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arXiv:1512.07236 · The Journal of Chemical Physics

Communication: Generalized canonical purification for density matrix minimization

Lionel A. Truflandier, Rivo M. Dianzinga, David R. Bowler

Published Mar 3, 2016• 24 citations
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Abstract

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.

DOI
10.1063/1.4943213

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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.
Generalizations of McWeeny's purification formula are developed within the formalism of the particle-hole duality from the theory of reduced density matrices. Each of the generalized purification formulas is expressed as a sum of the one-particle reduced density matrix (1-RDM) and a finite series in the product of the one-particle and the one-hole RDMs, a product which vanishes in the limit that the 1-RDM is idempotent. Two categories of purification formulas are explored: (i) formulas which treat the "occupied" and the "virtual" occupation numbers equivalently and (ii) formulas which treat these occupation numbers differently. The latter category includes and extends the purification formulas derived in the context of the 1,2-contracted Schrödinger equation [D. A. Mazziotti, J. Chem. Phys. 115, 8305 (2001)]. While the McWeeny purification minimizes the absolute error in the occupation numbers quadratically, the generalized purification formulas exhibit faster than quadratic convergence of the 1-RDM towards idempotency. Application of these purification formulas in existing algorithms for linear scaling will be explored and discussed including illustrative calculations on sodium wires of length 10, 20, 30, and 40 atoms.
A purification algorithm for expanding the single-particle density matrix in terms of the Hamiltonian operator is proposed. The scheme works with a predefined occupation and requires less than half the number of matrix-matrix multiplications compared to existing methods at low (10%) and high (>90%) occupancies. The expansion can be used with a fixed chemical potential, in which case it is an asymmetric generalization of and a substantial improvement over grand canonical McWeeny purification. It is shown that the computational complexity, measured as the number of matrix multiplications, essentially is independent of system size even for metallic materials with a vanishing band gap.
Despite the fact that idempotent one-particle reduced density matrices are pervasive in quantum chemistry, the understanding of a general energy functional of such idempotent density matrices for the ground state energy has been lacking. By a constrained search, we show the structure of the general functional, illuminating the contributions from various terms. For the examples of the "best idempotent density matrix" and Kohn-Sham idempotent density matrices, we contrast the functional forms and suggest how the best idempotent density matrix approach may be a good starting point for further development.
The notion of purification is generalized to treat correlated reduced density matrices. Traditionally, purification denotes the process by which a one-particle reduced density matrix (1-RDM) is made idempotent, that is, its eigenvalues are mapped to either 0 or 1. Purification of correlated RDMs is defined as the iterative process by which an arbitrary RDM is forced to satisfy several necessary N-representability conditions. Using the unitary decomposition of RDMs and the positivity conditions, we develop an algorithm to purify the 2-RDM. The algorithm is applied within the solution of the contracted Schrödinger equation CSE for the 2-RDM [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)]. Previous attempts to solve the CSE by powerlike methods have frequently produced divergent energies, but we show that the purification process eliminates the divergent behavior for systematic and reliable convergence of the contracted power method to the N-particle energy.
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.
An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.