Higher-order response in O(N) by perturbed projection

Abstract

Perturbed projection for linear scaling solution of the coupled-perturbed self-consistent-field equations [Weber, Niklasson and Challacombe, Phys. Rev. Lett. 92, 193002 (2004)] is extended to the computation of higher order static response properties. Although generally applicable, perturbed projection is developed here in the context of the self-consistent first and second electric hyperpolarizabilities of three dimensional water clusters at the Hartree-Fock level of theory. Non-orthogonal, density matrix analogues of Wigner's 2n + 1 rule are given up to fourth order. Linear scaling and locality of the higher order response densities under perturbation by a global electric field are demonstrated.

Canonical density matrix perturbation theory

“…(8), which is required to provide a variationally correct description of the energetics. We have not been able to find any explicit density matrix expressions for Wigner's 2n + 1 rule [16,18,[51][52][53][54][55] that are valid also at finite temperatures. A more detailed derivation of Eq (10) is given in the appendix.…”

“…With the density matrix formulation it is easy to utilize matrix sparsity from electronic nearsightedness [6,18,43,44] and it allows direct calculations of observables. The effective single-particle density matrix, P , at the electronic temperature T e , can be calculated from the Hamiltonian, H, using a recursive Fermi operator expansion [45][46][47][48],…”

“…Linear scaling quantum perturbation theory has so far mainly concerned properties at zero electronic temperature. Here we extend the idea behind linear scaling density matrix perturbation theory [16][17][18][19] to calculations of static response properties valid also at finite electronic temperatures with fractional occupation of the states. Our proposed canonical density matrix perturbation theory, which is applicable within effective single-particle formulations, such as tight-binding, Hartree-Fock or Kohn-Sham density functional theory, can be applied to calculate temperature dependent response properties from the solution of the coupled perturbed self-consistent field equations [1,23,24] as in density functional perturbation theory [2,25].…”

“…Initially, the development of linear scaling electronic structure theory was aimed at calculating ground state properties and not until recently has the focus shifted towards the computationally more demanding task of calculating the quantum response. A number of approaches to a quantum perturbation theory with reduced complexity have now been proposed and analyzed [11][12][13][14][15][16][17][18][19][20][21][22]. Linear scaling quantum perturbation theory has so far mainly concerned properties at zero electronic temperature.…”

Molecular-orbital-free algorithm for excited states in time-dependent perturbation theory

“…Recently, a number of groups 29,30,31,32,33,34 have achieved a linear scaling computational complexity for the ground state self-consistent field (SCF) problem in Hartree-Fock (or density functional theory) using "fast" algorithms for computation of the Fockian F and sparse matrix algebra (dropping of small elements) to exploit quantum locality of the density matrix P . If the transition densities in the time-dependent response equations also demonstrates quantum locality, then the same fast methods used for the ground state problem are applicable.…”