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Journal of the Physical Society of Japan

Orbital Optimization in the Resonating Hartree-Fock Approximation and Its Application to the One Dimensional Hubbard Model

Atsushi Ikawa, Shoji Yamamoto, Hideo Fukutome

Published May 15, 1993• 42 citations
Publisher

Abstract

A tractable direct optimization algorithm is developed to optimize orbitals in the Slater determinants (S-dets) in a resonating Hartree-Fock (Res HF) wave function. We reduce the variation space using the orbitals to put the first order energy variation in the steepest descent direction. The orbitals in the next iteration are determined so as to minimize the energy functional including up to the second order variation. This algorithm is applied to the one dimensional Hubbard model of half-filling. The optimized S-dets much deviate from the trial S-dets prepared from the HF calculations. The Res HF ground state generated with a few S-dets explains from 99.9 to 95.0% of the ground state correlation energy in all the correlation regimes. We have spin correlation functions with the correct short and long range behaviors and the lowest triplet and singlet spin excitations with correct dispersions, suggesting that the optimization of orbitals incorporates long range spin fluctuations and their mode-mode couplings.

DOI
10.1143/jpsj.62.1653

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A tractable direct optimization algorithm is developed to optimize orbitals in the Slater determinants (S-dets) in a resonating Hartree-Fock (Res HF) wave function. We reduce the variation space using the orbitals to put the first order energy variation in the steepest descent direction. The orbitals in the next iteration are determined so as to minimize the energy functional including up to the second order variation. This algorithm is applied to the one dimensional Hubbard model of half-filling. The optimized S-dets much deviate from the trial S-dets prepared from the HF calculations. The Res HF ground state generated with a few S-dets explains from 99.9 to 95.0% of the ground state correlation energy in all the correlation regimes. We have spin correlation functions with the correct short and long range behaviors and the lowest triplet and singlet spin excitations with correct dispersions, suggesting that the optimization of orbitals incorporates long range spin fluctuations and their mode-mode couplings.
A resonating Hartree-Fock (Res-HF) method is revisited and improved by the complete spin projection for the interacting Fermion system. This method approximates a many-body wave function by the superposition of nonorthogonal Slater determinants (S-dets). The nonorthogonality of the S-dets makes it possible to describe the large quantum fluctuations efficiently, since each S-det naturally includes the full-electron-excitation effects from other S-dets. The molecular orbitals in every S-det, as well as the superposition coefficients, are variationally determined. So far, however, the spin contamination, caused by the conventional half-projection, has been a serious obstacle to obtain the accurate wave functions for large-size fermion systems, especially in the intermediate and large correlation regimes. In this paper, we apply the complete spin projection method to the Res-HF calculations. As an example, the improved Res-HF method is applied to the one-dimensional Hubbard model. It will be shown that the complete spin projection improves the Res-HF wave functions significantly. In fact, the correlation energies explained by the improved Res-HF method are better than those by the variational Monte Carlo method in all the correlation regimes. The correlation structures at arbitrary fillings are also well described by the improved Res-HF method.
The electronic structure of the strongly correlated one dimensional Hubbard model is calculated by the resonating Hartree-Fock (Res HF) method. In the strong correlation regime with the Coulomb repulsion U >∼6, SDW solitons localize almost on the central two sites and only one kind of neutral soliton pair has a much smaller energy than the other pairs. As quantum fluctuations in a periodic chain with 12 sites, we consider up to three diradical breathers consisting of the low energy neutral soliton pair. The Res HF ground state can explain 77.8% ( U =6) and 80.1% ( U =8) of the exact correlation energy. This is much better than the Gutzwiller method to use the SDW as the reference state that explains about 50% of the correlation energy. The breather fluctuations explain the rapid decreases in the short distance of the spin and staggered spin correlation functions. Other correlation functions and the excited states due to the quantum fluctuations are studied.
Quantum fluctuations in the one dimensional half filled Hubbard model are investigated with the resonating Hartree-Fock method with the orbital optimization. The optimized Slater determinants include an inhomo-geneous spin density wave and configurations accommodating some neutral soliton pairs. The short and long range spin orders are destroyed by quantum translational and breathing motions of soliton pairs and gapless Gaussian type fluctuations, respectively. Our calculation suggests that the lowest singlet and triplet excitation branches are the translational modes of breathing neutral soliton pairs dressed with the Gaussian type fluctuations.
We develop a new general theory for large quantum fluctuations in a Fermion many-body system that cannot be described by fluctuations around the Hartree-Fock ground state but arises from resonance of different correlation structures. We start with an exact coherent state representation of a Fermion system on a unitary group. We show that the Hamiltonian in the coherent state representation has a close connection with the Hartree-Fock energy functional. From this, we can derive a new approximation called the resonating Hartree-Fock approximation in which a state is approximated by a superposition of non-orthogonal Slater determinants with different correlation structures. We derive the variation equations to determine a resonating Hartree-Fock wavefunction. We show that the resonance between degenerate broken symmetry Slater determinants may partially recover the symmetry. We discuss how to choose trial Slater determinants in a resonating Hartree-Fock wavefunction. We suggest that resonance of Slater determinats representing localized defects, such as solitons, polarons and breathers, produced in the long range order of the HF ground state may be the most important content of large quantum fluctuations in condensed matter systems.
Abstract In conventional Hartree-Fock calculations for molecular systems the optimization of a trial wavefunction by energy minimization with respect to the orbitals of a determinantal wavefunction is relatively easy. When Jastrow factors are included with a determinant as for typical QMC trial functions the task becomes more difficult. Optimization by minimizing the variance in local energies has been one alternative. This paper describes another approach which is, in fact, an analog of the Hartree-Fock method and provides an exact solution to the problem of optimizing the orbitals when a Jastrow function is present.
The electronic structure of the one dimensional (1D) Hubbard model is calculated by the resonationg Hartree-Fock (Res HF) method. As local fluctuations in the spin density wave (SDW) in the half filled case, we consider a soliton-antisoliton pair making quantum translational and breathing motions, i.e, breathers. The breather explains 76.5% (U = 3) and 70.1% (U = 4) of the ground state correlation energy, which is better than the result of the Gutzwiller method, and gives low energy excited states with the symmetries and energetic ordering characteristic in polyenes. The fluctuations by the breather also explain the spin and density correlations.