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Physical Review Letters

Idempotency-Conserving Iteration Scheme for the One-Electron Density Matrix

Dóra Kőhalmi, Ágnes Szabados, Péter R. Śurján

Published Jul 1, 2005• 14 citations
Publisher

Abstract

For the first order density matrix P of a noninteracting N-electron problem, an iterative formula is presented that preserves the trace and idempotency of P so that no purification is needed. Hermiticity--which may be slightly violated in the course of the iteration--gets restored when the iteration converges and the converged P corresponds to the exact solution. For sparse P, the energy is obtained by an O(N) procedure that needs no prior knowledge of the chemical potential. Illustrative calculations in tight-binding and ab initio Hartree-Fock levels are presented.

DOI
10.1103/physrevlett.95.013002

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For the first order density matrix P of a noninteracting N-electron problem, an iterative formula is presented that preserves the trace and idempotency of P so that no purification is needed. Hermiticity--which may be slightly violated in the course of the iteration--gets restored when the iteration converges and the converged P corresponds to the exact solution. For sparse P, the energy is obtained by an O(N) procedure that needs no prior knowledge of the chemical potential. Illustrative calculations in tight-binding and ab initio Hartree-Fock levels are presented.
A Lagrangian formulation for the constrained search for the N-representable one-particle density matrix based on the McWeeny idempotency error minimization is proposed, which converges systematically to the ground state. A closed form of the canonical purification is derived for which no a posteriori adjustment on the trace of the density matrix is needed. The relationship with comparable methods is discussed, showing their possible generalization through the hole-particle duality. The appealing simplicity of this self-consistent recursion relation along with its low computational complexity could prove useful as an alternative to diagonalization in solving dense and sparse matrix eigenvalue problems.
Two methods are suggested for performing ground-state electronic-structure calculations within the independent-electron approximation. Both methods involve the purification of a specified initial density matrix, which can be done either canonically (at a fixed electron count Ne){N}_{e})Ne​) or grand canonically (at a fixed chemical potential \ensuremath{\mu}). Linear system-size scaling is achieved either way, as we illustrate in example tight-binding calculations on carbon nanotubes.
Two possible routes are considered to arrive at a one-particle reduced density matrix formulation of electronic structure theory. In the first scheme, an extended Fock matrix H is defined that has twice the dimension of the one-particle basis set. The corresponding Green’s function, defined as the upper left block of (ω1−H)−1, yields the exact one-particle density matrix and energy. The poles of the Green’s function are precisely the ionization potentials and electron affinities of the extended Koopmans theorem. In the second scheme, a generalized Fock equation [F(ρ),ρ]=X is derived that is satisfied by the exact non-idempotent one-particle density matrix. The antisymmetric matrix X on the right-hand side is obtained from the irreducible part of the two-particle reduced density matrix, while F is the usual Fock matrix defined using the correlated one-matrix. The generalized Fock equation is a necessary condition but does not determine ρ uniquely. Alternatively, the one-matrix can be obtained from the irreducible part of the two-matrix directly, using a sum rule. The analysis leads to some additional desiderata and separability properties that may be imposed on traditional wave function based approaches. Possibilities for practical computational schemes are addressed briefly.
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.
To help simplify the many-electron problem, a new algorithm is presented which enables the generation of approximate ground-state electron densities and approximate ground-state energy differences of isoelectronic processes for interacting and noninteracting systems. This is accomplished starting exclusively from knowledge of the following: one isoelectronic ground-state ``seed'' density ${\ensuremath{\rho}}_{A}$, the external potential vA{v}_{A}vA​ which generates ${\ensuremath{\rho}}_{A}$, and the external potential vB{v}_{B}vB​ for which the ground-state energy EB{E}_{B}EB​ and density ${\ensuremath{\rho}}_{B}$ are sought. Use is made of a matching theorem in conjunction with trial first-order density matrices (one-matrices), the virial theorem, and a variation of a Legendre-transform idea. Furthermore, knowledge of the exact energy EA{E}_{A}EA​ enables the approximation of EB{E}_{B}EB​ using either one of two energy-difference formulas which are approximations to the exact integrated Hellmann-Feynman energy difference. Initial tests for noninteracting electrons are constructed.
Repeated computations on the same molecular system, but with different geometries, are often performed in quantum chemistry, for instance, in ab-initio molecular dynamics simulations or geometry optimizations. While many efficient strategies exist to provide a good guess for the self-consistent field procedure, which is usually the main computational task to be performed, little is known on how to efficiently exploit in this direction the abundance of information generated during the many computations. In this article, we present a strategy to provide an accurate initial guess for the density matrix within the self-consistent field iterations for parametrized Hartree-Fock problems where the nuclear coordinates are parametrized by a few user-specified parameters such as normal modes or collective variables in general. Our approach is based on an offline-stage where the Hartree-Fock eigenvalue problem is solved for some particular parameter values and an online-stage where the initial guess is computed very efficiently for any new parameter value. The method is based on non-linear approximations of density matrices, which belong to a non-linear manifold that is isomorphic to the Grassmann manifold, by means of the so-called Grassmann exponential and logarithm in order to map the manifold onto the tangent space and thus providing the correct geometrical setting accounting for the manifold structure when working with subspaces rather than functions itself. Numerical tests on different amino acids show promising initial results.