Interpolant polynomial technique applied to the PPP model. II. Testing the interpolant technique on the Hubbard model

One is often led, in quantum mechanics, to a perturbative solution of an eigenvalue problem, which is defined by a given Hamiltonian. The perturbative series for the energy which results will be a function of a coupling constant which appears in the Hamiltonian. In this article, the perturbative series for the energy of a state of a cyclic polyene ring which are valid for the small and large coupling limit of the model are used to construct algebraic functions. These algebraic functions are defined in terms of polynomials which are given as a function of the energy variable and coupling parameter and can be solved to give the energy as a function of coupling. It is found that relatively small polynomials give very good agreement with the exact values and that the accuracy of the results increases rapidly as the degree of the polynomial increases. The final goal of this and subsequent articles is to study energy levels in PPP models of planar conjugated hydrocarbons. In this article, we test an interpolant technique on the case of the one-dimensional Hubbard model, where an exact solution can be obtained by solving a system of nonlinear equations. In the case of the Hubbard model, the correlation effects are overestimated. Therefore, if the technique works for the Hubbard model, it is reasonable to assume that the technique would work even better for the PPP model. © 1996 John Wiley & Sons, Inc.     

Correlation effects in the PPP model of alternant π-electronic systems: two-point Padé approximant approach

A general procedure, based on the two-point Padé approximant technique, is formulated, enabling an interpolation of the low-lying energy levels of the PPP (Pariser–Parr–Pople) Hamiltonian over the divergency gap separating the asymptotic perturbation expansions in both strongly and weakly correlated limits. The methods for the determination of the first few terms of the perturbation expansions in both limits are also briefly outlined. The procedure is applied to the five lowest singlet energy levels of the PPP model of the benzene molecule and the results are compared with the exact solutions for this model. The advantages as well as the shortcoming of this approach are pointed out and illustrated on the model calculation for benzene. The applicability of this technique to larger π-electronic systems is discussed and a feasible procedure for its implementation is formulated.     

Multiple solutions of coupled-cluster equations for PPP model of [10]annulene

Multiple (real) solutions of the coupled-cluster (CC) equations (corresponding to the CCD, ACP and ACPQ methods) are studied for the Pariser–Parr–Pople (PPP) model of [10]annulene, C₁₀H₁₀. The long-range electrostatic interactions are represented either by the Mataga–Nishimoto potential, or Pople’s R⁻¹ potential. The multiple solutions are obtained in a quasi-random manner, by generating a pool of starting amplitudes and applying a standard CC iterative procedure combined with Pulay’s DIIS method. Several unexpected features of these solutions are uncovered, including the switching between two CCD solutions when moving between the weakly and strongly correlated regime of the PPP model with Pople’s potential.     

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.     

Relation between BCS Hamiltonian and Ginzburg–Landau equation

It is intended to derive the Ginzburg–Landau (GL) equation directly from the Bardeen–Cooper–Schrieffer (BCS) Hamiltonian. By the use of the Hubbard–Stratonovitch transformation, the electron–electron interaction composed of four fermion operators is eliminated to yield an auxiliary boson field. This is an effective field in which electrons behave as if they were free. In applying the path integral method, the electron field is integrated out to remain the Lagrangian for this boson field. The symmetry breaking and the phase transition of the system described by this field are discussed, and it is shown that this boson field turns out to be the GL order parameter. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 693–703, 1998     

Behavior of coupled cluster energy in the strongly correlated limit of the cyclic polyene model. Comparison with the exact results

Perturbation expansion for the ground-state energy in the strongly correlated limit of cyclic polyenes, as described by the Pariser–Parr-Pople (PPP ) and Hubbard Hamiltonians, is examined. Focus is placed on the first- and second-order contributions in terms of the resonance integral, and the performance of the approximate coupled pair theory corrected for the connected quadruple excitations (ACPQ ), as well as ACPTQ and ACPQ + T (ACPQ ) methods, which are two different versions of ACPQ approximately accounting for the triexcited clusters, is studied. The second-order contribution obtained with the ACPQ, ACPTQ , and ACPQ + T (ACPQ ) approaches is then compared with the exact result for the fully correlated limit as provided by the diagonalization of the nearest-neighbor Heisenberg Hamiltonian. It is shown that all three coupled pair theories yield vanishing first-order contribution, in agreement with the exact result. The best result for the second-order contribution for the cyclic polyenes considered in this article is provided by the ACPQ + T (ACPQ ) method. In this respect, all three coupled pair theories seem to deteriorate as the polyene size approaches infinity, in spite of yielding excellent energies. Comparison is also made with the results of alternant molecular orbital method.     

Molecular and electronic structures of bipolaron in poly-para-phenylene in terms of molecular orbital symmetry

The molecular structures of the model systems of the polaron and the bipolaron in poly-para-phenylene (PPP) were calculated by an ab initio molecular orbital (MO) method with fairly sophisticated approximations. The calculated models are monocations, dications, monoanions and dianions of biphenyl, para-terphenyl, para-quaterphenyl, para-quinquephenyl and para-hexaphenyl. The calculations show that the longer the PPP oligomer is, the stronger is the tendency to take on a non-planar twisting structure. This was accounted for by the combination of repulsions between proximate ortho-hydrogen atoms with resonance interactions between benzene π MOs. The magnitude of the resonance interactions was assessed by using the symmetry of benzene π MOs as well as an analytical Hückel solution of the π MO for polyene. In addition, negatively charged polarons and bipolarons were found to have a stronger tendency to take on a planar structure than positively charged ones. This result was also explained by comparing the benzene π HOMO with the benzene π LUMO. Received: 30 June 1998 / Accepted: 9 September 1998 / Published online: 23 February 1999     

Semiempirical Hamiltonians for spatially confined π-electron systems

A study of π-electron systems confined by impenetrable surfaces is presented. The study results in a nonempirical-based approach to obtain confinement-adapted semiempirical π-Hamiltonians including repulsive terms (PPP or Hubbard). The impenetrable surface confinement of a physical system involves changes in the boundary conditions that the eigenvectors of its differential Hamiltonian operator have to fulfill, while the Hamiltonian itself remains unchanged. However, if this Hamiltonian is written in second quantization language, then confinement only involves changes of the Hamiltonian scalar factors (integrals). Semiempirical Hamiltonian integrals are replaced by parameters; therefore, confinement involves only changes of these parameters. It is shown that confinement changes Coulomb (α_i) and exchange (β_ij), while repulsion (γ_ij) parameters remain unaffected. Next, the influence of confinement upon the electron correlation of (i) π-electron molecular systems, (ii) atoms, and (iii) an electron gas is discussed. The behaviour of the correlation energy vs. the confinement size is found to be different for each type of system. A neat explanation of this variety is given in terms of the Coulomb attractive fields of the systems. Some chemical confinement effects such as an increase in the reactivity of π-electron systems is also outlined. © 1996 John Wiley & Sons, Inc.     

Intermediate Hamiltonian Fock-space multireference coupled-cluster method with full triples for calculation of excitation energies

The intermediate Hamiltonian multireference coupled-cluster (CC) method with singles, doubles, and triples within the excited (1,1) sector of Fock space (FS) is implemented and formulated to calculate excitation energies (EEs). Due to the intermediate Hamiltonian formulation, which provides a robust computational scheme for solving the FS-CC equations, coupled to an efficient factorization strategy, relatively large basis sets and model spaces are employed permitting basis set converged comparisons of the calculated vertical EEs, which can be compared to the experimental data for the N(2) and CO molecules. The issue of charge-transfer separability is also addressed.     

Spin alignment in organic high-spin molecules as studied by ESR and the generalized Hubbard Hamiltonian approach

In order to examine the role of topological symmetry in spin alignment of organic high-spin molecules, a novel hydrocarbon, 3,4′-diphenylmethylenebis(phenylmethylene) (3,4′-DPBPM), has beeen synthesized and characterized by single-crystal ESR. This molecule has a triplet ground state with a closely located quintet excited state in contrast to its topological isomer, 3,3′-DPBPM, with a non-degenerate septet ground state. UHF SCF calculation of the generalized Hubbard Hamiltonian satisfactorily predicts these results.     

Spin-projected EHF method. III. Applications to π-electron systems

The surely convergent procedure to obtain extended Hartree–Fock (EHF ) solutions of the spin-projected scheme, for which the equations are given in Part II, is applied to eleven π-electronic systems with 2–10 electrons at the PPP level of integral approximations. The method takes into account a considerable part of the correlation energy. The symmetry properties of the EHF wave functions obtained are discussed together with some computational details. A simplified algorithm is also described.     

Diagrammatic valence-bond theory for finite model Hamiltonians

Valence bond (VB ) diagrams form a complete basis for model Hamiltonians that conserve total spin, S, and have one valence state, ?_p, per site. Hubbard and Pariser–Parr–Pople (PPP ) models illustrate ionic problems, with zero, one, or two electrons in each ?_p, while isotropic Heisenberg models illustrate spin problems, with only purely covalent VB diagrams. The difficulty of nonorthogonal VB diagrams is by-passed by exploiting the finite dimensionality of the complete basis and working with unsymmetric sparse matrices. We introduce efficient bit manipulations for generating, storing, and handling VB diagrams as integers and describe a new coordinate relaxation method for the ground and lowest excited states of unsymmetric sparse matrices. Antiferromagnetic spin-½ Heisenberg rings and chains of N ? 20 spins, or 2^N spin functions, are solved in C₂ symmetry as illustrative examples. The lowest S = 1 and 0 excitations are related to domain walls, or spin solitons, and studied for alternations corresponding to polyacetylene. VB diagrams with arbitrary S and nonneighbor interactions are constructed for both spin and ionic problems, thus extending diagrammatic VB theory to other topologies.     

取代苯体系的二阶非线性光学性质:动力学李代数方法

提出了一种动力学李代数方法来研究取代苯体系的非线性光学性质. 对于给定的PPP模型(Pariser-Parr-Pople)哈密顿量, 生成了一个动力学李代数. 依据这些代数元构造出演化算子作为群参数的函数, 通过求解一组非线性微分方程能够得到这些群参数. 再按照统计力学中的密度算子公式给出取代苯分子体系偶极矩的统计平均值. 于是导出二阶极化率的表达式. 与其他量子力学计算结果比较, 表明这种动力学李代数方法在预言有机共轭分子的非线性光学性质上同样有用.     

A generalised 17-state vibronic-coupling Hamiltonian model for ethylene

In a previous work [B. Lasorne, M. A. Robb, H.-D. Meyer, and F. Gatti, "The electronic excited states of ethylene with large-amplitude deformations: A dynamical symmetry group investigation," Chem. Phys. 377, 30-45 (2010); and ibid. 382, 132 (2011) (Erratum)], we investigated the electronic structure of ethylene (ethene, C(2)H(4)) in terms of 17 dominant configurations selected at the multiconfiguration self-consistent field level of theory. These were shown to be sufficient to recover most of the static electron correlation among the first valence and Rydberg states at all geometries. We also devised a strategy to build a 17-quasidiabatic-state matrix representation of the electronic Hamiltonian for curvilinear coordinates using dynamical symmetry. Here, we present fitted surfaces in the form of a generalised vibronic-coupling Hamiltonian model for two nuclear coordinates, CC bond stretching and torsion. Dynamic electron correlation is included into the electronic structure to improve the energetics of the Rydberg states at the multireference configuration interaction level of theory. The chemical interpretation of the adiabatic states of interest does not change qualitatively, which validates our choice of underlying quasidiabatic states in the model. The absorption spectrum is calculated with quantum dynamics and partially assigned. This first two-dimensional model shows a surprisingly good agreement with the experimental spectrum.     

Multi-reference Fock space coupled-cluster method in the intermediate Hamiltonian formulation for potential energy surfaces

The effective and intermediate Hamiltonian multi-reference coupled-cluster (CC) method with singles and doubles for the doubly ionized (0,2) sector of Fock space (FS) is formulated and implemented. The intermediate Hamiltonian realization of the (0,2) FS problem provides a robust computational scheme for solving the FS-CC equations free from the intruder state problem. By introducing an efficient factorization strategy, we obtain a very efficient tool that can be used for computing double ionization potentials but more significantly to describe multi-reference problems in CC theory, illustrated by twisted ethylene and the potential energy curve for F(2). The latter separates smoothly to two F atoms, while the former avoids the cusp behavior at the 90° dihedral. We also explore the double ionization potentials for several small molecules, H(2)O, CO, C(2)H(2), and C(2)H(4).     

Convergence radii of the perturbation expansions for the ground-state energies of finite Hubbard models

The ground state energies of finite Hubbard molecules are calculated by numerically solving the Lieb–Wu equations for a complex Hubbard repulsion parameter U. From the positions of the singular points located in the complex plane, the radii of convergence of the perturbation expansions for the ground state energies are determined.