Bond length alternation in cyclic polyenes. II. Unrestricted hartree–fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian, together with an empirical quasi harmonic σ-core potential is investigated using the unrestricted Hartree–Fock wave function employing different spatial orbitals for different spins. It is shown that in contrast to the restricted Hartree–Fock method, which favors bond alternation in large cyclic polyenes, the unrestricted Hartree–Fock method stabilizes the symmetric structures with equidistant internuclear separation. An assessment of the amount of correlation error recovered by the unrestricted Hartree–Fock procedure is examined and the qualitatively different behavior of the cyclic polyene models when described by restricted and unrestricted Hartree–Fock wave functions is discussed from this viewpoint.     

Bond length alternation in cyclic polyenes. I. Restricted Hartree–Fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian and its relationship to the singlet stability problem for symmetry adapted Hartree–Fock solutions for these systems is investigated using the restricted Hartree–Fock method. The σ-energy contribution is approximated by a quasiharmonic empirical potential. It is shown that the restricted Hartree–Fock energies favor the cyclic polyene distortion and an estimate of the distortion and of the stabilization energy for infinite linear polyenes is obtained.     

Bond length alternation in cyclic polyenes. V. Local finite-order perturbation theory approach

The finite-order many-body perturbation theory using the localized Wannier orbital basis is applied to the problem of bond length alternation in the Pariser–Parr–Pople model of cyclic polyenes C_N H_N, N = 4v + 2, which may be regarded as a simplified model of polyacetylene. Both the Møller–Plesset and the Epstein–Nesbet-type partitionings of the model Hamiltonian are employed. The localized orbital basis enables an efficient truncation of the perturbation theory summations over the intermediate states as well as an elimination of energetically unimportant diagrams, thus enabling one to obtain the fourth-order Møller–Plesset-perturbation energies with a relatively small computational effort even for large polyenes. The results obtained with the second-, third-, and fourth-order Møller–Plesset and with the third-order Epstein–Nesbet perturbation theories yield very similar bond length distortions (about 0.05 Å) and stabilization energies per site (about 0.04 eV) as obtained earlier with the RHF , one-parameter AMO , and delocalized orbital perturbation theories. The effects of truncation and diagram elimination in the fourth-order Møller–Plesset perturbation theory and the abnormal behavior of the second-order Epstein–Nesbet perturbation theory results in the localized Wannier basis near the instability threshold of the RHF solutions are discussed.     

Cluster analysis of the full configuration interaction wave functions of cyclic polyene models

The first two members of the cyclic polyene homologous series are studied over a wide range of the coupling constant using the Hubbard and Pariser–Parr–Pople model Hamiltonians. The full and various limited configuration interaction (CI ) correlation energies and wave functions are calculated exploiting the unitary group approach. The formalism for the cluster analysis of the exact wave function expressed through the unitary group formalism electronic Gelfand states is developed and applied to the full CI wave functions of the cyclic polyene models studied. It is shown that the connected tetraexcited clusters become essential in the fully correlated limit and that their contribution also significantly increases with electron number even for the coupling constant corresponding to the spectroscopic parametrization of the model Hamiltonians used.     

Hydrogen bonding in simple π-electron systems I. Pyridinium–pyridine

In this work, the application of the Pariser–Parr–Pople scheme to hydrogen-bonded systems containing π electrons has been examined. The potential energy curves for the movement of the hydrogen atom in the bond have been calculated, and the reliability of the results obtained from this method, applied in this manner, have been discussed.     

The relationship between the unrestricted and projected Hartree–Fock methods in a simple three-electron model system

Various forms of the unrestricted Hartree–Fock (UHF ) scheme are studied for a simple three-electron model system, represented by the PPP (Pariser–Parr–Pople) π-electron model of the allyl radical. Both spin and space symmetry are violated in the UHF trial wave function, either individually or simultaneously. A comparison with the projected Hartree–Fock (PHF ) schemes studied earlier is made and the effect of the order in which various symmetries are broken in both UHF and PHF schemes is studied. The effectiveness of various schemes follows from a comparison of the correlation energy and the wave function is obtained by various UHF (or projected UHF ) and PHF schemes, in the whole range of the coupling constant, with the corresponding quantities given by the exact solution of the model. Finally, the implications of the stability of the restricted HF solutions for the behavior of various single- and multiparameter UHF and PHF schemes are briefly outlined and exemplified on the studied model.     

Formulation of the coupled cluster theory with localized orbitals in correlation calculations on polymers

It is shown that in the framework of coupled cluster theory both the correlation energy per unit cell and quasi-particle band structures of polymers can be computed directly from matrix elements of the excitation operator and the two-electron integrals calculated in localized orbital basis. Further, it is described how to thake advantage of the localized nature of the orbitals applied. Ab initio test calculations on a finite model system similar to the Pariser–Parr–Pople Hamiltonian are presented.     

Symmetric and symmetry-broken Hartree–Fock solutions of one-dimensional models

Pariser–Parr–Pople Hartree–Fock crystal orbital calculations (both restricted and unrestricted versions) are performed for one-dimensional models of π-electron polymers. The π-electron band structures corresponding to symmetric and nonsymmetric solutions have been calculated. To investigate the influence of the form of the interelectronic interaction, the calculations were performed with the Mataga–Nishimoto (MN ) formula and a modified Mataga–Nishimoto (MMN ) formula for two-electron integrals. We have found that curves of the points of the minimum total energy per unit cell for the alternating models are very similar, but band structures and energy gaps are quite different when the MN formula was substituted with the MMN formula for two-electron integrals.     

π-Electron SCF MO calculations for disubstituted benzene derivatives containing two donor groups

The π-electronic structures and spectra of all possible disubstituted benzene derivatives containing fluoro, chloro, hydroxy, and amino groups have been calculated by the Pariser—Parr—Pople method. For all compounds the same starting parameters determined from the best fits to the spectra of the monosubstituted derivatives were used. The calculations give good results for singlet transition energies as well as for the ground state charge densities and bond orders.     

Full configuration interaction study of the ground state of closed-shell cyclic PPP polyenes

Full configuration interaction (FCI ) calculations are reported for the closed-shell cyclic polyenes C_NH_N (N = 6, 10, 14, 18) in the Pariser–Parr–Pople (PPP ) approximation, as a function of the hopping parameter β. A wide range of values of β is considered, from a highly correlated situation, β = 0, to a very weakly correlated limit, β = ?10 eV. An estimate of the role of higher than two-body connected cluster components was done, through a partial (i.e., limited to two-body terms in the exponent) cluster analysis performed on the FCI wave function. The comparison with the approximate coupled pair theory that accounts for quadruply excited clusters [see P. Piecuch and J. Paldus, Theor. Chim. Acta 78 , 65 (1990)] shows a good agreement in the whole range of the hopping parameter, particularly when the contribution of connected triples excitations is also taken into account. © 1994 John Wiley & Sons, Inc.     

Application of the many-body perturbation theory by using localized orbitals

Diagrammatic formulation of the many-body perturbation theory is investigated when both the occupied orbitals and the virtual ones are localized, i.e., they are unitary transforms of the canonical Hartree–Fock orbitals. All diagrams representing ground state correlation energy can be generated through fifth order. For cyclic polyenes C₆H₆ and C₁₀H₁₀ as model systems, the energy corrections are calculated in the Pariser–Parr–Pople approximation for a wide range of the coupling constant β^?1, through fourth order including some fifth order terms. The results are compared to those obtained by other methods: perturbation theory by using canonical orbitals and full CI. The effect of neglecting contributions from orbitals localized into neighboring sites is also studied.     

A generalized formula for the energies of alternant molecular orbitals. I. Homonuclear molecules

Using the method of alternant molecular orbitals (AMO ) it is shown that the energies of AMO 's (E_k), for any alternant homonuclear molecule having a singlet ground state, are connected with the energies of the MO 's (e_k) obtained by the conventional Hartree–Fock (HF ) method by the formula \documentclass{article}\pagestyle{empty}\begin{document}$ E_{k\alpha (\beta )} = \pm \sqrt {\Delta ^2 + e_k ^2 } $\end{document}, where Δ is the correlation correction. The formula is applicable in the semiempirical LCAO form used in the Pariser–Parr–Pople theory, by Hubbard's approximation of γ integrals.     

Midgap levels of photoexcited conductive polymers. II. Detailed analysis of trans-polyacetylene

We have studied in more detail aspects of the soliton-antisoliton pair in a trans-polyacetylene chain based on our previous study of the midgap levels associated with the photogenerated oppositely charged soliton-antisoliton pair in conductive polymers employing the concept of the molecular orbital interaction (Part I of this study, this issue). The intersoliton distance has been estimated to be about 10 Å from the Pariser–Parr–Pople method. We have found that the energy gap between the midgap levels is estimated to be 0.45 eV, being significantly related to an additional photoinduced absorption.     

Theoretical investigation of the π -electronic structure and spectra of protonated aromatic carbonyl compounds

The π-electronic structure and spectra of the protonated aromatic carboxylic acids, aldehydes and ketones have been calculated by the Pariser–Parr–Pople method. An essential modification was that the positive charge has been considered as delocalized within the substituent group. The best agreement for the cation of benzoic acid was obtained using a symmetric carboxy model with equal charges on both oxygen atoms. This model gave equally good results for the protonated fluoro-, chloro- and methylbenzoic acids, as well. The delocalized charge model was successfully applied in the calculation of aromatic aldehydes and ketones. The methyl group was treated both as a one- and as a two-centre substituent.     

MO-LCAO-Calculations on polymethines. V. PPP-type calculations and configuration analyses of simple prototype dyes in terms of molecular subsystems

In order to elucidate certain controversies in interpreting the π-electronic structure of some simple quinone and indigo dyes the Pariser–Parr–Pople SC _β,γ-wave-functions have been subjected to configuration analyses. Whereas 2,5-diamino-quinone (1) can be excellently represented by coupling of two trimethine-merocyanine chains, the analogous consideration is less appropriate with bispyrrolindigo (2). In this case the results of the configuration analyses indicate clearly the limited applicability of the Longuet/Higgins–Murrell–method.     

Localized bond model for molecular energy to fourth order in perturbation theory

The localized bond model of Malrieu, Diner, and Claverie is extended to fourth order in perturbation theory. Single, double, triple, and quadruple replacements from the doubly occupied bonding reference function are included utilizing a symmetric form of diagrammatic perturbation theory. The fourth order theory derived executes on a computer as quickly as does the third order theory. Results are examined utilizing the Pariser–Parr–Pople and CNDO/2 model Hamiltonians, and are compared with third order results and with either exact results where they are known, or with a configuration interaction of all singles and doubles. The influence of the initial hybridization, localization, and bond polarization is discussed. In general, the fourth order corrections are of comparable size to third order. Improvement in results appears to be marginal in the Nesbet–Epstein scheme in passing to fourth order because of the oscillating nature of the series; for Moller–Plesset theory errors are approximately halved. The relative energies as a function of modest geometry change about minima is about the same at third order as it is at fourth for most cases examined.     

Correlation effects in the low–lying excited states of the PPP models of alternant hydrocarbons. I. Qualitative rules for the effect of limited configuration interaction

Simple rules for an estimate of the correlation effects in the low-lying states of alternant hydrocarbons, as described by the Pariser–Parr–Pople Hamiltonian, are formulated. These rules are based on the alternancy and spin symmetry classification of states in both strongly and weakly correlated limits and on the valence bond characteristics of those states in the fully correlated limit. It is shown that the largest effect of the electron correlation will be found for the singlet “minus” states (using Pariser's classification of the alternancy symmetry species), a smaller effect for the triplet “plus” states, and a much smaller effect for the remaining states. These rules are exemplified by limited CI calculations including all monoexcited and all mono- and bi-excited configurations, respectively, for a number of π-electronic systems. In view of these rules the success of the PPP model in the monoexcited CI approximation may be understood.     

The low-lying excitation energies of polyenes investigated with a chain-length-dependent screened potential

The chain-length dependencies of 2Ag and 1Bu excitation energies as well as their unexpected inversion observed experimentally in trans-polyenes have so far been explained satisfactorily only in terms of configuration interactions within the standard Pariser–Parr–Pople (PPP ) parametrization scheme with at least double excitations involving prohibitively large computational labor for long polyenes. A simpler calculation allowing nonzero differential overlap and employing restricted (first-order) single-excitation configuration interaction with chain-length-dependent screened potential is shown to provide an adequate alternative for the studies of those basic spectroscopic features of polyene excited states. The screening factor is parametrized in accordance with the chain-length-dependent behavior of the electrical polarizabilities of polyenes derived within the standard PPP approximation scheme. © 1993 John Wiley & Sons, Inc.     

Relation of the charge and spin correlation functions to the structures of π electron wave functions in alternant hydrocarbons

We analyze, in the Pariser–Parr–Pople (PPP ) model of alternant hydrocarbons, how the charge and spin correlation functions (CF 's) are related to the structure of a CI wave function and the MO 's of the systems. The analysis is based on the fact that an uncorrelated electron present in an orbital does not contribute to the linked dynamically correlated parts of the CF 's. By using the fact, simple rules are deduced predicting the covalent or ionic nature of the correlation structure in a low-lying state with two correlated electrons. The rules predict that the singlet minus and triplet plus states are covalent, while singlet plus and triplet minus states are ionic, where the plus and minus mean the alternancy symmetry. The rules also give a prediction for the unknown charge and spin correlation structures between different sites.     

Hydrogen bonding in simple π-electron systems II. Pyridine–pyrrol

Previous work in this laboratory concerning the properties of hydrogen bonds in the base pairs of DNA [1–6] has led to considerable interest in the properties of hydrogen bonds in π-electron systems. The first paper in this series [7] has investigated the usefulness of the LCAO –MO –SCF method and the Pariser–Parr–Pople approximation as applied to this problem, by calculation on the ideal pyridine–pyridinium complex. In this paper, a relation with experiment will be established by comparison of the results obtained from this method of calculation with the properties of the experimentally observable pyridine–pyrrol system.