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International Journal of Quantum Chemistry

Reduced density matrices for valence-bond wave functions

Walter R. Thorson, J. H. Choi, R. B. Hake

Published Jun 18, 2009• 6 citations
Publisher

Abstract

First and second-order reduced density matrices, and their eigenvalues and eigenvectors, have been obtained for resonating valence-bond wave functions for N-membered rings of atoms with one orbital site and one electron per atom (one-dimensional lattice system). Such a wave function offers a good description of the electron correlation which dominates behavior at very low electron density (large atomic spacing). Results are compared briefly with the amo model.

DOI
10.1002/qua.560010654

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The Journal of Chemical Physics•Apr 26, 2013•39 citations
First and second-order reduced density matrices, and their eigenvalues and eigenvectors, have been obtained for resonating valence-bond wave functions for N-membered rings of atoms with one orbital site and one electron per atom (one-dimensional lattice system). Such a wave function offers a good description of the electron correlation which dominates behavior at very low electron density (large atomic spacing). Results are compared briefly with the amo model.
A systematic procedure is described for calculating the reduced density matrices of first and second order for a resonating valence bond wavefunction on square planar nets with Born-von Karman boundary conditions of finite period. The density matrices explicitly display the electron correlation in such a wavefunction, which is a model ground state for a two-dimensional Mott insulator. Because of the extremely large number of contributing valence bond structures, it is essential to utilize their symmetry under the space group for minimum computation. For the totally symmetric singlet state described, density matrix eigenvectors form basis vectors for irreducible representations. The irreducible representations spanned by pair states on the square plane lattice are listed, and a set of irreducible basis vectors for the two-particle density matrix is defined which clearly displays the nature of electron pair correlations in such a system (correlated geminals). The apparatus has been employed in a calculation described in the paper accompanying this one (succeeding article).
One- and two-particle reduced density matrices have been calculated for a ``resonating'' Heitler-London wavefunction, in a system which is a finite model of a two-dimensional square planar lattice structure (4×4 net of one-electron atoms with Born-von Karman boundary conditions). The Heitler-London wavefunction describes the electronic ground state of a system with a single narrow half-filled band (the Mott insulator). Various measures of electron correlation (pair correlation functions, etc.) are defined and calculated. The intent of these calculations is to provide ``empirical'' guidelines for many-body methods for calculation of Green's functions for one and two particles (reduced density matrices are initial values for the respective Green's functions). Results show that the Heitler-London description is incompatible with several assumptions and widely used models for the Mott insulator. In particular it is found that (a) pair correlations in the Heitler-London ground state remain local even when ``resonance'' of an extremely large number of valence-bond structures occurs; the long range antiferromagnetic ordering obtained in several models of the Mott insulator (the ``two-sublattice'' or ``different bands for different spins'' models) does not occur. (b) The density matrix for the Heitler-London state does not exhibit a BCS type ``pair condensation'', as has been conjectured to describe the Mott insulator in some recent work. Our results strongly suggest that if the Heitler-London description is appropriate, the ``Hubbard Hamiltonian'', which includes only intraatomic pair interactions, does not offer an adequate model of the Mott insulator.
One-electron wave functions of half-filled-band closed-shell systems have been decomposed into their orthogonal single-determinant valence-bond (VB) components (antisymmetrized products of orthogonal atomic orbitals). Despite the ordering effect of the Fermi hole among electrons of the same spin, favoring spin-density and charge-density waves, the relative independence between the \ensuremath{\alpha} and \ensuremath{\beta} parts of the wave function results in an overestimation of the weight of multiply ionic structures. Comparison is made with the internally correlated functions for different graphs and Hamiltonians (Hubbard, Pariser-Parr-Pople, and ab initio types). The scaling effect of the correlation does not appear as essentially dependent on the ionicity of the VB forms but exponentially linked to the mean energy of each VB component.
In this thesis we develop the variational two-electron reduced-density-matrix method for extended systems. Extended systems are represented in two ways: i) lattice models describing the dominant valence electronic structure with periodic boundaries to account for their extended nature and ii) a crystalline-orbital basis built from atomic orbitals using the generalization of molecular orbital theory to polymers. The first part of this thesis (Ch. 3-4) examines the performance of the variational 2-RDM method on lattice systems with tunable electron correlation. The first of these systems is the classic Hubbard model with linear and ladder lattice topologies. Because electron correlation functions, such as charge- and spin-ordering, are linear functions of the 2-RDM, the difference in electronic structure between one- and quasi-one-dimensional systems is accurately characterized. The second model contains only two-body interactions and is unique among typical spin models in that it does not have a mean-field reference wave function. The ground state wave functions from all Hamiltonians in the model have the same 1-electron reduced density matrix; consequently, one-electron theories are largely inapplicable. The superconducting eta-pairing ground states make the model a unique tool for demonstrating the necessary N-representability in highly correlated environments. The second part of this thesis (Ch. 5-6) develops a formalism for modeling materials by solving the full Schrodinger equation. Crystalline-orbital Hartree-Fock provides a set of orbitals and integral tensors for the variational 2-RDM method. We demonstrate that time-reversal symmetry, which is implicitly included in position space electronic structure calculations, must be explicitly included as an N-representability constraint on the 2-RDM when using a momentum space basis. The necessity of these equality constraints is demonstrated by the accurate recovery of the binding energy of two polymers and the symmetry of their natural orbital occupations.
Numerical results are presented for a generalized resonance-valence-bond state which includes both ionic and covalent contributions in each bond; non-nearest-neighbor bonds are also considered. Variational calculations have been performed and the space-group symmetry has been taken into account. The results for the Hubbard Hamiltonian are compared with the exact ones for the 2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2 and the 4\ifmmode\times\else\texttimes\fi{}4 lattices. The agreement is quite satisfactory for large values of the interaction UUU (for U=40U=40U=40 the overlap with the exact ground-state wave function is 0.999). Several correlation functions are also compared.
In this series, the n-body reduced density matrix (n-RDM) approach for nonorthogonal orbitals and their applications to ab initio valence bond (VB) methods are presented. As the first paper of this series, Hamiltonian matrix elements between internally contracted VB wave functions are explicitly provided by means of nonorthogonal orbital based RDM approach. To this end, a more generalized Wick's theorem, called enhanced Wick's theorem, is presented both in arithmetical and in graphical forms, by which the deduction of expressions for the matrix elements between internally contracted VB wave functions is dramatically simplified, and the matrix elements are finally expressed in terms of tensor contractions of electronic integrals and n-RDMs of the reference VB self-consistent field wave function. A string-based algorithm is developed for the purpose of evaluating n-RDMs in an efficient way. Using the techniques presented in this paper, one is able to develop new methods and efficient algorithms for nonorthogonal orbital based many-electron theory much easier than by use of the first quantized formulism.