Consequences of electron correlation for XPS binding energies: Representative case for C(1s) and O(1s) XPS of CO

In this paper, we present a study of the signs and the magnitudes of the errors of theoretical binding energies, BE's, of the C(1s) and O(1s) core-levels compared to BE's measured in X-Ray photoemission, XPS, experiments. In particular, we explain the unexpected sign of the error of the Hartree-Fock C(1s) BE, which is larger than experiment, in terms of correlation effects due to the near degeneracy of the CO(1π) and CO(2π) levels and show how taking this correlation into account leads to rather accurate predicted BE's. We separate the initial state contributions of this near degeneracy, present for the ground state wavefunction, from the final state near degeneracy effects, present for the hole state wavefunctions. Thus, we are able to establish the importance for the core-level BE's of initial state charge redistribution due to the π near-degeneracy. While the results for CO are interesting in their own right, we also consider whether our conclusions for CO are relevant for the analysis of XPS spectra of a wider range of molecules.


I. INTRODUCTION
Wavefunctions, WF's, have been used extensively to understand, predict, and assign the peaks in X-Ray photoelectron spectra, XPS, see Ref. 1 for a review of representative applications of WF theory to XPS and for a discussion of the theoretical formulations. It is possible to obtain reasonably accurate XPS binding energies, BE's, for core levels with the SCF method where the BE's are denoted as BE( SCF). This method involves calculating the WF's and total energies for the initial state configuration and for the final, core-hole, state configuration and taking the BE( SCF) as the di↵erence of the variational total energies of the initial and final states. The very large majority of WF calculations of core-level BE's are done using Hartree-Fock, HF, theory. For compounds containing light atoms, C, N, O, and F, the errors of HF BE( SCF) are typically of order 0.5-1.0 eV. [2][3][4] Similar accuracy can be obtained with BE( SCF) methods based on density functional theory, DFT; 2-7 indeed with an appropriate choice of density functional, the DFT BE( SCF) are somewhat more accurate than the HF BE( SCF). One expects the HF BE( SCF) to be smaller than the XPS experiment and this is normally the case; see, for example, Refs. 2-4. The expectation that the HF BE( SCF) will be smaller than experiment follows from a logic originally proposed by Mulliken. 8 The final, ionic state has fewer electrons, hence fewer electron pairs, than the initial state and, hence, one expects the correlation energy, that is the error of the HF energies, to be smaller in the final state than in the initial state. It is easy to see, given that the error of the higher energy state, the final state with the core hole, is smaller than the error of the initial state, that the calculated HF BE( SCF) will be smaller than that measured in the XPS experiment.
However, there are exceptions to this simple rule for the order of the error of HF SCF BE's and there are cases where BE( SCF) is larger than experiment. This means that the correlation energy of the hole-state, with one fewer electron, is larger than the correlation error of the initial state. One of the earliest exceptions found where the HF BE( SCF) is greater than experiment was for the lowest 2 S states of Ne + and Ar + and their isoelectronic ions. 9 For Ar, for example, the lowest 2 S state has the configuration . . . 3s 1 3p 6 and there is a nearly degenerate configuration with a 2 S multiplet that is . . . 3s 2 3p 4 3d 1 . 10 Since this configuration involves filling the 3s hole with a 3p electron and promoting another 3p electron to the higher energy, but still bound, 3d level, it has been described as a frustrated Auger configuration, FAC. 11 The static correlation introduced by mixing the two configurations is present only for the 3s-hole configuration but not for the initial configuration. When this static correlation e↵ect is taken into account, the SCF BE is now, correctly, smaller than the XPS experiment. 10 Furthermore, this static correlation is general and it also leads to corrections for the 3s-hole XPS of 3d transition metal cations. 11,12 However, for the special cases of Ne and Ar, the static correlation involves the same shell as the ionized electron. For CO, considered in this paper, the static correlation does not involve the same shell as the shell ionized in the XPS measurement.
For CO, the O(1s) BE( SCF) obtained with HF wavefunctions is, as expected, lower than the XPS BE by ⇠1 eV while the C(1s) BE( SCF) is larger than the XPS BE, also by ⇠1 eV, opposite to the expected order; see Ref. 13 and results in this paper. For CO, the static correlation does not involve the deep core, C(1s) and O(1s), shells but involves the higher lying valence levels, in particular the 1⇡ and 2⇡ shells where the 1⇡ is fully occupied and the 2⇡ is empty in the HF configuration 1 2 2 2 3 2 4 2 5 2 1⇡ 4 2⇡ 0 . (1) In order to determine these many-body e↵ects, we use Complete Active Space, CAS, 14 16 Furthermore, the use of multi-configuration WF's for the core-hole states may allow a di↵erent relaxation, or screening, in the hole state WF's than occurs for the HF wavefunctions and this will lead to di↵erent final state contributions to the BE's. In other words, while for the HF WF, the screening must be entirely through changes in the orbital character, for the multi-configuration WF's, there is an additional screening through changes in the mixing in the WF of the configurations where 1⇡ electrons are moved to the 2⇡ shell. Our objectives in this work are twofold. First, we wish to examine whether the inclusion of static, near degeneracy, electron many-body e↵ects leads to having both O(1s) and C(1s) SCF BE's smaller than the measured XPS BE's and that the error of the calculated BE's is reduced and is below 1 eV. The second objective is to determine the origin of the many-body changes in the BE's and to relate them to features of the many-body WF's. This analysis will include separation of the changes in the BE's due to many-body e↵ects into the initial state e↵ects that arise from the changes in the ground state WF and final state e↵ects due to changes in the many-body hole state WF's. To our knowledge, this type of analysis of the many-body contributions to core-level BE's has not been made before. The SCF and CASSCF WF's that we use directly allow the role of static correlation in the CO ⇡ space to be investigated; furthermore, they allow the separation of initial and final state contributions. 1 In our discussion above and in the remainder of the paper, we have used concepts and terminologies that merit further discussion in order to avoid ambiguity or uncertainty concerning our usage of them. In particular we have described a classification of electron correlation e↵ects as being static or dynamic and we have described a division of the contributions to core-level BE's in terms of initial and final state e↵ects. The classification of static as contrasted to dynamic electron correlation [17][18][19][20][21] is commonly used to distinguish the correlation e↵ects that arise from near degeneracy of di↵erent configurations from the dynamical correlation e↵ects that are required to obtain very precise energies. Although, there may be an uncertainty in the division of particular terms to be included in each classification, 18,19,21 our usage is very clear in that we are concerned with the correlation that arises from including the 1⇡ and 2⇡ orbitals of CO in the active space and, hence, the included correlation e↵ects are non-dynamical. In particular, we shall examine how this participation is di↵erent for the CO GS and for the C(1s) and O(1s) core-hole states. We are not concerned with finer distinctions of the inclusion of orbitals in the active spaces. The separation of initial and final state contributions to core-level BE's is standardly used in the interpretation of XPS BE's. 1 However, we want to be explicit about our usage of these distinctions and of the related concept of relaxation energies since they are important to relate the core-level BE's to the electronic structure of the material under study. 1 Initial state e↵ects are those which arise essentially from the electrostatic potential at the ionized atom in the WF of the GS. For closed shell HF WF's, the initial state BE's are rigorously given by Koopmans' theorem, 1-3 KT. However, as we show in Sec. II, the definition of initial state e↵ects has a rigorous generalization beyond HF to correlated WF's. In this generalization, one applies an annihilation operator to the GS WF, to obtain what we will call an FO, or Frozen Orbital, WF where no response to the core-hole is allowed and FO BE's arise entirely from the charge distribution of the GS WF. It is important to remark that while only the orbitals are fixed in the FO single configuration HF WF's, in the CAS CI WF's, both the orbitals and the configuration mixing coe cients are fixed in the FO CAS CI WF's. As we note in a later discussion, our use of FO WF's follows that introduced by Aberg 22 for the analysis of XPS satellites. The relaxation energy, E R , is also a rigorously defined quantity that is the di↵erence between the energy of the FO and the fully self-consistent SCF WF for the hole states. 1,2 In this sense, E R is a direct measure of the energetic importance of the response, or relaxation, of the WF orbitals and CI mixing coe cients to the presence of the core-hole; thus, E R is a pure final state e↵ect. 1 The distinction between initial state and final state e↵ects is especially important when one considers shifts of BE's between inequivalent atoms in a system, as between atoms at the surface and in the bulk of a solid, 1 or between di↵erent atoms in di↵erent molecules. [2][3][4] Here, it allows the chemical contributions to the BE shifts to be distinguished from those arising from the response to the presence of the core-hole. In our use in the present paper, we use the FO analysis of initial state e↵ects to decompose the contributions to the changes in the BE's between the HF and the CAS WF's. This decomposition allows us to distinguish the changes in the CAS BE's, which include static correlation e↵ects that arise from the changes in the charge distribution in the initial, GS, state from those that arise from the relaxation. We stress that the final state relaxation changes include changes in the CI mixing of the 2⇡ orbital in the final hole state CAS WF as well as changes in the orbitals. This will be discussed further when we present the results.
In Sec. II, the theoretical methodology and the computational methods used are presented and described in relationship to the scientific issues addressed. In particular, rigorous definitions are given for the separation of initial and final state e↵ects. In Sec. III, the theoretical BEs are presented and compared with the XPS BE's. Properties of the CAS wavefunctions are considered in connection with the shifts of the core-level BE's that arise due to these changes. Finally, in Sec. IV, we summarize our conclusions and stress the magnitude of the changes that may arise from static, near degeneracy, electron correlation e↵ects, and describe the features that indicate that such static correlation e↵ects may be present.

II. THEORETICAL METHODOLOGY AND COMPUTATIONAL DETAILS
The HF and CASSCF wavefunctions use spatial and spin equivalence restrictions and the CO ground state, GS, WF is a pure 1 ⌃ + while the hole-state WF's are pure 2 ⌃ + . For the GS CASSCF, the 4 ⇡ electrons are active and configurations are included where these electrons are distributed over 4, 1⇡ x , 1⇡ y , 2⇡ x , and 2⇡ y , active orbitals; this is denoted as CAS (4,4). For the core hole states, either the 1 , O(1s), or the 2 , C(1s), orbital is added to the active electron space but it is forced to be singly occupied in all configurations in the CAS wavefunction. Although the active spaces are increased for these hole state CAS WF's, we still refer to them as CAS (4,4) where (4,4) now refer only to the ⇡ active spaces. We shall use the 1⇡ and 2⇡ occupation numbers, N(1⇡) and N(2⇡), as a measure of the importance, especially the relative importance, of the CAS static correlation e↵ects for the GS and the core-hole states. For the HF WF's these occupation numbers are N(1⇡) = 4.00 and N(2⇡) = 0.00 for all states; for the CAS(4,4) WF's, the occupation numbers are the CAS natural orbital occupation numbers. 14 For these wavefunctions, we consider BE's for initial state e↵ects only, denoted BE(FO) for frozen orbital BE's, and BE( SCF) where both initial and final state e↵ects are included. 1 The formal mathematical definition of the FO WF for the core ion 22 is that an annihilation operator for the core orbital is applied to the initial, N electron GS WF, to yield the N 1 electron FO WF, (FO), for the core hole In Eq. (2), a i is an annihilation operator for the i-th orbital, where i is either O(1s) or C(1s), and N (GS) is the N electron wavefunction for the GS. The physical meaning of i (FO) is that, within the sudden approximation, SA, it is the wavefunction that is created at the instant of photoionization. 22 It was introduced to determine losses to satellites in the XPS spectra. 22,23 In the present case, we use (FO) to permit a decomposition of the BE's into contributions from initial state e↵ects and final state e↵ects. We stress that the i (FO) in Eq. (2) does not have to be limited to a single configuration but it can be a fairly general CI wavefunction and, hence, it can include a rather broad range of many electron, or electron correlation e↵ects. The only limitation in the definition of i (FO) is that the i-th core electron cannot be correlated in the GS WF; however, this correlation is not included in the CAS(4,4) WF since only the ⇡ electron static correlation is treated.
For the energies, the expectation value of the energy of (FO) gives the FO energy of the core ionized molecule, E(FO) = h (FO)|H| (FO)i and the SCF energy of the variational solution for the N 1 electron core ion, either HF or CAS (4,4), is E(SCF) = h (SCF)|H| (SCF)i. The respective BE's, BE(FO) and BE( SCF), are the di↵erences between the appropriate ionic and GS energies. For simplicity, we have dropped the index i, indicating core-hole, in this discussion. The di↵erence between BE(FO) and BE( SCF), defines the relaxation energy, E R , where E R must be 0 since it denotes the energy lowering due to the screening of the corehole. 1 The CO molecule is placed along the z-axis with C at (0,0,0) and O at (0,0,1.128 Å). The experimental equilibrium 24 R(C-O) is used for all SCF and CAS calculations in order to be able to directly compare the SCF and CAS BE's without introducing e↵ects due to using di↵erent R(C-O) for these two kinds of WF's. 3 It is appropriate to use the same R(C-O) for the GS and the core-hole states since the XPS process, neglecting vibrational excitations which may introduce a fine structure in the XPS, 25 is a vertical transition. 1 In order to allow a more nearly correct comparison to experiment, we have also estimated relativistic corrections to the directly calculated non-relativistic BE( SCF) following the procedure that we have used earlier for the N(1s) BE's. 3 For the open shell atoms, C, with configuration 2p 2 , and O, with configuration 2p 4 , we have calculated total HF and Dirac-Hartree-Fock, DHF, energies for the average of configurations 26 of these open shell atoms. The relativistic calculations were based on 4 component spinors and included scalar as well as spin-orbit relativistic e↵ects. The average of configuration HF and DHF energies were calculated for the GS and for the hole-states. The relativistic correction is the di↵erence of the HF and DHF BE's. These corrections increase the calculated non-relativistic BE( SCF) by 0.13 eV for C and 0.45 eV for O. The average of the configuration energies are used to avoid having to separate the relativistic corrections for the individual Russell-Saunders multiplets and J levels of the atoms; however, for these light atoms, we find that the corrections are quite similar for these di↵erent multiplets. The relativistic corrections obtained reflect only atomic e↵ects but they should give us reliable estimates of the relativistic corrections for the CO BE( SCF). To confirm that this procedure provides a very accurate estimate of the relativistic corrections to the O(1s) and C(1s) BE's, we have also determined BE's from relativistic DHF calculations for the CO molecule and compared them to non-relativistic BE's obtained with the same basis set. The relativistic BE's obtained from this direct procedure were identical, within 0.01 eV, to those obtained, as described above, using relativistic corrections from atomic calculations for C and O 1s ionizations. The same C and O Cartesian Gaussian basis sets were used for all calculations. The exponents were taken from the 10s and 6p basis sets optimized by Duijneveldt 27 with a single d exponent of 0.8 (C) and 1.25 (O) added. The basis sets were not contracted except for the innermost 3s functions that were contracted to a single function leading to basis sets with 8s and 6p basis functions. The logic of using only minimal contractions was to allow the basis set to represent, with comparable quality, both the ground and the core-hole states. 2,3 To a good approximation, when an electron is removed from a core shell, the electrons see a nucleus with an e↵ective charge that is larger by one; thus C with a 1s electron removed appears to be a N nucleus and O with a 1s electron removed appears to be a F nucleus. This is the equivalent core model proposed by Jolly. 28,29 It is necessary to use uncontracted or minimally contracted basis sets to have the necessary flexibility to accurately represent the orbitals for both the initial and the final, core-hole, configurations. 2,3 The HF WF's, the FO CAS WF's, and the orbital properties were calculated using the CLIPS suite of programs. 30 The CAS(4,4) WF's were computed using the CASSCF programs developed by B. Roos and P. E. M. Siegbahn at Lund and Stockholm. 14 The relativistic atomic and CO calculations were carried with the DIRAC program system. 31 For the 1s-hole state CAS WF's, we took a precaution to avoid variational collapse to the lowest ionic state rather than to the core-level ionic state. This involved several pairs of steps of fixing, or freezing, a subset of the orbital space while the remaining orbitals were varied. In the first step, the single occupied 1s orbital was frozen as determined for the HF WF and the other orbitals were variationally optimized. In the second step of this pair, the occupied orbitals, both the active, 1⇡ and 2⇡, and the inactive, doubly occupied , were frozen as determined in the first CASSCF step, and the singly occupied 1s orbital was allowed to mix with the virtual space. This pair of steps was repeated until there were no further changes in the CAS energies or orbitals; it was found that three pairs of steps were su cient to obtain convergence to the full CASSCF solution.

III. RESULTS AND DISCUSSION
In Table I, the C(1s) and O(1s) BE( SCF) for the SCF and CAS(4,4) WF's are compared with experiment. In this table, we also include the BE( SCF) with a relativistic correction  24 We also include the BE with a correction for relativistic e↵ects, denoted the rows HF-rel and CAS(4,4)-rel; see text. An error with respect to experiment less than zero indicates that the calculated BE is less than the experimental XPS value. All BE's and errors are in eV. added of 0.13 eV for the C(1s) and 0.45 eV for the O(1s) BE's. These relativistic corrections were determined from calculations on the C and O atoms, see Sec. II, and include only atomic e↵ects but they should be reliable estimates for the relativistic corrections for CO. The HF SCF BE, with relativistic corrections, for O(1s) is as expected too small by 0.6 eV but the HF SCF BE for C(1s) is too large by more than 1 eV, which is completely unexpected. This sign of the error for the C(1s) BE means that there is more correlation energy for the C(1s) core ionized state of CO + than there is for the neutral CO. As we noted in the Introduction, it is unexpected that a system with fewer electrons will have a larger correlation energy. In the following, we will show how this anomalous sign of the BE error can be explained by an important static correlation e↵ect for the 10 valence electrons in the GS and the core-hole states that we study. Based on the equivalent core model, 28,29 it is an acceptable approximation to view CO + with a C 1s-hole as equivalent to NO + and to view CO + with a O 1s-hole as equivalent to CF + . 1 An important near degeneracy e↵ect for the 14 electron isoelectronic series is that involving excitations from 1⇡ to 2⇡. When this near degeneracy is taken into account in a CAS(4,4) WF, we find that the predicted BE's for both C(1s) and O(1s) are both smaller than experiment, exactly as predicted and explained by Mulliken. 8 Also, the magnitudes of the errors of the CAS(4,4) BE's with the relativistic correction are substantially reduced to 0.33 eV for C(1s) and 0.32 eV for O(1s) indicating that we have included the key electron correlation e↵ects with our treatment of the 1⇡-2⇡ near degeneracy. In particular, once static correlation e↵ects are included the errors of the O(1s) and C(1s) BE's, with a relativistic correction included, are almost identical. Thus, the remaining dynamical correlation e↵ects will not introduce di↵erential changes between the C(1s) and O(1s) BE's. Of course, the remaining error in these BEs should be accounted for by inclusion of dynamical correlation e↵ects, which we do not consider since the static correlation reduces the errors in the BE( SCF) and because the di↵erential dynamical correlation e↵ects between the O(1s) and C(1s) BE's are small. However, it still remains to understand the origin of the changes between the HF and CAS BE's. In order to understand the nature of the static correlation for the ground state of CO, we give, in Table II, the properties of the 1⇡ and 2⇡ orbitals and the total dipole moments, µ, for the HF and CAS WF's. The properties in Table II  However, the 0.5 D change in the total dipole from the HF to the CAS(4,4) WF shows that there is a motion of charge from O towards C. This is fully consistent with our analysis of the occupation of the 2⇡ orbital, where the hzi shows that it is closer to C than to O, in the CAS(4,4) WF. In order to have an indication of the magnitude of the motion of charge, the 0.5 D change in µ is equivalent to moving 0.1 electrons from the position of the O nucleus to the position of the C nucleus. While this motion is not especially large, it could lead to changes in the core-level XPS BE's. The direction of the initial state e↵ect of this charge motion can be seen from considerations of the electrostatics involved in moving the charge from O to C. From this motion, the O becomes somewhat less negatively charged and the C somewhat less positively charged. The electrostatic e↵ect of a larger net charge, less negative or more positive, on an atom is to lead to an increase in the core-level BE's while a smaller net charge, less positive or more negative, leads to a decrease in the BE's; see Ref. 1 and references therein. Thus we would expect the motion of charge from the HF to the CAS WF's would decrease the initial state C(1s) BE because C has gained electrons, in the CAS(4,4) WF with respect to the HF WF, through the occupation of the 2⇡ orbital. Furthermore, the motion of charge would increase the initial state O(1s) BE because O has lost electrons through the 2⇡ occupation. The initial state change in the CAS(4,4) C(1s) BE is in the correct direction to change the sign of the error of the CAS(4,4) BE( SCF) from being larger than experiment to smaller than experiment; see Table I. The initial state change in the CAS(4,4) O(1s) BE is in the correct direction to reduce the magnitude of the error of the HF BE( SCF); see Table I.
In order to have a more complete understanding of the separation of the changes in BE's between the HF to CAS(4,4) WF's into contributions from initial and final state e↵ects, we present additional information in Tables III and IV. In  Table III, the initial state BE's, labelled BE FO, and the final state relaxation energies, E R see Eq. (3), for the HF and CAS WF's are given. The changes from HF to CAS values of BE(FO) and E R are also shown in Table III. For the calculation of the initial state CAS BE(FO), the GS CI expansion coe cients as well as the GS variational orbitals are used in the WF where the 1s electron is annihilated. In order to better understand the changes in E R , we present in Table IV information on the CAS(4,4) WF's for the O(1s) and C(1s) hole states that parallels the information in Table II for the GS. In particular, we give the N(1⇡), N(2⇡), and E for these CAS WF's; the di↵erences in the occupation numbers, N, will help us understand the changes of E R between the HF and CAS WF's. The data in Table III show that the direction of the changes of the BE(FO) is as expected from the electrostatic considerations discussed above. The CAS(4,4) O(1s) BE(FO) is 0.3 eV larger than the HF BE(FO) while the CAS(4,4) C(1s) BE(FO) is 0.75 eV smaller than the HF BE(FO). The magnitudes are not the same because the 1⇡ and 2⇡ orbitals have di↵erent spatial extents which change the potential of electrons in these orbitals at the respective nuclei. However, the changes in the C(1s) and O(1s) BE(FO) are both reasonably large and have exactly the direction expected on simple electrostatic considerations. On the other hand, the changes in the relaxation energy, E R , between the HF and CAS C(1s) and O(1s) BE's di↵er by an order of magnitude; large for the C(1s) BE and small for the O(1s) BE. Before we consider these di↵erences in the CAS E R , we discuss how the initial and final state contributions to BE( SCF) influence the C(1s) and O(1s) BE's. Since the relaxation energy, E R , lowers the BE, a positive change in E R for the CAS WF from the HF WF, (CAS-HF) > 0 in Table III, means that the relaxation contributes to a lowering of the CAS BE( SCF) with respect to the HF value. The total change in the CAS BE( SCF) with respect to the HF value is given by This equality can be seen in the data presented in Table I  and Table III Tables I and IV to understand the large di↵erence in the changes in the (CAS-HF) values for E R between the O(1s) and C(1s) hole state WF's. As we noted earlier, there are two contributions to E R for the CAS WF's. One is from the orbital relaxation and the other is from the change in the importance of the configuration mixing of excitations from the HF reference configuration. That this mixing is di↵erent for the GS and the hole-states can be seen from the ⇡ occupation numbers, N(1⇡) and N(2⇡), for the CAS WF; see Tables II  and IV Table III. The situation is quite di↵erent for the C(1s) CAS(4,4) WF where the ⇡ static correlation is significantly more important than for the GS; see Table I and Table IV. The change in energy for the C(1s) CAS WF is 3.5 eV or ⇠65% larger than the lowering of the total energy of the GS. Similarly, the O(1s) CAS N(1⇡) = 3.8 is significantly smaller than the GS value of 3.9, directly showing the larger importance of the mixing of the 2⇡ orbital in the CAS WF. Thus, the contribution of changes in the CI coe cients to the CAS E R , as well as changes in the orbitals, will be large and it is no surprise that the O(1s) (CAS HF) value for E R is an order of magnitude larger than the comparable value for the C(1s) hole state.

IV. CONCLUSIONS
We have investigated a physical mechanism that leads to HF XPS BE's being larger than observed XPS BE's. This direction of error is opposite to the general expectation that HF BE's will be smaller than experiment because the correlation error of HF WF is expected to be smaller for an N 1 electron ion than for the N electron initial state. We have considered the case of CO where the HF O(1s) BE( SCF), which includes initial state electrostatic and final state relaxation contributions, is as expected larger than the experimental XPS BE but where the HF C(1s) BE( SCF) is ⇠1 eV larger the XPS BE. We have shown that there is a static, near degeneracy correlation e↵ect in the ⇡ space that changes the order of the error of the theoretical BE's so that both the O(1s) and C(1s) BE's are smaller than experiment. The static correlation involves introducing configurations with occupation 1⇡ 4 n 2⇡ n in addition to the HF configuration 1⇡ 4 2⇡ 0 within a complete active space, CAS, multi-configuration self-consistent field formulation where both the orbitals and the configuration mixing coe cients are variationally optimized. We have shown that, when corrections for relativistic e↵ects are included, the errors of the O(1s) and C(1s) BE's are small, ⇠0.3 eV, and almost the same. In addition, we have identified the physical reason for the initial state contribution to the BE's where the 1⇡ and 2⇡ near degeneracy is taken into account as arising from a motion of electronic charge from O to C. From simple electrostatic considerations, this motion of charge with respect to HF wavefunctions leads to an increase of the O(1s) BE and a decrease of the C(1s) BE with respect to the HF values. Furthermore, we have shown that the di↵erence of the E R for the CAS wavefunctions from the HF wavefunctions depends on whether the C(1s) or the O(1s) BE is considered. We have related this to the di↵erent extent of the static correlation for the GS and the O(1s) and C(1s) hole state CAS wave functions. The ⇡ correlation is similar for the GS and the O(1s) CAS(4,4) WF's and the CAS and HF E R are quite similar. On the other hand, the ⇡ static correlation is significantly more important for the C(1s) CAS WF than for the GS WF. Thus for the O(1s) hole state, the CAS(4,4) E R is larger than the HF E R because, for the CAS case, the relaxation includes changes in the CAS CI coe cients as well as changes in the orbitals.
Although we have specifically investigated CO, our observations may have general validity in that errors of the HF core-level BE's compared with values from XPS measurements could identify cases where static correlation e↵ects are important for the GS WF and properties. A cornerstone of our logic for the core level ionizations of light atoms in molecules is that the GS and the core-hole configurations are iso-electronic in the valence region where the e↵ects of static correlation are treated. Thus if static correlation is important for the core-hole configurations, it will also be important, albeit perhaps with di↵erent magnitude, for the ground state configuration. If the HF BE( SCF) is larger than experiment, this shows that correlation e↵ects are larger for the N 1 electron core-hole state than for the N electron GS. It is possible, perhaps even likely, that this di↵erence of electron correlation will arise from static e↵ects, at least this was the case for CO. On the other hand, if the HF BE( SCF) is smaller than experiment by an amount much larger than average, as was the case for the O(1s) core-hole of CO, this is also an indication of possibly important static electron correlation e↵ects. In this case the correlation e↵ects are much smaller for the N 1 electron core-hole state than for the N electron GS. Again, in this case, it is reasonable to suggest that the di↵erent extent of electron correlation could arise from static e↵ects. An important caveat for the use of errors of the HF BE( SCF) to identify cases where static correlation e↵ects may be important is that suitably large basis sets with minimal contractions of the basis functions used to describe the core region should be used to calculate the WF's and the BE's. These are the basis sets used in the present work and in our prior studies of the 1s BE's of first row atoms in several molecules. [2][3][4] In our application to CO, we have shown that the static correlation that makes the CAS BE( SCF) smaller than experiment and reduces the magnitude of the error of the core-level BE's also reduces the error of the CO dipole moment and leads to a µ > 0 in agreement with experiment. Our arguments about using the errors of HF BE( SCF) to identify cases where a treatment of static correlation may be required can also be applied to our earlier analysis 2 of 19 N(1s) BE's in 17 di↵erent molecules. The errors of all the HF BE( SCF) were <0 showing that the HF BE's were all less than experiment and, without the relativistic correction that was estimated to be 0.2 eV, 3 they were, in magnitude, 0.5 eV with only 6 outliers with larger errors. 2 For these outliers, it might be worthwhile to investigate whether static correlation e↵ects might be important. Our arguments about the general utility of the errors of HF BE( SCF) to identify the need to treat static electron correlation are speculative. However, to our knowledge, this is the first time that evidence has been presented to show that the errors of calculated HF core-level BE's may indicate that there are static correlation e↵ects in the ground, initial state wavefunctions that need to be taken into account.