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arXiv:1807.05299 · Physical Review A

Orthogonalization of fermion k -body operators and representability

Volker Bach, Robert Rauch

Published Apr 9, 2019• 2 citations
arXivHTMLPDF

Abstract

The reduced $k$-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of $k$-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of $k$-body operators for all $k$.

DOI
10.1103/physreva.99.042109

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The reduced kkk-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of kkk-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of kkk-body operators for all kkk.
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of ppp particles for each value of ppp. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the NNN-representability problem is given, and two sufficient conditions and one necessary condition for NNN representability are given in terms of distances and norms.
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.
A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.
The N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy consisting of embedding an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices, and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both, simple models as well as molecular systems at finite temperature, demonstrating its robustness.
We demonstrated in the previous chapter (Chapter 1) that the evaluation of matrix elements between states represented by the Slater determinant can be evaluated unambiguously using Slater’s rule [1, 28, 40]. While the determinantal representation and the evaluation matrix elements between the states are quite straightforward, this procedure can eventually be ineffective and cumbersome with increasing size of the system. The occupation number formalism or second quantization technique introduced by Dirac in quantum field theory, in this context has been found to be quite formidable and elegant. This formalism not only allows us to represent the states and the operators in a compact and convenient form but also enables us to deal with systems containing variable numbers of particles without explicitly specifying the number of particles. It implicitly assumes the existence of an unspecified number of functions in a single-particle basis. These one-particle functions, which constitute the Hilbert space for N particles, are generally chosen to be spinorbital because of their orthogonality. In the case of fermions, the many-particle basis is the anti-symmetric product of N one-particle basis functions (Slater determinant). The vector space consisting of all possible anti-symmetric Hilbert spaces is referred to as the Fock space, in which the operators and functions are represented without specifying the number of particles.
A Gaussian operator representation for the many-body density matrix of fermionic systems, developed by Corney and Drummond [Phys. Rev. Lett. 93, 260401 (2004)], is used to derive approximate decoupling schemes for their dynamics. In this approach the reduced single electron density-matrix elements serve as stochastic variables which satisfy an exact Fokker-Planck equation. The number of variables scales as $\ensuremath{\sim}{N}^{2}$ rather than $\ensuremath{\sim}\text{exp}(N)$ with the basis set size, and the time-dependent Hartree-Fock approximation (TDHF) is recovered in the ``classical'' limit. An approximate closed set of equations of motion for the one- and two-particle reduced density matrices provides a direct generalization of the TDHF.