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The Journal of Chemical Physics

Reduced Density Matrices for Valence Bond Wavefunctions. II. Symmetry and Reduction Procedures for Square Planar Nets

Jong H. Choi, R. P. Messmer, Walter R. Thorson

Published Jul 1, 1972• 3 citations
Publisher

Abstract

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).

DOI
10.1063/1.1677954

Related papers

Reduced Density Matrices for Valence Bond Wavefunctions. III. Density Matrices for the 4×4 Square Planar Net

Jong H. Choi, Walter R. Thorson

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.

The Journal of Chemical Physics•Jul 1, 1972•12 citations

Reduced density matrices for valence-bond wave functions

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

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.

International Journal of Quantum Chemistry•Jun 18, 2009•6 citations

THE VARIATIONAL TWO-ELECTRON REDUCED-DENSITY-MATRIX METHOD FOR EXTENDED SYSTEMS

Nicholas C. Rubin

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.

•Jan 1, 2016•0 citations

CONCERNING THE SYMMETRY PROPERTIES OF THE REDUCED DENSITY MATRIX AND THE NATURAL SPIN ORBITALS AND SPIN GEMINALS (THE NATURAL ONE- AND TWO-ELECTRON FUNCTIONS)

Werner Kutzelnigg

The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)

•Oct 1, 1963•0 citations

Density matrices and density functionals : proceedings of the A. John Coleman symposium

Alan Coleman, Robert Erdahl, Vedene H. Smith

A Tribute to A. John Coleman - The Tame Mathematician.- Reduced Density Matrices: 1929-1989.- Some Aspects on the Development of the Theory of Reduced Density Matrices and the Representability Problem.- Representability Conditions.- On the Diagonal N-Representability Problem.- Fermion N-Representability Conditions Generated by a Decomposition of the 1-Particle Identity Operator onto Mutually Orthogonal Projection Operators.- The Unitarily Invariant Decomposition of Hermitian Operators.- Building Up N-Electron States with Symplectic Symmetry.- Time Dependent Antisymmetrized Geminal Power Theory Using a Coherent State Formulation.- Griffiths Inequalities for Fermion Systems.- Entropy of Reduced Density Matrices.- A Lower Bound to the Ground State Energy of a Boson System With Fermion Source.- Reduced Density Operators, Their Related von Neumann Density Operators, Close Cousins of These, and Their Physical Interpretation.- Theory and Practice of the Spin-Adapted Reduced Hamiltonians (SRH).- Variational Principle with Built-In Pure State N-Representability Conditions. The N-Electron Case.- Wigner Distributions as Representations of the Density Matrix.- Inter-Relationships Between Various Representations of One-Matrices and Related Densities: A Road Map and an Example ..- Current Problems in Density Functional Theory.- The Interface Between Reduced Density Matrices and Density Functional Theory.- The Physics Underlying the Langreth-Mehl Scheme for Non-Uniform Systems.- Understanding Energy Differences in Density Functional Theory.- Density Functional Calculations of Molecular Bond Energies.- Non-Local Effects on Atomic and Molecular Correlation Energies Studied with a Gradient-Corrected Density Functional.- An Evaluation of Local Electron Correlation Corrections and Non-Local Exchange Corrections to the Hartree-Fock-Slater Method from Calculations on Bond Energies and Electronic Spectra of Molecular Systems.- Correlation Energy Functionals of One-Matrices and Hartree-Fock Densities.- Some Remarks on Scaling Relations in Density Functional Theory.- Deduction of Semiempirical MO Methods from Density Functional Theory.- Charge and Spin Densities in Molecular Solids: Local Density Functional Calculations Versus Experiment.- A Functional of the Two-Particle Density Matrix for the Approximate Calculation of the Electronic Correlation Energy.- Extracules, Intracules, Correlation Holes, Potentials, Coefficients and All That.- The Exact Schroedinger Equation for the Electron Density.- Adiabatic Separation, Broken Symmetries and Geometry Optimization.- Asymptotic Results for Density Matrices and Electron Density in Atoms and Nearly Spherical Molecules.- An Algorithm for Calculating Isoelectronic Changes in Energies, Densities, and One-Matrices.- Atoms and Ions in the Limit of Large Nuclear Charge.- Improved Thomas-Fermi Theory for Atoms.- A Bond Energy from Quantum Mechanics.- Measured Electron Densities and Band Structure Calculations.- X-ray Orthonormal Orbital Model for Crystallography.

D. Reidel Pub. Co. eBooks•Jan 1, 1987•19 citations

The Study of Model Hamiltonians on Finite Clusters.

Duan Chen

We present two methods to study model Hamiltonians defined on clusters. One is the symmetrization of the Hilbert space basis according to irreducible representations of the symmetry group of the Hamiltonian. The other one adapts the Lanczos diagonalization method directly to the second quantized form of the Hamiltonian studied. By the symmetrization of the basis, one takes advantage of the symmetries of the Hamiltonian, and uses the group theory to find the invariant subspaces of the Hamiltonian in the complete Hilbert space. Then one needs only to work within the invariant subspaces. This method cuts down substantially the size of the matrix representation of the Hamiltonian to be diagonalized. Therefore, one could study relatively larger systems. As a result, all the eigenvectors and eigenvalues are obtained. Consequently, one could calculate thermodynamical quantities and various correlation functions, such as, the spin correlation, the charge correlation, the superconductivity pairing correlation, and the photoemission spectra. One the other hand, the Lanczos method, mentioned here in particular, is a specially developed computer program for the study of the model Hamiltonians. This program uses the key feature of the Lanczos method in diagonalization, allows one to use the second quantized form of the Hamiltonian directly without pregeneration or storing the matrix elements of the Hamiltonian. Two methods could be combined to take the full advantage of the symmetries of the Hamiltonian by input of the symmetrized wavefunction as the starting trial wavefunction in the Lanczos method. Here we illustrate the above two methods by applying to the Hubbard, and extended Hubbard models on a simple cubic cluster, and the lattice Anderson model on the two dimensional square lattice. Other interesting results for different systems are also briefly discussed here.

Civil War Book Review•Jan 1, 1989•0 citations
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.
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.
The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)
A Tribute to A. John Coleman - The Tame Mathematician.- Reduced Density Matrices: 1929-1989.- Some Aspects on the Development of the Theory of Reduced Density Matrices and the Representability Problem.- Representability Conditions.- On the Diagonal N-Representability Problem.- Fermion N-Representability Conditions Generated by a Decomposition of the 1-Particle Identity Operator onto Mutually Orthogonal Projection Operators.- The Unitarily Invariant Decomposition of Hermitian Operators.- Building Up N-Electron States with Symplectic Symmetry.- Time Dependent Antisymmetrized Geminal Power Theory Using a Coherent State Formulation.- Griffiths Inequalities for Fermion Systems.- Entropy of Reduced Density Matrices.- A Lower Bound to the Ground State Energy of a Boson System With Fermion Source.- Reduced Density Operators, Their Related von Neumann Density Operators, Close Cousins of These, and Their Physical Interpretation.- Theory and Practice of the Spin-Adapted Reduced Hamiltonians (SRH).- Variational Principle with Built-In Pure State N-Representability Conditions. The N-Electron Case.- Wigner Distributions as Representations of the Density Matrix.- Inter-Relationships Between Various Representations of One-Matrices and Related Densities: A Road Map and an Example ..- Current Problems in Density Functional Theory.- The Interface Between Reduced Density Matrices and Density Functional Theory.- The Physics Underlying the Langreth-Mehl Scheme for Non-Uniform Systems.- Understanding Energy Differences in Density Functional Theory.- Density Functional Calculations of Molecular Bond Energies.- Non-Local Effects on Atomic and Molecular Correlation Energies Studied with a Gradient-Corrected Density Functional.- An Evaluation of Local Electron Correlation Corrections and Non-Local Exchange Corrections to the Hartree-Fock-Slater Method from Calculations on Bond Energies and Electronic Spectra of Molecular Systems.- Correlation Energy Functionals of One-Matrices and Hartree-Fock Densities.- Some Remarks on Scaling Relations in Density Functional Theory.- Deduction of Semiempirical MO Methods from Density Functional Theory.- Charge and Spin Densities in Molecular Solids: Local Density Functional Calculations Versus Experiment.- A Functional of the Two-Particle Density Matrix for the Approximate Calculation of the Electronic Correlation Energy.- Extracules, Intracules, Correlation Holes, Potentials, Coefficients and All That.- The Exact Schroedinger Equation for the Electron Density.- Adiabatic Separation, Broken Symmetries and Geometry Optimization.- Asymptotic Results for Density Matrices and Electron Density in Atoms and Nearly Spherical Molecules.- An Algorithm for Calculating Isoelectronic Changes in Energies, Densities, and One-Matrices.- Atoms and Ions in the Limit of Large Nuclear Charge.- Improved Thomas-Fermi Theory for Atoms.- A Bond Energy from Quantum Mechanics.- Measured Electron Densities and Band Structure Calculations.- X-ray Orthonormal Orbital Model for Crystallography.
We present two methods to study model Hamiltonians defined on clusters. One is the symmetrization of the Hilbert space basis according to irreducible representations of the symmetry group of the Hamiltonian. The other one adapts the Lanczos diagonalization method directly to the second quantized form of the Hamiltonian studied. By the symmetrization of the basis, one takes advantage of the symmetries of the Hamiltonian, and uses the group theory to find the invariant subspaces of the Hamiltonian in the complete Hilbert space. Then one needs only to work within the invariant subspaces. This method cuts down substantially the size of the matrix representation of the Hamiltonian to be diagonalized. Therefore, one could study relatively larger systems. As a result, all the eigenvectors and eigenvalues are obtained. Consequently, one could calculate thermodynamical quantities and various correlation functions, such as, the spin correlation, the charge correlation, the superconductivity pairing correlation, and the photoemission spectra. One the other hand, the Lanczos method, mentioned here in particular, is a specially developed computer program for the study of the model Hamiltonians. This program uses the key feature of the Lanczos method in diagonalization, allows one to use the second quantized form of the Hamiltonian directly without pregeneration or storing the matrix elements of the Hamiltonian. Two methods could be combined to take the full advantage of the symmetries of the Hamiltonian by input of the symmetrized wavefunction as the starting trial wavefunction in the Lanczos method. Here we illustrate the above two methods by applying to the Hubbard, and extended Hubbard models on a simple cubic cluster, and the lattice Anderson model on the two dimensional square lattice. Other interesting results for different systems are also briefly discussed here.