Nonorthogonal density-matrix perturbation theory

Abstract

Density matrix perturbation theory [Phys. Rev. Lett. 92, 193001 (2004)] provides an efficient framework for the linear scaling computation of response properties [Phys. Rev. Lett. 92, 193002 (2004)]. In this article, we generalize density matrix perturbation theory to include properties computed with a perturbation dependent non-orthogonal basis. Such properties include analytic derivatives of the energy with respect to nuclear displacement, as well as magnetic response computed with a field dependent basis. The non-orthogonal density matrix perturbation theory is developed in the context of recursive purification methods, which are briefly reviewed.

\mathcal{O}(N) methods in electronic structure calculations

“…When m = 1 the two polynomials are x 2 and 2x − x 2 ; this has become known as the TC2 (trace-correcting second order) method. It has been extended to spin-unrestricted methods [228] and non-orthogonal bases [229] (discussed more fully in Sec. 4.1; the main change to the algorithm is to require the overlap matrix in density matrix products, so P 2 ⊥ becomes P SP where P ⊥ is the density matrix in an orthogonal basis, and to require the inverse of the overlap for initialisation), and compared to LNV [191], with some advantage found especially for high and low filling (though it is important to note that these iterative methods are not variational, and so forces cannot be calculated using the HellmannFeynman theorem).…”

“…Density matrix perturbation theory [379] (which has been extended to nonorthogonal basis functions [229]) uses the trace-correcting TC2 method to generate a sequence of density matrices (X (0) n for the unperturbed Hamiltonian). The expansion X n = X (0) n + ∆ n allows a recursive expression for ∆ n+1 to be derived in terms of ∆ n and X (0) n .…”

Notes on density matrix perturbation theory

“…Within the same spirit, Larsen et al 28 followed by Coriani et al 29 have derived and implemented, respectively, the response equations using the asymmetric Baker-Campbell-Hausdorff expansion 30 for the auxiliary density matrix. Within the field of density matrix purifications, Niklasson, Weber, and Challacombe have introduced a recursive variant of the DMPT [31][32][33] based on the purification spectral projection method detailed in Ref. 34.…”

“…Consequently, energy derivatives up to the order (k + 1) involves knowledge of the density matrices up to the order k. A more popular approach, 27,[44][45][46][47][48] for computing energy derivatives for k > 2, relies on the (2k + 1) Wigner rule, which states that E (2k+1) can be obtained from the kth-order perturbed wave functions. 42 It is noteworthy that McWeeny et al, 10,12,18,19 later reported by Niklasson et al, [31][32][33] have adapted the (2k + 1)th theorem to perturbed density matrices as inputs, up to order 4 with respect to the energy, without any support of the wave functions.…”

Real‐Space and Multigrid Methods in Computational Chemistry

“…SIESTA utilizes a localized, numerical atomic-orbital basis, as do the Seijo-Barandiaran Mosaico 208 and Ozaki OpenMX codes, 209,210 while ONETEP employs plane-wave ideas to generate a localized psinc basis. [123][124][125] Earlier linear-scaling developments are thoroughly summarized by Goedecker. Other real-space representations based on Lagrange functions 71,72 or discrete variable representations (DVRs) [68][69][70]211,212 appear to offer advantages over the FD and FE methods.…”