Spin-Peierls vs. Peierls distortions in a family of conjugated Polymers

Distortions in a family of conjugated polymers are studied within two complementary approaches, i.e. within a many-body Valence Bond (VB) approach using a transfer matrix technique to treat the Heisenberg model of the systems, and also in terms of the tight-binding band-theoretic model with interactions limited to nearest neighbors. The computations indicate that both methods predict the presence or absence of the same distortions in most of the polymers studied.

Distortions in a family of conjugated polymers are studied within two complementary approaches, i.e. within a manybody Valence Bond (VB) approach using a transfer matrix technique to treat the Heisenberg model of the systems, and also in terms of the tight-binding band-theoretic model with interactions limited to nearest neighbors. The computations indicate that both methods predict the presence or absence of the same distortions in most of the polymers studied. The recent discovery of the first inorganic spin-Peierls material, CuGeO 3 , [1] has engendered a renewed interest in Spin-Peierls systems, i.e. systems which present a structural distortion below the spin-Peierls temperature due to residual magnetoelastic couplings stabilizing the ground-state, in analogy to Peierls distortion [2] associated to an electron-soft phonon instability opening a band gap at the Fermi level. Recent experiments [3] suggest that this is not an isolated case, and the pronounced decrease of susceptibility observed [4] in α-NaV 2 O 5 is also due to a spin-Peierls transition.
The spin-Peierls transition was first observed in predominantly organic compounds as TTFCuBDT [5], TTF-TCNQ [6], (TMTSF) 2 PF 6 [7], or TTF-AuBDT [8]. Theoretically, it has been studied (see for instance [9,10,11,12,13,14] and Refs. therein) as a geometrical symmetry breaking for the lowest eigenstate of a Heisenberg Hamiltonian. Peierls and spin-Peierls phenomena are still a subject of discussion for many other polymers, since if a deviation occurs that lowers the chain's symmetry, then different symmetry-equivalent distorted ground-states may arise which correspond to different thermodynamic phases and, at sufficiently low temperature, the possibility of solitonic excitations and/or conduction could arise [15].
Furthermore, it has been argued [16] that under similar structural circumstances a Peierls distortion is predicted for the simple Hückel tight-binding model of πnetwork strips if and only if a spin-Peierls distortion is also predicted from Valence Bond (VB) theory (or the formally equivalent s = 1/2 Heisenberg model) at the simple resonance theoretic level. At this level of approximation the VB wave-functions are restricted to equally weighted superpositions of special covalent VB singlet states, i.e. of Kekulé structures [17], where every π-electron is coupled to a singlet state with one of their nearest-neighbors. These Kekulé structures may be partitioned into long-range ordered spin-pairing phases, the lowest-lying phase corresponding to the highest count of Kekulé structures contributing to it. Within this approach, a spin-Peierls distortion is predicted if there are two maximum-cardinality-degenerate Kekulé phases (see [18] and references therein). Then, this correspon-dence between Peierls and spin-Peierls instabilities implies that a zero-width band gap for a π-network polymer is predicted if and only if there are two such cardinalitydegenerate Kekulé phases. The question then arises as to whether this correspondence is maintained when going beyond the resonance theoretic approximation.
For instance, the dimerization in polyacetylene has traditionally been interpreted in terms of band theory [19,20] as a Peierls distortion. Recently, however, this dimerization has also been successfully explained [21] with a Heisenberg-like Hamiltonian model [22] as a spin-Peierls distortion, using both cluster-expanded wave-functions and perturbation theory. This clusterexpanded many-body treatment of distortions has also been applied to the polyacene polymer [23], which earlier has been extensively studied from the independent particle point of view, since it exhibits an accidental zerowidth band gap at a simple tight-binding level (see [23,24] and references therein), and a new quasi-degeneracy has been predicted.
Thence, the comparison between the independentparticle and the many-body VB treatments for degeneracy and symmetry-breaking in polymers deserves further analysis. It is our purpose here to investigate the groundstate symmetries and degeneracies for several conjugated polymers using both a simple many-body VB framework and a simple tight-binding model. The rationale for these simplest models (with just nearest-neighbor interactions) is that they reveal distortive responses which qualitative dominate over the otherwise harmonic responses (e.g., associated with the σ electrons. That is, these simplest models should reveal dominant qualitative features, which should persist independently of parameterization. The polymers we focus our attention on are: polyaceacene (PAA), poly(benz[m,n])anthracene (PBA), and polyperylene (PPR) (Fig. 1). All these systems exhibit a zero-width band gap at the simplest tight-binding level. So far, very few experimental results are available. Only PBA [25], and PPR [26] have already been synthesized. Theoretically PPR has been treated from the independent particle point of view [27,28], and also using the valence effective hamiltonian technique [29], while as far as we know PBA has not been previously treated. PAA has been discussed in the literature, mostly from an independent particle point of view [28,30,31,32,33] and less frequently from a resonance-theoretic approach [15,16], though it has not been synthesized yet. PAA can be seen, together with polyacetylene and polyacene, as the first members of a family of poly-trans-polyacetylenes, graphite being the final member of the family. All these can be thought of as special cases of ladder materials [34] as already had been pointed out in Refs. [15,32,33].
Within the many-body VB framework, we will consider the antiferromagnetically-signed spin- 1 2 Heisenberg model (for more general derivations of this model than those based on degenerate perturbation expansions see for instance [22,35] and references therein). Adequate many-body wave-function ansätze provide variational upper bounds to the ground-state energy. Two different kinds of variational localized-site cluster expanded ansätze have been considered: first a Resonating VB (RVB) ansatz , where the trial wave-function is a weighted superposition over all singlets constructed as products of singlet pairs each involving two (not necessarily nearest-neighbor) sites at a time; and second a Néel-state-based ansatz , where a Néel state is the zeroth order wave-function from which the trial wave-function is generated. We evaluate the matrix elements for each ansatz with a transfer-matrix technique introduced previously [21,23,36,37,38,39]. For the tight-binding band theory calculations we consider the so-called translationally adapted Hückel model limited to nearest neighbors. This paper is organized as follows: in section II the description of the polymers, their symmetries and relevant distortions are given. In section III we introduce briefly the translationally adapted Hückel model. In section IV a description of the VB method is given in terms of the Heisenberg Hamiltonian and the trial wave-functions are presented. Also, the technique to compute the physical magnitudes based on a transfer matrix is introduced and applied to obtain the ground-state energy of the systems. Results are presented and discussed in section V. And finally our conclusions can be found in section VI.

II. DESCRIPTION OF THE POLYMERS AND THEIR SYMMETRIES
The systems studied are polymeric strips of finite width and infinite length (L → ∞) (see Fig. 1). They are constructed with fused benzene rings, and can be seen as cut from the 2-dimensional graphite or honeycomb lattice. Each site of the lattice is taken to represent an sp 2 -hybridized carbon atom with one π-orbital perpendicular to the plane of the lattice and with one π-electron per site. These strips are presumed to be translationally symmetric along L, with periodic boundary conditions; so that the strips may be divided into unit cells or eventually reduced unit cells, when the space group of the strip contains operations involving glide reflections. The space group of the strips include, along with the primitive translation, rotations C n , reflections σ, and combination of rotations and reflections (improper rotations), coordinate inversion i, and screw-rotations and glide-reflections, C s , i.e. a combination of an improper two-fold rotation or reflection with a nonprimitive translation of half a unit cell which by themselves do not leave the lattice invariant (see for instance Fig. 2 and Table I).
Of special interest are minimal subsets of symmetry operations, whose removal lead to: (i) a band gap opening at the Fermi level, when analyzed from the bandtheoretic point of view; (ii) the lifting of the degeneracy of Kekulé phases, if seen from the resonance-theoretic treatment. If a zero gap occurs at k = π (as is frequently the case for benzenoid polymers) then such a minimal subset will be so as to no more than double the size of a unit cell.
When a symmetry is broken, there is a distortion parameter ∆ l associated to the stretching or shortening of the bond l, with bonds numbered as in Figs. 4, 5 and 6. Two symmetry elements are chosen to label the interesting distortions for every polymer as shown in Tables  II, III and IV, where appropriate constraints on the ∆ l , imposed by the different symmetry breakings, are also shown. The distortions are classified as to symmetric (+1) or antisymmetric (-1) with respect to these two selected symmetry elements.
The PAA polymer is formed by benzene rings sharing four consecutive edges with neighboring rings as shown in Fig. 1(a). It can also be seen as a trimer of non-dimerized parallel all-trans polyacetylene chains. The 6-site unit cell can be broken into two 3-site reduced unit cells, defined as the region between dashed lines in Fig. 1(a). In the band picture, there is a half-filled band and, consequently, a zero-width band gap is predicted, regardless of distortions which preserve the glide-reflection symmetry. In the simplest VB picture, i.e. resonance theory, there are two maximum-cardinality degenerate Kekulé phases. For instance, defining M as the number of "double bonds" crossed by an oblique line (see Fig. 3), there are two Kekulé phases M = even equivalent to two M = odd which do not mix because of the cyclic boundary conditions of the strip and they are degenerate since they each contain essentially a single Kekulé structure. A distortion that could open the band gap at the Fermi level and lift the degeneracy of the Kekulé phases requires the destruction of the glide-reflection symmetry. The distortions to be considered are then those which are antisymmetric with respect to interchange of the two reduced unit cells in a new unit cell.
PBA is formed by a polyacene strip where added benzene rings have been, top and bottom, alternatively fused on (see Fig. 1(b)). A reduced unit cell can be defined for this system between the dashed lines in Fig.  1(b), with two 7-sites reduced unit cells per unit cell. It is a half-filled band system and, like PAA, a zerowidth band gap is predicted. Resonance theory, following Ref. [16], predicts two maximum-cardinality degenerate Kekulé phases. As in the PAA polymer, the interesting distortions that could open the band gap and lift the degeneracy are those that are antisymmetric under operations which interchange the two types of reduced unit cells.
The PPR polymer is formed by fused benzene rings as drawn in Fig. 1(c). The unit cell containing 10 sites is defined between the dashed lines in the graph, and there is no smaller reduced unit cell for this system. The space group is generated by the point group D 2h and the translation operations along the strip (see Table I and Fig. 2(c)). Differently from the rest of the polymers here, there is no glide-reflection symmetry operation for PPR. Furthermore, it does not have an odd number of π-electrons per reduced unit cell so that it does not correspond to a half-filled band system. Nevertheless, there is an accidental degeneracy at the Hückel level of approximation, so that it has a zero-width band gap anyway (see next section). Correspondingly, resonance theory predicts two maximum-cardinality degenerate Kekulé phases [16]. A totally symmetric distortion will also be considered for this system (see Table IV).

III. TRANSLATIONALLY ADAPTED HÜCKEL MODEL
The Hückel model is the simplest tight-binding model: c + niσ (c niσ ) are the creation (annihilation) electron operators on site i of unit cell n with spin σ and β ni,mj is the "Hückel resonance integral" (or hopping integral) between sites i and j in unit cells n and m, respectively. < ni, mj > indicates that the sum is restricted to nearest neighbors. Considering the translational invariance symmetry of the system, we can define translationally symmetry adapted states e ikn | n, j , k = 2πn k L , n k = 0, 1, · · · , L − 1. (2) The matrix elements of the Hamiltonian between these new states are Diagonalizing the hamiltonian matrix elements, the energy bands ε(k) are finally obtained. Symmetry breaking can be considered taking β ni,mj as where ∆ ni,mj (| ∆ ni,mj |≪ 1) is the distortion parameter, as introduced in the previous section, that measures the strength of the distortion between sites ni and mj.

IV. VALENCE BOND METHOD
Within the VB picture we attempt here to go beyond resonance theory when solving the Heisenberg Hamiltonian: J ni,mj is the "exchange integral" between nearestneighbor sites ni and mj and S ni denotes the spin operator for one electron on site ni.
with ∆ ni,mj being the distortion parameter associated to the bond between sites ni and mj when there is a symmetry breaking.
While solving the Hückel model is an easy task, solving the Heisenberg hamiltonian is in general a non-trivial problem. In order to obtain, along with the appropriate approximate wave-functions, good variational upper bounds to the ground-state energy of this model, E(∆), for the polymer systems, we consider two different types of cluster-expanded ansätze that depend on variational parameters, each of which describes the local features of the system. Since our polymers are bipartite systems with total spin zero, we have considered a Néel-statebased ansatz and a RVB ansatz . These ansätze were introduced in Ref. [23] and we shall make here a brief description of them. Related ansätze have also been successfully considered by other authors [40,41] when solving the s = 1/2 Heisenberg hamiltonian for the square lattice.
A. Néel-state-based ansatz (NSBA) The cluster expanded wave-function ansatz in this section is based upon the Néel state as a zeroth-order wavefunction, where A and B denote the two sets of sites such that each member of one set is a nearest neighbor solely to (some) sites of the other set, and α(i) (β(i)) indicate that the spin of the electron on site i is +1/2 (−1/2). A lowering of the energy, with respect to that of the Néel state, occurs for an ansatz defined within a subspace spanned by | Φ N and the states obtained when applying to | Φ N the XY terms, S ± ni S ∓ mj , of the Heisenberg operator, an arbitrary number of times in an "unlinked" way. These additional states which are to be mixed with the Néel state can be generated in terms of the nearest-neighbor pair excitation operator: where the x ni,mj are scalars to be optimized and S − ni and S + mi are spin raising and lowering operators on site ni From that, the Néel-state-based wave-function ansatz (NSBA) will be a cluster-expansion in terms of P excitations acting on the Néel state where U indicates that only unlinked terms are to be retained from the Taylor series expansion. Namely, | Ψ N is a wave-function where the Néel state is mixed with states that differ from it by an arbitrary number of couples of disjoint pairs of neighboring spins that have been flipped, each state in the superposition being weighted by the product of the variational parameters associated to the flips in that state.

B. Resonating Valence Bond ansätze
In this approach we start with a one-bond-range RVB (1BR-RVB), that plays the fundamental zerothorder role for the more elaborated three-bond-range RVB (3BR-RVB) ansatz following.

One-Bond-Range RVB ansatz
A 1BR-RVB | Ψ 1 is a weighted superposition of Kekulé states, i.e. nearest neighbor VB states, where every site ni is spin paired to one of its neighbors mj. It can be written as I is the identity operator, U 0 indicates that the terms to be retained are those where each site appears once and only once, and the weighting factor of a Kekulé state in Ψ 1 is a product of variational parameters x ni,mj associated to the singlet pairs ni, mj in the Kekulé state considered.

Three-Bond-Range RVB ansatz
The 3BR-RVB is a weighted superposition of all the VB structures within a phase with each spin-pairing between A-and B-sublattice sites separated by no more than 3 bonds. In the usual form for cluster expanded wave-functions, it may be viewed as generated from the 1BR-RVB as follows: The XY terms, S ± ni S ∓ mj , of the Heisenberg Hamiltonian acting on the 1BR-RVB wavefunction of Eqn. (11) yield "long-bonded" states with pairings among three-bond distant neighbors, along with "neighbor-bonded" states already in Ψ 1 . These "longbonded" states can be directly generated by the "recoupling" of two neighboring bond-singlets in Ψ 1 (see Ref. [23]). We may denote byq ef the operator related to such a recoupling between two bond-singlets e and f . From | Ψ 1 we may build the 3BR-RVB allowing an arbitrary number of recouplings of two simply neighboring bondsinglets, i.e. unlinked pairs with one and only one site in a pair being a nearest neighbor to a site in the other pair. Then, the overall 3BR-RVB excitation operator above the 1BR-RVB wave-function might be viewed to be (12) with the x ef being variational parameters, and where < e, f > indicates that the sum is restricted to simply neighboring bond-singlets. The corresponding ansatz would then be where again U indicates that only unlinked terms are to be retained. That is, in the Taylor series expansion of e Q one retains only products ofq ef such that no pair index (e or f ) shares any vertices with another pair index in the product. And Q and Ψ 1 are to be optimized simultaneously.

C. Expectation-value calculations by the Transfer-Matrix Technique
The ground-state energy is computed as a function of variational parameters for each of the above introduced ansätze assuming translational symmetry and cyclic boundary conditions along L. The way our ansätze are chosen allows us to deal with the systems locally, so that one can define a transfer matrix [23] that describes the local features and reduces the computation of Eqn. (14) to products of "small" matrices [21,23,36,37,38,39,42]. Let's suppose there are imaginary vertical lines cutting the strip on translationally equivalent positions (including improper translations). We can define the ansatz -dependent "local states" according to every possible local spin-pairing/spin-flip pattern around a given position determined by one of the imaginary vertical lines, and ultimately use this to compute Ψ | Ψ . Thence these local states contain the contributions from both the bra and the ket . From the assumed translational symmetry, local states in every position are to be the same. Now, labelling these local states by e t , t ranging over the whole set of local states, we let the transfermatrix element denote a weighted sum over the various ways a local state e s may succeed a local state e t . The weight of every contribution is obtained by considering the variational parameters associated to the way e t evolves to e s , and, eventually, additional factors coming from Pauling's superposition rules [43]. The overlap is then evaluated in terms of the T matrix: For L → ∞, the largest eigenvalue Λ of T dominates, and the overlap reduces to The hamiltonian expectation value over | Ψ can be obtained in a similar way introducing a "connection" matrix, C, defined according to where c measures the range of the interaction within the ansatz . In our case c = 2. And, the matrix element is a weighted sum over the various ways a local state e s may succeed a local state e t after c transfer-matrix-steps when the Hamiltonian operators per unit cell are present. In the long-length limit Eqn. (14) reduces to: where (Λ, l | and | Λ, r) are left and right eigenvectors corresponding to the maximum eigenvalue Λ of T . This expression is a function of the variational parameters associated to Ψ and an upper bound to the exact groundstate energy is obtained. Implementation of a suitable numerical optimization yields a best upper bound. The energy expression (20) can be readily generalized when considering possible distortions. The connection matrix per unit cell can be understood as a sum of matrices C ni,mj , each one concerning two body interactions between neighboring sites ni and mj, weighted by the factor 1 + ∆ ni,mj that modifies its interaction strength. Then and the energy expectation evaluation is reduced to some "simple" matrix manipulations.

V. RESULTS AND DISCUSSION
Computations based on band theory at a Hückel tightbinding level of approximation (see section III), and within VB theory with the cluster expanded 1BR-RVB ansatz and NSBA (see section IV) were carried out for all polymer systems here described. For the PAA also the 3BR-RVB has been used. In this case the 1BR-RVB ansatz contains only one Kekulé structure, so it is especially appropriate to use the 3BR-RVB wave-function and go beyond a single Kekulé-structure approximation. This circumstance differs with the rest of the polymers, where the number of Kekulé structures in the corresponding 1BR-RVB is big. The different VB upper bounds to the energy of the undistorted polymers are presented in Table V, together with that for polyacetylene. The lowest upper bound to the ground-state energy for the undistorted system is given by the NSBA.

PAA
The highest occupied Hückel tight-binding band and the lowest unoccupied band cross at k = π. Taking into account the perturbation ∆ ni,mj in the Hückel resonance integral, β ni,mj = β(1 + ∆ ni,mj ), only the totallyantisymmetric distortion with respect to C s and σ v labelled as C 3 (see Table II) opens a gap at k = π. But the leading term of the energy lowering △E versus ∆ is ∼ ∆ 2 , as it is the positive phonon energy contribution to be added. So band theory at this low level of approximation predicts neither a presence nor absence of a C 3 distortion for this system, the result depending on the final balance between these two contributions to the energy. Nevertheless, if interactions with more distant π-centers are included, although small, linear terms in ∆ are argued to arise [16] and then the distortion is favored.
Still within band theory, this system has also been studied by other authors at different levels of approximation. Kertesz [31] and Tanaka [32] suggest a totally antisymmetric distortion, though leading to a quadratic small gap that could be suppressed by interchain interactions. The tight-binding SCF-MO method at the level of CNDO/2 (complete neglect of differential overlap) calculations suggests that the Peierls distortion does not take place so one can expect metallic behavior [30], while Bozović [28] combining tight-binding calculations with group-theoretical arguments predicts distortions of type B (see Table II) as favored. Therefore, within band theory, predictions about the opening or not of a band gap at the Fermi level, or the distortion driving it, depend crucially on the level of approximation.
Let us consider now the many-body VB method. The ground-state energy has been obtained using the NSBA and both the 1BR-and the 3BR-RVB ansätze of section IV, as a function of ∆ for the different distortions A, B 1 , B 2 , C 1 , C 2 , C 3 (see Table II). Transfer and connection matrices of dimensions 14 × 14 (for the NSBA) and 60 × 60 (for the 3BR-RVB ansatz ) were needed in order to carry out computations. The energy for the different distortions when the NSBA is used has been plotted as a function of ∆ in Fig. 7(a), while results obtained with the 3BR-RVB ansatz are presented in Fig. 7(b). Plots from the 1BR-RVB ansatz are not given, since they are qualitatively identical to those from the 3BR-RVB ones. Comparing NSBA and RVB ansätze, it can be seen that the ordering of △E for the different distortions is the same in any case, the strongest lowering corresponding to the C 3 distortion.
Nevertheless, while the energy response to A, B and C distortions is linear for the RVB ansätze, clearly predicting a C 3 distorted ground-state, in the NSBA case they still go as ∼ ∆ 2 . Fitting the results in a parabolic curve, it is obtained that △E ∼ −1.923∆ 2 . Again a distortion is not clearly predicted with our NSBA. A comparison of the coefficients coming from this term and those from the phonon energy should be made in order to decide whether this ansatz is able to predict or not to predict a C 3 distortion. This ambiguity of prediction in some sense rationalizes earlier contradictory results: via numerical band theory by Yamabe et al. [30], predicting an undistorted ground-state, and via band/group-theoretic considerations by Bozović [28], predicting a B distortion.
Although the RVB ground-state energy is higher than the NSBA, its predictions on ground-state instabilities are based upon the known global-singlet character of the ground state along with its local-singlet character, leading to asymptotically orthogonal and noninteracting phases responding essentially independently to distortions. Relaxation of this local-singlet character would imply the inclusion of pairings between distant sites, leading to undesirable long-range correlations of the type in the Néel state. Then we expect a C 3 distorted groundstate as predicted by RVB. Furthermore, NSBA at this lower level, with only two-site excitations, does not always seem sensitive to instabilities as at higher order, such as for polyacetylene [21], then we expect that the distortion could be clearly predicted when going to a higher order NSBA too. Also it can be argued that inclusion of slightly longer-range interactions (as between next-nearest neighbors) in the hamiltonian will increase the 'frustration' and the NSBA energy whereas the RVB expectations will change but little. Thus there is a tendency to invert the energy ordering of these states. Still another argument favoring RVB predictions is that the NSBA is not a pure singlet, as the ground state is known to be. Also the RVB type ansatz accords more closely to a classical organic chemical view of these polymers.

PBA
The lowest occupied Hückel tight-binding band and the highest unoccupied one cross at k = π, so it is a zerowidth band gap system. From all the possible distortions considered in Table III, only the totally antisymmetric distortions C 1 and C 2 open a gap with an energy dependence linear in ∆. Therefore, band theory predicts that the system will distort. In the VB picture the possible distortions in PBA have been studied with the 1BR-RVB ansatz . For this system we only carried out calculations with this ansatz for two reasons i) the 1BR-RVB ansatz gives already a good upper bound to the ground-state energy because there is mixing of Kekulé states and ii) the dimension of the transfer and connection matrices for the 3BR-RVB and the Néel-state-based ansätze grow substantially with respect to the 1BR-RVB one. In Fig.  8 the energy of the 1BR-RVB ansatz has been plotted as a function of ∆ for the distortions A 1 , A 2 , C 1 and C 2 classified in Table III. The most favored distortions are the totally-antisymmetric ones C 1 and C 2 , in particular C 1 with a dependence ∼ ∆. This result agrees with the predictions given from band theory, concluding that complementary approaches lead to the same kind of distortions for this system.

PPR
PPR is not a half-filled band system but the Hückel model predicts an accidental zero-width band gap at k = 0. A, B, C and D distortions (see Table IV) have been considered. The distortions C 1 and C 2 open a gap at k = 0 weakly, with an energy dependence △E ∼ ∆ 2 . But the totally symmetric distortion D 1 opens a band gap with an energy response linear in ∆. This result agrees with the predictions given by Bozović [28] and Tanaka et al. [27]. In Fig. 9(a) the Néel-state-based energy obtained, using 5 × 5 transfer and connection matrices, is plotted as a function of ∆ for various possible distortions (see Table IV). Clearly the totally symmetric distortion, D 1 is favored with a linear energy dependence on ∆. Also in Fig. 9(b) the 1BR-RVB energy is plotted for the various distortions as a function of ∆ and results agree with the NSBA energy, namely that the D 1 distortion is the most favored one with a linear dependence in ∆. As in PBA, the 1BR-RVB ansatz gives already a good upper bound due to the mixing of Kekulé states.
Band theory and the many-body VB method predict the same distortional behavior for this system, i.e. the system is unstable to a totally symmetric D 1 distortion. Some evidence exists for polyperylene synthesis [26], but further experimental information on the structure (and properties) of this system is still needed.

VI. CONCLUSIONS
We have presented, both with the simple Hückel tight-binding band theory and with a Heisenberg model hamiltonian (or, equivalently, the VB model), a study of the ground-state nature of a family of polymers: polyaceacene, poly(benz[m,n]anthracene), and polyperylene. We have focused our attention on correspondences between Peierls and spin-Peierls instabilities predictions, when analyzed from these two complementary approaches.
Upper bounds to the energy of the Heisenberg model in each case have been obtained with two alternative localized-site cluster-expanded wave-functions, i.e. RVBtype ansätze and a Néel-based ansatz . We have shown that simple expressions of the physical magnitudes we were interested in, were easily obtained by using the transfer-matrix technique of ref. [23].
From our results, it is concluded that the RVB wavefunctions considered, which are restricted to 1BR type for all the systems other than PAA, do not give our best upper bound to the ground-state energy of the undistorted systems. Nevertheless, they are relevant for studying such phenomena as the spin-Peierls instability and elementary excitations as hole excitations or excitonic excitations as already pointed out [38]. Moreover the RVB ansätze have a global-singlet character and a local-singlet character, precluding long-range order of the type of the Néel state, and generally improve relative to Néel-based ansätze upon inclusion of higher-order (frustrative) terms in an elaborated Heisenberg model.
The Néel-state-based ansatz gives a fairly good upper bound to the ground-state energy for all the systems considered. For the nearest-neighbor model considered, this ansatz always yields lower energy than the RVB ones for undistorted systems. We have shown that with such a simple Néel-state-based wave-function, the corresponding energy is notably lower than the energy of the Néel state, while computations remain fairly simple. The Néel-state-based ansatz predicts for the polymers studied, the same distortions as the RVB description, except for the case of polyaceacene where this ansatz in our current simple considerations does not show whether the distortion is going to take place or not, although the strongest lowering of the energy also correspond a totally antisymmetric distortion.
From the Heisenberg hamiltonian, or equivalently from the VB model, we have obtained that: 1. PAA shows a totally antisymmetric distortion from the RVB, while NSBA is not conclusive, depending on the balance between electronic energy lowering and the phonon energy contribution.
3. PPR is unstable to a totally symmetric distortion.
Within the band-theoretic picture, the Hückel tightbinding model has been studied for all the same polymers. Results obtained for our π-network system are as follows: 1. PAA could show a totally antisymmetric distortion at a simple Hückel level, depending on the balance between electronic energy lowering and the phonon energy contribution. Other approximations already in the literature [28,30] yield contradictory results.
3. PPR shows a totally symmetric distortion.
Comparing band theory and the Heisenberg model results, it can be concluded that predictions of these two models based on opposite (or complementary) limits seem to lead to similar consequences under similar structural circumstances, i.e. both approaches predict the presence or absence of the same instability to symmetry for the polymers. It is to be noted that the band-theoretic results depend crucially on the level of approximation, as it is observed in the study of polyaceacene, where this picture at different levels of approximation gives rise to different predictions. On the other hand the Heisenberg model has proven to give predictions consistent from one level to another. Even in the case of PAA, where NSBA cannot make a clear prediction as it happens with band theory at its lower level, NSBA still shows the strongest lowering of the energy for the very same distortion suggested by RVB. Since the NSBA at this lower level, with only two-site excitations, does not always seems so sensitive to instabilities as at higher orders, such as it happens with polyacetylene [21]. That is, the distortion sometimes seems to only occur with a higher order NSBA, in agreement with RVB results. Therefore, it seems that the VB model, which includes correlation explicitly, gives a good description of these benzenoid systems, predicting spin-Peierls distortions whenever a Peierls distortion is also predicted. These results modify earlier suggestions (see ref. [44]) that inclusion of correlation "a posteriori", as a perturbation, diminishes the distortion. That is, we find any diminishment does not go to zero in the (strong correlation) Heisenberg-model limit, and indeed the RVB results indicate a stronger response to distortions (at least at the undistorted point on the potential energy hypersurface).
It has been shown that this treatment is computationally feasible specially for quasi-one dimensional systems where the transfer-matrix technique proves to be a powerful tool of computation. It is important to note that the results are developed in terms of quantities which remain finite as the strip-length goes to infinity. It is of some interest to compare the computational effort involved in the tight-binding approach versus that involved in our transfer-matrix cluster-expansion approach (for ei-ther RVB or Néel-state-based wavefunctions). The matrices H(k) of eqn. 3 and T of eqn. 15 arise in these respective approaches and are both finite independently of L → ∞. Both types are to be diagonalized, but there are some differences: 1. Typically T increases in size much more rapidly with unit-cell "width" than does H(k) (though these behaviours are reversed if the unit-cell "length" is considered instead).
2. The total energy requires sampling many of the L → ∞ H(k) matrices (varying smoothly with wavevector k) whereas for given parameters there is but one T matrix to treat.
3. The optimal total energy for the cluster expansions entails treating T matrices for numerous variational parameter values whereas there is no much repetition with the H(k).
Notably if one goes beyond the tight-binding method to Hartree-Fock (or density-functional) approaches this last noted difference no longer occurs. Evidently the computational effort via either SCF or our cluster expansion is roughly comparable (at least for linear polymers with modestly sized unit cells). The analysis carried out in this paper would require experimental testing. Though the synthesis of some of the systems considered like PAA seems quite difficult to achieve, there are hopes in this direction. Finally, some aspects of this treatment are not restricted only to the model Hamiltonian and the ground-state ansätze presented but can be applied to any system with effective short-range interactions if described by a localized-site cluster expanded ground-state wave-function. and B2 for ∆1 = 0 and ∆2 > 0. For C distortions we identify subcases: C1 for ∆1 > 0 and ∆0 = 0; C2 for ∆1 = 0 and ∆0 > 0; and C3 for ∆1 > 0 and ∆0 < 0.
Ground-state Heisenberg energy per site in J units for the family of π-network polymers studied. PA stands for polyacetylene. The first row corresponds to the energy obtained with a single Kekulé structure | K . | Ψ1 stands for the 1BR-RVB ansatz of Eqn. (11), | Ψ3 is the 3BR-RVB ansatz of Eqn. (13), | ΦN is the Néel state, and | ΨN the Néel-state-based ansatz of Eqn. (10). The last row corresponds to the exact ground-state energy which is known only for the 1D case.