e: dC:d?:d?mnaV?d ?:d ?:d ?t??d?:d ??t? 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R. Moreira,1,2 and Josep Maria Bofill1,3 1Institut de Qu?mica Te?rica i Computacional, Universitat de Barcelona, IQTCUB, C/ Mart? i Franqu?s 1, E-08028 Barcelona, Spain 2Departament de Qu?mica F?sica, Universitat de Barcelona, C/ Mart? i Franqu?s 1, E-08028 Barcelona, Spain 3Departament de Qu?mica Org?nica, Universitat de Barcelona, C/ Mart? i Franqu?s 1, E-08028 Barcelona, Spain (Received 28 January 2013; accepted 1 April 2013; published online 2 May 2013) A comparison model is proposed based on the L?wdin partitioning technique to analyze the differences in the treatment of electron correlation by the wave function and density functional models. This comparison model provides a tool to understand the inherent structure of both theories and its discrepancies in terms of the subjacent mathematical structure and the necessary conditions for variationality required for the energy functional. Some numerical results on simple molecules are also reported revealing the known phenomenon of ?overcorrelation? of density functional theory methods. ? 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4802200] INTRODUCTION The fundamental mathematical formulation of the non- relativistic many electron problem for a system defined by n electrons and N nuclei in interaction is the time indepen- dent Schr?dinger equation. The corresponding exact solutions provide the essential quantum-mechanical description of each electronic state in terms of the different n electron wave func- tions ?(r1s1, . . . , rnsn). The time independent Schr?dinger equation for a system defined by n electrons and N nuclei in interaction can be written as a Rayleigh-Ritz quotient given by E [?] = ??| H? |?? ?? | ?? (1) in which the Hamiltonian operator is defined as H? = T? + V? + W? (2) with T? (r1, . . . , rn) = ?12 n ? i=1 ?2(i), V? (r1, ? ? ? , rn;R1; ? ? ? ;RN ) = n ? i=1 N ? I=1 ?ZI |RI ? ri | = n ? i=1 v(ri ;R1; ? ? ? ;RN ), W? (r1, . . . , rn) = n?1 ? i=1 n ? j=i+1 1 ? ?ri ? rj ? ? = n ? i>j=1 1 rij . In this expression, the first term stands for the kinetic energy of the electrons, the second arises from the external poten- tial v(ri; R1; . . . ; RN) generated by the nuclei and 1/rij is the two electron interaction. This Hamiltonian operator de- fines, along with its boundary conditions, an elliptic second order differential equation and the wave functions must sat- isfy some specific conditions to be an acceptable variational solution of Eq. (1). Since the non-relativistic many-electron Hamiltonian does not act on spin coordinates, anti-symmetry and spin restrictions must be imposed ad hoc in the wave function,?(r1s1, . . . , rnsn), so as to satisfy the Pauli principle and spin symmetry requirements of the quantum mechanical state of the system. Defining ? 1 and ? 2, the usual one- and two-electron den- sity matrices directly obtained from the n-electron wave func- tion ?(r1s1, . . . , rnsn), as ?1(r1; r1?) = n ? rn ? ? ? ? r2 ? sn ? ? ? ? s1 ?(r1s1, ? ? ? , rnsn) ???(r1?s1, ? ? ? , rnsn)ds1 ? ? ? dsndr2 ? ? ? drn, (3) ?2(r1, r2; r1?, r2?) = n(n? 1)2 ? ? rn ? ? ? ? r3 ? sn ? ? ? ? s1 ?(r1s1, r2s2, ? ? ? , rnsn) ???(r1?s1, r2?s2, ? ? ? , rnsn)ds1 ? ? ? dsndr3 ? ? ? drn, (4) the most compact expression for the energy of the n-electron system in the field of N fixed nuclei can be written as E[?1, ?2] = ?12 ? r1=r1 ? [? ? ?T ?1(r1; r1?)]dr1 + ? r1 v(r1;R1; . . . ;RN )?1(r1)dr1 + ? r2 ? r1 ?2(r1, r2) r12 dr1dr2 (5) in which the many-electron quantities ? 1(r1) and ? 2(r1, r2) are the diagonal elements, ? 1(r1) = ? 1(r1; r1) and ? 2(r1, r2) = ? 2(r1, r2; r1, r2), of the spinless one- and two-electron den- sity matrices (Eqs. (3) and (4)), respectively.1?3 Notice that the former is the one-electron density ?(r) commonly used 0021-9606/2013/138(17)/174107/5/$30.00 ? 2013 AIP Publishing LLC138, 174107-1 Downloaded 08 Sep 2013 to 161.116.100.92. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 174107-2 Caballero, Moreira, and Bofill J. Chem. Phys. 138, 174107 (2013) in electronic structure theory. Therefore, an accurate predic- tion of the energy of a given system in a given electronic state requires a reasonable estimate of ?(r1s1, . . . , rnsn). It is cus- tomary to expand ?(r1s1, . . . , rnsn) in a known basis set and to find the expansion coefficients using the variational method and with all necessary and sufficient constraints (spin and space symmetries) to prevent the variational collapse.4 This is the basis of the so-called Full Configuration Interaction (FCI) method which provides the exact variational solution for the energy functional of the electronic system defined in Eq. (5) in a given basis set. Equation (5) is the starting point of Density Functional Theory (DFT) which aims to replace both ? 1(r1; r1?) and ? 2(r1, r2) by the one-electron density, ? 1(r1). For the ground state, this wish is justified by the celebrated Hohenberg-Kohn (HK) theorems, which state that the exact ground state total energy of any many-electron system is given by a universal, unknown, functional of the electron density only.5 Rigorously speaking, only the second term of the right hand side part of Eq. (5) is an explicit functional of the diagonal one-electron density matrix, ? 1(r1). The first term, which corresponds to the kinetic energy, is an explicit functional of the complete one-electron density matrix, ? 1(r1; r1?). The major contribu- tion to the electron-electron term comes from the classical electrostatic ?self-energy? of the charge distribution, which is also an explicit functional of the diagonal one-electron density matrix.1 The remaining contribution of the electron- electron term is still unknown. However, the two-electron density ? 2(r1, r2) cannot be factorized in terms of ? 1(r1; r1?) for the exact energy of the exact ground-state, even if this state has a strong closed-shell nature. Hence, information regarding ? 1(r1) and ? 2(r1, r2) is required to reconstruct the energy of the system provided that the spin is introduced ad hoc to ful- fill the Pauli principle. In DFT, the last term of Eq. (5) and the non-diagonal part of the electron kinetic energy term are usu- ally added into a so-called ?exchange-correlation? functional which also depends on the one-electron density only (EXC[?]). The definition of EXC[?] is the basis for the practical use of DFT. Since EXC[?] is a functional of the density and that the first and the last terms of the right hand side part of Eq. (5) de- pend on ? 1(r1), it is possible to define a universal functional which is derivable from the one-electron density itself and without reference to the external potential v(ri; R1; . . . ; RN). Hence, DFT offers a way to eliminate the connection with the n electron wave function working in terms of the density function ?(r) alone. In addition, since the first HK theorem states that there exists a one-to-one mapping between the ex- ternal potential v(ri; R1; . . . ; RN), the particle density ? 1(r1) (or ?(r)) it follows that ?(r) determines the exact non- relativistic Hamiltonian (Eq. (1)) and hence one may, incor- rectly, claim that ?(r) does also determine the ground state wave function ?(r1s1, . . . , rnsn). Therefore, following McWeeny2 one should reformulate DFT extending Levy?s constrained search6 to ensure not only that the variational procedure leads to a ? 1(r1) which de- rives from some wave function ?(r1s1, . . . , rnsn) (the N- representability problem)4 but also that ?(r1s1, . . . , rnsn) be- longs to the appropriate irreducible representation of the spin permutation group Sn (the Pauli principle). The above propo- sition, corresponding to Eq. (1), can be written in a mathe- matical form by rewriting Eq. (5) as E[?1, ?2] = min ???1 derived from??Sn ? ? ? ? 1 2 ? r1=r1 ? [? ? ?T ?1(r1; r1?)]dr1 + ? r1 v(r1;R1; . . . ;RN )?1(r1)dr1 + 1 2 ? r1 ? r2=r2 ? ?1(r1)(1? P12)?1(r2; r2?) r12 dr1dr2 ? ? ? + min ?2 derived from? ? Sn {ECorrelation[?2(r1, r2)]}, (6) which clearly shows the one-to-one relation between the one-electron density matrix, ? 1(r1; r1?), and the main part of the energy E including the explicit dependence of the electron-electron correlation on ? 2(r1, r2). Notice that if ECorrelation[? 2(r1, r2)] in Eq. (6) is forced to be zero, one obtains another form of the well-known Hartree-Fock (HF) energy expression. In DFT, according to the HK theorem, ultimately ECorrelation[? 2(r1, r2)] is also assumed as a func- tion of the one-electron density only and, if this is written in terms of the electron density, one obtains the Kohn-Sham (KS) equations,7 provided the non-diagonal terms of the ki- netic energy and those arising from the permutation operator are all included in EXC[?]. In the HF method, the energy is obtained through a variational iterative procedure which in- volves the non-local Fock operators.8 In DFT, the variational problem possesses the same mathematical structure of the HF problem and it can also be solved iteratively leading to the KS equations.7 The current implementation of DFT based meth- ods differ in the particular way to model the unknown EXC[?] term. Taking into account this comparison and within the lan- guage of DFTmodel, Eq. (6) can be written in a more compact form as E[?] =EKS[?]+ EXC[?], (7) where EKS[?] is the KS energy and accounts the kinetic, nu- clear potential, and Coulomb terms, whereas EXC[?] accounts for the exchange term plus correlation energy. This correlation energy is the extra energy term not contained in the EKS[?] plus exchange terms. In this paper, we propose a comparison scheme to es- tablish an equivalence between the wave function and DFT methods. The aim of this comparison model is to compare different energy functionals defined by the same external po- tential v(ri; R1; . . . ; RN) generated by the N fixed nuclei and using the same basis set to describe the system of n electrons. To this end, we use the L?wdin partitioning technique9,10 constructed using the exact non-relativistic FCI solution of Eq. (1), to establish a comparison model between DFT and wave function theories. We apply this analysis to simple and well-defined molecular systems, namely, H2O, SH2, NH3, CH4, NH4+ (closed shell singlet ground state) and CH2 and O2 (triplet ground state) to provide some numerical results. Downloaded 08 Sep 2013 to 161.116.100.92. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 174107-3 Caballero, Moreira, and Bofill J. Chem. Phys. 138, 174107 (2013) THE L?WDIN PARTITIONING TECHNIQUE AND ITS RELATED FUNCTION To compare different energy functionals to describe a system of n electrons moving under the effect of a given ex- ternal potential v(ri; R1; . . . ; RN) generated by N fixed nuclei, we start from the definition of the fundamental Rayleigh-Ritz quotient given in Eq. (1) and we split the energy functional of the system as E = Eref + ECorr . (8) By exploring simple forms of the component functionals, it is possible to establish some equivalences in the subjacent mathematical structure between different energy functionals. We emphasize that this equivalence does not mean equality and our intention is to provide a comparison criterion for wave function and DFT based energy functionals. To establish a formal mathematical expression for the energy of the system as in Eq. (8), we use the L?wdin partitioning technique of a secular equation9,10 applied to the FCI electronic Hamilto- nian (for a given number of electrons, n, and a given basis set) to solve the time independent Schr?dinger equation given in Eq. (1). We split the FCI electronic Hamiltonian secular equa- tion of dimension K through I and II subspaces (K = KI + KII) as follows: ? HI,I HI,II HII,I HII,II ?? c(i)I c(i)II ? = Ei ? c(i)I c(i)II ? . (9) For any eigenvalue, Ei, for which the components of the cor- responding eigenvector (c(i))T = (cI(i) cII(i))T ?= (0I 0II)T, being 0I and 0II the zero vectors of the subspaces I and II, respec- tively, the solutions of the secular equation given in Eq. (9) are equivalent to the solutions of the partitioned secular equation [HI,I ? HI,II(HII,II ? EiIII,II)?1HII,I]c(i)I = Eic(i)I , (10) where III,II is the identity matrix in the II subspace. Notice that in Eq. (10) the vector cI(i) is un-normalized. For simplicity, in the present analysis we take the subspace I of dimension one with HI,I = H11 = Eref, i.e.: only a Configuration State Function (CSF) defines this subspace to represent singlet or triplet electronic states of representative systems. The rest of the CSFs of the FCI space provide the basis of the II sub- space. Now we define the L?wdin function f (E) which can be seen as the ?eigenvalue? of the one-dimensional matrix [HI,I ? HI,II(HII,II ? E III,II)?1HII,I], more explicitly, f (E), can be written as a ?Rayleigh-Ritz? quotient of this one- dimensional matrix with the one-dimensional vector, d, with a coefficient d = 1 due to normalization, f (E) = dT [HI,I ? HI,II(HII,II ? EIII,II)?1HII,I]d dT d = 1 d = d = 1. (11) Notice that in the present case HI,II = (HII,I)T is a vector of dimension K ? 1,HII,II is a matrix of dimension (K ? 1)? (K ? 1), and finally HI,I is an element of the H matrix. The do- main of E is E ? (??,?). The set of K values of the L?wdin function such that f (E) takes the value of E, f (E) = E = Ei is an eigenvalue of the secular equation (9). In this case, d = cI(i) [(cI(i))T(cI(i))]?1/2 if c(i) is a normalized vector. Notice that in the present case cI(i) is also of dimension one. The func- tion f (E) is a non-increasing function of E. When the function f (E) is represented in front of E, the horizontal asymptote of f (E) tends to the value of the matrix element, HI,I = H11, i.e.: lim E??? f (E) = HI,I = Eref . This limit coincides with the HF energy if the FCI electronic Hamiltonian has been constructed using the set of orbitals that makes stationary the one-CSF en- ergy functional, Eref [? 1] = EHF [? 1] = HI,I [? 1]. Notice that in Eq. (11) theHI,II = (HII,I)T vector andHII,II matrix are func- tions of the one-electron components of the two-electron den- sity matrix, see, e.g., Ref. 11. In summary, at the point where f (E) = E = Ei the variational condition required in Eq. (9) for the eigenstate i is satisfied, the L?wdin function given in Eq. (11) defines an energy functional that can be expressed more explicitly as E[? 1, E] = f [HI,I[? 1], HI,II, HII,II, E] = f [EHF[? 1], HI,II, HII,II, E]. At the value E = 0, the function f (E) takes the value, f (0)= dT[HI,I ?HI,II(HII,II)?1HII,I]d, with d= d = 1, and can be seen as a parametric functional of ? 1 and the one-electron components of the two-electron density matrix. This expres- sion resembles Eq. (7) provided that it is constructed using the set of orbitals that optimizes the functional EHF = HI,I and adding the kinetic and the exchange terms of HI,I to the second term, namely, ?HI,II(HII,II)?1HII,I. The resulting ex- pression for this second term resembles the EXC[?] functional in DFT. Notice that the EXC[?] functional does not depend on the energy itself. With this consideration the function f (E) = f [HI,I[? 1], HI,II, HII,II, E] at E = 0 can be used to com- pare the wave-function theory with the DFT energy functional E[?]. If one represents or plots f (E) in front of E, the energy of any DFTmethod should be located on the vertical axis f (0). The reason, as noted above, is because the EXC[?] of any DFT method is equivalent to the second term of the f (0) L?wdin function where in this case E does not appear explicitly in the function. So far it seems the most licit and appropriate way to compare both models, namely, the wave-function theory, also so-called ab initiomethods, and the DFT. The functions repre- senting both theories, namely, E[?] in Eq. (7) and the L?wdin function f (E) at E = 0 from Eq. (11) are equivalent if they are build using the same basis set with the same external poten- tial v(ri; R1; . . . ; RN), the same number of electrons, n, and in addition in DFT the set of orbitals is the KS while in wave function is the HF. With these two premises the comparison is well established and permitted. NUMERICAL ANALYSIS OF SIMPLE MOLECULAR SYSTEMS In this section, we compute the above described L?wdin partitioning function (cf. Eq. (11)) for representative, simple, stable, and well-defined molecular systems in their electronic ground state and analyze the behavior of several EXC[?] func- tionals by comparison. For simplicity, we have taken the op- timized geometry using the singles and doubles CI (SDCI) wave function over the RHF (ROHF for triplet states) single CSF reference and the 6-31++G?? basis set.12 In all cases, this geometry has been used to extract all the roots of the FCI expansion in a minimal STO-3G basis set (K = N(FCI)) using Downloaded 08 Sep 2013 to 161.116.100.92. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 174107-4 Caballero, Moreira, and Bofill J. Chem. Phys. 138, 174107 (2013) TABLE I. Relevant values for the L?wdin function (dimension of the FCI space, eigenvalues of the FCI matrix, etc.) and energies obtained using HF and several standard parameterizations of the EXC[?] functional for LDA,17,18 GGA (PW91PW91,19 BLYP,20,21 and PBEPBE22), meta-GGA (revTPSS,23 PKZBPKZB,24 and VSXC25) and hybrid (B3LYP21,26 and PBE027) approximations. In all cases, the basis set is STO-3G and the geometries correspond to CISD/6-31++g?? optimised structures.12 (Boldface italics are used to separate the different blocks: FCI dimensions and eigenvalues (rows 1?7), f (E = 0) (row 8), E(HF/ROHF) = H11 (row 9), and DFT based calculations (rows 10?18)). H2O NH3 SH2 CH4 NH4+ CH2 O2 K = N (FCI) 70 616 382 1436 1436 148 106 E1 ?75.012114 ?55.517543 ?394.354092 ?39.805451 ?55.946702 ?38.472608 ?147.747895 E2 ?74.419386 ?54.989548 ?393.780008 ?38.973091 ?55.099684 ?37.943644 ?147.146834 E3 ?74.015740 ?54.931878 ?393.610959 ?38.917079 ?55.065056 ?37.888937 ?146.901982 . . . EK?1 ?28.052661 ?16.773511 ?179.589921 ?9.0789790 ?13.342182 ?11.697235 ?103.312082 EK ?27.401570 ?16.607329 ?177.750980 ?8.9003961 ?13.139050 ?11.366848 ?101.543057 f (E = 0) ?74.961208 ?55.450762 ?394.311791 ?39.722529 ?55.863139 ?38.427856 ?147.630696 E(HF/ROHF) = H11 ?74.962674 ?55.453330 ?394.311615 ?39.726846 ?55.866554 ?38.429892 ?147.632382 E(SVWN5) ?74.731653 ?55.289280 ?393.511068 ?39.616730 ?55.709647 ?38.235051 ?147.197578 E(PW91PW91) ?75.278919 ?55.752563 ?394.908835 ?40.006167 ?56.167937 ?38.617965 ?148.238832 E(BLYP) ?75.277026 ?55.744244 ?394.899925 ?39.994248 ?56.157526 ?38.614300 ?148.245053 E(PBEPBE) ?75.225100 ?55.706750 ?394.755311 ?39.966679 ?56.123036 ?38.581657 ?148.141707 E(revTPSS) ?75.326254 ?55.796958 ?394.970071 ?40.051700 ?56.210114 ?38.666296 ?148.328874 E(PKZBPKZB) ?75.193024 ?55.688394 ?394.529979 ?39.958448 ?56.104226 ?38.573009 ?148.079450 E(VSXC) ?75.349301 ?55.814583 ?395.109922 ?40.059908 ?56.228813 ?38.668603 ?148.359499 E(PBE0) ?75.245260 ?55.725543 ?394.816928 ?39.984130 ?56.141393 ?38.599394 ?148.155159 E(B3LYP) ?75.312120 ?55.784317 ?394.951857 ?40.038517 ?56.197663 ?38.646414 ?148.273295 the Graphical Unitary Group Approach (GUGA) algorithm13 and the molecular symmetry except for degenerate groups, where a suitable non-degenerate subgroup has been consid- ered. The FCI eigenvalues and eigenvectors have been used to construct the L?wdin function described above using the space generated by the set of CSFs of the same space and spin symmetry of the reference RHF/ROHF wave function. The Kohn-Sham DFT energies using the same CISD/6-31++G?? optimized geometry and the STO-3G basis set have been cal- culated using representative local, GGA, meta-GGA, and hy- brid EXC[?] functionals. CI calculations have been carried out using the GAMESS 2012 code14,15 whereas the DFT calcula- tions have been carried out using the GAUSSIAN 09 code.16 In Table I, we report the full set of results for the set of simple molecules (H2O, NH3, SH2, CH4, NH4+ (closed shell singlet ground state) and CH2 and O2 (triplet ground state)) for which the L?wdin function defined in Eq. (11) has been constructed using the FCI eigenpairs in a STO-3G minimal basis set. The L?wdin function f (E) for the present systems has the general structure shown in Figure 1 with the corresponding value of K (dimension of the FCI matrix). For the minimal ba- sis set, all electronic states are bound (i.e., Ei < 0 for i = 1, K with K = N(FCI)) and, hence, the value f (0) appears after the last vertical asymptote (aK?1). However, for a general case, the f (0) value can be located between two vertical asymptotes ai and ai+1. This fact does not have any relevant consequence for the discussion below. We note that each vertical asymptote ai is an eigenvalue of the HII,II matrix. In the present case, we have only K ? 1 vertical asymptotes because this is the di- mension of the HII,II matrix if the dimension of the full H is K = N(FCI). When the E variable takes a value E = ai, then f (E) function goes to infinity at this point. Finally, we recall that as E tends to ??, then f (E) goes to HI,I = H11 = Eref. In this case, each branch located within two consecutive ver- tical asymptotes, lets say, ai and ai+1, cuts the straight line f (E) = E one time and only one and corresponds to an eigen- value of the full H matrix. The last branch cuts the straight line f (E) = E at the point E = EK corresponding to the high- est eigenvalue, cuts the vertical axis f (0) and from this point it goes to Eref as E tends to+?. All these features show that the overall behavior of the calculated L?wdin function throughout the whole E horizontal axis is well defined. FIG. 1. Qualitative representation of the L?wdin function f (E) for a FCI problem with K = 8 CSFs and H11 = Eref, to show the general behaviour of this function. In this example, the horizontal asymptote corresponds to Eref (i.e., single configuration representations: RHF for closed shell systems, ROHF or GVB for open shell singlets and triplets). The points E1 ? E2 ? E3 ? ? ? ? ? E8 correspond to the set of eigenvalues of the full H matrix of dimension K = 8 (i.e., those in which f (E) = E). The K ? 1 = 7 vertical asymptotes a1 ? a2 ? a3 ? ? ? ? ? a7 correspond to the eigenvalues of the K ? 1 reduced H matrix, namely, HII,II. See text for more explanation details of this plot. Downloaded 08 Sep 2013 to 161.116.100.92. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 174107-5 Caballero, Moreira, and Bofill J. Chem. Phys. 138, 174107 (2013) In Table I, we also report the energy of the systems (same n, same external potential v(ri; R1;. . . ;RN), and same basis set) obtained using HF and several standard parameteriza- tions of the EXC[?] functional. For all the systems described above, the HF energy is always above the exact FCI energy, as known from the variational nature of the wave function ex- pressed in terms of CI expansions. In general, the DFT ener- gies obtained using the selected functionals show important differences depending on the particular form of the exchange- correlation functional considered. The local density approxi- mation (LDA) approach provides a value of the energy of the ground state that is above the exact FCI energy value. Inter- estingly enough, the LDA energy value is also above the HF energy and close to the value of the branch of the L?wdin function that cuts the f (E) axis at E = 0. However, the GGA, meta-GGA, and hybrid DFT energy values are all below the exact FCI value. As a general rule, the differences between energy values provided by the GGA, meta-GGA, and hybrid DFT functionals are system dependent and larger for systems with larger number of electrons. In view of the previous dis- cussion, the equivalence between E [?] and the expression of the L?wdin function at E = 0 provides an argument to explain this dispersion of results in terms of the complexity of the par- ticular functional (larger number of parameters leads to lower energy) and the lack of variationality with respect to E. This comparison model suggests that the present DFT approaches require a revision to include an additional variational term in addition to the density ?(r), playing the role that the E vari- able plays in the L?wdin function. CONCLUSIONS The results exposed in the ?The L?wdin partitioning technique and its related function? and ?Numerical analysis of simple molecular systems? sections reveal that the L?wdin function of Eq. (11) provides a tool to establish a compari- son model between the wave function and the density func- tional theories. This model reveals the inherent structure of both theories in order to be compared appropriately and the necessary conditions for variationality required for any en- ergy functional to describe an electron system. The DFT func- tionals can be seen as special cases of the L?wdin function for the corresponding n electron problem but do not satisfy all necessary and sufficient conditions on ?(r) and also on EXC[?] term. The latter implies that, if we are given any den- sity function ?(r) which are not correctly derivable from a spin and space symmetry adapted wave function, there will always exist at least a functional which is capable of detect- ing this fact, in the sense that ?(r) will yield to an energy for this functional which is lower than its minimum energy. Ignoring the type of restrictions imposed by the spin and space symmetry when making a variational calculation on such a system leads to an energy minimum that will exhibit the phenomenon of ?overcorrelation.? That is, the correla- tions expressed by the ?(r) function will be inconsistent with a ? 1(r1; r1?) function derivable from a spin and space adapted symmetry wave function?(r1s1, . . . , rnsn), even for a closed- shell system. ACKNOWLEDGMENTS Financial support has been provided by the Spanish Min- isterio de Econom?a y Competitividad (formerlyMinisterio de Ciencia e Innovaci?n) through Grant Nos. CTQ2011-22505 and FIS2008-02238 and by the Generalitat de Catalunya through Grant Nos. 2009SGR-1472, 2009SGR-1041, and XRQTC. Part of the computational time has been provided by the Centre de Supercomputaci? de Catalunya (CESCA) which is also gratefully acknowledged. 1P.-O. L?wdin, Adv. Chem. Phys. 2, 207 (1959). 2R. McWeeny, Philos. Mag. B 69, 727 (1994). 3P.-O. L?wdin, Adv. Phys. 50, 597 (2001). 4C. Garrod and J. K. Percus, J. Math. Phys. 5, 1756 (1964). 5P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 6M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979). 7W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965). 8R. McWeeny, Methods of Molecular Quantum Mechanics, 2nd ed. (Academic Press, Oxford, 1992). 9P.-O. L?wdin, J. Math. Phys. 3, 969 (1962). 10P.-O. L?wdin in Perturbation Theory and its Application in Quantum Mechanics, edited by C. H. Wilcox (Wiley, New York, 1966), p. 255. 11B. O. Roos, P. R. Taylor, and P. E. M. Siegban, Chem. Phys. 48, 157 (1980). 12Optimized molecular geometries at CISD/6-31++G?? level of theory in a suitable symmetry group/subgroup using the HF or ROHF or- bitals: H2O (1A1 ground state in C2v (d(O?H) = 0.95837 ?, a(H?O?H) = 105.6778?)), SH2 (1A1 ground state in C2v (d(S?H) = 1.33088 ?, a(H?S?H) = 92.9297?)), CH4 (singlet 1A1 ground state optimized in C2v (d(C?H)= 1.08527 ?, a(H?C?H)= 109.4712?)), NH3 (singlet 1A1 ground state optimized in C2v (d(N?H) = 1.00967 ?, a(H?N?H) = 107.9010?)), NH4+(singlet 1A1 ground state optimized in C2v (d(N?H) = 1.01972 ?, a(H?N?H) = 105.6778?)), CH2 (3B1 ground state in C2v (d(C?H) = 1.07593 ?, a(H?C?H) = 132.3511)), and O2 (3B1g ground state opti- mized in D2h (d(O?O) = 1.2215 ?)). 13W. Duch and J. Karwowski, Comput. Phys. Rep. 2, 93 (1985). 14M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, J. Comput. Chem. 14, 1347 (1993). 15M. S. Gordon and M. W. Schmidt, in Theory and Applications of Computational Chemistry: The First Forty Years, edited by C. E. Dykstra, G. Frenking, K. S. Kim, and G. E. Scuseria (Elsevier, Amsterdam, 2005), Chap. 41, pp. 1167?1189. 16M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN 09, Revision B.01, Gaussian, Inc., Wallingford, CT, 2010. 17J. C. Slater, The Self-Consistent Field for Molecular and Solids, Quantum Theory of Molecular and Solids (McGraw-Hill, New York, 1974), Vol. 4. 18S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980). 19J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992); 48, 4978 (1993) (Erratum). 20A. D. Becke, Phys. Rev. A 38, 3098 (1988). 21C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). 22J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996); 78, 1396 (1997) (Erratum). 23J. M. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003). 24J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev. Lett. 82, 2544 (1999). 25T. Van Voorhis and G. E. Scuseria, J. Chem. Phys. 109, 400 (1998). 26A. D. Becke, J. Chem. Phys. 98, 5648 (1993). 27C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 (1999). Downloaded 08 Sep 2013 to 161.116.100.92. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions For Peer Review Only ??????? ?? ??? ????? ?????? ?????????? ??? ????????? ?? ???????? ????? ??? ?????? ???????????? ?????????? ???????? ????????? ??????? ?????????? ??? ????? ?????????? ????? ??????? ????? ????? ???? ????????? ?? ??? ??????? ??? ???????? ???? ?? ???????? ??????? ?????? ??????????? ?? ?????????? ??? ??? ???????? ???????? ??????? ??????????? ?? ?????????? ??? ??? ?????? ?????????? ????? ??????????? ?? ?????????? ??? ??? ?????? ????????? ????????????? ??????? ??????? ?????????? ??????? ?????? ??????????????????? ??????????? ???????? ?????????? ????? URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics For Peer Review Only 1 ??????? ?? ??? ????? ?????? ?????????? ??? ????????? ?? ???????? ????? ??? ?????? ???????????? ????????? Marc Caballero,a,b Ib?rio de P. R. Moreira,a,b Josep Maria Bofilla,c,* aInstitut de Qu?mica Te?rica i Computacional, Universitat de Barcelona, IQTCUB, C/ Mart? i Franqu?s 1, E-08028 Barcelona, Spain bDepartament de Qu?mica F?sica, Universitat de Barcelona, C/ Mart? i Franqu?s 1, E-08028 Barcelona, Spain cDepartament de Qu?mica Org?nica, Universitat de Barcelona, C/ Mart? i Franqu?s 1, E-08028 Barcelona, Spain *Corresponding author. Email: jmbofill@ub.edu (Version: August 29, 2013) ?????????? A comparison model based in the L?wdin partitioning technique is used to analyze the differences between the wave function and density functional models. This comparison model provides a tool to understand the structure of both theories and its discrepancies in terms of the subjacent mathematical structure and the variationality required for the energy functional. It is argued that the density functional theory can be compared to the wave-function theory. The wave-function theory provides an explicit form of the exact energy functional for a fermion system from the Full Configuration Interaction approach. The density functional theory can be seen as special cases of L?wdin function that do not satisfy all variational conditions on and also on the term. This analysis shows that ignoring the restrictions imposed by the spin and space symmetry requirements of the solutions when making a variational calculation implies that the correlations expressed by the function will be inconsistent with a function derivable from a spin and space symmetry adapted wave function , even for a closed-shell system. The comparison scheme also provides a new insight in order to achieve a consistent description of the molecular electronic structure of both ground and excited states. Some numerical results are reported. ????????? wave-function theory, density functional theory, L?wdin partition technique, variational methods, comparison model. ?(?) EXC ?[ ] ?(?) ? 1(?1;?1 ') ?(?1s1,?,?nsn ) Page 1 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 2 ?? ????????????? The quantum many-electron system defined by n electrons and N nuclei in interaction is one of the central problems in chemistry and physics. The fundamental mathematical formulation of the non-relativistic n electron problem is the time independent Schr?dinger equation for this system and the corresponding exact solutions provide the essential quantum-mechanical description of each electronic state in terms of the different n electron wave function ?(?1s1,?, ?nsn ). For a system of n electrons and N nuclei in interaction, the time independent Schr?dinger equation (in the Born-Oppenheimer approximation) can be written as a Rayleigh-Ritz quotient given by E ?[ ] = ? H? ?? ? (1) in which the Hamiltonian operator is defined as H? = T? + V? + W? (2) with In this expression, the first term stands for the electrons kinetic energy of the electrons, the second arises from the external potential v(?i;?1; ??? ;?N) generated by the nuclei and 1/rij is the two electron interaction. This hamiltonian operator defines, along with its boundary conditions, an eliptic second order differential equation and the wave-function must satisfy some specific conditions to be an acceptable solution of Eq. (1), namely: a) must be bounded and continuous, b) the partial derivatives with respect to spatial coordinates must be continuous, and c) the function | |2 must be integrable. Since the non-relativistic many-electron Hamiltonian does not act on spin coordinates, anti-symmetry and spin restrictions must be imposed ad hoc to restrict the solutions (i.e. the wave functions ?(?1s1,?,?nsn )) to T? (?1,?,?n ) = ? 12 ?(i ) 2 i=1 n? V? (?1,?,?n;?1;?;?N ) = ?ZI? I ? ?iI =1 N? i=1 n? = v(?i;?1;?;?N ) i=1 n? W? (?1,?,?n ) = 1?i ? ?jj= i+1 n? i=1 n?1? = 1riji> j=1 n? ?(?1s1,?,?nsn ) ?(?1s1,?,?nsn ) Page 2 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 3 the set of functions that satisfy the Pauli principle and spin symmetry requirements of the quantum mechanical state of the system. In the wave-function theory (WFT) formalism, the most compact expression for the energy of the n-electron system in the field of N fixed nuclei can be written as (3) in which the many electron quantities ?1(?1) and ?2(?1,?2) are the diagonal elements, ?1(?1) = ?1(?1;?1) and ?2(?1,?2) = ?2(?1,?2;?1,?2), of the spinless one- and two-electron density matrices, respectively [1-3] The function ?1(?1) = ?1(?1;?1) corresponds to an observable, the one electron density ?(?), and is commonly used in electronic structure theory. In WFT, large efforts are devoted to obtain accurate prediction of the energy of a given system in a given electronic state using a reasonable estimate of ?(?1s1,?,?nsn ). It is customary to expand ?(?1s1,?,?nsn ) in a known basis set and to find the expansion coefficients using the variational method and with all necessary and sufficient constraints (spin and space symmetries) to prev nt the variational collapse [4]. This mathematical requirement is essential to avoid to converge to a solution with no physical meaning. This is the basis of the so-called Full Configuration Interaction (FCI) method which provides the exact solution for the energy functional of the electronic system defined in Eq. (3) in a given basis set. Indeed, for a finite basis set this is the exact solution and has been extensively used as a benchmark for quantum chemical methods [5,,,,9]. Density Functional Theory (DFT) propose a different approach which aims to replace both ?1(?1;?1?) and ?2(?1,?2) in Eq. (3) by the one-electron density, ?1(?1). For the ground state, this wish is justified by the celebrated Hohenberg-Kohn theorems (HK), which state that the exact ground state total energy of any many-electron system is given by a universal, unknown, functional of the electron density only [10]. Rigorously speaking, only the second term of the right hand side part of Eq. (3) is an explicit functional of the diagonal one-electron density matrix, ?1(?1). The first term, which E ? 1,? 2[ ] = ? 12 ???T? 1 ?1;?1?( )?? ??d?1?1=?1?? + + v(?1;?1;?;?N )? 1 ?1( )d?1 ?1 ? + + ? 2 ?1,?2( )r12 d?1 d?2?1??2? Page 3 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 4 corresponds to the kinetic energy, is an explicit functional of the complete one-electron density matrix, ?1(?1;?1?). The major contribution to the electron-electron term comes from the classical electrostatic ?self energy? of the charge distribution, which is also an explicit functional of the diagonal one-electron density matrix [1]. The remaining contribution of the electron-electron term is an explicit functional of ?2(?1,?2). It is important to stress the well known fact that the two-electron density ?2(?1,?2) cannot be factorized in terms of ?1(?1;?1?) in the expression of the exact energy of the exact ground-state, even for a closed-shell system. In DFT, this and the non-diagonal part of the electron kinetic energy term are usually added into a so-called ?exchange- correlation? functional which also depends on the one-electron density only (EXC[?]). The definition of EXC[?] is the basis for the practical use of DFT. Since EXC[?] is a functional of the density, it is possible to define a universal functional which is derivable from the one-electron density itself and without reference to the external potential v(?i;?1; ??? ;?N). Hence, DFT offers a way to eliminate the connection with the n electron wave function working in terms of the density function ?(?) alone. In addition, since the first HK theorem states that there exists a one-to-one mapping between the external potential v(?i;?1; ??? ;?N), the particle density ?1(?1) (or ?(?)) it follows that ?(?) determines the exact non relativistic Hamiltonian (Eq. (1)) and hence one may, incorrectly, claim that ?(?) does also determine the ground state wave function ?(?1s1,?,?nsn ). However, one must advert that, using the exact non-relativistic FCI wave function, information regarding ?1(?1) and ?2(?1,?2) is required to reconstruct the energy of the system provided that the spin is introduced ad hoc to fulfill the Pauli principle. It is interesting to reformulate the exact energy functional expressed in Eq. (3) to provide a general expression to compare the WFT and DFT theories using a common language. To this end, following McWeeny [1] one should reformulate DFT extending Levy?s constrained search [11] to ensure not only that the variational procedure leads to a ?1(?1) which derives from some wave function ?(?1s1,?,?nsn ) (the N-representability problem)4 but also that ?(?1s1,?,?nsn ) belongs to the appropriate irreducible representation of the spin permutation group Sn (the Pauli principle). The above proposition, corresponding to Eq. (1), can be written in a mathematical form by rewriting Eq. (3) as Page 4 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 5 E ? 1,? 2[ ] = min??? 1 derivedfrom ??Sn ? 1 2 ??? T? 1 ?1;?1?( )?? ??d?1 ?1=?1? ? +????? + v(?1;?1;?;?N )? 1 ?1( )d?1 ?1 ? + + 12 ? 1 ?1( ) 1? P12( )? 1 ?2;?2?( ) r12 d?1 d?2?2=?2???1? ? ?? ?? + + min? 2 derivedfrom ??Sn ECorrelation ? 2 ?1,?2( )?? ??{ } (4) which clearly shows the one-to-one relation between the one-electron density matrix, ?1(?1;?1'), and the main part of the energy E and the explicit dependence of the electron- electron correlation on ?2(?1,?2). Notice that if ECorrelation[?2(?1,?2)] in Eq. (4) is forced to be zero one obtains another form of the well-known Hartree-Fock energy expression. In DFT, according to the HK theorem, ultimately ECorrelation[?2(?1,?2)] is also assumed as a function of the one electron density only and, if this is written in terms of the electron density, one obtains the Kohn-Sham equations [12] provided the non-diagonal terms of the kinetic energy and those arising from the permutation operator are all included in EXC[?]. In the Hartree-Fock method, the energy is obtained trough a variational iterative procedure which involves the non-local Fock operators [13]. In DFT the variational problem possesses the same mathematical structure of the Hartree-Fock problem and it can be also solved iteratively leading to the Khon-Sham (KS) equations [12]. The current implementation of DFT based methods differ in the particular way to model the unknown EXC[?] term. Taking into account this comparison and within the language of the DFT model, Eq. (4) can be written in a more compact form as E [?]=EKS[?] + EXC[?] (5) where EKS[?] is the Khon-Sham energy and accounts for the kinetic, nuclear potential and Coulomb terms, whereas EXC[?] accounts for the exchange term plus correlation energy. This correlation energy is the extra energy term not contained in the EKS[?] plus exchange terms. In this work, we analyze the mathematical structure of the exact energy functional for fermions derived from FCI by extending the analysis that we introduced in a previous work [14] where we have established a comparison scheme between the WFT and DFT methods to compare energy functionals defined by the same external potential Page 5 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 6 v(?i;?1; ??? ;?N) and using the same basis set to describe the system of n electrons. To this end, we use the L?wdin partitioning technique [15, 16] constructed using the exact non-relativistic FCI solution of Eq. (1) using different sets of basis functions (orbitals) defined in a minimal basis set. We apply this analysis to simple and well defined molecular systems, namely, H2O, SH2, NH3, CH4, NH4+ (closed shell singlets) and CH2 and O2 (triplet ground state) to provide some numerical results that show de inherent structure of the energy functional and the dependence of the L?wdin function with respect to the orbitals used to construct the FCI space. ?? ??? ?????? ???????????? ????????? ??? ??? ??????? ????????? In a previous work [14] we have proposed a comparison scheme to establish an equivalence between the WFT and DFT methods. The aim of this comparison model is to compare different energy functionals defined by the same external potential v(?i;?1; ??? ;?N) generated by the N fixed nuclei and using the same basis set to describe the system of n electrons. To this end, we split the energy functional of the system as E =Eref + ECorr . (6) By exploring simple forms of the component functionals it is possible to establish some equivalences in the subjacent mathematical structure between different energy functionals. We emphasize that this equivalence does not mean equality and our intention is to provide a comparison criterion for WFT and DFT based energy functionals. For this purpose we use the L?wdin partitioning technique of a secular equation [15, 16] applied to the FCI electronic Hamiltonian in a given basis set and number of electrons, n, to solve the time independent Schr?dinger equation given in Eq. (1). We split the FCI electronic Hamiltonian secular equation of dimension K through I and II subspaces (K = KI + KII) as follows: ?I ,I ? I ,II ?II ,I ?II ,II ? ? ?? ? ? ?? ?I(i) ?II(i) ? ? ?? ? ? ?? = Ei ? I(i) ? II(i) ? ? ?? ? ? ?? (7) For any eigenvalue, Ei, for which the components of the corresponding eigenvector (?(i))T = (?I(i) ?II(i))T ? (?I ?II)T, being ?I and ?II the zero vectors of the subspaces I and II respectively, the solutions of the secular equation given in Eq. (7) are equivalent to the solutions of the partitioned secular equation Page 6 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 7 ?I ,I ?? I ,II ? II ,II ? Ei?II ,II( )?1?II ,I?? ??? I(i) = Ei?I(i ) (8) where ?II,II is the identity matrix in the II subspace. For simplicity, in the present analysis we take the subspace I of dimension one with ?I,I = H11 = Eref, i.e.: only a Configuration State Function (CSF) defines this subspace to represent singlet or triplet electronic states of representative systems. The rest of the CSFs of the FCI space provide the basis of the II subspace. Now we define the L?wdin function f (E) which can be seen as the ?eigenvalue? of the one-dimensional matrix [?I,I ? ?I,II(?II,II ? E ?II,II)-1?II,I], more explicitly, f (E), can be written as a ?Rayleigh-Ritz? quotient of this one dimensional matrix with an one-dimensional vector, say ?, with a coefficient d = 1 due to normalization, f (E) = ?T[?I,I ? ?I,II(?II,II ? E ?II,II)-1?II,I]? ?T? = 1 ? = d = 1 (9) Notice that in the present case ?I,II = (?II,I)T is a vector of dimension K?1, ?II,II is a matrix of dimension (K?1)?(K?1), and finally ?I,I is an element of the ? matrix. The domain of E is E ? (-?, ?). The set of K values of the L?wdin function such that f (E) takes the value of E, f (E) = E = Ei corresponds to the set of K eigenvalues of the secular equation given in Eq. (7). In this case ? = ?I(i) [(?I(i))T(?I(i))]-1/2 if ?(i) is a normalized vector. The function f (E) is a non-increasing function of E. When the function f (E) is represented in front of E, the horizontal asymptote of f (E) tends to the value of the matrix element, ?I,I = H11 i.e.: limE??? f E( ) =?I ,I = Eref. This limit coincides with the Hartree-Fock energy if the FCI electronic Hamiltonian has been constructed using the set of orbitals that makes stationary the one-CSF energy functional, Eref [?1] = EHF [?1]= ?I,I [?1]. However, the remaining term in Eq. (9), namely, the ?I,II = (?II,I)T vector and ?II,II matrix, are very complex functions that can be expressed in terms of the one-electron components of the two-electron density matrix. Hence, at the point where f (E) = E = Ei the variational condition required in Eq. (7) for the eigenstate i is satisfied, the L?wdin function given in Eq. (9) defines an energy functional that can be expressed more explicitly as E[?1, E] taking into account that the ?I,II = (?II,I)T vector and ?II,II matrix can be expressed in terms of the one-electron components of the two- electron density matrix . The details of these relations are described in the next section. ? 2 (?1,?2;?1 ',?2 ') Page 7 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 8 ?? ?????? ??????????? ???? ??? ?????? ????????? ???????? At the point where f (E) = E = Ei the variational condition required in the FCI equation for the eigenstate i is satisfied, the L?wdin function defines an energy functional that can be expressed explicitly as (10) and represents the simplest expression for the energy functional that can be derived from the FCI that satisfies all necessary and sufficient variational conditions to be a solution of the Non Relativistic-Time Independent-Schr?dinger Equation for n electrons and N nuclei defined by Eq. (1) in a given basis set. The simplest form of the energy functional in the previous expression, namely E[?1, E], can be derived as follows. Following Roos et al. [17] the ?I,II = (?II,I)T vector and the ?II,II matrix are functions of the one-electron components of the two-electron density matrix (a fourth rank tensor) for electronic state L given by L? 2( ) ij,kl = cIL I Eij K K Ekl J cJL ?? jkcIL I Eil J cJL?? ??I ,J,K? (11) where and is the creation operator of an alpha electron in orbital i (resp. the anihilation operator of an alpha electron in orbital j) and I, J, and K are CSFs of the FCI space defined in a given basis set. The term (12) is the ij,kl component of the two-electron density matrix formed by the I,J CSFs. Now, defining the one electron density matrix (a rank two tensor) for state L as L?1 = cIL I Eij J cJL I ,J ? = cIL ?1IJ( ) ij cJLI ,J? (13) where (14) E ? 1,E[ ] = f ?I ,I (? 1),?I ,II ,?II ,II ,E?? ?? = f EHF (? 1),?I ,II ,?II ,II ,E?? ?? Eij = ai?+ aj?? + ai?+ aj?? ai?+ (resp. aj?? ) ? 2IJ( )ij,kl = I Eij K K Ekl J ?? jk I Eil J?? ??K? ? 1IJ( )ij = I Eij J Page 8 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 9 is the ij component of the one-electron density matrix formed by the I,J CSFs, the ij,kl component of the two-electron density matrix formed by the I,J CSFs given in Eq. (12) can be rewritten as ? 2IJ( )ij,kl = F ?1IJ( )kl{ }( ) = ?1IK( )ij ?1KJ( )kl ?? jk ?1IJ( ) il??? ???K? . (15) Using the above definitions the second order reduced density matrix element ij,kl for the electronic state L can be rewritten as L? 2( )ij,kl = cIL ? 2IJ( )ij,kl cJLI ,J? = cIL ?1IK( )ij ?1KJ( )kl cJL ?? jkcIL ?1IJ( ) il cJL??? ???I ,J ,K? . (16) With these premises, the simplest expression for the energy functional for a fermion system that takes into account all necessary and sufficient variational conditions (and antisymmetry) can be reduced to . (17) This energy functional corresponds to the exact energy derived from the FCI and explicitly depends on the one electron density matrix, the set of component one-electron density matrices defined in Eq. (14) necessary to build the two-electron density matrix, and E. At the value E = 0 the above function f (E) given in Eq. (17) reduces to the form f 0( ) = f ?I ,I ?1( ),?I ,II ,?II ,II , 0?? ??= ?T ?I ,I ??I ,II ?II ,II( )?1?II ,I?? ??? (18) with ? = d = 1, and defines a parametric functional of ?1 through ?I,I and the set {?1IJ} through ? ?I,II(?II,II)-1?II,I. This expression resembles Eq. (5) provided that it is constructed using the set of orbitals that optimizes the functional EHF = ?I,I and adding the kinetic and the exchange terms of ?I,I to the second term, namely, ? ?I,II(?II,II)-1?II,I. The dependence of the set of orbitals is analyzed below although the FCI energy is invariant with respect to any unitary transformation of orbitals [13]. The resulting expression for this second term resembles to the EXC[?] functional in DFT. Notice that the EXC[?] functional does not depend on the energy itself. With this consideration the function f (E) = f [?I,I[?1], ?I,II, ?II,II, E ] at E = 0 can be used to compare the WFT with the energy functional, E[?], of the DFT. If one represents or plots f (E) in front of E, the f (E) = f ? I,I (? 1),?I ,II ,?II ,II ,E?? ?? Page 9 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 10 energy of any DFT method should be located on the vertical axis f (0). The reason, as noted above, is because the EXC[?] of any DFT method is equivalent to the second term of the f (0) L?wdin function where in this case E does not appear explicitly in the function as occurs in DFT. So far it seems the most licit and appropriated way to compare both models, namely, the WFT, also so-called ab initio methods, and the DFT. The functions representing both theories, namely, E[?] in Eq. (5) and the L?wdin function f (E) at E = 0 from Eq. (9) are equivalent if they are build using the same basis set with the same external potential v(?i;?1; ??? ;?N), the same number of electrons, n, and in addition in DFT the set of orbitals is the KS while in WFT is HF. With these two premises the comparison is well established and permitted. ?? ????????? ???????? ?? ?????? ????????? ???????? In this section, we compute the above described L?wdin partitioning function (cf. Eq. (9)) for representative, simple, stable and well-defined molecular systems in their electronic ground state and analyze the behavior of several EXC[?] functionals by comparison. For simplicity, we have taken the optimized geometry using the singles and doubles CI (SDCI) wave function over the RHF (ROHF for triplet states) single CSF reference and the 6-31++G** basis set. In all cases, this geometry has been used to extract all the roots of the FCI expansion in a minimal STO-3G basis set using the Graphical Unitary Group Approach (GUGA) algorithm [18] and the molecular symmetry except for degenerate groups, where a suitable non-degenerate subgroup has been considered. To this end, different sets of orbitals have been used to solve the FCI problem and construct the L?wdin function for all the molecular systems considered. The FCI eigenvalues and eigenvectors have been used to construct the L?wdin function described above. The Kohn-Sham DFT energies using the same CISD/6-31++G** optimized geometry and the STO-3G basis set have been calculated using representative local, GGA, meta-GGA and hybrid EXC[?] functionals. CI calculations have been carried out using the GAMESS 2012 code [19, 20] whereas the DFT calculations have been carried out using the GAMESS 2012 and the Gaussian09 code [21]. Regarding the molecular geometry of the systems used in the reported calculations, for H2O the geometry has been optimized in C2v for the singlet A1 ground state (d(O-H)= 0.95837 ?, a(H-O-H)= 105.6778o) and the L?wdin function has been Page 10 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 11 calculated using this geometry and FCI/STO-3G singlet wave functions of A1 symmetry generated using the GUGA algorithm (N(FCI) = 70 CSF's) using the HF orbitals. We proceed similarly for SH2 (A1 ground state in C2v symmetry with d(S-H) = 1.33088 ?, a(H-S-H) = 92.9297o; N(FCI) = 382 CSF's), CH4 (singlet A1 ground state optimized in C2v (d(C-H) = 1.08527 ?, a(H-C-H) = 109.4712o; N(FCI) = 1436 CSF's), NH3 (singlet A? ground state optimized in Cs (d(N-H)= 1.00967 ?, a(H-N-H)= 107.9010o; N(FCI) = 616 CSF's), NH4+( singlet A1 ground state optimized in C2v (d(N-H)= 1.01972 ?, a(H- N-H)= 105.6778o; N(FCI) = 1436 CSF's). We also included two molecules having a triplet ground state: CH2 (triplet B1 ground state optimized in C2v (d(C-H)= 1.07593 ?, a(H-C-H)= 132.3511o; L?wdin function calculated using this geometry and FCI/STO-3G triplet wave functions of B1 symmetry with N(FCI) = 148 CSF's) and O2 (B1g ground state optimized in D2h (d(O- O)= 1.2215 ?); L?wdin function calculated using this geometry and FCI/STO-3G singlet wave functions of B1g symmetry with N(FCI) = 106 CSF's). In Table 1 we report a set of WFT and DFT energy values for the set of simple molecules described above for which the L?wdin function defined in Eq. (9) has been constructed using the FCI results in a STO-3G minimal basis set. Here we report the energy of the systems (same n, same external potential v(?i;?1; ??? ;?N), and same basis set) obtained using HF and several standard parameterizations of the EXC[?] functional for LDA [22, 23], GGA (PW91PW91 [24], BLYP [25, 26], and PBEPBE [27]), meta- GGA (revTPSS [28], PKZBPKZB [29], and VSXC [30]) and hybrid (B3LYP [26, 31] and PBE0 [32]). It should be emphasized that exactly the same FCI energies are obtained using different orbitals to construct the FCI space as expected from the fact that the same atomic basis set is used in all the cases. In Table 2, we provide some relevant values for the L?wdin functions constructed from the FCI matrix using different sets of orbitals to analyze the dependence of the H11 and the f(0) values on the set of basis fuctions used to solve the FCI problem. In particular, we have used the RHF/ROHF set of orbitals (orbital basis set A), symmetrized Huckel orbitals (orbital basis set B), the Kohn-Sham orbitals from S-VWN [22, 23] LDA functional (orbital basis set C), BLYP [25, 26] GGA functional (orbital basis set D) and the PBE0 [32] hybrid functional (orbital basis set E). Finally, we also used the natural orbitals corresponding to the FCI ground state to show the Page 11 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 12 relevance of the exact one electron density matrix in defining the wave function of the ground state (orbital basis set F). In all cases, the L?wdin function f (E) for the present systems has the general structure shown in Figure 1 with the corresponding value of K (dimension of the FCI matrix). For the minimal basis set, all electronic states are bound (i.e.: Ei < 0 for i = 1, K with K = N(FCI)) and, hence, the value f (0) appears after the last vertical asymptote (aK). However, for a general case the f (0) value can be located between two vertical asymptotes ai and ai+1. This fact does not have any relevant consequence for the discussion below. We note that each vertical asymptote ai is an eigenvalue of the ?II,II matrix. In the present case we have only K?1 vertical asymptotes because this is the dimension of the ?II,II matrix if the dimension of the full ? is K = N(FCI). When the E variable takes a value E = ai then f (E) function goes to infinity at this point. The f (E) function is a non-increasing function of E. Finally we recall that as E tends to ?? then f (E) goes to ?I,I = H11 = Eref. In this case, each branch located within two consecutive vertical asymptotes, lets say, ai and ai+1, cuts the straight line f (E) = E one time and only one. This straight line passes through the point (0, 0). The point where a branch cuts the straight line is an eigenvalue of the full ? matrix. The last branch cuts the straight line f (E) = E at the point E = EK corresponding to the highest eigenvalue of this matrix, and also cuts the vertical axis f (0) and from this point it goes to Eref as E tends to +?. The f (E) function when E tends to ?? takes the values Eref (which corresponds to EHF when the HF orbitals are used (orbital set A), and to the expectation value of the Hamiltonian using the Huckel determinant, the KS determinant or the natural determinant for the ground state (orbital sets B to F)). In other words, = H11 = Eref. All these features show that the overall behavior of the calculated L?wdin function throughout the whole E horizontal axis is well defined. Regarding the particular values of the energies for the systems reported in Table 1, we emphasize that for each system the same number of electrons n, the same external potential v(?i;?1; ??? ;?N), and same basis set has been used to obtain the energy using the FCI, HF and several standard parameterizations of the EXC[?] functional. For all the systems described above, the HF energy is always above the exact FCI energy, as known from the variational nature of the wave function expressed in terms of CI expansions. In general, the DFT energies obtained using the selected functionals show important differences depending on the particular form of the exchange-correlation limE??? f E( ) =? I ,I Page 12 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 13 functional considered. The LDA approach provides a value of the energy of the ground state that is above the exact FCI energy value. A detailed analysis of the values reported in Table 2 shows that the L?wdin function has essentially the same form but shows a the dependence of the H11 and the f(0) values on the set of basis fuctions used to solve the FCI problem. The expected values of the energy H11 = Eref using the sets of orbitals C to F are slightly above the HF energy, whereas a larger difference is observed when using the symmetrized Huckel orbitals. This is in line with the fact that the HF determinant provides the lowest variational energy using a single determinant description. Interestingly enough, the LDA energy value is also above the HF energy and close to the value of the branch of the L?wdin function that cuts the f (E) axis at E = 0. However, the GGA, meta-GGA and hybrid DFT energy values are all below the exact FCI value. As a general rule, the differences between energy values provided by the GGA, meta-GGA and hybrid DFT functionals are system dependent and larger for systems with larger number of electrons. A similar picture of the L?wdin function is obtained using different basis sets with a small variation of the f (0) value (except perhaps when using the roughly approximated symmetrized Huckel basis set). Also, a significant change of the weight of the reference determinant in the ground state is observed for the differents orbital sets, with the larger value obtained using the natural orbitals of the FCI ground state. This value provides an estimation of the contribution of the one electron density matrix to the FCI wave function. In view of the previous discussion, the equivalence between E [?] and the expression of the L?wdin function at E = 0 provides an argument to explain this dispersion of results in terms of the complexity of the particular functional (larger number of parameters leads to lower energy) and the lack of variationality with respect to E. This comparison model suggests that the present DFT approaches require a revision to include an additional variational term in addition to the density ? ??, playing the role that the variable E plays in the L?wdin function. From this comparison scheme it is clear that the variable E controls the variational requirements in order to achieve a consistent description of the molecular electronic structure of both ground and excited states. The lack of an explicit dependence on E in current DFT based approaches (along with the difficulties to treat open shell states) explains the limitations of this theory to describe excited states. Finally, this formal mathematical correspondence also suggests that DFT functionals can be seen as effective single determinant approaches in the sense Page 13 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 14 used by Clementi and coworkers to define their semi-empirical density functional approximations to obtain correlation energies from HF wave-functions [33]. ?? ??????????? The results exposed in the previous sections reveals that the L?wdin function of Eq. (9) provides a tool of comparison model between the wave function and the density functional theories. This model provides an explicit form of the exact energy functional for a fermion system and reveals the inherent structure of both theories in order to be compared. The DFT functionals can be seen as special cases of L?wdin function ???? ?? ??? ??????? ??? ????????? ??? ?????????? ?????????? ?? ? ? ?. The latter implies that, if we are given any density function ? ? ? which are not correctly derivable from a spin and space symmetry adapted wave function, there will always exist at least a functional which is capable of detecting this fact, in the sense that ? ? ? will yield to an energy for this functional which is lower than its minimum energy. Ignoring the type of restrictions imposed by the spin and space symmetry adapted when making a variational calculation on such a system leads to an energy minimum that will exhibit the phenomenon of "overcorrelation". That is, the correlations expressed by the ? ?? function will be inconsistent with a ?1(?1;?1') function derivable from a spin and space symmetry adapted wave function , even for a closed-shell system. From this analysis, the increasing empiricism in defining current EXC based on the statistical performance is clearly limited by the form of the exact energy functional. Extension of DFT to include the dependence with E to restore variationality may improve existent functionals to describe the ground and excited states. ???????????????? Financial support has been provided by the Spanish Ministerio de Econom?a y Competitividad (formerly Ministerio de Ciencia e Innovaci?n) through grants CTQ2011-22505 and FIS2008-02238) and by the Generalitat de Catalunya through grants 2009SGR-1472, 2009SGR-1041 and XRQTC. Part of the computational time has been provided by the Centre de Supercomputaci? de Catalunya (CESCA) which is also gratefully acknowledged. ?(?1s1,?,?nsn ) Page 14 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 15 Page 15 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 16 ?????? ?? Qualitative representation of the L?wdin function f (E) for a FCI problem with K = 8 CSF's and H11 = Eref, to show the general behaviour of this function. In this example, the horizontal assimptote corresponds to Eref (i.e. single configuration representations: RHF for closed shell systems, ROHF or GVB for open shell singlets and triplets). The points E1 ? E2 ? E3 ? ... ? E8 correspond to the set of eigenvalues of the full ? matrix of dimension K = 8 (i.e.: those in which f (E) = E). The K - 1 = 7 vertical assimptotes a1 ? a2 ? a3 ? ... ? a7 correspond to the eigenvalues of the K - 1 reduced ? matrix, namely, ?II,II. See text for more explanation details of this plot. Page 16 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 17 ????? ?? Relevant energy values for the FCI energies and energies of representative EXC[?] functionals. In all cases, the basis set is STO-3G and the geometries correspond to CISD/6-31++g** optimised structures (see text for details). The same sets of FCI eigenvalues are obtained independently of the orbitals used to construct the determinants discussed in Table II. ??? ??? ??? ??? ???? ??? ?? ? ? ? ????? 70 616 382 1436 1436 148 106 ?? ?????????? ?????????? ??????????? ?????????? ?????????? ?????????? ????????????? -74.419386 -54.989548 -393.780008 -38.973091 -55.099684 -37.943644 -147.146834?? -74.015740 -54.931878 -393.610959 -38.917079 -55.065056 -37.888937 -146.901982??? ???? -28.052661 -16.773511 -179.589921 -9.0789790 -13.342182 -11.697235 -103.312082?? -27.401570 -16.607329 -177.750980 -8.9003961 -13.139050 -11.366848 -101.543057 ?????????? -74.962674 -55.453330 -394.311615 -39.726846 -55.866554 -38.429892 -147.632382 ???????? -74.731653 -55.289280 -393.511068 -39.616730 -55.709647 -38.235051 -147.197578 ??????????? -75.278919 -55.752563 -394.908835 -40.006167 -56.167937 -38.617965 -148.238832 ??????? -75.277026 -55.744244 -394.899925 -39.994248 -56.157526 -38.614300 -148.245053 ????????? -75.225100 -55.706750 -394.755311 -39.966679 -56.123036 -38.581657 -148.141707 ?????????? -75.326254 -55.796958 -394.970071 -40.051700 -56.210114 -38.666296 -148.328874 ??????????? -75.193024 -55.688394 -394.529979 -39.958448 -56.104226 -38.573009 -148.079450 ??????? -75.349301 -55.814583 -395.109922 -40.059908 -56.228813 -38.668603 -148.359499 ??????? -75.245260 -55.725543 -394.816928 -39.984130 -56.141393 -38.599394 -148.155159 ???????? -75.312120 -55.784317 -394.951857 -40.038517 -56.197663 -38.646414 -148.273295 Page 17 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Fo r P eer Re vie w O nly 18 ?? ??? ??R ele van tv alu es for the L? wd in fun cti on usi ng dif fer ent ss ets of orb ita lst oc ons tru ctt he FC Iw ave -fu nct ion s. Th ee xac ten erg ya nd the coe ffic ien to ft he do mi nan tC SF in the gro und -st ate wa ve- fun cti on .In all cas es, the bas is set is ST O- 3G and the geo me trie sc orr esp ond to CIS D/6 -31 ++ g** op tim ise ds tru ctu res (se ete xt for det ails ). ??? ?? ??? ??? ?? ?? ??? ?? ? ? ? ? ?? ? ?? ? ?? ? ?? ?? ?? ? ? ? ?? ?? ??? ??? ? ? ? ??? ?? -75 .01 21 14 -55 .51 754 3 -39 4.3 540 92 -39 .80 545 1 -55 .94 670 2 -38 .47 260 8 -14 7.7 478 95 ?? ?? ??? ?? ? ??? ?? ? -74 .96 12 08 -55 .45 076 2 -39 4.3 117 91 -39 .72 252 9 -55 .86 313 9 -38 .42 785 6 -14 7.6 306 96 ? ? ??? ??? -74 .96 26 74 -55 .45 333 0 -39 4.3 116 15 -39 .72 684 6 -55 .86 655 4 -38 .42 989 2 -14 7.6 323 82 ? ? 0.9 866 96 0.9 832 85 0.9 843 08 0.9 786 63 0.9 803 82 0.9 846 50 0.9 64 67 5 ?? ?? ?? ?? ? ??? ?? ? -74 .78 62 24 -55 .33 971 2 -39 4.2 223 58 -39 .66 865 0 -55 .62 018 9 -38 .38 355 8 -14 7.4 257 54 ? ? ??? ??? -74 .79 80 71 -55 .35 297 0 -39 4.2 248 13 -39 .68 159 3 -55 .64 073 4 -38 .39 168 3 -14 7.4 322 19 ? ? 0.9 159 02 0.9 473 72 0.9 522 31 0.9 689 63 0.9 241 57 0.9 725 71 0.8 88 46 1 ?? ?? ?? ?? ??? ?? ? -74 .96 04 33 -55 .44 899 1 -39 4.3 067 86 -39 .71 803 8 -55 .86 145 5 -38 .42 476 5 -14 7.6 287 81 ? ? ??? ??? -74 .96 19 94 -55 .45 189 0 -39 4.3 070 12 -39 .72 309 9 -55 .86 524 5 -38 .42 725 1 -14 7.6 304 06 ? ? 0.9 867 16 0.9 827 93 0.9 834 80 0.9 776 85 0.9 800 36 0.9 838 24 0.9 64 70 2 ?? ?? ?? ? ??? ?? ? -74 .96 03 40 -55 .45 022 9 -39 4.3 088 81 -39 .72 121 6 -55 .86 255 6 -38 .42 638 0 -14 7.6 287 70 ? ? ??? ??? -74 .96 18 29 -55 .45 292 9 -39 4.3 090 88 -39 .72 583 9 -55 .86 611 3 -38 .42 860 4 -14 7.6 303 66 ? ? 0.9 867 32 0.9 832 33 0.9 841 10 0.9 784 85 0.9 803 32 0.9 841 34 0.9 64 64 5 ?? ?? ?? ? ??? ?? ? -74 .96 07 65 -55 .45 027 9 -39 4.3 098 20 -39 .72 126 9 -55 .86 279 3 -38 .42 693 6 -14 7.6 298 08 ? ? ??? ??? -74 .96 22 21 -55 .45 292 4 -39 4.3 100 09 -39 .72 578 1 -55 .86 627 2 -38 .42 907 7 -14 7.6 313 82 ? ? 0.9 867 74 0.9 831 79 0.9 841 19 0.9 783 79 0.9 802 75 0.9 844 13 0.9 64 79 2 ?? ?? ??? ??? ??? ?? ? -74 .96 09 33 -55 .45 058 7 -39 4.3 112 63 -39 .72 255 2 -55 .86 287 3 -38 .42 751 6 -14 7.6 303 74 ? ? ??? ??? -74 .96 23 66 -55 .45 316 7 -39 4.3 114 33 -39 .72 688 0 -55 .86 630 1 -38 .42 956 4 -14 7.6 319 21 ? ? 0.9 868 59 0.9 833 72 0.9 844 42 0.9 786 64 0.9 804 87 0.9 848 85 0.9 64 83 6 Pa ge 18 of 21 UR L: htt p:/ /m c.m an us cri ptc en tra l.c om /ta nd f/tm ph Mo lec ula r P hy sic s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Fo r P eer Re vie w O nly 19 Pa ge 19 of 21 UR L: htt p:/ /m c.m an us cri ptc en tra l.c om /ta nd f/tm ph Mo lec ula r P hy sic s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 For Peer Review Only 20 ?????????? [1] R. McWeeny, Philos. Mag. B ??, 727 (1994). [2] P.-O. L?wdin, Adv. Phys. ??, 597 (2001). [3] The standard definition of the one- and two-electron density matrices, ?1 and ?2, from the n-electron wave function is given by and [4] C. Garrod and J.K. Percus, J. Math. Phys. ?, 1756 (1964). [5] R.J. Harrison and N.C. Handy, Chem. Phys. Lett. ??, 386 (1983). [6] R.J. Harrison, J. Chem. Phys. ??, 5021 (1991). [7] C.W. Bauschlicher, P.R. Taylor, J. Chem. Phys. ??? 2779 (1986). [8] C.W. Bauschlicher, S.R. Langhoff, P.R. Taylor, Adv. Chem. Phys. ??, 103 (1990). [9] F. Illas, J. Rubio, J.M. Ricart and P.S. Bagus, J. Chem. Phys. ??, 1877 (1991). [10] P. Hohenberg and W. Kohn, Phys. Rev. ???, B864 (1964). [11] M. Levy, Proc. Natl. Acad. Sci. (USA) ??, 6062 (1979). [12] W. Kohn and L. Sham, Phys. Rev. ???, A1133 (1965). [13] R. McWeeny, Methods of Molecular Quantum Mechanics (2nd edition) Academic Press, Oxford (1992). [14] M. Caballero, I. de P. R. Moreira and J. M. Bofill, J. Chem. Phys. ???, 174107 (2013). [15] P.-O. L?wdin, J. Math. Phys. ?, 969 (1962). [16] P.-O. L?wdin in Perturbation Theory and its Application in Quantum Mechanics, edited by C. H. Wilcox (Wiley, New York, 1966), p. 255. [17] B.O. Roos, P.R. Taylor, and P.E.M. Siegban, Chem. Phys. ??, 157 (1980). [18] W. Duch and J. Karwowski, Comput. Phys. Report ?, 93 (1985). [19] M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J.H. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen, S.J. Su, T.L. Windus, M. Dupuis, J.A. Montgomery, J. Comput. Chem. ??, 1347 (1993). ?(?1s1,?,?nsn ) ? 1 ?1;?1'( ) = n ? ? ? ?1s1,?,?nsn( )?* ?1's1,?,?nsn( )ds1?dsn d?2?d?n s1 ? sn ? ?2 ? ?n ? ? 2 ?1,?2;?1',?2'( ) = n n ?1( )2 ? ? ? ?1s1,?2s2,?,?nsn( ){s1?sn??3??n? ?* ?1's1,?2' s2,?,?nsn( )}ds1?dsnd?3?d?n Page 20 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 For Peer Review Only 21 [20] M.S. Gordon, M.W. Schmidt in Theory and Applications of Computational Chemistry,the first forty years, edited by C.E. Dykstra, G. Frenking, K.S. Kim, G.E. Scuseria, editors Elsevier, Amsterdam, 2005, Chapter 41, pp 1167-1189. [21] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G.A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H.P. Hratchian, A.F. Izmaylov, J. Bloino, G. Zheng, J.L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J.A. Montgomery, Jr., J.E. Peralta, F. Ogliaro, M. Bearpark, J.J. Heyd, E. Brothers, K.N. Kudin, V.N. Staroverov, T. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J.C. Burant, S.S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J.E. Knox, J.B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, R.L. Martin, K. Morokuma, V.G. Zakrzewski, G.A. Voth, P. Salvador, J.J. Dannenberg, S. Dapprich, A.D. Daniels, O. Farkas, J.B. Foresman, J.V. Ortiz, J. Cioslowski, and D.J. Fox, Gaussian 09, Revision B.01, Gaussian, Inc., Wallingford CT, 2010. [22] J. C. Slater, The Self-Consistent Field for Molecular and Solids, Quantum Theory of Molecular and Solids, Vol. 4 (McGraw-Hill, New York, 1974). [23] S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. ??, 1200 (1980). [24] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B ??, 6671 (1992). Erratum: ibid. ??, 4978 (1993). [25] A. D. Becke, Phys. Rev. A ??, 3098 (1988). [26] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B ??, 785 (1988). [27] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. ??, 3865 (1996). Erratum: ibid. ??, 1396 (1997). [28] J. M. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys. Rev. Lett. ??, 146401 (2003). [29] J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev. Lett. ??, 2544 (1999). [30] T. Van Voorhis and G. E. Scuseria, J. Chem. Phys. ???, 400 (1998). [31] A. D. Becke, J. Chem. Phys. ??, 5648 (1993). [32] C. Adamo and V. Barone, J. Chem. Phys. ???, 6158 (1999). [33] G. Corongiu and E. Clementi, Theor. Chem. Acc. ???, 209 (2009). Page 21 of 21 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 REGULAR ARTICLE The reaction between HO and (H2O)n (n 5 1, 3) clusters: reaction mechanisms and tunneling effects Javier Gonzalez ? Marc Caballero ? Antoni Aguilar-Mogas ? Miquel Torrent-Sucarrat ? Ramon Crehuet ? Albert Sole? ? Xavier Gime?nez ? Santiago Olivella ? Josep M. Bofill ? Josep M. Anglada Received: 5 July 2010 / Accepted: 13 September 2010 / Published online: 30 September 2010 ? Springer-Verlag 2010 Abstract The reaction between the HO radical and (H2O)n (n = 1, 3) clusters has been investigated employ- ing high-level quantum mechanical calculations using DFT-BH&HLYP, QCISD, and CCSD(T) theoretical approaches in connection with the 6-311 ? G(2df,2p), aug-cc-pVTZ, and aug-cc-pVQZ basis sets. The rate con- stants have also been calculated and the tunneling effects have been studied by means of time?dependent wavepac- ket calculations, performed using the Quantum?Reaction Path Hamiltonian method. According to the findings of previously reported theoretical works, the reaction between HO and H2O begins with the formation of a pre-reactive complex that is formed before the transition state, the formation of a post-reactive complex, and the release of the products. The reaction between HO and (H2O)2 also begins with the formation of a pre-reactive complex, which dis- sociates into H2O?HO ? H2O. The reaction between HO and (H2O)3 is much more complex. The hydroxyl radical adds to the water trimer, and then it occurs a geometrical rearrangement in the pre-reactive hydrogen-bonded com- plex region, before the transition state. The reaction between hydroxyl radical and water trimer is computed to be much faster than the reaction between hydroxyl radical and a single water molecule, and, in both cases, the tun- neling effects are very important mainly at low tempera- tures. A prediction of the atmospheric concentration of the hydrogen-bonded complexes studied in this work is also reported. Keywords Atmospheric chemistry ? Hydroxyl radical ? Water clusters ? Reaction mechanism ? Tunneling effects 1 Introduction Hydroxyl radical (HO) is a very important species in sev- eral fields of chemistry. In the earth atmosphere, it plays a central role in the degradation processes of air pollutants such as carbon monoxide or volatile organic compounds (VOCs) [1]. In atmospheric conditions with low NOx concentrations, hydroxyl radical can destroy one ozone molecule producing molecular oxygen and hydroperoxyl radical, which destroys a second ozone molecule yielding molecular oxygen and recycling the hydroxyl radical [1?6]. Moreover, it also interacts with atmospheric gas-phase water, and it can be taken up by atmospheric aerosols or water droplets, so that it can oxidize soluble tropospheric pollutants [7?9]. In biological systems, hydroxyl radical is also a powerful oxidant. It is one of the well-known Published as part of the special issue celebrating theoretical and computational chemistry in Spain. J. Gonzalez ? M. Torrent-Sucarrat ? R. Crehuet ? S. Olivella ? J. M. Anglada (&) Departament de Qu??mica Biolo`gica i Modelitzacio? Molecular, Institut de Qu??mica Avanc?ada de Catalunya, IQAC?CSIC, c/Jordi Girona, 18, 08034 Barcelona, Spain e-mail: anglada@iqac.csic.es M. Caballero ? A. Aguilar-Mogas ? A. Sole? ? X. Gime?nez Departament de Qu??mica F??sica, Universitat de Barcelona, c/Mart?? i Franque?s, 1, 08028 Barcelona, Spain J. M. Bofill (&) Departament de Qu??mica Orga`nica, Universitat de Barcelona, c/Mart?? i Franque?s, 1, 08028 Barcelona, Spain e-mail: jmbofill@ub.edu M. Caballero ? A. Aguilar-Mogas ? A. Sole? ? X. Gime?nez ? J. M. Bofill Institut de Qu??mica Teo`rica i Computacional, Universitat de Barcelona (IQTCUB), c/Mart?? i Franque`s, 1, 08028 Barcelona, Spain 123 Theor Chem Acc (2011) 128:579?592 DOI 10.1007/s00214-010-0824-5 ??reactive oxygen species??, which play an important role in oxidative stress, aging, and cell damage [10?13]. In envi- ronmental chemistry, hydroxyl radical is a highly oxidant molecule reacting by addition to double bonds or by abstracting hydrogen atoms. Its chemistry is directly rela- ted to waste water treatments, and it is associated with the peroxone chemistry as one of the advanced oxidation processes [14?17]. H2O? HO? HO? H2O ?1? Among the reactions involving hydroxyl radical, the reaction with water (reaction 1) has become a prototype for the hydrogen atom abstraction reactions by free radicals. This reaction is also relevant for atmospheric purposes because it can be associated with the isotopic composition of the atmospheric water [18]. In the troposphere, the concentration of hydroxyl radical is close to 10-7 molecule cm-3 [3], and a typical gas-phase concentration of H2O is 6.95 9 1017 molecule cm-3, corresponding to 50% of relative humidity at 298 K, so reaction 1 can easily take place. Unfortunately, the direct measurement of the rate constant of reaction 1 is not possible because it is a silent reaction regarding the formation of the products. However, Dubey et al. [18] investigated the deuterium-labeled reactions, and they suggested an activation energy Ea = 4.2 ? 0.5 kcal mol-1 over the temperature range 300?420 K. Several theoretical studies have also been published in the literature regarding reaction 1 [18?25], reporting an energy of activation close to 10 kcal mol-1, but pointing out an important contribution from quantum mechanical tunneling effect. In the present work, we are considering the theoretical study of reaction 1 along with reactions of hydroxyl radical with a cluster of two and three water molecules. Our aim is twofold: (a) Firstly, to investigate how additional water molecules affect reaction 1. It has been recently shown in the literature that water vapor produces a catalytic effect in several reactions of atmospheric relevance [26?34], and therefore it is relevant to have a deeper knowledge of the role that water vapor plays in gas-phase reactions involving hydrogen atom abstraction processes; (b) Secondly, to analyze the quantum chemical tunneling effect in the studied reactions and consequently its influence on the reaction rate. In this regard, it is known that complex structures involving hydrogen transfer processes might lead to enhanced tunneling [35?37]. In addition, it is worth noting that water clusters exhibit a network of hydrogen bonds, whose continuous rearrangement is inherently of quantum mechanical origin. Zero-point energy issues add to the former phenomena, thereby restricting the available energy flows and substantially altering the ensuing H-atom motion, when compared to classical mechanics predictions [38]. Therefore, this quantum effect is studied here with emphasis on multidimensional issues and, in particular, on specificities involving water clusters of increasing size, by means of an efficient quantum reaction path methodology. As far as we know, there are no data yet in the literature regarding the reaction between hydroxyl radical and the water dimer and trimer, despite its potential importance in the chemistry of the troposphere and in climate change. Pfeilsticker and coworkers [39] detected water dimers in the atmosphere by near-infrared absorption spectroscopy, and they suggested an atmospheric concentration of 6 9 1014 molecule cm-3 at 292 K; Dunn et al. [40] predict a dimer concentration of 9 9 1014 molecule cm-3 and Goldman et al. [41] predict concentrations up to 1.7 9 1015 molecule cm-3 at high relative humidity. In these studies, it has also been speculated that water trimers may also exist in the troposphere, with a predicted con- centration of 2.6 9 1012 molecule cm-3. Overall, these data stress the potential role that water dimer and trimer can play in the atmosphere. 2 Technical details of the calculations We employed density functional BH&HLYP method [42] with the 6?311 ? G(2df,2p) basis set [43, 44] to optimize all stationary points investigated in this work. At this level of theory, we also computed the harmonic vibrational frequencies to verify the nature of the corresponding sta- tionary points (minimum or saddle point) and to provide the zero-point vibrational energy (ZPE) and the thermal contributions to the enthalpy and Gibbs energy. Moreover, we performed intrinsic reaction coordinate calculations (IRC) [45?48] to ensure that the transition states connect the desired reactants and products. More accurate relative energies have been obtained by performing single-point CCSD(T) [49?52] energy calculations at the optimized geometries using the more flexible aug-cc-pVTZ basis set [53, 54]. The reliability of our results has been checked by per- forming two additional sets of calculations on the water dimer and trimer as well as on the prototype reaction 1. The first one involves single-point CCSD(T) energy cal- culations using the aug-cc-pVQZ basis set at the BH&HLYP optimized geometries. The second one, involves re-optimization of some selected stationary points employing the QCISD method [55] with the 6- 311 ? G(2df,2p) basis set and then carrying out single- point CCSD(T) energy calculations at the optimized geometries using the aug-cc-pVTZ basis set. In this case, we checked that the harmonic vibrational frequencies obtained at QCISD and BH&HLYP levels of theory compare well, which is necessary for the study of the tunneling effects. 580 Theor Chem Acc (2011) 128:579?592 123 Moreover, for reaction 1, we also carried out additional CASSCF calculations [56] to investigate the possibility of finding valley-ridge inflection (VRI) points along the potential energy surface (PES). In this case, we employed the 6-311 ? G(2df,2p) basis set and an active space con- sisting in three electrons and three orbitals. Furthermore, to get a reliable energy position of the VRI point, we performed single-point energy calculations at the CCSD(T) level of theory with the aug-cc-pVTZ basis set for the transition state and the VRI point located at the CASSCF level of theory. All the DFT, QCISD, and CCSD(T) calculations have been performed by using the Gaussian 03 suite of programs [57], whereas the CASSCF calculations have been done using the GAMESS program [58]. The tunneling dynamics have been studied by means of time-dependent wavepacket calculations, performed using our recently developed Quantum?Reaction Path Hamilto- nian method (Q?RPH) [59]. A coherent state wavepacket is propagated in time along a potential energy profile, obtained from the IRC associated with each of the present reactions, under a position-dependent mass term, which accounts for the effect of the transversal vibrational modes. It is thus an effective one-dimensional reaction?path implementation of the nuclear dynamics [60]. Thanks to the reduced-dimen- sional nature of the method, it favorably scales with an increase in the number of degrees of freedom, being suitable for multidimensional dynamics studies of large polyatomic systems [61, 62], including quantum effects. The variable mass term is found to depend on gradients and hessians along the reaction path, so that it is fully computed prior to the propagation step and stored along with the potential energy profile [63]. The Q?RPH method has proven to reliably describe the multidimensional dynamics of several polyatomic reactions such as H ? H2 and, remarkably, F ? H2 (including the low-energy resonances), by means of the above effective one-dimensional, variable mass method. The transmission factor, i.e., the Fourier transform of a suitable wavepacket?s autocorrelation function, has been used to provide quantitative estimations of the tunneling effect. Calculations are made defining a sufficiently large spatial grid and then obtaining the time evolution with small time increments, by means of an Askar?Cakmak numerical method [64]. The number of grid points and the time increment are used as convergence parameters to ensure stability of results. 3 Results As usual in many reactions of atmospheric interest, all reactions investigated in this work begin with the formation of hydrogen-bonded complexes occurring before the tran- sition states, the formation of post-reactive hydrogen- bonded complexes, and the release of products. Moreover, all reactions considered in this work are symmetric, so that pre-reactive and post-reactive complexes are the same species and the reactants and products too. In what follows the pre-reactive complexes are labeled by the prefix CR followed by a number and the transition states are labeled by the prefix TS followed by a number too. In many cases, it occurs that there are isomers of a given stationary point that differ in the relative orientation of dangling hydrogen atoms. To distinguish between these isomers, we appended lower case letters to the acronym of the corresponding stationary point. 3.1 The water dimer and trimer The water dimer and trimer have already been reported in the literature, and we considered them as reactants in the present study. The results regarding their relative stabilities are contained in Table 1, whereas Fig. 1 shows the most relevant geometrical parameters. As shown in Table 1 we checked the reliability of our calculations by performing geometry optimizations at the BH&HLYP and QCISD levels of theory, and the computed bond lengths obtained using both methods differ in less than 0.03 A?. At the optimized geometries, we carried out single-point energy calculations at the CCSD(T) level of theory with the aug- cc-pVTZ and aug-cc-pVQZ basis sets for de dimer and with the aug-cc-pVTZ for the trimer. The water dimer (H2O)2 has CS symmetry. Its electronic state is X1A?, and we computed a binding energy of 2.86 kcal mol-1 in an excellent agreement with the 3.3 kcal mol-1 reported in the literature [40, 65]. The water trimer has two isomers having a three- membered ring structure, and we labeled them as (H2O)3-a and (H2O)3-b. Both differ in the orientation of the dangling hydrogen atoms (two toward one side and the third toward the opposite side in (H2O)3-a; and the three pointing to the same side in (H2O)3-b, see Fig. 1). The probability that the water trimer is formed by a collision of three water mol- ecules is very low, and therefore we assumed that it is formed by reaction between a water dimer and a water molecule. Thus, taking (H2O)2 ? H2O as reactants, the computed binding energies are 7.73 and 7.40 kcal mol-1 for (H2O)3-a and (H2O)3-b, respectively, which compares very well with the 7.26 kcal mol-1 reported in the litera- ture [40]. 3.2 The reaction between hydroxyl radical and a single water molecule As pointed out previously, there are several theoretical studies in the literature dealing with this reaction [18, 19, 21?24]. Therefore, we considered only the reaction path Theor Chem Acc (2011) 128:579?592 581 123 having the lowest energy barrier, and we will focus on the main trends regarding the reaction mechanism. For this reaction, we also checked the reliability of our calculations by performing geometry optimizations at BH&HLYP and QCISD levels of theory, and the com- puted bond lengths obtained using both methods differ in less than 0.04 A?. At the optimized geometries, we carried out single-point energy calculations at the CCSD(T) level of theory with the aug-cc-pVTZ and aug-cc-pVQZ basis sets. Table 2 contains the relative energies of reaction 1, and Fig. 2 shows an energy profile along with the most relevant geometrical parameters of the stationary points. The reac- tion begins with the barrierless formation of a pre-reactive hydrogen-bonded complex (CR1), occurring before the transition state (TS1), followed by the formation of a post- reactive hydrogen-bonded complex (CR1) and the release of the products. CR1 has CS symmetry (X2A?), and it is formed by interaction between the hydrogen of the hydroxyl radical and the oxygen of water. The computed hydrogen bond length is 1.898 A?, and its binding energy is computed to be 3.78 kcal mol-1. Our results are in excel- lent agreement with other experimental and theoretical results from the literature [27, 66?71]. TS1 has C2 sym- metry (2B), and it involves a hydrogen atom transfer (HAT) mechanism. The hydrogen atom being transferred is placed midway between the two oxygen atoms (dOH = 1.150 A??). The process corresponds to the homolytic breaking and forming of the OH bonds, and it involves an adiabatic energy barrier of 12.53 kcal mol-1, relative to CR1, which is in good agreement with other results from the literature [19, 21, 22, 24]. Table 2 shows that the energy barrier computed at different levels of theory used in this work differs in less than 0.26 kcal mol-1. This provides further support to the reliability of our results. Table 1 Zero-point energies (ZPE in kcal mol-1), entropies (S in a.u.), and relative energies, ZPE-corrected energies, enthalpies, and Gibbs energies (in kcal mol-1) for the formation of the water dimer and trimer Compound Methoda ZPE S DE D(E ? ZPE) DH(298 K) DG(298 K) H2O ? H2O A 27.88 90.0 0.00 0.00 0.00 0.00 B 27.27 90.1 0.00 0.00 0.00 0.00 C 27.27 90.1 0.00 0.00 0.00 0.00 (H2O)2 A 30.10 69.4 -5.23 -3.01 -3.52 2.62 B 29.50 69.7 -5.20 -2.97 -3.45 2.63 C 20.59 69.7 -5.09 -2.86 -3.34 2.75 (H2O)2 ? H2O A 44.05 114.4 0.00 0.00 0.00 0.00 B 44.05 114.4 0.00 0.00 0.00 0.00 (H2O)3-a A 47.38 80.0 -11.07 -7.73 -9.00 1.24 B 47.38 80.0 -11.00 -7.67 -8.93 1.31 (H2O)3-b A 46.88 83.8 -10.24 -7.40 -8.34 0.77 B 46.88 83.8 -10.19 -7.36 -8.29 0.82 a Method A stands for values computed at the CCSD(T)/aug-cc-pVTZ//BH&HLYP/6-311 ? G(2df,2p) level with ZPE, enthalpy, and Gibbs energy corrections obtained at the BH&HLYP/6-311 ? G(2df,2p) level; Method B stands for values computed at the CCSD(T)/aug-cc-pVTZ// QCISD/6-311 ? G(2df,2p) level with ZPE, enthalpy, and Gibbs energy corrections obtained at the QCISD/6-311 ? G(2df,2p) level; Method C stands for values computed at the CCSD(T)/aug-cc-pVQZ//QCISD/6-311 ? G(2df,2p) level with ZPE, enthalpy, and Gibbs energy corrections obtained at BH&HLYP/6-311 ? G(2df,2p) level Fig. 1 Selected geometrical parameters obtained at BH&HLYP and QCISD levels of theory for the optimized structures of water dimer and trimer. Values in parenthesis correspond to QCISD-optimized geometries 582 Theor Chem Acc (2011) 128:579?592 123 With respect to the PES of the HO ? H2O reaction, there is a controversy in the literature regarding the features of the pre-reactive hydrogen-bonded complex. Unresolved question concerns whether the pre-reactive complex is the HOH?OH species, as suggested by Hand et al. [22], or the global minimum H2O?HO (CR1), as pointed out by Masgrau et al. [19] and by Uchimaru et al. [21]. To bring more light to this point, we also considered the possibility that along the PES, it could exist a branching of the reaction that could occur through a VRI point [72]. We note that the steepest descent from the transition state in mass-weighted Cartesian coordinates is the simplest and widely used representation of a reaction path, which is well known as the IRC [48]. However, it should be taken into account that the IRC does not bifurcate, and due to this fact this representation of a reaction path is not well adapted to tackle the problem of branching reaction paths. Neverthe- less, the VRI points gain importance when tackling the problem of reaction path branching. In fact, the possible existence of a VRI point along the PES in many cases explains the mixture of products observed experimentally, and now it is considered in many mechanistic studies [73]. From strictly theoretical point of view, the VRI points may form a manifold in the configuration space of the chemical species. This manifold can have the dimension N-2, if the configuration space of the PES has dimension N [74]. The characteristic attribute of a VRI point is that at least an eigenpair of the Hessian at this point has zero eigenvalue. This eigenvalue changes its sign when going along the gradient, where the corresponding eigenvector is orthogo- nal to the gradient. This means that a valley changes into a ridge or vice versa. The above two conditions, namely eigenvalue zero and zero overlap between the corre- sponding eigenvector with the gradient, characterize the N-2 dimensional manifold. In addition to the VRI points another possibility emerges, that is, the IRC meets in a point a direction with zero curvature of the PES orthogonal to the gradient. There, the gradient is not orthogonal to one of the eigenvectors of the PES, in the general case, the eigenvalues are not zero. The zero curvature of the PES along the level line comes from a suitable linear combi- nation of the eigenvectors. This point is labeled as valley- ridge transition point (VRTp). Notice that the features of a Table 2 Zero-point energies (ZPE in kcal mol-1), entropies (S in a.u.), and relative energies, ZPE-corrected energies, enthalpies, and Gibbs energies (in kcal mol-1) for the reaction between HO and H2O Compound Methoda ZPE S DE D(E ? ZPE) DH(298 K) DG(298 K) HO ? H2O A 19.5 87.5 0.00 0.00 0.00 0.00 B 19.0 87.6 0.00 0.00 0.00 0.00 C 19.5 87.5 0.00 0.00 0.00 0.00 CR1 A 21.5 67.3 -5.90 -3.85 -4.49 1.54 B 21.2 67.0 -5.87 -3.72 -4.39 1.77 C 21.5 67.3 -5.83 -3.78 -4.42 1.61 TS1 A 18.8 60.2 9.27 8.55 7.00 15.14 B 18.4 60.3 9.15 8.56 7.02 15.17 C 18.8 60.2 9.48 8.75 7.21 15.35 a Method A stands for values computed at the CCSD(T)/aug-cc-pVTZ//BH&HLYP/6-311 ? G(2df,2p) level with ZPE, enthalpy, and Gibbs energy corrections obtained at the BH&HLYP/6-311 ? G(2df,2p); Method B stands for values computed at the CCSD(T)/aug-cc-pVTZ// QCISD/6-311 ? G(2df,2p) level with ZPE, enthalpy, and Gibbs energy corrections obtained at the QCISD/6-311 ? G(2df,2p); Method C stands for values computed at the CCSD(T)/aug-cc-pVQZ//BH&HLYP/6-311 ? G(2df,2p) level with ZPE, enthalpy, and Gibbs energy corrections obtained at the BH&HLYP/6-311 ? G(2df,2p) Fig. 2 Schematic reaction profile and selected geometrical parame- ters, at BH&HLYP and QCISD levels of theory, for the stationary points of the H2O ? HO reaction. Values in parenthesis correspond to QCISD-optimized geometries Theor Chem Acc (2011) 128:579?592 583 123 VRI point are stronger than those of a VRTp point. As will see below, the present PES possesses this type of point. In the PES associated with the present reaction mecha- nism, the IRC curve leads directly from the transition state TS1 to the corresponding deeper valley. The IRC curve does not follow the crest of the ridge that leads to another transition state, and it leads to one minimum that corre- sponds to the global H2O?HO (CR1) minimum. In other words, the located IRC leaves the asymmetric ridge, and it goes to a minimum. However, this IRC path crosses an equipotential line or contour line of the PES with null curvature and meets with the border of the ridge region. At the CASSCF level of theory, the VRTp is characterized by a HO?HOH length of 1.458 A? and a HO?H?OH angle of 151 degrees. Taking into account the CCSD(T) energies computed at the CASSCF geometries, we predict that the VRTp is located at 3.99 kcal mol-1 below TS1. The bor- der of the ridge region is defined by the set of points such that the projected Hessian matrix [(I - PT) H (I - P), where the P matrix is build by the gradient direction and the six zeros corresponding to translations and rotations] has eigenvectors with null eigenvalues in addition to these seven mentioned directions. These eigenvectors with null eigenvalues are orthogonal to the gradient vector. Notice that these eigenvectors are not eigenvectors of the full Hessian matrix. At each point of the border line of the ridge region, the direction of the equipotential line has null curvature, and this direction is a linear combination of a subset of eigenvectors of the full Hessian matrix. The existence of a VRTp has important implications since both the HOH?OH complex and the global minimum H2O?HO (CR1) can be reached through this reaction. 3.3 The reaction between hydroxyl radical and a water dimer Figure 3 displays a schematic representation of the energy profile of the reaction, along with the most relevant geo- metrical parameters of the corresponding stationary points. Table 3 contains their relative energies. For each of the stationary points, we found two isomers differing in the relative orientations of the dangling hydrogen atoms. These two isomers are distinguished from each other by appending the letter a, and the results displayed in Table 3 show that the two isomers of each stationary point are almost energetically degenerate. For the sake of clarity, we have drawn in Fig. 3 only one of these two isomers for each stationary point, and the discussion along the text will refer to only one of these isomers. The potential energy profile schematized in Fig. 3 shows that, starting at the (H2O)2 ? HO reactants, the reaction begins with the formation of a hydrogen bond complex CR2, for which we computed a binding energy of 7.42 kcal mol-1. This complex has a three-membered ring structure where the two water molecules and the hydroxyl radical are held together by three hydrogen bonds. The calculated geometrical parameters compare quite well with those reported recently by Tsuji et al. [75]. Then, the reaction can proceed in two different ways, namely (a) dissociating into the H2O?HO hydrogen bond complex plus H2O and (b) proceeding through three different tran- sition states. In this case, the exit channels are the same species than in the reactant channels. Regarding to path (a), the CR2a complex has a geo- metric orientation that allows to dissociate directly into Fig. 3 Schematic reaction profile and selected geometrical parameters, at BH&HLYP level of theory, of the stationary points of the (H2O)2 ? HO reaction 584 Theor Chem Acc (2011) 128:579?592 123 H2O?HO ? H2O products, and the computed reaction energy is -0.83 kcal mol-1. This process is tagged in the present work as reaction 2, and it is very important from atmospheric purposes as it contributes to the formation of the hydrated hydroxyl radical in the atmosphere. ?H2O?2 ? HO ! CR2a ! H2O. . .HO? H2O ?2? Regarding to path (b), we found three different elementary reactions. The first one occurs through TS2a, which has a three-membered ring structure. This reaction involves the homolitic breaking and forming of the O3?H and H?O1 bonds, respectively, so that the hydroxyl radical abstracts one hydrogen atom from a water molecule through a HAT mechanism. Figure 3 shows that the hydrogen atom being transferred is slightly closer (0.956 A?) to the oxygen atom of the water moiety than to the oxygen atom of the radical (1.118 A?) and that the transition state is stabilized by two hydrogen bonds. From an energetic point of view, our calculations predict this transition state to lie 5.58 kcal mol-1 above the energy of the reactants water dimer plus hydroxyl radical (see Table 3). At this point, it is worth comparing this value with the 8.75 kcal mol-1 computed for the transition state of the H2O ? HO reaction described in the previous section (see Table 2). Consequently, the addition of a second water molecule produces a relative energy stabilization of the transition state of 3.17 kcal mol-1, which can be attributed to the energy stabilization produced by the formation of two hydrogen bonds in TS2a. However, looking at the energy barrier relative to the CR2a complex, the computed value is 13.07 kcal mol-1, which is even slightly larger than that reported for the naked reaction (12.40 kcal mol-1, see Table 2). Consequently, it seems that the possible catalytic effect originated by the hydrogen bond stabilizations occurring in the transition state is counteracted by the enhanced stability of the pre-reactive complex, in a similar way as reported by Allodi et al. [70] for the methane oxidation by hydroxyl radical. The second and third elementary reactions occur through TS3 and TS4, which lie very high in energy (about 18 kcal mol-1 above the energy of the (H2O)2 ? HO reactants; see Table 3; Fig. 3), so that it is expected that they do not play any role. However, these elementary reactions are interesting from a mechanistic point of view. The anal- ysis of TS3 wave function indicates that the unpaired elec- tron is mainly located over the hydroxyl radical moiety. It does not participate at all in the process, and the reaction mechanism involves a triple proton transfer process. In TS4, the different atoms are oriented in such a way that the unpaired electron of the hydroxyl radical moiety interacts with one oxygen atom of a water molecule, so that it occurs a transfer of an electron from this oxygen atom (O2) to the oxygen of the hydroxyl radical (O1), and this originates a simultaneous transfer of two protons (from O2 to O3 and from O3 to O1, see Fig. 3). This elementary reaction cor- responds to a proton coupled electron transfer mechanism (pcet), and similar processes have been described recently in the literature for the gas-phase oxidation of formic acid by hydroxyl radical and for the same process assisted by a single water molecule [25, 27, 76]. 3.4 The reaction between hydroxyl radical and water trimer Figure 4 contains a schematic energy profile of the reaction between (H2O)3 and HO radical, along with the most rel- evant geometrical parameters of the main stationary points. Table 4 contains the relative energies. Figure 4 shows that the reaction begins with the formation of a complex between the water trimer and the hydroxyl radical (CR3), with a computed binding energy of 4.45 kcal mol-1. This complex retains the three-membered ring structure of the Table 3 Zero-point energies (ZPE in kcal mol-1), entropies (S in a.u.), and relative energies, ZPE-corrected energies, enthalpies, and Gibbs energies (in kcal mol-1) for the reaction between HO and (H2O)2 Compound ZPE S DE D(E ? ZPE) DH(298 K) DG(298 K) H2O?H2O ? OH 35.7 111.9 0.00 0.00 0.00 0.00 H2O?HO ? H2O 35.5 112.3 -0.66 -0.83 -0.96 -1.07 CR2 38.5 79.7 -10.28 -7.42 -8.67 0.93 CR2a 38.1 77.9 -9.94 -7.49 -9.03 1.12 TS2 35.5 76.2 6.27 6.06 4.28 14.95 TS2a 35.4 76.7 5.85 5.58 3.86 14.36 TS3 34.8 66.9 18.86 18.03 14.90 28.31 TS3a 34.7 67.0 18.45 17.52 14.41 27.80 TS4 36.2 67.7 17.79 18.29 15.38 28.55 TS4a 35.8 68.2 17.60 17.74 14.90 27.94 Values computed at the CCSD(T)/aug-cc-pVTZ//BH&HLYP/6-311 ? G(2df,2p) level with ZPE, enthalpy, and Gibbs energy corrections obtained at the BH&HLYP/6-311 ? G(2df,2p) Theor Chem Acc (2011) 128:579?592 585 123 water trimer, and the hydrogen atom of the radical interacts with one of the oxygen atoms of (H2O)3 cluster. As in the previous section, for each stationary point there are three isomers differing in the relative orientations of the dangling hydrogen. They are almost degenerate in energy and, for the sake of clarity, we will consider the lowest energy one along the discussion. The reaction goes on through TS51 to form the CR4 complex, which has a four-membered ring structure (see Fig. 4). It is stabilized by four hydrogen bonds, and it lies 8.57 kcal mol-1 below the sum of the energies of the water trimer and the hydroxyl radical reactants. Our computed geometrical parameters compare quite well with those reported recently by Tsuji et al. [75]. After this complex, the reaction can proceed through three different reaction paths. The three elementary reactions have the same elec- tronic features as described in the previous section for the reaction of the water dimer with the hydroxyl radical, and they will be not discussed here. The first reaction (HAT process) goes through TS6, and it involves the homolytic breaking and forming of the O4-H and H-O1 bonds, respectively. Our calculations predict this transition state to lie 5.27 kcal mol-1 above the reactants energy, and the computed energy barrier is 13.84 kcal mol-1 relative to CR4. As discussed in the previous section, TS6 lies lower in energy, relative to the separate reactants, than TS1 of the Fig. 4 Schematic reaction profile and selected geometrical parameters, at BH&HLYP level of theory, of the stationary points of the (H2O)3 ? HO reaction. The reaction products are the same species than the reactants, and therefore they are not explicitly drawn Table 4 Zero-point energies (ZPE in kcal mol-1), entropies (S in a.u.), and relative energies, ZPE-corrected energies, enthalpies, and Gibbs energies (in kcal mol-1) for the reaction between HO and (H2O)3 Compound ZPE S DE D(E ? ZPE) DH(298 K) DG(298 K) (H2O)3 ? OH 52.9 122.6 0.00 0.00 0.00 0.00 (H2O)2?HO ? H2O 52.5 124.7 4.71 4.23 4.25 3.61 CR3a 54.8 99.3 -6.27 -4.45 -4.86 2.07 TS5a 53.9 99.9 -4.55 -3.55 -4.03 2.74 CR4 55.3 93.4 -10.91 -8.57 -9.42 -0.72 TS6 52.3 89.2 5.90 5.27 3.87 13.82 TS6a 52.1 90.4 6.38 5.53 4.24 13.83 TS6b 52.2 89.8 6.25 5.52 4.18 13.94 TS7 50.1 78.4 14.63 11.78 8.86 22.03 TS8 51.3 78.6 17.45 15.85 13.04 26.14 Values computed at the CCSD(T)/aug-cc-pVTZ//BH&HLYP/6-311 ? G(2df,2p) level with ZPE, enthalpy and Gibbs energy corrections obtained at the BH&HLYP/6-311 ? G(2df,2p) a For the enthalpic, and Gibbs energy corrections, see Footnote 1 1 The CR3 ? CR4 path is very flat and we have failed to find TS5 at BH&HLYP level of theory We have optimized it, and CR3 too, using the B3LYP functional and their corresponding ZPE, entropy and enthalpy corrections have been employed in this case. 586 Theor Chem Acc (2011) 128:579?592 123 naked reaction, due to the extra stabilization originated by the hydrogen bond interactions. However, the energy bar- rier is also slightly larger because of the stability of the pre- reactive complex, counteracting, in part, thus the possible catalytic effect. The transition structure of the second reaction path (TS7) lies 11.78 kcal mol-1 above the energy of the reactants, and the process involves the simultaneous transfer of four protons. The third reaction path goes through TS8, and it lies 15.85 kcal mol-1 above the energy of the reactants. In the same way as discussed for TS5 above, this process involves the transfer of an electron from oxygen atom (O4) to the oxygen of the hydroxyl radical (O1), and this originates a simultaneous transfer of three protons (from O2 to O1, from O3 to O2, and from O4 to O3), through proton-coupled electron transfer mechanism. The large energy barrier computed for these processes suggests that they will not play any role in the chemistry of the atmosphere. 3.5 Tropospheric concentration of the (H2O)2, (H2O)3, H2O?HO, (H2O)2?HO, and (H2O)3?HO complexes The calculations carried out in this work allow us to obtain the equilibrium constants for the formation of the water adducts reported in this work. With these values, we can estimate the atmospheric concentration of water dimer and trimer and the adducts formed between these complexes and the hydroxyl radical. This information is very inter- esting for atmospheric purposes, and therefore we consid- ered three different conditions of relative humidity, namely 25, 50, and 75%, and temperatures ranging between 278 and 308 K. Reactions 3?8 have been considered for esti- mating the concentration of these species and, for the complexes containing hydroxyl radical, we considered a HO concentration of 1.0 9 107 molecules cm-3. H2O? H2O? ?H2O?2 ?3? ?H2O)2 ? H2O? ?H2O?3-a ?4? ?H2O)2 ? H2O? ?H2O?3-b ?5? HO? HO? H2O. . .HO?CR1? ?6? ?H2O)2 ? HO? ?H2O)2. . .HO?CR2? ?7? ?H2O)3-b? HO? ?H2O)3. . .HO?CR4? ?8? In Table 5, we collected the computed equilibrium constants of these reactions, and Table 6 contains the estimated concentration of the (H2O)2, (H2O)3, H2O?HO, (H2O)2?HO, and (H2O)3?HO complexes. Our calculations predict that the dimer concentration ranges between 5.95 9 1013 and 1.06 9 1016 molecule cm-3, whereas the trimer concentration ranges between 1.59?1012 and 1.21 9 1015 molecule cm-3. These values compare with the estimated values between 6 and 9 9 1014 molecule cm-3 for the dimer [39, 40] although Goldman et al. predict higher concentrations, up to 1.7 9 1015 molecule cm-3, at higher relative humidity [41]. Regarding the concentrations of the (H2O)n?HO com- plexes, the results displayed in Table 6 show that our cal- culations predict atmospheric concentrations of H2O?HO in the 1.01 9 104?8.78 9 104 molecule cm-3 range, whereas the concentrations of the (H2O)2?HO and (H2O)3?HO complexes are predicted to be much smaller, up to 1.14 9 103 molecule cm-3. For the H2O?HO, our results compare very well with the 5.5 9 104 mole- cule cm-3 predicted by Allodi et al. at 298 K [70] but for the (H2O)2?HO and (H2O)3?HO complexes our calculations differ in about one order of magnitude from those predicted byAllodi et al. In analyzing these values, it should be pointed out that they have been obtained considering a population of the hydroxyl radical of 107 molecule cm-3, and therefore predicted concentration of these complexes will increase several orders of magnitude if local conditions imply higher concentrations of hydroxyl radical. 3.6 Tunneling dynamics and rate constants The energy profiles shown in Figs. 2, 3, and 4 evidence that the present symmetric HAT proceeds across barrier heights of several kcal mol-1. This means, as it is well known, that tunneling transmission is instrumental for the accurate description of the reaction mechanism. In addition, the pre- Table 5 Calculated equilibrium constants (Keq in cm3 molecule-1) for the formation of the (H2O)2, (H2O)3; H2O?HO, (H2O)2?HO, and (H2O)3?HO complexes at different temperatures a For the reaction see text Reactiona 278 K 288 K 298 K 308 K 3 5.55 9 10-21 4.60 9 10-21 3.87 9 10-21 3.30 9 10-21 4 8.61 9 10-20 5.06 9 10-20 3.09 9 10-20 1.95 9 10-20 5 1.72 9 10-19 1.05 9 10-19 6.68 9 10-20 4.37 9 10-20 6 9.75 9 10-21 7.61 9 10-21 6.05 9 10-21 4.89 9 10-21 7 8.87 9 10-21 5.41 9 10-21 3.40 9 10-21 2.21 9 10-21 8 6.83 9 10-19 3.84 9 10-19 2.24 9 10-19 1.36 9 10-19 Theor Chem Acc (2011) 128:579?592 587 123 reactive complexes also play a determining role in the form of interferences (resonances) that may be reflected in the transmission factor. Quantifying the above tunneling, resonance features requires performing some sort of quantum dynamics study. The present systems involve a large number of coupled degrees of freedom, so that solving exactly the nuclear dynamics problem proves as a daunting task. To alleviate this problem, some of the authors recently developed a quantum dynamics methodology, based on restricting the molecular motion to the reaction path, so that the compu- tational requirements are reduced to a minimal amount. Specific details of the methodology are given elsewhere [59]. Here, it suffices to understand that a properly defined wavepacket has been time evolved along the reaction profile, as provided by the reaction path, and the energy- dependent transmission factor is computed afterward, by means of a suitable Fourier transform of a reactants auto- correlation function. Results of the tunneling dynamics are shown below, restricted to one- and two-water cases, for the sake of deeming clearer the ensuing comparisons. Since in all the cases, the HAT mechanism is energetically favored (see Figs. 2, 3, 4), we limited ourselves to it. This process has proven to have an important tunneling contri- bution for reaction 1 as it has been pointed out in previous studies of the literature [19, 21]. Figure 5 shows the energy-dependent transmission factor, for the reaction between OH radical and one water molecule. Fixed-mass calculations are compared with variable mass, since this comparison provides interesting clues on the role played by perpendicular modes in the reaction dynamics. Fixed-mass calculations show an important tunneling contribution to transmission, as well as antitunneling at energies above the barrier. These are rather expected results, considering the fact that, in the present process, a light atom, hydrogen, is transmitted between two heavy centers (H2O and O). In addition, inspection of the potential energy profile along the IRC clearly tells that one should expect tunneling to be important, since the barrier develops a rather thin profile, becoming thinner as energy is increased. This has been done by considering the geometries and frequencies obtained at BH&HLYP/6-311 ? G(2df,2p) level of theory and the energies computed at CCSD(T)/aug-cc-pVTZ level of theory. However, tunneling appears even more enhanced, when one considers variable?mass results. One might then say that the influence of perpendicular modes is, rather surprisingly, to enhance reactivity below the Table 6 Estimated tropospheric concentrations (in molecule cm-3) of the (H2O)2, (H2O)3, H2O?HO, (H2O)2?HO, and (H2O)3?HO com- plexes at different relative humidities (RH) and temperatures (T) T RH = 25% RH = 50% RH = 75% H2O (H2O)2 (H2O)3 H2O (H2O)2 (H2O)3 H2O (H2O)2 (H2O)3 278 1.04 9 1017 5.95 9 1013 1.59 9 1012 2.07 9 1017 2.38 9 1014 1.27 9 1013 3.11 9 1017 5.36 9 1014 4.30 9 1013 288 1.92 9 1017 1.70 9 1014 5.08 9 1012 3.84 9 1017 6.79 9 1014 4.06 9 1013 5.76 9 1017 1.53 9 1015 1.37 9 1014 298 3.48 9 1017 4.68 9 1014 1.59 9 1013 6.95 9 1017 1.87 9 1015 1.27 9 1014 1.04 9 1018 4.21 9 1015 4.29 9 1014 308 5.99 9 1017 1.18 9 1015 4.47 9 1013 1.20 9 1018 4.73 9 1015 3.58 9 1014 1.80 9 1018 1.06 9 1016 1.21 9 1015 H2O?HO (H2O)2?HO (H2O)3?HO H2O?HO (H2O)2?HO (H2O)3?HO H2O?HO (H2O)2?HO (H2O)3?HO 278 1.01 9 104 9.28 7.24 2.02 9 104 3.71 9 101 5.79 9 101 3.03 9 104 8.35 9 101 1.96 9 102 288 1.46 9 104 1.52 9 101 1.32 9 101 2.92 9 104 6.08 9 101 1.05 9 102 4.39 9 104 1.37 9 102 3.55 9 102 298 2.10 9 104 2.49 9 101 2.44 9 101 4.21 9 104 9.95 9 101 1.95 9 102 6.31 9 104 2.24 9 102 6.58 9 102 308 2.93 9 104 3.87 9 101 4.21 9 102 5.85 9 104 1.55 9 102 3.36 9 102 8.78 9 104 3.48 9 102 1.14 9 103 These estimations have been done according Eqs. 3?8 The concentration of the hydroxyl radical employed for the formation of the (H2O)n?HO complexes is 1.0 9 107 molecules cm-3 Fig. 5 Calculated transmission factor for the reaction between HO and H2O. The vertical black line corresponds to the classical energy barrier 588 Theor Chem Acc (2011) 128:579?592 123 potential energy barrier. This is in contrast to numerous previous studies on tunneling, where the inclusion of more degrees of freedom normally inhibits tunneling. In addi- tion, oscillations in the transmission factor, as energy is increased, are more intense in the variable?mass case, indicating that interference phenomena are enhanced by the inclusion of perpendicular modes. This is a suggestive result, whose origin may be traced back to the symmetric character of the present reactions and their Heavy?Light? Heavy nature. Tunneling contribution to reaction, for the reaction between OH and two water molecules, is analyzed in Fig. 6. Q?RPH transmission factor, as a function of energy, is shown for cis and trans configurations of the water dimer. Results show, first, that cis?trans differences are only an energy shift, the qualitative shape being fairly equal. This is the reason why we do not show results for all the isomers in all cases, limiting ourselves to this case. Second, the contribution to tunneling is again remarkable, but the most outstanding feature is the oscillating nature of the transmission factor in the threshold and post-threshold regions. These oscillations are much more pronounced than the previous case. It is clearly an influence of the deeper CR1 well (ca. 7.5 kcal mol-1 for the two-water case, versus ca. 3.8 kcal mol-1 well depth for the one-water reaction). Further comparison between one- and two-water reac- tions evidences that, whereas tunneling is of similar importance for both reactions, antitunneling is remarkably larger in the one-water case. It is shown by the slower trend to unity, in the large energy regime, when comparing Figs. 5 and 6. This feature is interesting, since this large antitunneling appears only in the variable mass case of the one-water reaction, i.e., only when the perpendicular modes to reaction are taken into account. The primary conclusion is then that energy sequestering (from that available to reaction) is more effective for one-water than for two-water reactions. The explanation for such behavior might resort to a kind of entropic effect: energy flow concentrates more easily in a small number of non-reacting modes than in a larger number of them. In addition, the higher symmetry content, of the two-water TS, also sug- gests a much higher difficulty in concentrating the energy flow in non-reacting modes, when compared to the one- water reaction. Once analyzed the behavior of the transmission factor along the reaction path, we computed the rate constant of the reaction. To do this, we considered the reactions to occur at 1 atm. of pressure, and we applied the steady-state approximation to the pre-reactive complex, and the rate constant is given by Eq. 9. kTOT ? Keq ? k2 ? jtun ?9? where Keq is the equilibrium constant, k2 is the unimolecular rate constant, and jtun is the tunneling contribution that has been computed by Eq. 10. jtun?T? ? R1 0 T?E? exp ? EkBT   dE R1 Vb exp ? E kBT   dE ?10? We have to mention here that applying Eq. 9 is just a limiting case, and the effect of low pressures would require a different treatment that is beyond the scope of the present work. In Table 7, we collected the computed rate constant. For the H2O ? HO reaction, our computed values range between 2.76 9 10-17 and 6.04 9 10-17 cm3 mole- cule-1 s-1 in the 200?400 K range of temperatures and compare very well with the results reported by Masgrau et al. [19] and Uchimaru et al. [21], the differences with the last one being mainly attributed to the fact that they use a reaction profile beginning in the reactants without taking into account the pre-reactive complex. It is worth noting here the importance of the tunneling contribution, mainly at low temperatures, as it was pointed out in previous works on this reaction [19, 21]. Although for the case of the reaction between HO and (H2O)2 we report rate constants, we pointed out in a previous section that the pre-reactive complex CR2 dis- sociates into H2O?HO ? H2O and therefore, we can conclude that the reaction through TS2 would not occur at all. The reaction between hydroxyl radical and water trimer is much more complex, and k2 is computed using Eq. 11 according to the unified statistical model [77], and the Fig. 6 Calculated transmission factor through TS2 and TS2a for the reaction between HO and (H2O)2. The vertical black lines correspond to the classical energy barriers Theor Chem Acc (2011) 128:579?592 589 123 1 k2 ? 1 kTS5 ? 1 kTS6 ?11? results displayed in Table 5 show that in this case, the reaction is much faster than reaction 1, with values ranging between 1.25 9 10-13 and 8.33 9 10-16 cm3 mole- cule-1 s-1 in the 200?400 K range of temperatures. Here, it is important to note that the tunneling contribution is very similar to those occurring in reaction 1. 4 Conclusions The reactions between HO and (H2O)n (n = 1, 3) have been investigated using high-level theoretical methods, and the results obtained allow us to highlight the following points. Regarding the water dimer and trimer, our calculations predict a binding energy of 2.86 kcal mol-1 for (H2O)2 relative to H2O ? H2O and 7.67 kcal mol-1 for (H2O)3 relative to (H2O)2 ? H2O. Our calculations predict atmospheric concentration for the (H2O)2 complex in the range between 5.95 9 1013 and 1.06 9 1016 molecule cm-3 and for the (H2O)3 complex in the range between 1.59 9 1012 and 1.21 9 1015 mole- cule cm-3; depending on the atmospheric conditions of relative humidity and temperature. For the H2O?HO complex, we predict atmospheric concentrations in the 1.01 9 104 ? 8.78 9 104 molecule cm-3, depending on the atmospheric conditions and considering a hydroxyl radical population of 107 molecule cm-3. The concentra- tion of the (H2O)2?HO and (HO)3?HO is predicted to be about two orders of magnitude smaller. All the reactions considered are symmetric, and they begin with the formation of a pre-reactive hydrogen-bon- ded complex. For the reaction of hydroxyl radical with a single water molecule, our calculations predict that the pre-reactive hydrogen-bonded complex CR1 has a binding energy of 3.78 kcal mol-1, whereas the transition state lies 12.53 kcal mol-1 above CR1. Very interestingly, we found a VRI point along the reaction path, which is ener- getically located close to 4 kcal mol-1 below the transition state, which allows branching the reaction path. The computed rate constant for this reaction range between 2.76 9 10-17 and 6.04 9 10-17 cm3 molecule-1 s-1 in the 200?400 K range of temperatures. For the reaction between hydroxyl radical and water dimer, our calculations predict the pre-reactive complexes (CR2 and CR2a) to have binding energies close to 7.5 kcal mol-1. Then, the reaction can proceed in two different ways: (a) dissociating directly into the H2O?HO Table 7 Computed rate constants for the reaction between hydroxyl radical and water, water dimer, and water trimer Keq are in cm3 molecule-1, K2 are in s-1, then total constants KTOT are in cm3 molecule-1 s-1 stun is the tunneling correction K2 stands for kTS1, kTS2, kTS5, and kTS6, respectively a stun applies to reaction through kTS6 Reaction OH ? H2O Temperature Keq kTS1 stun KTOT 200 1.72 9 10-19 5.09 9 10-6 3.15 9 107 2.76 9 10-17 250 2.19 9 10-20 1.32 9 10-2 5.33 9 104 1.54 9 10-17 298 6.05 9 10-21 2.04 1.40 9 103 1.73 9 10-17 350 2.32 9 10-21 9.92 9 10 1.28 9 102 2.95 9 10-17 400 1.21 9 10-21 1.58 9 103 3.16 9 101 6.04 9 10-17 Reaction OH ? (H2O)2 Temperature Keq kTS2 stun KTOT 200 1.90 9 10-17 2.24 9 10-4 1.05 9 107 4.47 9 10-14 250 2.59 9 10-19 4.04 9 10-1 2.78 9 104 2.91 9 10-15 298 1.57 9 10-20 4.98 9 10 9.47 9 102 7.40 9 10-16 350 1.77 9 10-21 2.05 9 103 1.02 9 102 3.70 9 10-16 400 3.70 9 10-22 2.9 9 104 2.76 9 10 2.97 9 10-16 Reaction OH ? (H2O)3 Temperature Keq kTS5 kTS6 stuna KTOT 200 4.50 9 10-16 6.14 9 1011 1.24 9 10-5 2.24 9 107 1.25 9 10-13 250 4.40 9 10-18 1.20 9 1012 3.89 9 10-2 4.01 9 104 6.86 9 10-15 298 2.24 9 10-19 1.89 9 1012 6.95 1.11 9 103 1.73 9 10-15 350 2.27 9 10-20 2.73 9 1012 3.83 9 102 1.08 9 102 9.37 9 10-16 400 4.45 9 10-21 3.56 9 1012 6.76 9 103 2.77 9 10 8.33 9 10-16 590 Theor Chem Acc (2011) 128:579?592 123 complex and H2O, requiring an energy of ca 6.6 kcal - mol-1 and (b) proceeding through three different elemen- tary reactions. In this case, the more favorable path occurs through TS2a, which is predicted to lie 13.07 kcal mol-1 above CR2a, and it involves a hydrogen atom transfer mechanism. The high energy barrier computed for TS2a along with the rate constant computed for this elementary reaction indicated that the reaction between HO and (H2O)2 will produce H2O?HO and H2O, and it will contribute to the atmospheric formation of hydrated hydroxyl radical, which has an important significance in the chemistry of the atmosphere. The reaction with hydroxyl radical and water trimer begins with the addition of the HO radical to the three- membered ring of the water trimer forming the CR3 complex. Then, it occurs a geometrical rearrangement to form the four-membered ring CR4 complex, which is computed to be 8.57 kcal mol-1 more stable than the reactants. Then, the reaction can go on through three dif- ferent reactions paths. The more favorable one occurs through TS6, which is predicted to lie 5.27 kcal mol-1 above the separate reactants, and it involves a hydrogen atom transfer mechanism. This reaction is predicted to be faster than the reaction between hydroxyl radical and one water molecule, which has a rate constant in the range 1.25 9 10-13 ? 8.33 9 10-16 cm3 molecule-1 s-1 in the 200?400 K range of temperatures, a fact that is attributed to the complexity of the whole reaction path. For all the reactions considered, the transmission via tunneling effect is found to be very important, especially at low temperatures. Other features that also appear to be relevant from a dynamical point of view and that deserve more studies are the enhancement of tunneling due to the inclusion of perpendicular modes via the Q-RPH, the remarkable antitunneling for the reaction between HO and H2O, and the oscillations in the transmission factor for the reaction of the HO radical and (H2O)2. The latter is attributed to an increment of resonances due to the deeper well of the corresponding CR1 compared with the HO ? H2O case. Acknowledgments This research has been supported by the Gen- eralitat de Catalunya (Grant 2009SGR01472) and the Spanish Di- reccio?n General de Investigacio?n Cient??fica y Te?cnica (DGYCIT, grants CTQ2008-06536/BQU and CTQ2008-02856/BQU). The cal- culations described in this work were carried out at the Centre de Supercomputacio? de Catalunya (CESCA), at the Computational Center of CTI?CSIC, and at the cluster of workstations of our group. Antoni Aguilar-Mogas and Marc Caballero gratefully thank to Min- isterio de Ciencia e Innovacio?n for a predoctoral fellowship. Javier Gonza?lez and Miquel Torrent-Sucarrat acknowledge CSIC for a JAE- DOC contract. References 1. Wayne RP (2000) Chemistry of atmospheres, 3rd edn. Oxford University Press, Oxford 2. Wennberg PO, Hanisco TF, Jaegle? 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Miller WH (1976) J Chem Phys 65:2216?2223 592 Theor Chem Acc (2011) 128:579?592 123 A Relation Between the Eikonal Equation Associated to a Potential Energy Surface and a Hyperbolic Wave Equation Josep Maria Bofill,*,?,? Wolfgang Quapp,*,? and Marc Caballero?,? ?Departament de Qu?mica Organ?ica, Universitat de Barcelona, Mart? i Franques?, 1, 08028 Barcelona, Spain ?Institut de Qu?mica Teor?ica i Computacional, Universitat de Barcelona (IQTCUB), Mart? i Franques?, 1, 08028 Barcelona, Spain ?Mathematisches Institut, Universitat? Leipzig, PF 100920, D-04009 Leipzig, Germany ?Departament de Qu?mica F?sica, Universitat de Barcelona, Mart? i Franques?, 1, 08028 Barcelona, Spain ABSTRACT: The potential energy surface (PES) of a molecule can be decomposed into equipotential hypersurfaces. We show in this article that the hypersurfaces are the wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, or the steepest descent, or the steepest ascent lines of the PES. The energy seen as a reaction coordinate plays the central role in this treatment. 1. INTRODUCTION The potential energy surface (PES)1?3 is the basic element of the chemical reaction path and of theories of chemical dynamics. The PES is a continuous function with respect to the coordinates of the nuclei, having also continuous derivatives up to a certain order not specified here, but required by the operations which are to be carried out. The PES can be seen as formally divided in catchments associated with local mini- mums.1 The minimums are associated with chemical reactants and products. The first order saddle points or transition states (TSs) are located at the deepest points of the boundary of the basins. According to these definitions, both points, TS and minimums, correspond to stationary points of the PES, but they differ in the structure of the Hessian matrix. Two minimums of the PES can be connected through a TS via a continuous curve in the N-dimensional coordinate space, describing the coordinates of the nuclei. The curve characterizes a reaction path (RP). One can define many types of curves satisfying the above requirement. This is the reason for the variety of RP models. One of the most used RP models is the steepest descent (SD) from the TS to the reactant or product. The SD reaction path in mass weighted coordinates is normally called the intrinsic reaction coordinate (IRC).4?6 The discussion of a coordinate independent definition of SD curves was already given.7 The SD curves and in particular the IRC path are in fact orthogonal trajectories to the contour hypersurfaces, V(q) = ? = constant, if the corresponding metric relations are used, see ref 7. In this paper, we will assume N orthogonal and equidistant coordinates, q, thus Cartesians of the n atoms with N = 3n, for simplicity only. Then the metric matrix reduces to the unity matrix. In the determination of the SD curves, the relation between the gradient field and the associated orthogonal trajectories is relevant. At this point it is important to remember that the Hamilton?Jacobi equation or eikonal equation describes a relation between the contour of a surface and curves.8 In addition the difference between two contour hypersurfaces is related to a functional depending on some arguments that characterize the SD curves. The connection between the field of SD curves of a PES and the picture of the Hamilton?Jacobi theory was discussed by Bofill and Crehuet.8 From a theoretical point of view (however not numerically) the SD curves and the inverse ones, the steepest ascent (SA) curves, are equivalent. (Intensive numerical treatments of IRC following procedures are known.9) The SA curves emerging from a minimum of the PES can be seen as traveling in an orthogonal manner through the contour hypersurfaces of this PES. In addition, it should be noted that the construction of the contour hypersurface, V(q) = ? = constant, such that all points satisfying this equation possess the same equipotential difference with respect to another contour hypersurface and specifically with the value of the PES in the minimum, is similar to the construction of the Fermat?Huygens of propagation of the cone rays. Notice that the construction of the Fermat? Huygens of propagation of rays and Hamilton?Jacobi theory are strongly connected.10 Using this analogy, we will develop a wave equation theory for contour hypersurfaces of the PES. 2. THE EIKONAL EQUATION The eikonal equation of SD curves in a PES domain is8,11,12 ? ? =V V Gq q q( ) ( ) ( )Tq q (1) where ?qT = (?/?q1,...,?/?qN) and G(q) is the square of the gradient norm at the point q. The equation means that, if a function G(q) is given, then we search for a potential V(q) fulfilling this eikonal condition. In eq 1 the PES function V(q) represents the minimal total geodetic distance, and G(q) is the geodetic distance function at each point q of the PES domain. This geodetic distance function is defined at the beginning of the problem, and the solution V(q) to the above problem represents the total geodetic distance, which is the smallest obtainable integral of G(q), considered over all possible curves throughout the computational domain from a start point to a Special Issue: Berny Schlegel Festschrift Received: June 23, 2012 Published: August 22, 2012 Article pubs.acs.org/JCTC ? 2012 American Chemical Society 4856 dx.doi.org/10.1021/ct300654f | J. Chem. Theory Comput. 2012, 8, 4856?4862 final point. The integral of G(q) on the SA curve joining these two points is8,11?14 ?=V G d dt d dt tq q q q( ) ( ) ( / ) ( / ) d t t T SA f 0 (2) where t is the parameter that characterizes the SA curve and q = qSA(t). From a practical point of view, first order nonlinear partial differential eq 1 is solved for this total geodetic distance function V(q) first, and the actual stationary or cheapest path from qSD(tf) to qSD(t0) is obtained by starting at the final point and integrating a trajectory backward along the gradient field ?qV(q). Eikonal eq 1 is an example of the general static Hamilton?Jacobi equation. In other words, eikonal eq 1 tells us that as the parameter t evolves, the coordinates q = qSA(t) evolve, and the contour hypersurface with constant potential energy V(q) = ? changes, through the coordinates q. A point of this contour hypersurface is linked to a point of the neighborhood contour hypersurface. This set of points defines a SD curve that makes the integral functional, eq 2, extreme. One can establish some analogies between the propagation of light through a medium having a variable index of refraction and the present problem. Just as the light rays are given as extremal paths of least time, now the SD curves are extremal paths of the PES. 3. AN ANSATZ OF A WAVE EQUATION The SA curves starting at a minimum are a set of curves traveling in an orthogonal manner through the contour hypersurfaces of the PES. Notice the important fact that each SA curve cuts each member of the family of contour hypersurfaces in one and only one point. Additionally, the SA curves are strictly monotonic in ? between stationary points. Thus, we can establish a one-to-one relation between a point of the curve and the energy value of the member of the family of contour hypersurfaces. In other words the SA curve, q(t), can be expressed as q(?) being ? the energy of the contour hypersurface at the point q(t).13 The family of contour hypersurfaces is geodesically equidistant. In the present understanding the distance is the energy difference. It is known that a family of energy equidistant contour hyper- surfaces is the solution of eikonal eq 1.10 But the construction of a solution of eikonal eq 1 as a contour hypersurface with constant potential energy is similar to the Fermat?Huygens principle for the construction of wave fronts.8,11,15 The aim of this paper is to find the equation that governs the propagation of the ?wave? associated with the SA curves. The unique possibility is a second order partial differential equation such that its associated characteristic equation is eikonal eq 1, which is related with the PES. Furthermore, characteristic curve solutions of eq 1 are just the SA curves.8 Consequently, let us consider a wave equation in N + 1 dimensions, q1,...,qN, and ? ? ? ? ? ?? ? ?? =Gq q q( , ) ( ) ( , ) 0q 2 2 2 (3) where ?q2 = ?qT?q = ?2/?q12 + ...+ ?2/?qN2. Note that we treat ? as an independent variable. ?(q,?) is, for the time being, an abstract field in any medium with slowness 1/G(q)1/2, which also emerges in eq 1. Equation 3 looks like a wave equation where the time variable, t, is replaced by the variable ? and where the factor 1/G(q)1/2 plays the role of the velocity of the corresponding wave solution. The concept of ?medium? is used here by comparison with the propagation of waves associated with rays of light that propagate in a (maybe inhomogeneous) medium. Of course, eq 3 is of hyperbolic type by signature ++...+? since G(q) > 0, thus outside of stationary points. Note that the factor, G(q), changes the character of eq 3 into a differential equation with variable coefficients. The solution of such equations is usually assumed to be more difficult than that of equations with constant coefficients. However, in this case, a particular solution will become easier, see section 4. First, we look for the characteristic manifold of eq 3, see ref 10, part II, chapter VI, paragraph 2. The solutions of hyperbolic equations are ?wave-like?. If a disturbance is done in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to the fixed ?energy? coordinate, ?, the disturbances have a finite propagation speed. They travel along the so-called character- istics of the equation. The method of characteristics is a technique to solve this type of partial differential equation. Though it is usually applied to first order equations, the method of characteristics is valid also for our second order hyperbolic equation. The idea is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface. To find here the characteristics, we have to treat the quadratic form, which is connected with eq 3 ? ? ?? =? = G q( ) 0 i N q 1 2 2 i (4) where ?qi means the derivation to qi of the characteristic function ?, and the last derivation to energy, ?, with number (N + 1) is symbolized by the index ?. We note that the factor G(q) creates problems since it becomes zero at stationary points. The shapes of the solution of eq 4, the characteristics, are N-dimensional hypersurfaces. They are conoids in regions outside of a stationary point of the PES. The global behavior is distorted by the zero of the gradient at a stationary point. On the other hand, the characteristic equation of eq 3 has an associated characteristic plane ? ? ? ? ? | ? ? | = = = ? ? ? ? ? ? ? ?v v v q q(( ) , ( )) / 0T q q q q q 0 0 0 0 (5) We call the vector r the resulting vector of the normalization of the vector (?qT?,??/??). An element of the r vector, say ri, is the cosine of the angle between the normal vector to the characteristic plane and the qi axis. Because the characteristic equation, eq 4, is homogeneous with respect to the vector (?qT?,??/??), we can replace it by the normalized vector r and get in eq 4 Using the normalization condition of the r vector, the left-hand side of the equation can be transformed in the following manner ? = ? + =G r G rr r q q( ) 1 (1 ( )) 0T v vq q 2 2 (7) From this, we find = + r G q 1 (1 ( ))v 1/2 (8) Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct300654f | J. Chem. Theory Comput. 2012, 8, 4856?48624857 which is the cosine of the angle between the v axis and the normal vector to the characteristic plane. Using the forward conoid expression, we obtain the radius of the ?circle? of the conoid, for an intersection hyperplane to a fixed v, = + G G r r q q ( ) ( ) (1 ( )) T q q 1/2 1/2 1/2 (9) A schematic view to a straightforward cone with apex qo is given in Figure 1. A surface ?(q,v) = 0, such that at every point (q,v) the gradient vector (?qT?,??/??) has the same direction as the r vector, is a characteristic surface. The Cauchy?Kowalewsky theorem breaks down at such points. A trivial conclusion is the following. If dq/dt is the tangent of a steepest ascent (descent) curve, then the two relations hold = +r d dt d dtq q1/(1 ( / ) ( / ))v T 1/2 (10) = +d dt d dt d dt d dt r r q q q q ( ) [( / ) ( / )/(1 ( / ) ( / ))] T T T q q 1/2 1/2 (11) which determine the direction of the tangential hyperplane of a characteristic surface, ?(q,v), through the curve. We develop a one-dimensional example (N = 1), and we use x = q1 for the coordinate. The test function is a double well potential = ?V x x( ) ( 1)2 2 (12) The gradient of the function is ?xV(x) = 4x(x2 ? 1), and G(x) is the square of the gradient. The stationary points of the surface are located at x = 0 and ?1, point 0 being a first order saddle point with V(0) = 1. The two other points are the minimums with V(?1) = 0. We have = ? +r x x1/(16 ( 1) 1)v 2 2 2 1/2 (13) So, for both x = 0 (saddle point) and x = 1 (minimum), it is rv = 1 and rx = 0. For the coordinate x, we have the following relation on the (0,1) interval = ? ? +r x x x x4 ( 1)/(16 ( 1) 1)x 2 2 2 2 1/2 (14) The development of the vector (rx,rv) is shown in Figure 2. The opening angle of the conoid changes along the profile. At the stationary points, the cone apex is a pure vector head. Its opening angle is zero. The vector is ?energy-like? along the energy axis, v. The vector is orthogonal to the gradient, which is zero. It is obvious that only nonstationary points with G(q) > 0 are useful points for any treatments. In analogy, if the profile is a minimum energy path (MEP) in a higher dimensional example, say for V(x,y) = (x2 ? 1)2 + y2, and we move a tiny step away from the SP, then we can calculate the steepest descent from SP to a minimum, as well as have the steepest ascent, and vice versa. Note, at values x1 = 0.27 and x2 = 0.84, the characteristic direction crosses the gradient. The gradient has an ?energy-like? direction inside the interval (x1,x2). (One could speculate that along ?energy-like? gradients a steepest descent method works well; however, if the gradient is ?space- like,? then such a method can tend to zig-zag.) To solve eq 4, we try a function ?(q,v) in the form16 ? = ???vq( , ) e 1i V q[ ( )] (15) Its substitution into eq 4 fulfills this equation, because the derivations of ? lead directly to eikonal eq 1. If we put ? = constant = 0, we can obtain the solution for the variable ?, with ? = V(q), which is exactly its definition; thus it is correct. The wavefront of eq 3 develops along its characteristic manifold along a curve in the (N + 1) space, which is described by a SA curve with V(q) = ? for the current t value. We can say that the progression of wave eq 3, which is a hyperbolic partial differential equation, is regulated by eikonal eq 1, a first order partial differential equation. A general rigorous proof of this result is based on the theory of characteristics of partial differential equations.10,17,18 4. A MORE REFINED WAVE EQUATION The function ?(q,v) from eq 15 does not fulfill the first ansatz of wave eq 3 because of the chain rule. There emerges an additional term besides the eikonal terms of eq 1. However, we can attempt an extension of eq 3 via a ?friction? term ? ? ? ?? ? ?? + ? ? = ? ? ? ? ? ?G Traceq H q( ) ( , ) 0q2 2 2 (16) H is the Hessian of the PES at the current point q; thus H = ?qgT with the gradient of the PES, g. If F is any function of one real variable, F(x), with first and second continuous derivations, then ? ?? = ?F Vq q( , ) ( ( )) (17) is a solution of extended wave eq 16. The proof is straightforward. The solution is a kind of generalized plane Figure 1. Schematic forward cone in three dimensions, if G(q) = 1 is used for simplicity. Figure 2. Function V(x) and vectors of the characteristic direction (rx,rv) along the profile. Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct300654f | J. Chem. Theory Comput. 2012, 8, 4856?48624858 wave with the fixed phase function, Z(q,?) = ? ? V(q), and the waveform function F, see ref 10, part II, chapter VI, paragraph 18. Note that the solution, eq 17, is simpler than some spherical wave solutions for wave equations with constant coefficients,19 and ref 10 part II, chapter VI, paragraph 13, section 4, because we can drop the so-called ?distortion? factor. The principal part of eq 16 is the same as in eq 3; thus the characteristic manifold here is also the same. Since the friction term is of low order, eikonal eq 1 is not affected. Note that the plus sign in the ?friction? term with the Trace H coefficient is somewhat ?unphysical? if Trace H > 0. The solution, eq 17, of the refined wave equation allows the application of deep conclusions concerning the Huygens principle, see section 7 below. In the general theory of hyperbolic differential equations, one uses the normal form of the wave equation. We will now derive it. The part of second order in eq 3, or eq 16, is ? ? ? ? ?? ? ??Gq q q( , ) ( ) ( , )q 2 2 2 (18) Note that the factor G(q) depends only on the space variables, q, but not on the potential variable, v. The so-called metric matrix in eq 18 is = ?= + ? ? ? ? ? ?g G E 0 0 q ( ) ( ) i j i j N T , , 1,..., 1 (19) E is the N-dimensional unit matrix, and 0 is the N-dimensional zero vector. Because the metric matrix here is only a diagonal matrix, its inverse matrix is simply = ?= + ? ? ? ? ? ?g G E 0 0 q ( ) 1/ ( )i j i j N T, , 1,..., 1 (20) (Compare a remark in the Introduction: the metric matrix in q space is a simplification.) The absolute value of the determinant of the inverse matrix is ? = 1/G(q). Now, we can write the v part of eq 18: ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ?? ? ? ? + + ? ? ? ? ? ? ? ? ? ? ? ? G G G g q q q q q ( ) ( ) ( ) ( , ) 1 ( , )N N 1/2 1/2 1, 1 (21) For the space variables, q, we have ? ?= ?? ? ? == = ? ? ?? ? ? ??G q V q G V Vq q q( ) ( ) , thus ( ) 2 i N i k i N q q q 1 2 1 i i k (22) Because it holds for the coefficients gi,j = ?i,j, for i,j = 1,...,N, we can try an ansatz ? ? ? ? ? ? ? ? ? ? ?? ? ? ??q q q1 ( , ) i i (23) If we differentiate the factor ?? inside the formula, with the product rule we get ?? ? ? ??? + ? ? ?= ? ? ? ? ? ?q G G V V q q q q q( , ) ( ) 1 2 1 ( ) 2 ( , ) k i N q q q k 2 2 1/2 3/2 1 i i k (24) Consequently, we get for the second order part of eq 18 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? = ?? ? ? ? ? ? ? ? + ? ? ? + + = ? ? ? ? ? ? ? ? ? ? ? ? ? ? G q g q g G V q q q q q q q H q ( , ) ( ) ( , ) 1 ( , ) 1 ( , ) 1 ( ) (( ) ) ( , ) i i j j N N i N T i i q q 2 2 2 , 1, 1 1 (25) where Hi means the ith column of H. The first two parts of the right-hand side build the normal form of a second order wave equation with (N + 1) variables and with variable coefficients. For such a normal form, we have an integration theory which guarantees the existence of exactly one forward (and one backward) fundamental solution.20 But we stop here and look for other properties of the phase function V(q) = ?. 5. A CHARACTERISTIC INITIAL VALUE PROBLEM No stationary point should emerge between a q(t0) on a hypersurface for ?0 and the corresponding q(tf) on an upper hypersurface at energy ?f. A wave equation with a second order part like eq 18 allows a characteristic initial value problem posed for the characteristic surface in section 3. If we put the value ? = V(q) into solution, eq 17, we get ? ? ?| = ? =?= F V Fq q( , ) ( ( )) (0)V q( ) (26) everywhere on the characteristic surface, because for ? = V(q) we are also on the characteristic surface ?(q,?) = 0 with eq 15. Without a loss of generality, we can assume F(0) = 0. We treat a double-conoid (DC) between two points Po = (q(t0),?0) and Pf = (q(tf),?f), see Figure 3, compare Figure 56 in ref 10, chapter VI, paragraph 6, section 1. The corresponding defining formula is eq 4. We can multiply eq 18 by (?2?v) everywhere in the DC, and we have ?? ? ?? =?? = G q2 ( ( ) ) 0v i N q q 1 i i (27) where the indices mean the corresponding derivation. It is equal to ? ?? ? ? ?? + =? ? ? ? = = G q( ( ) ) 2 ( ) ( ) 0 i N q q i N q 2 1 1 2 i i i (28) Figure 3. Three-dimensional schematic representation of two intersecting conoids, one forward conoid with the apex in Po and one backward conoid with the apex in Pf. They form by their curve of intersection a double conoid: all points in the interior of both conoids between Po and Pf. Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct300654f | J. Chem. Theory Comput. 2012, 8, 4856?48624859 The Variational Structure of Gradient Extremals Josep Maria Bofill,*,?,? Wolfgang Quapp,? and Marc Caballero?,? ?Departament de Qu?mica Organ?ica and ?Departament de Qu?mica F?sica, Universitat de Barcelona, C/Mart? i Franques?, 1, 08028 Barcelona, Catalunya, Spain ?Institut de Qu?mica Teor?ica i Computacional, Universitat de Barcelona (IQTCUB), C/Mart? i Franques?, 1, 08028 Barcelona, Catalunya, Spain ?Mathematisches Institut, Universitat? Leipzig, PF 100920, D-04009 Leipzig, Germany ABSTRACT: The gradient extremals can be taken as a representation of reaction paths. We prove that this type of curve possesses a variational nature. We report the conditions such that these curves have the character of a minimal curve. Finally we discuss the relations between the points of these curves being turning points with respect to other special points of the potential energy surface, like the valley-ridge inflection points. 1. INTRODUCTION The potential energy surface (PES) is the basic tool of the mechanistic and dynamical studies of the chemical reactivity. Some reactions with a diradical character approach the extreme case where the flatness of some regions of the PES precludes the definite assignment of distinct minima, saddle points (SPs), and lowest-energy pathways on the way from a reactant to several products. An example is the PES associated with the mechanism of the rearrangement vinylcyclopropane? cyclopentene.1 In these examples the non-intrinsic reaction coordinate (IRC) pathways emerge. The IRC is the most widely used curve to represent a reaction path (RP). The dynamic behavior of these reactions has been studied by several authors observing a deviation from that predicted by the well- known transition-state theory (TST) which is a statistical, dynamical theory.2?5 For these reactions the RP matched by the bulk of trajectories joining the reactant and product regions is different from the IRC pathway.6 As a starting point, we explain this behavior. One hypothesizes a PES such that an utter flat intermediate region has one entrance and two exits. It is quite unlikely that a particular entrance will be dynamically coupled with equal strength to the two exits, even when the symmetry properties of the PES would appear to make the two exits equivalent.2 We associate this dynamical situation with the mechanism that a reagent A can go to two products B and C. We explain this nonsymmetric rearrangement bifurcation, A ? B and A ? C, in terms of the PES model saying that in between the two SPs of the entrance to this region there exists a nonsymmetrical bifurcation implying that there is not a monkey SP.7 An example of this PES may be represented by the function = ? ? ? ? + + +? ? ? ? ?? ? ? ? ? ? ? ??V x y x xy x y x y( , ) 1 3 ( 3 ) ( ) 1 40 7 4 3 2 4 4 (1) This PES has two adjacent first-order SPs. In the left panel of Figure 1, the PES of eq 1 is depicted. There are two SPs, which we name the SPab for the pathway from A to B and SPbc for the pathway from B to C. By inspection of the two SPs, we note that SPab is higher in energy units than SPbc. In this PES, the SPs may be nodes of a RP, however, from the left minimum A to the minimum C top right, no direct steepest descent (SD) exists, or anything that is equivalent to an IRC path. The SD from the two SPs leads to the minimum B at the right bottom. The fat curves in the right panel of Figure 1 are gradient extremals (GE).8?18 The combination of the GE branches from minimum A to SPab, further to SPbc, and then further to minimum C can be seen as a static model of a RP, which indeed connects the two minimums A and C. This is a non-IRC path and matches the cloud of dynamic trajectories that, starting in the minimum A, end in the region associated with the minimum C, see Figure 2. In the left panel of Figure 2 we show a set of classical trajectories at a given time. The starting point of the trajectories is the minimum A. At this time, some of them reside in the initial region, but a big portion of these trajectories are located in the region associated to the minimum B. A subset of a few trajectories crosses the region of the transition state SPbc arriving to the region of the minimum C. The classical trajectories were computed using a code described in ref 19. In the central and right panels we show two different times of a Gaussian wave packet propagation on the same PES. At the initial time the Gaussian wave packet is centered in the region of the minimum A. In the central panel it is shown the time that an important portion of the Gaussian wave packet is located in the region of the PES where not only the valley bifurcates but also the wave packet spreads in this region. In the right panel we show the behavior of this quantum propagation after a long time. An important portion of the Gaussian wave packet resides in the region of the minimum A, whereas the reminder resides mainly in the bifurcation region, around the minimum B, but a small part resides in the region of the minimum C. As in classical simulation, the transformation from A to B is favored before A to C. For this example we can say that the IRC is the curve representing the RP being more favored according to the classical and quantum dynamic models, whereas the GE curve represents the RP being less favored according to these models. Received: November 10, 2011 Published: January 23, 2012 Article pubs.acs.org/JCTC ? 2012 American Chemical Society 927 dx.doi.org/10.1021/ct200805d | J. Chem. Theory Comput. 2012, 8, 927?935 The quantum propagation was computed using the MCTDH program.20?22 From the classical dynamic theory, the RP model, which always is represented by a curve, can be envisaged as the curve that matches in the best way the average in position of the cloud of trajectories after a long time. In the present case, both the IRC and the GE play this role, because both match the average of the two different clouds of trajectories after a long time.23 From a quantum point of view this relation between quantum dynamics and RP model is more difficult. However, in some cases we can say that the curve representing a RP can be seen as the curve that matches in the best way the maximum of the square of the wave function with respect to the position after a long time.24 Nevertheless, in the present study we are interested in the variational properties of the GE curves. The GEs are curves that usually run along valley floors or ridges of a PES. More rigorously and specifically speaking, the GEs of a PES, V(q), are defined as the curves, q(t), where t is the curve parameter, which cuts at each point a member of the isopotential hypersurfaces of this PES, V(q(t)) = v(t), and the square of the gradient norm, ?qTV(q)?qV(q) = g(q)Tg(q), is stationary at each point of this curve in respect of the variations of q within the member of isopotential hypersurfaces that is cut by the curve at this point.11,16 Because one can use GEs as model reaction pathways, we treat this kind of curves in this letter. We note that also another sort of curves, Newton trajectories (NT),25?29 is well adapted to the connection of adjacent SPs, compare refs 30 and 31 and references therein. As explained before in studies on dynamics of chemical reactions, examples of nonstatistical dynamical behavior in large organic systems involve cases in which transient intermediates occupy plateau regions or at best shallow minimums on the PES. These regions are accessible by GEs, and due to this fact, a GE can be seen as a representation of some average of a representative ensemble of classical trajectories. In principle it is plausible to believe that intermediates in deeper minimums show some statistical behavior due to the higher density of vibrational states at energies near their exit channels. A rapid intramolecular vibrational energy redistri- bution should often be present. But examples of this kind suggest that nonstatistical dynamics can persist even in these cases. The RP model is roughly defined as a curve in the coordinate space, which connects two minimums by passing the SP, the Figure 1. Left panel: PES associated to the mechanism of the reaction A ? B + C. The mechanism for this reaction consists of two elementary reactions, namely, A? B and A? C. Each product is associated with different minimums in the PES. Right panel: GE in the enlarged center of the left panel. VRI: The valley from minimum B separates into two valleys over the SPs and a ridge between. The VRI point is crossed by the GE, which connects the two adjacent SPs. Figure 2. Left panel: A representative set of exact classical trajectories on the PES given in eq 1. A static RP, which is represented by a curve, can be understood as the curve that matches the average position of the cloud of trajectories after a long time. The IRC can be seen as an approximation to the average position of the set of trajectories that goes from the minimum A to B, whereas the GE curve is the approximation from A to the minimum C. Central panel: Quantum propagation of a Gaussian wave packet at the time that spreads into the bifurcation region of the PES. Notice that an important portion of the Gaussian wave packet resides in the region of the minimum A. Right panel: The propagation of the Gaussian wave packet at a latter time. Portions of the wave packet reside in the bifurcation region and the minimums as well. The portion of wave packet in the minimum B is bigger than that the minimum C. Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct200805d | J. Chem. Theory Comput. 2012, 8, 927?935928 transition structure (TS) of a PES. If the RP curve is totally confined in a valley floor, then the RP takes the category of a minimum energy path (MEP). It is a secondary question, how a RP ascends to the SP and descends from it to a minimum. This looseness makes a variety of RP definitions possible, the most widely used being the IRC, the NT and the GE type of curves. A parametrization t of the RP curve q(t) = (q1(t), . . ., qN(t))T is called reaction coordinate. In the last years the use of the variational theory of calculus of variations in the analysis and derivation of the different type of curves satisfying the features of the RP model has been reviewed.31?46 The variational analysis of a RP curve representation provides, at least from a mathematical point of view, the nature and the features of the associated curves to this representation, like the extremal properties do, and what type of curves satisfy the minimum conditions. As noted before, the GE model has been studied in detail by the works of Ruedenberg and Jensen,17,18 however in this article we present a study based on the theory of calculus of variations47 to give the grounds and features of this type of curve. In this article we analyze the variational nature of the GE curves and their implications. Also the tangent of a GE curve is derived through a perturbation method widely used in quantum mechanics. The characterization of a GE curve as a maximum or minimum is reported. The relation between special points of the GE curve in respect of special points of the PES is discussed. Finally some conclusions are given. 2. LAGRANGE?BOLZA VARIATIONAL PROBLEM AS A THEORETICAL BASIS OF THE GE CURVES 2.1. The First-Order Variational Condition. We show here that, in contrast to a remark in ref 44, the GE curves are extremal curves that belong to the problem of the theory of calculus of variations called Lagrangian or Bolza problem.48,49 Briefly, within the possible generalizations of the simplest problems in the calculus of variations, one of the most important is the well-known problem of Lagrange and its generalizations, the so-called Bolza problem. In this type of variational problem, the extremal curve affording a stationary value to the fundamental integral is required to satisfy certain subsidiary conditions. The same requirements are also imposed on the curves of comparison being necessary to evaluate and analyze the necessary and sufficient extremal conditions. The definition of a GE curve implies that as the curve evolves, the hypersurface v changes as a function of t, the parameter that characterizes the curve. In order to formulate the present problem, it is important to take into account that the arguments of the functional are subject to the boundary conditions and an additional condition. These conditions refer to the entire course of the arguments of the functional leading to an essential modification of the Euler differential equations. We are facing a variational problem with a finite subsidiary condition. The specific formulation consists in making the integral ? ?= = ? ? I F t t tq q g g( ) ( , )d 1/2 d t t t t T 0 0 (2) stationary in comparison to curves q(t) which satisfy in addition to the boundary conditions, q(t0), q(t?), a subsidiary condition of the form = ? =G t V t v tq q( , ) ( ( )) ( ) 0 (3) where as before g = ?qV(q(t)). Geometrically speaking, we want to determine a curve q(t) lying on the PES by the extremal requirement. This problem can be formulated in a more compact form in the following way: ? ? ? = = ? ? = ? ? ? ? ? ? I L t t F t t G t t t V t v t t q q q q q g g q q ( ) ( , )d [ ( , ) ( ( )) ( , )]d {1/2 ( ( ))[ ( ( )) ( )]}d t t t t t t T 0 0 0 (4) where ? is the Lagrangian multiplier which depends on t through q. The extremal curve satisfies the following set of equations at each point ? ? ? ? ? ? =V vHg q q q g 0[ ( ) ] ( ) ( )q (5) where we have dropped the dependence on t and H = ?q gT is the Hessian matrix at the point q of the PES. Equation 5 is the resulting Euler?Lagrange equation of the variational problem (eq 4). Substituting eq 3 into eq 5 we get ? ? =Hg q g 0( ) (6) It means that the gradient is an eigenvector of the Hessian. It is the ?coining? property of GEs. To eliminate the Lagrange multiplier ?(q) from eq 6, we first multiply it from the right by gT/gTg and its transposed form from the left by g/gTg, subtracting and taking into account that H = HT, we obtain ? = ? =Hgg g g gg g g H HP PH O T T T T (7) where P is the projector onto the g subspace and O is the zero matrix. This equation is necessary but not sufficient; it must be combined with the auxiliary condition that the eigenvector is normal | | | | =? ?g g g g( ) ( ) 1T1 1 (8) where |g| = (gTg)1/2. Multiplying this condition from the left by g |g|?1 and from the right by gT |g|?1, we get = = =gg g g gg g g PP P gg g g T T T T T T (9) The condition 9 is called idempotency being characteristic of the projectors like P introduced in eq 7. We emphasize that eqs 7 and 9 are equivalent to the eq 6 and the normalization condition, (eq 8). Finally, from eq 9 we have,O = P ? PP = (I ? P) P, where I is the unity matrix and (I ? P) is the projector that projects to the orthogonal subspace of the subspace spanned by the g vector. If we multiply eq 7 from the left by (I ? P) and from the right by g we obtain, using the last equality ? = ? = ? ? ?? ? ? ??I P HPg I gg g g Hg 0( ) T T (10) Equations 9 and 10 are equivalent to eq 6 and the normalization condition, (eq 8). Equation 6 plus the normalization condition (eq 8) or eqs 7 and 9 or eqs 10 and 9 are equivalent forms to provide the necessary and sufficient conditions for the stationarity of the functional (eq 4). In ref 17, eq 10 was used as the starting point to implicitly obtain the curve solution, q(t), the GE. Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct200805d | J. Chem. Theory Comput. 2012, 8, 927?935929 2.2. Tangent Derivation of the Extremal Curve, GE, from an Application of Perturbation Theory. In the variational problem (eq 4), the integrand does not involve the tangent argument, t = dq/dt. Thus, the set of Euler?Lagrange equations reduces to the simple form given in eqs 6 and 8 or their equivalent form 7 and 9. Either of these two sets of equations determines the GE curve q = q(t) implicitly. We note that in this case the boundary values, q0 = q(t0) and qf = q(tf), cannot be prescribed arbitrarily if the problem should have a solution.47 On the contrary, one has to look for a solution starting at a q0 and take the value qf from there. The tangent vector, t, is necessary to integrate the curve but it does not appear in the expression. Here we use the perturbation theory due to McWeeny which is widely used in quantum mechanics50 to implicitly derive the tangent of the GE from the eqs 7 and 9. First we assume that the current point, q = q(t), is located on a GE, so that the eqs 7 and 9 are satisfied. The basic idea is to start at a point located on a GE in the absence of any first-order variation in q and to seek the necessary first-order changes in the gradient to maintain the GE condition when the perturbation in q is applied. In this manner the first-order variation in the gradient due to the first-order perturbation in q is the tangent of the GE. With eq 7, the problem of perturbation theory merely consists of restoring the GE condition when, due to a perturbation or variation in q, H ? H + ?H, and other quantities, change accordingly. This problem is easily solved using the properties of projection operators. Let us assume that all quantities appearing in eqs 7 and 9 can be expanded in powers of a perturbation parameter t, so that = + ? + ? + = + ? + ? + = = t t d dt t t d dt t t t t H H H H H H H ( ) ( ) 1 2 ... ( ) ( ) ... q q q q 0 0 0 2 2 2 0 0 0 (1) 0 2 0 (2) 0 0 (11a) = + ? + ? + = + ? + ? + = = t t d dt t t d dt t t t t P P P P P P P ( ) ( ) 1 2 ... ( ) ( ) ... q q q q 0 0 0 2 2 2 0 0 0 (1) 0 2 0 (2) 0 0 (11b) where H0 and P0 are the unperturbed Hessian and projector matrices evaluated at the point q0 = q(t0) belonging to the extremal curve. From the eqs 7, 11a, and 11b separating the orders, it follows readily that + ? + =H P H P P H P H O( ) ( )0 0(1) 0(1) 0 0 0(1) 0(1) 0 (12) will determine the first-order change in P0 with the auxiliary equation + =P P P P P0 0(1) 0(1) 0 0(1) (13) which arises from eq 9. This auxiliary condition requires that P0(1) be of the form = = ? + ? =t P P I P M P P M I Pd d ( ) ( )T q q 0 (1) 0 0 0 0 0 0 0 (14) where the matrix M0 is = = = ? ? ? ? ? ? ? ? ? ?t M H q g g g H t g g gd d ( ) ( )T T T T q q 0 0 0 0 1 0 0 0 0 0 1 0 0 (15) where g0 and t0 are the gradient and the tangent vectors evaluated at q0 = q(t0), respectively. The form of the M0 matrix results from the application of the directional derivative, d/dt = [?q ](dq/dt) = [?q ]t, on the normalized gradient vector, being ?qT = (?/?q1, ..., ?/?qN). On substituting eq 14 in eq 12 and multiplying from the left and right by (I ? P0) and P0, respectively, we obtain ? ? ? ? + ? = I P H I P M P I P M P H P I P H P O ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 0 0 0 (1) 0 (16) where we have noted that H0P0 ? P0H0 = O and that (I ? P0) P0 = O. The eq 16 incorporates all quantities and completely determines the first-order change in P0(1), which in turn determines the tangent vector, t0, of the gradient extremal at q0. A solution is obtained most conveniently by multiplying eq 16 from the right by g0 |g0|?1, then we have ? ? ? + ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?I g g g g F g H g H g g g H t 0 T T T T 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 (17) where we have applied that (I ? P0)H0P0 = O, since q0 is a point of the GE. F0 is the third energy derivative tensor with respect to q evaluated at q0. The ?F0g0? symbolism is used to indicate a square matrix that is a contracted product of a three- index array with a vector yielding a two-index array; thus, ?? ? = = gF g F( )ij k N ijk k0 0 1 0 0 (18) where g0k is the k element of the g0 vector. The term ?F0g0? arises from the last term of the left-hand side part of eq 16, that is = = ? ? = ? ? = ? ? ? ? ? ? ? ? d dt H g H g F t g F g t q q 0 (1) 0 0 0 0 0 0 0 0 0 (19) where the directional derivative and eq 18 have been used. The eq 17 was derived for the first time by Sun and Rudenberg.17 We represent the projector (I ? P0) = V0 V0T being V0 a rectangular matrix of dimension N ? (N?1) containing the N?1 eigenvectors of H0 orthonormal to g0 |g0|?1 and the tangent vector as t0 = g0 |g0|?1 eg0 + V0 e0N ? 1, where e0N ? 1 is a vector of dimension N?1. Substituting this in eq 17 we obtain after multiplying from the left by V0T ? ? ? | | = ? ? + ? ? ? ? ? ? ? ? ? ?eV F g g g V F g H g H g g g H V eT g T T T N0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 (20) Notice that V0T H0 V0 = DH0N?1 = {h ij0 ?ij}i,j=1N?1 , V0T H0 g0/ |g0|?1 = 0N?1 and V0T V0 = IN?1 because we are in a point of a GE. The vector 0N?1 is the zero vector of dimension N?1, while IN?1 is the unitary matrix of dimension (N?1) ? (N?1). When the determinant of the square matrix of the right-hand side Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct200805d | J. Chem. Theory Comput. 2012, 8, 927?935930 part of eq 20 is different from zero, then the vector e0N?1 is obtained as a function of eg0, and the latter is evaluated by a normalization of the vector e0T = (eg0, (e0N?1)T). This completely determines t0 the tangent vector of the GE at q0. 2.3. Special Points of GE Curves. We analyze the form of the solutions of the eq 20. In case that the determinant of the square matrix appearing on the right-hand side part of eq 20 is equal zero, a careful analysis of this equation should be taken into account to evaluate e0. For a deeper analysis of this situation see refs 17 and 18. Such points where the gradient extremal has a det(V0T[ + H02 ? ?0H0]V0) = 0 being ?0 = g0TH0g0/(g0T g0) can be either a turning point (TP), a point where the curve touches the isopotential energy contour tangentially, or a bifurcation point (BP), a point where two GEs cross. The structure of the tangent vector e0 is obtained in these points by transforming the eq 20 in the set of coordinates that diagonalizes the square matrix appearing on the right-hand side part of this equation, ? ? + ? = = ? ? ? ? ? ? ? ?V F g H g H g g g H V V C V U C UT T T T D T 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 (21) Where U0 is the unitary matrix of dimension (N?1) ? (N?1) such that it diagonalizes the V0TC0V0 matrix and C0D = {c0ij, ?ij}i,j=1N?1, then eq 20 is transformed to =eb C wg D0 0 0 0 (22) being b0 = ?U0TV0T?F0g0?g0|g0|?1, and w0 = U0Te0N?1, both vectors of dimension N?1. As explained above, if det(V0TC0V0) ? 0 then the tangent vector in the original coordinates is t0 = eg0(g0 |g0|?1 + V0U0(C0D)?1b0), where eg0 is computed by a normalization of t0 vector. If det(V0TC0V0) = 0, then at least one element of the diagonal matrix C0D is equal zero. Let us assume that cii0 = 0, then if the i element of the b0 vector, bi 0, is different from zero, the solution of eq 22 plus the normalization condition implies that eg0 = 0 and w0T = (01, ..., 1i, ..., 0N? 1), and from this, the tangent takes the following form e0T = (e g0, (eN ? 10 )T) = (0, w0T U0T) = (0, (ui0)T), where ui0 is the i column vector of the U0 matrix. The resulting normalized tangent vector in the original coordinates is t0 = V0ui0, where it does not have the component in the eigenvector pointing in the same direction to the gradient vector, thus eg0 = 0. Due to this fact, at this point the GE does not cross the isopotential energy contour. It touches tangentially this isopotential contour. This point is a TP for this GE, and we say that the curve is characteristic at this point.51,52 An example of two TPs is the PES depicted in Figure 1. The points at ?(?0.25, ?0.80) and ?(0.0, 1.0) are TPs of the central GE curve. In case that both cii0 = 0 and bi0 = 0, then eq 22 plus the normalization condition gives two solutions: one is that which touches the isopotential energy contour being the expression of the tangent the same as the one given above. The second solution is that which crosses the isopotential energy contour. In this case w0T = eg0(c110 /b10, ..., 0i, ..., cN?1N?10 /bN?10 ) and from this the tangent e0 takes the form e0T = (eg0, (eN?10)T) = eg0 (1, w0TU0T), where e g0 is computed by a normalization of e0 vector. The resulting normalized tangent vector in the original coordinates takes the form t0 = e0g (g0|g0|?1 + V0U0w0). The GE curve with this tangent is a noncharacteristic curve at this point, it transverses the isopotential energy contour because eg0 ? 0. Since in a point where both cii0 = 0 and bi0 = 0, two GE curves coincide at this point one with tangent t0 = V0ui0 and the other with tangent t0 = e 0g (g0|g0|?1 + V0U0w0), this point is called bifurcation point (BP) for this type of GE curves. It is interesting to notice that the structure of eq 20 is very close to the basic equation to integrate Newton Trajectory curves as one can see by an inspection of eq (31) of ref 31, see also refs 29 and 53. The basic differences between both equations lie in that the vector of the left-hand side part of eq 20 depends on the Hessian matrix rather than a contracted product of a three- index array with the gradient vector and that the square matrix of the right-hand side part is only a function of the Hessian matrix. An example of a GE bifurcation is shown in Figure 3, on the well-known Mu?ller?Brown PES (mb-PES).54 At point ?(0.06, 2.06) a bifurcation of the GE curve takes place. Note that the two eigenvalues of the Hessian matrix at this node = ?? ? ? ? ? ? ?H(0.06 2.06) 1902.96 389.644 389.644 1902.96 are totally different: they are 1513.31 and 2292.6. By the way, the GE curve starting orthogonally to the minimum valley meets itself at the BP. There is also its TP at the slope of the bowl. Now we show by an example that a degeneracy of eq 6 involving the eigenvector that points in the same direction to the gradient vector does not affect the behavior of the actual GE curve. Let us assume that on a point at the GE curve the Hessian is degenerated, but det(H) ? 0, and one of the degenerated eigenvectors coincides with the gradient direction at this point. Nevertheless, the derivation of the tangent of the curve due to Ruedenberg?s formula, eq 17, is true and the GE curve behaves ?normally?. In Figure 4 we again demonstrate this with the mb-PES. We again show the GE curves of Figure 3 and the level lines of two eigenvalues: The value of each of them is equal to 313.1 units. The two systems cross exactly at the GE curve near the deep minimum. There the GE curve does not show any anomaly. Finally, we say that the perturbational method due to McWeeny50 has been used until now to derive the well-known tangent vector that characterizes the GE curve. Using the same method one can evaluate the first-order gradient correction through the GE curve, g(q(t)). For this task, first we insert Figure 3. GE curves (fat dashes) on the mb-PES. Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct200805d | J. Chem. Theory Comput. 2012, 8, 927?935931 eq 15 into eq 16, multiply the equation from the right of the resulting expression by g0 |g0|?1, and assume that q0 is a point on a GE curve, ? ? + ? | | = | | + ? | | = + ? | | + | | = + ? | | = + ? | | ? ? ? ? ? ? ? ? ? = ? ? ? t t t t t t t e t t t t h e g q P P g g g g V V H t g g V V H g g V e g g V H e g g v g ( ( )) ( ( ) ) ( ) ( ( ) ( )) ( ( ) ) ( ( ) ( ) ) T T g N D N N i N i ii N i 0 0 0 (1) 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 (23) where we have taken that (I ? P0) = V0V0T and vi0 and (e0N?1)i are the i column vector and the i element of the V0 matrix and the e0N?1 vector, respectively. Notice that due to the structure of this particular perturbation problem, it is irrelevant that the eigenvector which points in the same direction like the gradient vector be degenerate or not in respect of the others eigen- vectors of the Hessian matrix, since the division by differences between eigenvalues does not appear as it usually appears in the application of perturbation methods. In the same way, one has the corresponding expression to compute the first-order eigenvector correction to each eigenvector of the subset, {vi0}i=1N?1. Because I = P0 + (I ? P0) = P0 + Q 0 then ? dQ 0/dt = dP0/dt = ?Q0(1) = P0 (1). Substituting g(q(t)) by vi(q(t)), P0 by Q 0, P 0(1) by ?Q 0(1) and g0 by vi0 in eq 23, we obtain the desired expression. The projector Q 0 is represented by V0V0T . 2.4. The Extremal Sufficient Conditions. Conjugate Points of GE Curves. The set of Euler differential eqs 6 is a necessary condition for an extremum. However, a particular extremal curve satisfying the boundary conditions can furnish an actual extremal, let us say with the character of a minimum, only if it satisfies certain additional necessary conditions that take the form of inequalities, normally denoted as ?2I ? 0. The formulation of such inequalities together with their refinement into sufficient conditions is an important part of the theory of calculus of variations.47,55 To prove this we first replace in the integral I(q) of eq 4, the argument q by qC(t,?) = q(t) + ? p(t), being ? a small number. The curve q(t) must be an extremal and qC(t,?) is an arbitrary curve both satisfying the eq 3. The functions p(t) are variations of the extremal curve q(t) and satisfy the equation: = ? = ? = =P t G t Vp p q p q p g( , ) ( , ) ( ) 0T T Tq q (24) with p(t0) = 0. In this way the comparison curves satisfy the subsidiary condition. Second, we expand I(q) by the Taylor theorem until the second order in ? ? = + ? = + ?? + ? ? + ? H I I I I O q p q q p q p ( ) ( ) ( ) ( , ) 2 ( , ) ( ) 2 2 2 (25) Since I(q) is stationary for the extremal curve q(t), dH(?)/ d? |?=0 = ?I(q,p) vanishes, where the Euler?Lagrange eq 6 follows. A necessary condition for a minimum is d2H(?)/d2?|?=0 = ?2I(q,p) ? 0. The derivative ?2I(q,p) is expressible in the form ? ? ? ? = ? ? = ? ? + ? ? = ? ? ? I L t t t t q p p q p p Fg H H p p Cp ( , ) [ ( , )] d [ ] d d t t T T t t T t t T q q 2 2 0 0 0 (26) where we have dropped the dependence on t. This integral is evaluated along the GE and ? is just that given in eq 6. The condition ?2I(q,p) ? 0 implies a problem of Lagrange in the tp- space of precisely the same type as the original problem in the tq space. The integral to be minimized is ?2I(q,p), and the equation of the condition corresponding to eq 3 is the eq 24. To this tp problem we apply the Euler?Lagrange equation. The condition 24 can be introduced without the use of the multiplier rule if we consider that p(t) = (I ? P) m(t), where m(t) is a vector of dimension N. We represent the projector (I ? P) = VVT being as before V, the matrix of dimension N ? (N?1) containing the set of N?1 eigenvectors of H orthogonal to the g|g|?1 vector. With this consideration we write p(t) = VVTm(t) = Vn(t), being n(t) a vector of length N?1. Substituting this form of p(t) vector in eq 26 we have, ? ?? = = = ? ? ? I t t Iq p p Cp n V CVn q n( , ) d d ( , ) t t T t t T T2 2 0 0 (27) where we have again dropped the dependence on t. From eq 27 we conclude that the GE will extremalize the integral (eq 4) with a minimization character if it applies along the curve det(VTCV) ? 0. More concisely, the minimum value of the integral of eq 26, or equivalently eq 27, defines the accessory problem of the variational problem under consideration. This variational tp problem of eq 26, which is the same variational tn problem of eq 27, is solved by the application of the Euler?Lagrange equations on the functional integral 27. These Euler?Lagrange equations are known as Jacobi equations of the accessory variational problem.56 The accessory problem 27 affects the Figure 4. GE curves (fat dashes) and two level lines of eigenvalues (continuous line and thin dashes) at exactly 313.1 units. The two level lines cross on the GE curve. Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct200805d | J. Chem. Theory Comput. 2012, 8, 927?935932 existence of extreme values of the fundamental integral (eq 4). The application of the Euler?Lagrange equation on the integral functional 27 results in the next expression: = ?V CVn 0T N 1 (28) If det(VTCV) ? 0 along the GE between t0 and t?, then the unique solution of eq 28 is n(t) = 0N?1, which implies p(t) = Vn(t) = 0. From this we infer that the integral ?2I(q,p), or, that is, the same like ?2I(q,n), vanishes if n is the solution of the Jacobi eq 28, which possesses zero at t = t0. In addition if det(VTCV) > 0 (det(VTCV) < 0) along this interval, then the GE curve extremalizes the integral (eq 4) with the character of a minimum (maximum). The value of integral (eq 4) evaluated using a different path, qC(t,?), is higher (lower) than that evaluated on the GE path q(t). If det(VTCV) = 0 at any point of the GE curve, say t = t1, then the procedure is close to that which follows eq 21. We diagonalize the matrix VTCV of Jacobi eq 28 at this point. As before, let U be the unitary matrix of dimension (N?1) ? (N?1), such that VTCV = UCDUT, where CD = {cij?ij}i,j=1N?1, then eq 28 is transformed to = ?C z 0D N 1 (29) being z(t1) = UTn(t1). Because det(VTCV) = 0, then at least one element of CD diagonal matrix is equal zero. Let us assume that cii = 0 then two solutions of the z(t1) vector exist both of the form z(t1)T = (01, ..., zi, ..., 0N?1), one with zi = 0 and other with zi ? 0. In the first case p(t1) = Vn(t1) = VUz(t1) = 0 and in the second case p(t1) = Vn(t1) = VUz(t1) = Vuizi ? 0, where ui is the i column vector of the U matrix. This result implies that in a point of the GE curve, q(t1), so that det(VTCV) = 0 there exist two solutions of the Jacobi eq 28, namely, p(t1) = 0 and p(t1) ? 0, making the value of the integrand of eq 27 equal to zero at this point of the GE curve, q(t1). According to the discussion that follows eq 20, a point of the GE curve so that det(VTCV) = 0 can be either a TP or a BP. In the case of a TP, the two solutions of eq 29, namely, p(t1) = 0 and p(t1) ? 0 coincide with the GE curve since in the last case the structure p(t1) is equal to the structure of the tangent vector t at this point. The same occurs for the BP, be- cause for p(t1) = 0 the arbitrary curve qC(t1) coincides with both GE curves that meet at this point. The other solution, p(t1) ? 0, the resulting arbitrary curve coincides with the GE curve that at the BP touches tangentially the isopotential contour hypersurface, since the form of the vector p(t1) is the same to the tangent vector t of this GE curve. For this reason in these cases both solutions of eq 29 make the integrand of eq 27 equal zero at this point. Once the value of the integral ?2I(q,p) has been evaluated along a GE curve, we now explore the effect of the existence on the GE curve of a point so that det(VTCV) = 0. Let us assume a GE curve starting at the point q0 = q(t0) being this point a stationary point with the character of a minimum of the PES and ending at a first-order SP located at qf = q(tf). At the point q1 = q(t1) with t0 < t1 < tf of this GE curve is the det(VTCV) = 0 but at each point of the subarc within t0 and t1 is the det(VTCV) > 0. Let the point be a turning point for the GE under consideration, and after this turning point, the GE enters into a region where det(VTCV) < 0 until the last point qf. Notice that C is a continuous matrix function on q. In this case we say that one can find an arbitrary curve, not necessarily a GE curve, joining the points q0 and qf such that the integral (eq 4) takes a lower value in respect of the initial GE curve. In Figure 5, being equivalent to Figure 1 of the ref 17, exists two GE paths joining the points M2 and TP1 as well as the points M2 and TP2. The value of the integral (eq 4) for the GE curve that coincides with the IRC path has a lower value in respect of the GE curve that evolves through the TP located at ?(?1.00, 0.25). The preceding explanation is supported by numerical results calculated with the Mathematica program.57 At point ?(?0.5, 0.5) on the direct GE path between the minimum M2 and TS1 of the mb-PES, the value of det(VTCV) = 1.0 ? 106 is greater than zero. Notice that in the present case, since mb-PES is a two- dimensional PES, the V matrix is in fact a vector of dimension two, vT = (vx, vy) and due to this fact, VTCV = vTCv, is already a number; in other words, det(VTCV) = vTCv. In contrast, the value at the point ?(?1.00, 0.25) on the GE path that goes through the mountains vTCv = ?200.00, it is less than zero. The same argument can be used if the q1 point is a BP. If the branch that achieves the point qf after the BP enters into a region where det(VTCV) < 0 until the last point qf then, as before, one can find an arbitrary curve, which is not necessarily a GE curve, so that the integral (eq 4) takes a lower value in respect of the original GE curve. The conclusion is: A point on a GE curve so that det(VTCV) = 0 reminds us of a Jacobi conjugate point of the curves that extremalizes the functional integral where the tangent argument appears explicitly.55,56 We say that these points of the GE curve are like Jacobi conjugate points if they represent a change on the sign of the determinant of the VTCV matrix function in the evolution of the curve. Notice that the existence of Jacobi conjugate points in a curve extremalizing a functional integral where the tangent argument appears explicitly implies that the functional integral of the corresponding accessory problem cannot be reduced to a quadratic functional. From this follows that the value of the original functional integral evaluated using any curve can be lower than that of the value of the extremal curve.55 In the present variational problem the accessory variational problem is already quadratic. Thus the existence of Jacobi-like conjugate points does not imply that the accessory problem can be reduced in a quadratic form. In the present case, these points may represent a change of sign of the quadratic functional nTVTCVn, therefore it follows that from this point any curve can reduce the functional integral of eq 4. We emphasize that the Jacobi conjugate point concept takes its full complete relevance when the extremal curves under consideration form a field of extremals, like SD curves.39 The GE curves do not form a field of extremals in the PES region. There exist N GE curves Figure 5. GE curves (fat dashes) on the mb-PES between SP1, minimum M2, and SP2 as well as turning points TP1 and TP2. Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct200805d | J. Chem. Theory Comput. 2012, 8, 927?935933 only if the dimension of the PES is N. Nevertheless, we say that TP and BP points of a GE arc curve in some aspects resemble the original Jacobi conjugate point concept, and due to this fact, we say that these points can be seen as Jacobi-like conjugate points. 3. RELATION BETWEEN TPs OF A GE CURVE AND VALLEY-RIDGE INFLECTION POINTS Regarding Figure 1, a valley-ridge inflection point (VRI) emerges in the GE subarc that goes from the TP located at the point ?(?0.2, ?0.6) and to the TP located at the point ?(0.0, 1.0). The VRI point is related to the existence of a bifurcation in the PES. The VRI is a feature of the PES and does not always have relations with the nature of the curve.58 In Figure 4 of the mb-PES, looking at the GE subarc being orthogonally to the GE RP at the minimum M2, which goes from the TP1 located at the point?(?1.00, 0.25) to the TP2 located at the point ?(0.6, 0.6), a point of this subarc is the minimum M2. This GE subarc is orthogonal to the GE RP at the stationary point, M2. As before, a stationary point is a feature of the PES and does not always have relations to the nature of the curve going through it. However the most important is a possible relation between TPs of a GE and VRIs of the PES. In Figure 5 of ref 59 is shown the TPs of the different GEs and the VRI points of the mb-PES. With this observation, we enunciate the next proposition. A GE touches at its TP, q(tTP), a isopotential hypersurface of the full PES. At other GE points, it crosses a family of isopotential hypersurfaces transversally. We may assume that at q(tTP) and the next points q(t) of the GE, with tTP < t, the family of isopotential hypersurfaces is pseudoconvex with the pseudoconvexity index:60 ? = >g Ag g g 0 (or vice versa) T T (30) If along the GE the index ? changes the sign then there is a VRI point. In eq 30 the A matrix is the adjoint matrix of the Hessian matrix H, and it satisfies the relation, AH = I det(H). The proof of the proposition is the following: At the transition point of the GE through the contour valley-ridge, we have to find a zero eigenvector of the Hessian matrix lying in the tangential plane of the corresponding isopotential hypersurface. Because this point belongs to a GE, the gradient is an eigenvector of the Hessian matrix, and due to this fact, the gradient vector is orthogonal to the eigenvector with null eigenvalue. This is nothing more that the definition of a VRI point.60 Notice that it is not necessary that a TP point of the GE curve exists before the GE transverses a contour valley-ridge inflection. In Figure 5 of ref 59 exists an isolated VRI point on a GE curve, but no TP of this GE exists near to this VRI point. On the other hand, there can be two consecutive TPs of a GE without a VRI point in between, see again Figure 5 of ref 59. Finally we remark that if a GE having a VRI point in a subarc, then it does not imply that in this subarc det(VTCV) ? 0 holds, or vice versa. In other words, the VRI point does not affect the extremal character of the subarc. Let us assume a GE subarc without TP lifting a valley region with det(VTCV)) ? 0 and entering in a ridge region through a VRI point, where in this new region the GE has det(VTCV)) ? 0. This result is a direct consequence of the discussion of Section2.4. 4. DISCUSSION, SUMMARY, AND CONCLUSION The GE curve can be used to determine the TS and intermediates of any reaction mechanism starting at the reactant minimum. Examples are demonstrated for methyleneimine (H2C?NH) computed using a CASSCF calculation,13 the electrocyclic reaction of cyclobutene to butadiene,9 or the formation of amino- acetonitrile.61 As reported in the Introduction Section, the GE path is also a type of curve that can be used as a representation of a RP in the cases that fail the IRC as a representation. Due to the importance of this type of path, we have proved the variational nature of the GE path. GEs are curves where the norm of the gradient has a local extremum on isopotential hypersurfaces of the PES, V(q(t)) = v(t). An example is discussed in ref 53, where the IRC is going down across a ridge, but the GE represents the MEP which is parallel nearby. The PES of that example is a modified PES of an alanine dipeptide rearrangement.62 The GE paths are extremal curves of a variational problem that is formulated in expression (eq 4). The tangent of this type of curves has been derived using the perturbation theory due to McWeeny that is widely used in quantum mechanics. In addition their extremal sufficient conditions are studied and reported being summarized as follows. If the curve starts at a minimum of the PES with det(VTCV) > 0, with eq 27, and ends at a first-order stationary point, the extremal curve achieves its condition of a minimal curve. However, if this curve has a TP or a BP, then from this point to the end point may be det(VTCV) < 0, and the GE curve looses its minimum character, see Figure 1. If det(VTCV) < 0 from the TP to the end point, then other curves exist. Not necessarily a GE curve joins the initial and final point so that the integral (eq 4) takes a lower value. The TP and the BP of a GE curve can be seen as a Jacobi conjugate point of this type of curves. Nevertheless, this equivalence should be taken carefully since the GE curves do not form a field of curves covering the PES region as occurs with the SD curves or with NTs. In the later cases the concept of Jacobi conjugate points takes its important relevance. The missing ?field? property may be an advantage of the GE calculation. In ref 30 was found a VRI point of the ring closure of the allyl radical, however that point is located after a small ring-opening. The search with NT failed because it was done in the false direction. But a search of the minimal GE along the minimum valley of the allyl radical found that point. The VRI points are important on PESs where the RP bifurcates.58 The relation between the VRI points and the GE curve is also analyzed. VRIs are the type of points related to the curve which leaves a valley region and enters into a ridge region of the PES, or vice versa. The behavior of the GE path can be used as a way to take information on the topology of the PES region where the reaction mechanism takes place.63 ? AUTHOR INFORMATION Corresponding Author *E-mail: jmbofill@ub.edu. Telephone: 34 93 402 11 96. Notes The authors declare no competing financial interest. ? ACKNOWLEDGMENTS Financial support from the Spanish Ministerio de Ciencia e Innovacion?, DGI project CTQ2011-22505 and, in part from the Generalitat de Catalunya projects 2009SGR-1472 are fully acknowledged. M.C. gratefully thanks the Ministerio de Ciencia e Innovacion? for a predoctoral fellowship. ? REFERENCES (1) Doubleday, C. J. Phys. Chem. A 2001, 105, 6333. (2) Carpenter, B. K. Angew. Chem., Int. Ed. 1998, 37, 3340. (3) Doubleday, C.; Li, G.; Hase, W. L. Phys. Chem. Chem. Phys. 2002, 4, 304. Journal of Chemical Theory and Computation Article dx.doi.org/10.1021/ct200805d | J. Chem. Theory Comput. 2012, 8, 927?935934 (4) Carpenter, B. K. J. Phys. Org. Chem. 2003, 16, 858. 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Theory Comput. 2012, 8, 927?935935 Search for conical intersection points (CI) by Newton trajectories Wolfgang Quapp a,?, Josep Maria Bofill b,d, Marc Caballero c,d aMathematisches Institut, Universit?t Leipzig, PF 100920, D-04009 Leipzig, Germany bDepartament de Qu?mica Org?nica, Universitat de Barcelona, c/Mart? i Franqu?s, 1, 08028 Barcelona, Spain cDepartament de Qu?mica F?sica, Universitat de Barcelona, c/Mart? i Franqu?s, 1, 08028 Barcelona, Spain d Institut de Qu?mica Te?rica i Computacional, Universitat de Barcelona (IQTCUB), c/Mart? i Franqu?s, 1, 08028 Barcelona, Spain a r t i c l e i n f o Article history: Received 28 February 2012 In final form 22 May 2012 Available online 30 May 2012 a b s t r a c t Recently, valley?ridge inflection points on the potential energy surface for the ring opening of the cyclo- propyl radical have been determined using Newton trajectories (NTs) [W. Quapp, J.M. Bofill, A. Aguilar- Mogas, Theor. Chem. Acc. 129 (2011) 803]. This Letter is the report about the utilization of NTs for the search for conical intersection (CI) points. These points play a main role in the understanding of intersec- tions of different electronic surfaces which open the door for photochemical reactions. We explain the reason why Newton trajectories can find CI points, and report a CI seam on the CASSCF (3,3) surface of the allyl radical ring closure. ? 2012 Elsevier B.V. All rights reserved. 1. Introduction: the theory of Newton trajectories and the attraction of NTs by CI points The concept of the potential energy surface (PES) is the basic ground of many theoretical chemistry models [1]. It is here an n-dimensional surface over the n internal coordinates ?n ? 3N ? 6? for an N-atomic molecule. Intersections of two or more electronic PES play a main role in the understanding of photochemical reac- tions. The key here is the event of a conical intersection (CI) [2,3]. Up to date, the calculation of CI points is a difficult job which has to include all the involved electronic PESs for a constrained optimi- zation, see references [4?6] and further references therein. From a general point of view, the methods to locate CI points can be classi- fied according to the type of the objective function used for this task. Whereas in reference [4] the restrictions are imposed through a set of Lagrange multipliers, in the references [5,6] the restrictions are given implicitly by using the energy difference or its square form between the electronic states of the energy as objective function that are assumed to have at least a contact point. Usually, CI points are not directly connected with the concepts of a reaction path (RP) and its more restrictive definition minimum energy path. On the ground state PES, the RP is defined as a contin- uous curve in the coordinate space, which connects two minimums of the PES by passing through a first order saddle point (SP), also called transition structure (TS). One special definition of an RP is the reduced gradient following (RGF) [7?14] and its equivalent def- inition, the so-called Newton trajectory (NT) [15,16]. In this Letter, we extend the application of NTs to search a CI point on the ground state PES. An NT can be calculated by an Euler?Branin-step method following along the direction of the vector field Ag of the so called Branin differential equation [17] dx?t? dt ? ?A?x?t?? g?x?t??; ?1? where A is the adjoint matrix to the Hessian, its desingularized inverse, and g is the gradient of the PES. t is a curve length param- eter, x is the current point. A second definition of the NT is given by the projector equation Prg?x?t?? ? 0: ?2? The projector can be defined by a dyadic product with a normalized search direction, r Pr ? E? r rT ; ?3? where E is the unit matrix. Then the curve can be calculated numer- ically by the derivation of the projector equation along the curve parameter t giving also the tangent of the NT [9]. The Eq. (2) means that along an NT the gradient always points into the r-direction. Stationary points of the PES have a zero gradient. Thus, NTs into all directions can start from here if one adequately selects the sign of Eq. (1) according to the index of the stationary point [8,9,18]. This means that the search direction of an NT, r, is optional from here. Thought inversely, this is the reason why stationary points are attractive points for NTs from all directions. An SP of index 2 on a PES is also the crossing point of infinitely many NTs, see Figure 1. It is an attractor if we take the positive sign in Eq. (1). There on the left hand side the summit surface ?x2?5 y2 is shown, on the 0009-2614/$ - see front matter ? 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.05.052 ? Corresponding author. Fax: +49 341 97321 99. E-mail address: quapp@uni-leipzig.de (W. Quapp). URL: http://www.mathematik.uni-leipzig.de/MI/~quapp (W. Quapp). Chemical Physics Letters 541 (2012) 122?127 Contents lists available at SciVerse ScienceDirect Chemical Physics Letters journal homepage: www.elsevier .com/locate /cplet t upper left hand panel, and on the lower line the field Ag is shown. The summit is at the point (0,0); it is an SP of index 2. Starting an NT uphill anywhere around the summit it will find the summit it- self. Analogously, a funnel surface of a typical peaked ?n? 2? CI event [2,3] is shown on the right hand side of Figure 1. It is the sur- face?sqrt?x2 ? 5y2?. Here, an analogous vector field Ag emerges. At this point it is important to emphasize that both types of problems, finding an SP of index two, a summit, or a peaked ?n? 2? conical intersection point, are quite close, both conceptually and computa- tionally. This is a hint that the NT method too finds CI points as well as summits. However, there are differences to the summit case: (i) The two components of the gradient in the branching sub- space do not go to zero around the peacked ?n? 2?-CI point, but hold any slope, and their direction is indefinite at the CI point. In Figure 1 right panel the gradient is 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ? 5y2p ?x;5y? T : ?4? At x ? 0 and y approximates zero, then the gradient points into the y-direction with value ? ffiffiffi5p , but at y ? 0 and x approximates zero, then the gradient points into the x-direction with value ?1. (ii) The two lowest Hessian?s eigenvalues are indefinite values at the CI point because of the kink in the energy of any path- way passing the CI point. The Figure 1 right panel is a simpli- fied case. The two eigenvalues of the Hessian are 0;?5 ?x2 ? y2?= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 ? 5y2p 3   : ?5? The first eigenvalue belongs to the x-axis and concerns the points outside the apex; because the sign of the gradient direction jumps at the apex, the eigenvalue here is indefinite as well. If the point ?x; y? moves to (0,0) then the second eigenvalue tends to minus infinity. In practical calculations, like the one reported here in this Letter, the two lowest Hessian?s eigenvalues are large negative values near putative CI points. (The Hessian is computed by finite differences.) For the left summit case, on the other hand, the two negative eigen- values describe the curvature transversally, as well as longitudinally to the summit. In a neighbourhood of the CI point, this may also hold for the PES around this point. But at the peak, the usual differ- ential calculation breaks down. In a path calculation on the PES, we can follow an NT uphill to a CI point. The indicator to arrive at the CI region is the increase of the value jAgj which itself becomes very large at the CI point. We find factors of 105 and larger against ??normal?? NTs. The reason is the eigenvalues, li, of the A matrix, which go with the eigenvalues of the Hessian, the kj j ? 1; . . . ;n [18] li ? Yn j?i kj: ?6? The stopping criterium of the CI search, for the simple 2-dimen- sional case, would be a zigzagging of the predictor at the CI point. This can be indicated. In the high-dimensional praxis on the CASSCF surface, indefinite eigenvalues of the Hessian do not emerge. The quick increase of the jAgj value is a first sign that the NT is attracted by a CI point. Then anywhere happens a saturation. The CI seam is reached. Of course, a second indicator is a bad convergence of the CASSCF calculation. (We stop the calculation by hand.) (iii) For an SP of any index, the gradient has to be zero. Thus, the SP is an NT-attractor in all dimensions if we take the appropriate sign in Eq. (1). The CI kink concerns a two- dimensional subspace of the configuration space. For higher dimensional problems it is recommended first to estimate the two vectors, vd; vc, that define the linear subspace called branching subspace, Sb, and its orthogonal complement called the tangent intersection subspace, Sti [4]. The two basis vectors of the Sb subspace are the difference vector, vd, and the coupling vector, vc, vd ? rx?W1;1?x? ?W2;2?x?? ?7? vc ? rx?W1;2?x?? ?8? whereWi;j?x? is the ij element of the matrix L?x? of the correspond- ing Lagrange problem [4]. The set of vectors that defines the basis of the Sti subspace is taken orthogonally to the vd and vc vectors [4]. If the latter two vectors are collected in a rectangular matrix, Tb ? ?vd vc?, and if the set of vectors of the Sti subspace is collected in a Tti submatrix, then both rectangular submatrices build the transformation matrix, T ? ?TbTti? which transforms the given x coordinates into new ones. Notice that T itself depends on the x coordinates. The process is summarized through the transformation [4], Dq ? ?TbTti?TDx: ?9? One could apply the NT method in the new coordinate space. However, following NTs which lead to a CI point on the seam, auto- matically also lead to vectors in the subspace Tb. Because these are two negative eigenvalues of the original Hessian of the ground state at, or near the crossed CI point. The mathematical proof of the state- ment is based on L?wdin?s partitioning technique [19] and will be outlined in a next paper. A technical problem is the following: Near CI points the Newton?Raphson method for corrector steps often fails, if we use the derivation of Eq. (2). Then we only work with predictor steps along Eq. (1), and we omit the corrector steps. The resulting NT is named quasi-NT, because it can deviate a little from the true one, especially if it is strongly curved. Such predictor-only steps work properly, because the A matrix is the desingularized inverse of the H matrix. It is calculable if the H matrix is given. Configuration interactions can take place in a sloped kind [2,3]. Such events are not found by the funnel search which we report here, because they do not show the pattern of the A g vector field which we need here. However, a quasi-NT search can also do well for sloped CI seams. If a seam is crossed then the vector A g usually jumps to another direction. The corresponding quasi-NT will show a sharp kink. We can study such nodes and do a corresponding 2 1 0 1 2 x 2 1 0 1 2 y 2 1 0 1 2 x 2 1 0 1 2 y 2 0 2x 2 0 2 y 80 40 0 2 0 2x 2 0 2 y 7.55 2.50 Figure 1. Summit surface on the left hand side, and funnel on the right: both have analogous Ag fields shown below. W. Quapp et al. / Chemical Physics Letters 541 (2012) 122?127 123 control calculation for upper surfaces. Examples can be found in Figs. 8, 9, 11 and 12 of reference [20]. 2. Attraction of quasi-NTs by CI points on the PES of the allyl radical ring closure In organic chemistry the prediction of stereochemistry of the electrocyclic ring closure reactions has been a long-standing ques- tion. For cyclic molecular structures in their ground state with an even number of electrons the reactions are governed by the Wood- ward?Hofmann rules [21]. The rules either predict a conrotatory or disrotatory stereochemistry evolution depending on the orbital diagram associated to the system. For the cyclopropyl radical, a system with an odd number of electrons, the Woodward?Hofmann rules predict that both the conrotatory and disrotatory stereo- chemistry evolution are nominally forbidden [22]. In the study [22] a highly asynchronous transition structure with C1 symmetry was identified. In a later study at the B3LYP/6-311G (2d) level of theory [23], the calculation of the intrinsic reaction coordinate was carried out from the C1-TS to the allyl radical. The unsymmet- ric pathway avoids the crossing of CI points which are expected on the PES [24]. For the purpose of this Letter, our consideration of the allyl radical is purely a test system for a search of CI points. Figure 2 shows the atom labels of the allyl radical that will be used. The GAMESS-US program is utilized for the quantum chemical calcula- tions by the CASSCF (3,3)/6-31G (d,p) method [25]. The NT-pro- grams (in Fortran) can be downloaded from the web-page http:// www.math.uni-leipzig.de/$quapp/SkewVRIs.html. Any tracing along NTs uses the curvilinear internal coordinates in the full dimension of n ? 3N ? 6 ? 18. At every node of an NT, the metric matrix is taken from the GAMESS-US program to convert co- and contra-variate objects, as well as the energy, the gradient and the Hessian of the lowest electronic PES. In the z-matrix, the distances are given in ?, and the angles and the dihedrals are given in de- grees, where in the GAMESS-US part Bohr and radiant units are used. z-matrix allyl radical C1 c1 c2 1 r1 c3 1 r2 2 c2c1c3 h4 1 hc1 2 hcc1 3 dih1 h5 2 hc2 1 hcc2 3 dih2 h6 3 hc3 1 hcc3 2 dih3 h7 2 hc4 1 hcc4 3 dih4 h8 3 hc5 1 hcc5 2 dih5 In the foregoing papers [20,26,27], we confirmed a valley-ridge inflection (VRI) point at the allyl radical end of the SP col. It was named VRIca to characterize the direction from the cyclopropyl bowl, the symbol is the lower case letter c, over SPca into the allyl radical bowl, the symbol is the lower case letter a. NTs are the curves which are well adapted to search for VRIs because they bifurcate there. The method of the search of VRI points is described in Ref. [28]. A VRI system of a bifurcating, singular NT consists of four branches. The character of the single branches determines the character of the VRI point. All VRI points can be classified into different main classes. A valley starting from an SP can bifurcate downhill and the two branches can lead to two valleys with their corresponding minimums. Between the two valleys a ridge emerges. However, the VRIca does not have this character. There is another possibility that a ridge on the PES bifurcates into two ridges, and between the two ridges emerges a valley. Then the VRI point is a ridge-pitchfork (rpVRI) bifurcation. The VRIca is of this character. The corresponding ridge from below is guessed to come from one of the allyl SPs, named SPaa. They have a turned methylene group at one end, where the other methylene remains in the plane of the C-atoms [20,27]. If the methylene 1 group is turned out off the C-plane then the SP gets the index (2,4) where in the contrary case, if the methylene 2 group is turned, the index will be (3,5). Additionally to the VRIca, we found VRI points near the SPs in the allyl radical bowl, the SPaa. These new VRIs are on the ridge over the SPaa. The are named VRIac to accentuate their local connection to the allyl bowl. The character of the points found is either rpVRI or of the mixed type where ridges of a different index cross [20]. Using the PES of the very floppy allyl radical as an example, we tried to make relaxed surface scans of diverse dihedrals to generate a 2D surface. But we were not successful to create the usual, mean- ingful level lines in 2D subspaces of the configuration space in the past. There are too many dimensions where the PES changes. How- ever, NTs are curves in the full-dimensional configuration space. They characterize a skeleton of valley- and ridge-lines. Their projection into 2-dimensional planes can be used to generate an imagination of the full PES. This is done in Figure 3. We report a new VRI point of the character ac 2;4, compare Ref. [20] which is here depicted by VRIlow. Its side branches lead to CI points where the central branch exhausts a valley. A further VRI point, VRIup, is found upwards in the PES region, which is connected with VRIlow over the central branch. Its central branch then leads uphill to the CI seam. The representation is the following.We use the projection of one and the same full-dimensional quasi-NT to the both dihedrals, dih2 or -dih5, and the ring closure angle, in the left panel, as well as to the dih3 or -dih4, and the ring angle in the right panel. So, every panel contains two pictures of the singular quasi-NT of interest. The abscissae is the ??reaction coordinate??. It is the angle between the C-atoms, compare Figure 2, which describes the ring closure of the allyl radical, from the right hand side to the left hand side. The singular quasi-NT is a fat curve. In both panels further bullets are included for dihedrals of the following special points: VRIlow; VRIup (see Table 1) and the points SPaa; SPca, and VRIca (see Ref. [20]). (For the latter all four bullets are included in every panel.) Their values on the abscissae are also highlighted by grid lines. All bullets on one upright grid line belong to one of the special points. Their descriptions emerge once on the grid line. The order of the bullets for SPca and VRIca is, from top to bottom, dih2, -dih5, dih3 and -dih4. Note, that the order of the bullets for the special points on the right hand side is another. The new VRI points, VRIlow and VRIup, of Figure 3 (see coordi- nates in Table 1) are a very special case where the two outer Figure 2. Ring closure of the allyl radical: numbering of atoms in the z-matrix. 124 W. Quapp et al. / Chemical Physics Letters 541 (2012) 122?127 branches of the lower bifurcating quasi-NT were attracted, after the bifurcation and after a long way over the PES, nearby by a CI seam. The central branch of the upper VRI also leads to this CI seam. Usually, the three branches of a singular NT wander after the bifurcation into different regions of the PES; that is even the character of a bifurcation to create dividing valleys or ridges. The character of the lower VRIlow point is rpVRI, a ridge pitch- fork. The ridge of index one from SPaa 2;4 bifurcates into two outer forks being also ridges of index one, and into one central branch being a valley line. All three branches climb uphill on the energy surface, however the quasi-NTs of the outer branches later meet the seam at CI points symbolized by a ???. One branch touches the seam and is then ??reflected??, thus goes again downhill to the SPaa region (the reflected part is not shown in Figure 3). The other branch touches the seam and scans then along the seam. The scan- part is built by the bold nodes in Figure 3, which are somewhat not smooth. After a long piece gliding along the seam, it also turns downhill to the SPaa, by a turn near 106 of the ring angle. A puta- tive CI point is given in Table 1. The quasi-NT of the central branch of VRIlow starts as a valley line uphill. Then it forms a repeated circular pathway beingmore or less inside the outer branches. At the end of the valley, it meets the sec- ond VRI point, VRIup. The character of the upper VRIup point is also of rpVRI. The central branch comes uphill as a valley and goes further uphill to the CI seam as a ridge. If one uses the plus sign in Eq. (1) for the funnel search then the calculation scans along a seam going up- hill in energy. (A minus sign would repel nodes from the summit.) Thus in this case the quasi-NT scan goes off from the usually searched minimum energy conical intersection. In the case of the seam in Figure 3 we use the other method: we go along A g direc- tion in Eq. (1) but always use the sign ??forward??, into the same ??halfspace??. It is the usual strategy to follow a quasi-NT. The ? sign can then be used to decide to go uphill or downhill on the PES. Using this and sliding along the seam, the method glides a little along the seam, but then it jumps over the seam and goes down the other side to the SPca. (The piece of the NT is not shown in the picture.) One side branch from the upper VRIup also returns as a ridge line into the valley region as a circular curve. After some circles over as well as below the central branch, it breaks out and goes down to SPaa. The other side branch turns up as a ridge and goes directly any- where into the PES mountains (not shown further). We try to explain first the behaviour of the outer branches of VRIlow, as well as the central branch of VRIup: they suffer under a quick increase of the jAgj value. The two first eigenvalues of the Hessian from an output of GamessUS at the crossing of the central branch with the seam are: k1 ? ?1:75; k2 ? ?0:051, so to say, that aremoderate values. In themiddle of the seam, there is a point with k1 ? ?34:44; k2 ? ?2:16 in the internal coordinates of GamessUS: in Bohr and radiant units. The absolute values are considerably large values. Usual eigenvalues in these units are % ?0:1. A bizarre observation is that also the largest eigenvalue on the seam is very large: k18 ? 11:44. We do not have an explanation for this number. The corresponding values for the CI point of Table 1 are: k1 ? ?24:40; k2 ? ?3:182 and k18 ? 6:956, where the minimum energy point at the end of the seam has: k1 ? ?14:61; k2 ? ?3:77 and k18 ? 3:55. The connection, using VRIlow as a knot, of the SP of index one, SPaa, and a CI point playing the role of a summit, an SP of index 90 95 100 105 110 115 120 125 c2c1c3 60 80 100 120 140 160 180 di h5 di h2 CI SPaa_5 SPaa_2 VRI_low VRI_up SPca VRIca 90 95 100 105 110 115 120 125 c2c1c3 0 25 50 75 100 125 150 175 di h4 di h3 SPaa_4 SPaa_3 VRI_lowVRI_up SPca VRIca CI CI Figure 3. Singular NT in the region over the transition state, SPaa , along a ring closure of the allyl radical. The PES construction is based on the functional energy CASSCF (3,3)/ 6-31G (d,p). Bullets with a cross of grid lines are the VRI points, VRIlow and VRIup , see text. The ? symbols indicate the CI seam reached by different branches of the singular quasi-NT. The bold, but not smooth curve is a part of the seam, scanned by one quasi-NT. Left: coordinates are the angle between the C-atoms, and the two dihedrals dih2 and -dih5. Right: the same quasi-NT for the ring-angle and the two dihedrals dih3 and -dih4. (For comparison the old known VRIca point is added. It is at the end of the SPca col, see Ref. [20].) Table 1 Coordinates of VRIlow , of VRIup , and an adjacent CI point in the order of the z-matrix (with energy in hartree, Eh). W. Quapp et al. / Chemical Physics Letters 541 (2012) 122?127 125 two, by the outer branches of the quasi-NT of VRIlow shows an index difference by one. It does not violate the index theorem for NTs [29,30] that a regular NT (without a VRI point) cannot connect two stationary points of the same index. We guess that the CI points of the seam have a peaked topology [31], because we find two negative eigenvalues of the Hessian there. A further hint for a peaked topology is the following: differ- ent steepest descent pathways from the putative CI points go into accidental directions. It is a hint that such a point is like a summit. For a sloped CI, the direction would be fixed. The found CI points are not symmetric in the dihedrals. However, the distances r1 and r2 cross near some CI points their symmetry line at r1 ? r2 ? 1:492 8. One should compare other symmetric CIs found in Refs. [32,33]. We guess that the growing string of the special branches of the singular quasi-NT meet the CI seam (by accident) anywhere and then we can jump over it, or we can scan along the CI seam with the quasi-NT. The condition for meeting the seam at all is that the search direction of the current quasi-NT fulfills the equation 1=Norm ? ??vc vcT ? vd vdT ?g? ? r; ?10? being r the selected direction by the actual NT with Eq. (1). 1=Norm is a normalization constant, and vc, vd are the vectors of the branching Tb subspace, see Eqs. (7) and (8). Using the ??forward?? strategy along the seam by a search direction with formula (10), we can descend to a putative minimum energy point on the conical intersection seam at 106? of the ring angle, see Figure 3. The energy difference of the first seam crossing point of the three branches of the singular quasi-NT, and the minimal node, is from ?116:28 Eh to ?116:37 Eh, see Figure 4. At all, one should keep in mind that a quasi-NT slips from one true NT to another, step by step, and it can therefore end up at a different search direction than the NT it started from (though our predictor step length is very small). Figure 3 suggests that the branch connecting VRIup with SPaa does not be- long to the same NT as the branch that connects SPaa and VRIlow. However it emerges to a miracle that the central branch connects the VRIlow with the VRIup. A control calculation at a putative CI point of the seam, see Table 1, was done. A CASCI calculation of the corresponding first excited electronic surface, eA2B, at this configuration confirms the conjecture to be at a CI point: It is based on the natural orbitals extracted from the CAS (3,3) (quartet) set. The energy difference between the two doublet states is 0.5 kcal/mol, it is an excellent value. The description of the three states is state 1: energy ? ?116:328 448 481; S ? 0:5, state 2: energy ? ?116:327 647 388; S ? 0:5, state 3: energy ? ?116:315 182 186; S ? 1:5, and SZ ? 0:5, space sym ? A holds in all three cases. The central branch of the lower VRI point, VRIlow, needs a second explanation, as well as the existence of the upper VRIup crossed by one singular NT. The lower central branch circulates five and a half times around a valley of the PES, up to the upper VRI point. The circular curves implicate the possibility that an NT can be quite convoluted on a hilly PES. In each case the uphill pathway is a valley line going up to a turning point, where the quasi-NT turns and goes parallel downhill. Then this pathway is a ridge line going down back to the initial region. And so on, five and a half rounds up to VRIup. There is a simplified analogy to a circular NT on a simple 2-dimensional surface, see the right panel of Figure 5 which may be compared with Fig. 3.11 of Ref. [30], the so-called dent of a 0 50 100 150 200 nodes 116.45 116.425 116.4 116.375 116.35 116.325 116.3 116.275 En er gy VRI_low VRI_up seam SPaa SPca Figure 4. Energy profile along an outer branch of the singular quasi-NT in VRIlow , along a ring closure of the allyl radical, up to the CI seam, then gliding downhill along the seam, and then going downhill back to SPaa. (The last piece of the quasi- NT is not shown in Fig. 3.) Bullets depict the energy of special points, like VRIlow; VRIup; SPca and SPaa. 2 1 0 1 2 x 1 0 1 2 3 y CI SP VRI VRI niMniM SP SP 2 1 0 1 2 x 1 0 1 2 3 y Figure 5. NTs connect stationary points. Left: A ridge to a CI point at (0,2). Two singular NTs connect the two minimums with the CI point which plays the role of a summit. Right: Compact NTs emerge inside the dent of a thumb on the ridge. A further singular NT emerges with two VRIs on the y-axis (bold dashes). 126 W. Quapp et al. / Chemical Physics Letters 541 (2012) 122?127 thumb. Maybe that the example can explain the behaviour of the circular pieces of the NT in Figure 3. The formula of a 2-dimen- sional toy surface on the left hand side is sl?x; y? ? x2 ?x2 ? 2? 1:5 y? ? y2 ?y? 4?2 ? 25 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ? 3 ?y? 2?2 q ?11? and additionally, for the right panel a dent is pressed into the ridge on the y-axis: sr?x; y? ? sl?x; y? ? 25 exp??3:48 x2 ? 4 ?y? 1:093?2?: ?12? The level lines of the PES are thin lines, where families of NTs are represented by bolder lines. In the left panel, a ??usual?? surface is drawn. Here two singular NTs of the mixed type connect the two lower minimums with the CI point at the top crossing one of the VRI points, and the other branches connect different SPs. On the right panel, an additional dent is pressed into the ridge which leads from the lower SP to the CI point. Now the VRIs are moved a little, and new compact NTs emerge. The dent on the slope of the ridge surface is filled by a pair of families of compact NTs, and a new singular NT (the dashes on the y-axis) divides the families. The singular NT crosses a lower VRI point at % ?0:0;0:2?, and a further, upper VRI point at % ?0:0;1:7?. Both VRIs are rpVRIs. The inner, circular NTs inside the dashed, singular NT do not cross stationary points or VRIs. They are compact curves. Note, the molecular case on the PES in Figure 3 may be a little more complicated. There are 18 dimensions for the PES. The central branch of the singular NT is a spiral line which cannot be in a 2-dimensional plane. Note further that the CI points first found must not be a minimum, or an SP on the high-dimensional seam, compare the Refs. [31,34]. 3. Conclusion Up to date, the calculation of CI points is a difficult job, see ref- erences [4?6] and further references therein. We have ??simply?? used the numerical following along quasi-NTs on the ground adia- batic PES and this leads to peaked CI points quite in analogy to the finding of SPs of index 2 [8,9]. The coordinates of a pututive CI point on the allyl radical PES are given in Table 1. We think that we did not find the exact CI, however the point is assumed to be near the apex of the corresponding conoid. In Figure 3 we depict the projec- tion of quasi-NTs into 2-dimensional planes. Even though quasi- NTs are curves in the full-dimensional configuration space, we assume that we can use their projections for four different, but important dihedrals to generate an imagination of the PES. NTs characterize a skeleton of valley- and ridge-lines. They bifurcate where valleys or ridges of the PES bifurcate. In the case discussed here, we find that a long valley exists on the ridge on the allyl radical PES (without an intermediate minimum), if one leaves the SPaa uphill in a corresponding ridge direction. The lower entrance into the valley goes through a VRI point. The upper departure gate is also a VRI point like in Figure 5. On the high-dimensional PES, the pathway of the central branch leaves the valley and later crosses the CI seam, as well as the two lower side branches. At the top near the former valley, a long CI seam is placed. The new avenue to find the structures is the Newton trajectory method. Acknowledgments Financial support from the Spanish Ministerio de Econom?a y Competitividad, DGI project CTQ2011-22505 and, in part from the Generalitat de Catalunya projects 2009SGR-1472 is fully acknowledged. M.C. gratefully thanks the Ministerio de Econom?a y Competitividad for a predoctoral fellowship. We thank two referees for many helpful suggestions and comments. References [1] D. 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Quapp et al. / Chemical Physics Letters 541 (2012) 122?127 127 Locating transition states on potential energy surfaces by the gentlest ascent dynamics Josep Maria Bofill a,b,?, Wolfgang Quapp c, Marc Caballero b,d aDepartament de Qu?mica Org?nica, Universitat de Barcelona, Mart? i Franqu?s, 1, 08028 Barcelona, Spain b Institut de Qu?mica Te?rica i Computacional, Universitat de Barcelona, (IQTCUB), Mart? i Franqu?s, 1, 08028 Barcelona, Spain cMathematisches Institut, Universit?t Leipzig, PF 100920, D-04009 Leipzig, Germany dDepartament de Qu?mica F?sica, Universitat de Barcelona, Mart? i Franqu?s, 1, 08028 Barcelona, Spain a r t i c l e i n f o Article history: Received 13 June 2013 In final form 26 July 2013 Available online 2 August 2013 a b s t r a c t The system of ordinary differential equations for the method of the gentlest ascent dynamics (GAD) has been derived which was previously proposed [W. E and X. Zhou, Nonlinearity 24, 1831 (2011)]. For this purpose we use diverse projection operators to a given initial direction. Using simple examples we explain the two possibilities of a GAD curve: it can directly find the transition state by a gentlest ascent, or it can go the roundabout way over a turning point and then find the transition state going downhill along its ridge. An outlook to generalised formulas for higher order saddle-points is added. ? 2013 Elsevier B.V. All rights reserved. 1. Introduction The concepts of the potential energy surface (PES) [1,2] and of the chemical reaction path are the basis for the theories of chemi- cal dynamics. The PES is a continuous function with respect to the coordinates of the nuclei. It is an N-dimensional hypersurface if N = 3n and n is the number of atoms. It must have continuous derivatives up to a certain order. The PES can be seen as formally divided in catchments associ- ated with local minima [1,3]. The first order saddle points or tran- sition states (TSs) are located at the deepest points of the boundary of the basins. Two neighbouring minima of the PES can be con- nected through a TS via a continuous curve in the N-dimensional coordinate space. The curve characterises a reaction path. One can define many types of curves satisfying the above requirement. The reaction path model widely used is the steepest descent (SD). There exist a large number of proposed methods that in princi- ple reach a TS when the minimums associated to the reactant and product are known. See Ref. [4] and references therein. There are also methods that find the TS when only one minimum is known. In this case, the problem is much more cumbersome because the initial data are just the geometry coordinates of the minimum, however, the direction of the search is open. As in the first case many algorithms have been developed for this type of problem [4]. A great number of these algorithms are based in a generalisa- tion of the Levenberg?Marquardt method [5?7] that basically con- sists of a modification of the Hessian matrix to achieve both, first the correct spectra of the desired Hessian at the stationary point, and second to control the length of the displacement during the location process. The first proposed algorithm within this philoso- phy is due to Scheraga [8] and from than up to now the list is very large [9?20]. None of the methods are foolproof, each of them has some problems. Recently E and Zhou [20] have proposed an ap- proach called the ?gentlest ascent dynamics? method. This method can be seen as a new reformulation of the method proposed some time ago by Smith [12,13] under the name ?image function?. The method is based on the generation of an image function that is a function which has its minima at exactly the points where the ori- ginal PES has its TSs and moreover by an application of a minimum search algorithm to this image function. The converged point should correspond to a TS in the actual PES. Helgaker [14] modified the algorithm by the trust radius technique. Sun and Ruedenberg [21] analysed the method concluding that image functions do not exist for general PES so that a plain minimum search is inappropri- ate for them. A nonconservative field gradient of the image func- tion exists. The global structure of the image gradient fields is considerably more complex than that of gradient fields of the ori- ginal function. However, the image gradient fields appear to have considerably larger catchment basins around TSs. Besal? and Bofill [18] showed that the Smith algorithm is a special type of the Levenberg?Marquardt method. In this Letter we show the connection between the Smith meth- od [12,13] and that described by E and Zhou [20] to find TSs, and additionally, the mathematical basis of this algorithm is discussed. Finally, some examples are reported. 0009-2614/$ - see front matter ? 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2013.07.074 ? Corresponding author at: Departament de Qu?mica Org?nica, Universitat de Barcelona, Mart? i Franqu?s, 1, 08028 Barcelona, Spain. Fax: +34 933397878. E-mail address: jmbofill@ub.edu (J.M. Bofill). Chemical Physics Letters 583 (2013) 203?208 Contents lists available at ScienceDirect Chemical Physics Letters journal homepage: www.elsevier .com/ locate /cplet t 2. Background of the method Let us denote by V(q) the PES function and by qT = (q1, . . ., qN) the coordinates. The dimension of the q vector is N. The superscript T means transposition. At every interesting point q the PES func- tion admits a local gradient vector, g(q) =rqV(q), and a Hessian matrix,H(q) =rqrqTV(q). The family of image functions of V(q), la- belled by W(q), is defined by the differential equation [21] f?q? ? Uvg?q? ? I? 2 v?q?vT?q? vT?q?v?q?   g?q? ?1? where f(q) is the image gradient vector, Uv is the Householder orthogonal matrix constructed by an arbitrary vector v(q) being in principle a function of q, and I is the unit matrix. The Householder orthogonal matrix is a reflection at v(q). It has the property that Uv = UvT, and it is the result of the difference between the projectors (I ? Pv) and Pv, because it holds trivially Uv = I ? 2Pv = (I ? Pv) ? Pv, being Pv the projector that projects into the subspace spanned by the v-vector [22]. If the derivatives of v(q) with respect to q are non- vanishing, the image Hessian matrix, F(q) = UvH(q), is not obtained by the differentiation of f(q). Taking into account Eq. (1), this differ- entiation results in rqfT?q? ? rq?gT?q?Uv? ? FT?q? ? gT?q?rqUv ? FT?q? ? 2gT?q?rq v?q?vT?q? vT?q?v?q?   : ?2? The term in brackets is usually not zero and not symmetric, and this non-symmetry is due to the effect of the differentiation on the Pv projector. In other words, rqfT?q?   ij ? rqfT?q?   ji ? rqfT?q?   ij ? rqfT?q?  T ij ?0 i?j: ?3? The inequality of Eq. (3) implies that the image gradient field defined by Eq. (1) is not integrable to an ?image PES? W(q). More explicitly, W?q1? ?W?q0?? Z t1 t0 fT?q??dq=dt?dt ?4? where dq/dt is the tangent of an arbitrary curve joining the points q0 = q(t0) and q1 = q(t1). Due to Eq. (3), this gradient field vector should be considered as a nonconservative force field. From this fact it follows an image of the PES function does, in general, not exist [21,23]. From Eq. (3) it is easy to see that at the stationary points, where g(q) = 0, the inequality is transformed to an equality if the v-vector is an eigenvector of the Hessian matrix. Note that if {h, v/(vTv)1/2} is an eigenpair of the H(q) matrix, then F(q) = UvH(q) = (I ? 2vvT/ (vTv)) H(q) = H(q) - 2vvT/(vTv) h = H(q) - h 2vvT/(vTv) = H(q) - H(q) 2vvT/(vTv) = H(q) (I ? 2vvT/(vTv)) = H(q)Uv = FT(q). As pointed out by Sun and Ruedenberg [21], the image functions do exist until the second order in the vicinity of its stationary points for any PES taking v as an eigenvector of the Hessian matrix. Due to this fact the SD curves of the quadratic image function are approximations to the gradient image curves of W(q) being the image potential of V(q). With the previous analysis of the general nonexistence of an image PES, we can take the image gradient field given in Eq. (1) to define the field of SD curves as, dq dt ? ?f?q? ? ?Uvg?q? ? ???I? Pv? ? Pv ?g?q? ?5? where t is the parameter that characterises the SD curve, q(t). If Eq. (5) is multiplied consecutively from the left by the set of (N-1) linear independent orthogonal vectors to the v-vector, we see that it cor- responds to a curve which is energy descending along this set of directions on the actual PES, whereas is ascending on the v-vector direction. This property makes the set of curves defined in Eq. (5) suitable for the location of a TS from a minimum. This observation is supported by the fact that Eq. (5) can be rewritten as dq dt ? ???I? Pv? ? Pv ?g?q? ? ??I? Pv?g?q? ? Pvg?q? ? ??I? Pv?g?q? ? l v ?vTv?1=2 ?6? being l = vTg(q)/(vTv)1/2, where the definition of Pv has been used. Eq. (6) is the basic equation of the string method proposed for the location of reaction paths and TSs [24]. The v-vector in this method is the current tangent of the path. Because we are interested to find TSs from minimums of the PES we can use the nonconservative property of the gradient image field to modify the v-vector, during the location process. For this purpose, we first consider that at the minimum, as well as at the TS, the last term of the right hand side part of Eq. (2) is equal zero due to g(q) = 0. Second, at the TS, the Hessian matrix, H(q), possesses only one eigenpair with negative eigenvalue. The associated Raygleigh-Ritz quotient of this eigenpair with a negative eigenvalue is the lowest that the Hessian matrix can achieve at this point being equal to the corresponding eigenvalue [25]. The Raygleigh-Ritz quotient for a given vector v and matrix H is defined as, k(v) = vTHv/(vTv). The structure of this eigenvector is unknown. To find the TS one should, however, ensure that during the research process the path walks through the PES (given until second order) such that the character of the surface becomes closer to a first order saddle point. Taking into account these two consid- erations we transform Eq. (3) imposing that g(q) = 0 at each point of the search and multiplying the resulting equation from the left by (I ? Pv) and from the right by Pv, 1 2 ?I? Pv? rqf T?q?   ? rqfT?q?  T  Pv g?q??0 ? 12 ?I? Pv??H?q?Uv ? UvH?q??Pv ? ??I? Pv?H?q?Pv: ?7? The effect of this multiplication by the projectors, (I-Pv) and Pv, is that the resulting Eq. (7), multiplied from the right by the v-vec- tor, is the gradient of the Rayleigh?Ritz quotient with respect to this vector. If the v-vector is an eigenvector of the H(q) matrix then the right hand side part of Eq. (7) is equal zero because every eigenvector extremises the corresponding Rayleigh?Ritz quotient. We will denote the Rayleigh?Ritz quotient by kq(v) to indicate its dependence on q through the Hessian matrix. If a v-vector makes the gradient of the Rayleigh?Ritz quotient equal zero then this vec- tor is an eigenvector of the H(q) matrix and the value of the Ray- leigh?Ritz quotient coincides with the corresponding eigenvalue of the H(q) matrix. These properties suggest that a v-vector can be changed following the SD direction of the Rayleigh?Ritz quo- tient gradient with respect to v, dv dt ? ? vTv 2 rvkq?v? ? ??I? Pv?H?q?Pvv ?8? Eq. (8) is also a function of q through the Hessian matrix. The Rayleigh?Ritz quotient of the new v-vector obtained from, v? v + dv/dt Dt, will be lower with respect to the previous one and, in addition, Eq. (5) will give us a new energy ascent direction and a set of orthogonal N-1 energy descent directions. Eq. (8) searches for either the lowest positive or the single negative Ray- leigh?Ritz quotient if it exists, whereas Eq. (5) determines points on the PES along the action of an increase of the energy in the v- vector, and a decrease along the set of orthogonal directions to this vector. The specific action of Eq. (5) defines the type of points of the 204 J.M. Bofill et al. / Chemical Physics Letters 583 (2013) 203?208 PES so that its structure until the second order on q resembles a first order saddle point. In turn, Eq. (8) finds v-vectors with lowest or possibly negative values of the Rayleigh?Ritz quotients of the corresponding Hessian matrix. These two coupled actions on the Eq. (5) and Eq. (8) describe a gradually or gentlest form to find the TS located on the boundary of the basin which contains the start minimum. Eqs. (5) and (8) are coupled since both depend on the v-vector explicitly on q through the gradient vector and the Hessian matrix respectively. The integration of Eqs. (5) and (8) implies the solution of the differential equation dq dt dv dt ! ? ? I? 2 vvT vTv   g?q? I? vvT vTv   H?q?v 0 B @ 1 C A ?9:a? with the initial conditions q?t0? v?t0?   ? q0g?q0?   ?9:b? where q0 is a slightly distorted point near the minimum. The gradi- ent at point q0 is selected automatically as the initial v-vector. Eqs. (9) are the basic expressions of the algorithm proposed by E and Zhou [20] which are reviewed in this study. Note that the system (9) is a system of differential equations. Corresponding to the initial conditions (9b) there emerges a full family of solution curves passing every point of a given region. This is in contrast to the long known gradient extremals (GE). They also realize a shallowest ascent idea; however, they are special solution curves of the equation Hg = kg, where k is an eigenvalue of H. Thus the GEs pass only points where the gradient itself is an eigenvector of H [2,26?31]. GEs do not cover a region. 3. Behaviour and analysis The set of Eqs. (9) has been integrated using the explicit Runge? Kutta method of order 8(5,3) [32]. We have used two-dimensional PES models to analyse the behaviour of the gentlest ascent path. The first PES model used for this purpose is the Wolfe-Quapp PES [33,34]. The equation of this surface model is V?x; y? ? x4 ? y4 ? 2x2 ? 4y2 ? xy? 0:3x? 0:1y ?10? We look for the gentlest ascent dynamics if we start from the min- imum located at the point (1.124,?1.486) with energy in arbitrary units ?6.369. We take the corresponding gradient vector there as a starting point (1.2,?1.5) and as the initial v-vector, according to Eq. (9.b). The curve line is depicted in Figure 1. It starts from the min- imum. The curve ends at the TS located in (0.093,0.174) with en- ergy ?0.644. It follows the valley joining these two stationary points. Figure 1 also shows the behaviour of the v-vector. Initially the v-vector has the same direction to the tangent of the curve but along the initial sub-arc of the curve the vector points to the direction where the energy increases very fast, and finally the vec- tor corrects its direction towards the direction of the transition vec- tor. The transition vector is the eigenvector of the Hessian matrix at the TS with the negative eigenvalue. At the end point, namely the TS, the v-vector is the vector which produces the lowest value of the Rayleigh?Ritz quotient of the Hessian matrix. It is the eigen- value of the transition vector. This behaviour conforms to that one which is explained in the previous section. The curve is deter- mined by the condition that the v-vector minimises the kq(v) within the limits imposed by the Mini-Max eigenvalue theorem [25]. A second curve is computed on the same PES to get the same TS revealing new features of this type of GAD curves. In this case the curve starts near the minimum located at the point (?1.174,1.477), with energy ?6.762. The behavior of this curve is drawn in Figure 2. The path follows the valley in the direction where kq(v) takes the possibly lowest value. The path arrives a highest energy value at a turning point and starts to descend along a ridge, and it ends again at the TS located at (0.093,0.174) with energy ?0.644. As shown in Figure 2, on a large sub-arc from the minimum the v-vector has the same direction as the tangent of the curve. Of course, the large arc with the turning point at its highest energy does not represent the ?gentlest ascent?, seen from the TS and the valley nearby. So, the name of the method, GAD, is a suggestive name but it is not rea- lised in every case. However, note that the TS is in a side valley, but the method finds the TS, in general. At the point (1.849,0.635) the curve achieves the highest energy value and starts its descents. This point is a turning point (TP) of the curve, where the next equation is satisfied Figure 1. Behaviour of the GAD curve on the Wolfe-Quapp PES model [33,34]. The curve starts at point (1.2,?1.5) which is near to the minimum located at the point (1.124,?1.486). The curve evolution ends at the TS located at point (0.093,0.174). The set of v-vectors generated during the search is indicated by the set of bold arrows. Figure 2. Behaviour of the GAD curve on the Wolfe-Quapp PES model [33,34]. The curve starts at the point near the minimum located at (?1.174,1.477). The curve evolution ends at the TS located at the point (0.093,0.174). At the point (1.849,0.635) the curve achieves the highest energy so this point is the TP. The curve leaves the valley where the starting minimum is located at the point (1.842,0.768). This point is a VRT point. The set of v-vectors generated during the search are indicated by the set of bold arrows. J.M. Bofill et al. / Chemical Physics Letters 583 (2013) 203?208 205 dV dt ? g T dq dt ? ?g TUvg ? ?gT?I? Pv?g? gTPvg ? 0 ?11? where Eq. (5) has been used. In general a TP appears on the gentlest ascent path when the square of the gradient norm in the subspace spanned by the v-vector is equal to the square of the gradient norm in the complementary subspace spanned by the set of linear inde- pendent vectors orthogonal to the v-vector. This result is identical to the equality gTg/2 = gTPvg, and with the definition of Pv we can rewritten it as, 1/21/2 = gTv/(gTgvTv)1/2, concluding that the GAD curve has a TP at the point where the v-vector and the gradient of the actual PES form an angle of p/4 radians. The curve leaves the valley where the starting minimum is located, at the point TP (1.842,0.768) and it enters to a ridge region. Because this point sat- isfies the next equations gTA?q?g gTg ? 0 ?12:a? A?q?g ?gTg?1=2 ?0 ?12:b? where A(q) is the adjoint matrix of the Hessian H(q) at the q point, we conclude that this point is a valley?ridge transition point (VRT) rather than a valley?ridge inflexion point (VRI) [35-37]. In a VRI point both equations are equal zero. From the minimum to the VRT point the curve shows positive values of the Rayleigh quotient, gTAg/(gTg), whereas they are negative from the VRT point to the TS. This quotient is taken as a convexity criterion of a PES region where a point is located. Positivity of the quotient means that the point is in a valley, otherwise it is on a ridge [38]. The result supports the above comment, namely, that one sub-arc of the curve is located in the deep valley from the minimum to the VRT point. At this point the curve turns into the direction of the ridge following the lowest value of kq(v). But this corresponds to the v-vector orthogonal to both, the ascent of the valley and the descending ridge directions. On the sub-arc located on the ridge at each point the gradient vector increases its orthogonality with respect to the current v-vector. The curve evolves along a direction of decreasing energy. We can write dV/dt = gTdq/dt = ?gTUvg = 2gTg((gTv)2/(gTg vTv) ? 1/2) % ?gTg < 0 since vTg % 0. In general the curve evolves so that the energy de- creases at each step if the gradient vector g and the v-vector form an angle within the open domains (p/4,3p/4) or (5p/4,7p/4) radians. In general we can mark the following features of a GAD curve. When along the sub-arc of the gentlest ascent curve its gradient vectors define an angle with the corresponding v-vectors within the open domains (p/4,3p/4) or (5p/4,7p/4) radians, then the sub-arc shows an energy decrease. If the angle is within the open domains (3p/4,5p/4) or (7p/4,p/4) radians it shows an energy growth. When the angle is equal to p/4, 3p/4, 5p/4, or 7p/4 radians the curve is stationary with respect to the variation of the energy, the TP situation. The same type of calculations with the PES proposed by Neria et al. [39] and modified by Hirsch and Quapp [38] confirms the same facts. The equation of this surface model is V?x; y? ? 0:06?x2 ? y2?2 ? xy? 9exp???x? 3?2 ? y2? ? 9exp???x? 3?2 ? y2?: ?13? The central point is the TS located at (0,0) with energy ?0.002. The minima are located at (2.71,?0.15) and (?2.71,0.15) with en- ergy ?5.24. The surface has gorges around the points (2,?2) and (?2,2) [38]. We start the gentlest ascent path at the point (2.6,?0.2) near to a minimum. Its evolution goes over the gorge, it leaves the valley where the minimum is located and it enters the ridge, but then it enters again to the valley region of the min- imum. The ascending energy behaviour ends at the point (0.871,?1.428) being a TP. On the sub-arc between the points TP and TS a VRT point exists at (0.002,?1.28) corresponding to the descending energy. At this point the GAD curve enters a ridge that touches the TS. Figure 3 depicts this behaviour, the evolution of the v-vector is also shown. On the M?ller?Brown PES [40], the GAD method exhausts its possibilities. The global minimum is located at a very deep valley. The coordinates are (?0.56,1.44). We use two different starting points. The first starting point of the GAD method is at the point (?0.54,1.4) near to the minimum. The GAD trajectory leads to the next TS, see Figure 4a where it is the dotted line. It goes over a TP but that is normal. In Figure 4a we show the behavior of the gradient extremal curve (GE) [2,26?31] for comparison. The GEs are the bold dashed lines. A part of the GAD on the ridge coincides with the GE. However, the coincidence is for different reasons. For the GAD case, in the ridge region the v-vector is orthogonal to the gradient and due to this fact the trajectory shows a decrease in en- ergy. In other words, dV/dt is reduced, it is, dV/dt = ?gTg < 0 since vTg = 0. Recall that the v-vector evolves in this process to reach the eigenvector of the Hessian matrix of lowest eigenvalue. Analyz- ing the GE in the region of coincidence, the gradient at each point of this curve is by construction an eigenvector of the Hessian cor- responding to a positive eigenvalue. The eigenvalue is positive be- cause its eigenvector comes from the evolution of the Hessian at the TS labeled as SP1, in direction orthogonal to the col. For the GE the variation energy is given by, dV/dt = (gTg)1/2eg, being eg a component of the GE tangent equation [28,31] and depending of its sign, the curve increases or decreases in energy. One can con- clude that in the region of coincidence the v-vector is orthogonal to the gradient vector, an eigenvector of the Hessian matrix. Thus, the v-vector is in this region an eigenvector of the Hessian. For a second trajectory we use the point (?0.58,1.427) for the start, also near the minimum, however, we get a totally different result, see Figure 4b. At the beginning, the GAD trajectory, a contin- uous line, goes along the deep valley of the minimum, uphill like the GE, which in Figure 4b is again represented by dashed lines, see also Figure 2 of Ref. [28], or Figure 5 of Ref. [41]. The initial evolution of GAD goes in the direction where the gradient and Figure 3. Behaviour of the GAD curve on the Neria?Fischer?Karplus PES model [39] modified according to that proposed by Hirsch and Quapp [38]. The curve starts at point (2.6,?0.2) near the minimum located at (2.71,?0.15). The curve evolution ends at the TS located at point (0,0). At point (0.871,?1.428) the curve achieves the highest energy value. The point is the TP. The curve leaves the valley where the starting minimum is located at the point (0.002,?1.28). This point is a VRT point. The set of v-vectors generated during the search is indicated by the set of bold arrows. 206 J.M. Bofill et al. / Chemical Physics Letters 583 (2013) 203?208 the v-vector form an angle lower than p/4 radians. In fact it coin- cides with the GE, which in turn implies that the v-vector is paral- lel to the gradient. In the region where both curves coincide the increasing energy along the GAD curve is dV/dt = (gTv)2/vTv > 0. The behavior ends at a TP which emerges near (?2.8,0) in a region which is far away from the border line of the searched valley?ridge transition. In Figure 14 of Ref. [38] the borders between valleys and ridges of the M?ller?Brown PES are shown. After the TP the trajec- tory turns into the left large side valley of the M?ller?Brown sur- face. There is no stationary point, so GAD cannot find any one. The region is a kind of a trap, a ?dead? valley. The GE passes the re- gion and leaves it to the mountains at the left hand side, but the GAD trajectory goes on and back, from one TP to the next, and again back, and only by an accident, it later leaves the region. How- ever, then it immediately leads to the TS2 of the M?ller?Brown surface. It seems to be accidentally which TS is found. After the ?chaotic? evolution, the GAD trajectory finally catches a TS. 4. A different view of the theory and generalisation Mathematically the above method can be reformulated as Max v Min w fU?v;w?jv ?vTq;w?ZTq and ZTv? ?z1j . . . jzN?1?Tv?0N?1g where the v-vector satisfiesMin v ?vTH?q?v vTv 8 < : ?14? where U(v,w) is the PES function, V(q), expressed as a function of (v,w) coordinates being the w vector of dimension N-1. Finally, the Z matrix is formed by a set of N-1 linear independent vectors orthogonal to the v-vector. A suggested solution of the conditions is formulated in Eq. (9). As noted, the method is appropriate for a first-order saddle point search, and can be easily generalised to the search of higher order saddle-points. The importance of second order saddle-points is known [42,43]. If I is the order of the saddle- point, where I 6 N, then the set of Eqs. (9) takes the form dq dt dv1 dt .. . dvI dt 0 B B B B B @ 1 C C C C C A ? ? I? 2 X I i?1 Pvi ! g?q? I? X I i?1 Pvi ! H?q?v1 ? X I i?1i?1 Pvi ! H?q?v1 .. . I? X I i?1 Pvi ! H?q?vI ? X I i?1i?I Pvi ! H?q?vI 0 B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C A ?15:a? with the initial conditions q?t0? v1?t0? .. . vI?t0? 0 B B B B @ 1 C C C C A ? q0 g?q0? .. . pI0 0 B B B B @ 1 C C C C A ?15:b? where Pvi = viviT/(viTvi) for i = 1, ..., I, and {p0i}i=2I is a set of I-1 orthog- onal vectors which are orthogonal to the vector g(q0). If I = N then the identity matrix I in Eq. (15.a) should be substituted by the res- olution of identity I ? X N i?1 Pvi : ?16? The variation energy through this curve is given by dV dt ? ?g T dq dt ? ?g T?I? 2 X I i?1 Pvi ?g ? 2 X I i?1 ?gTvi?2 gTgvTi vi ? 12 ! gTg ?17? where the first N components of Eq. (15.a) have been used. Eq. (17) tells us that when the sum of the square of the components of the projected gradient vector in the subspace spanned by the set of vi- vectors is higher than 1/2 then the curve evolves in the direction of increasing the energy. If it is lower than 1/2 then evolves in the direction of a decrease of energy. When the sum is equal to 1/2 the curve is at a TP. 5. Conclusion In this study is reviewed the GAD method for the location of the saddle points which surround a given minimum. It behaves like a growing string method. The integration of the curve can be done by any method suitable to integrate a system of first order ordinary differential equations. The GAD constructs a curve supported by a Min SP1 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 x y Min SP2 3 2 1 0 0 1 2 3 x y ? ? Figure 4. (a) Behaviour of the GAD trajectory, dotted line, in the M?ller?Brown PES model [40]. The curve starts at the point (?0.54,1.4) near to the minimum located at (?0.56,1.44). It leads to the TS labeled as SP1. The GE is represented by fat, dashed lines, for comparison. (b) ?Chaotic? GAD trajectory, continuous lines, starting at point (?0.58,1.427). The curve goes through the large side valley at the left hand side by repeated arcs. At last it leaves the valley at an accidental point and finds the TS labeled as SP2, see text. The GE is represented by fat, dashed lines. J.M. Bofill et al. / Chemical Physics Letters 583 (2013) 203?208 207 set of generated vectors. The numerical examples reported also show that GAD has a large convergence region. 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