Efficient representation of the linear density‐density response function

Abstract

We present a thorough derivation of the mathematical foundations of the representation of the molecular linear electronic density-density response function in terms of a computationally highly efficient moment expansion. Our new representation avoids the necessities of computing and storing numerous eigenfunctions of the response kernel by means of a considerable dimensionality reduction about from 10 3 to 10 1 . As the scheme is applicable to any compact, self-adjoint, and positive definite linear operator, we present a general formulation, which can be transferred to other applications with little effort. We also present an explicit application, which illustrates the actual procedure for applying the moment expansion of the linear density-density response function to a water molecule that is subject to a varying external perturbation potential.

Reduced eigensystem representation of the linear density‐density response function

“…In this section, we recall the specific notation for the expression of linear operators that we recently introduced. [50] The derivation closely follows the previous version, but differs in the final expression because the basis functions are now allowed to be non-orthogonal for the image as well as the domain of the linear operator.…”

“…We recently generalized the moment expansion to any compact, self-adjoint, and positive-definite linear operatorT : [50] Assuming a finite (orthonormal) basis {| P 1 i, | P 2 i, …, | P N i} for a subspace of the domain ofT, we showed recently that there is a specific basis set {| ξ 1 i, | ξ 2 i, …, | ξ N i} for the (finite) image ofT which fulfills a partial orthogonality condition with respect to the domain basis:…”

“…In a recent paper, we demonstrated that the density response of a molecule due to the perturbing potential of a neighboring molecule can be described within a few moment expanded states. [50] In this article, we reuse the design of the experiment in order to compare the convergence of the density response with respect to different numbers of employed moment expanded and reduced eigenstates: We choose the perturbative effect of one water molecule on an adjacent one as an elementary example. For simplicity, we use the Hartree potential using the partial charges of one of the most common force field water models (TIP3P) as perturbation.…”

“…[29,30,[47][48][49][50] In particular, we generalized these theorems to arbitrary linear, self-adjoint, positive-definite, and compact operators. [50] Within this article, we derived the reduced eigensystem representation as a completion of our development effort toward an efficient representation combined with an efficient determination of the static linear density-density response function. Due to its truncated eigensystem like shape, the enormous reduction of the dimensionality becomes obvious in comparison to the regular eigensystem representation.…”

“…[47] In order to tackle this problem, we developed a more efficient, iterative algorithm (referred to as direct moment expansion), which needs a single DFPT calculation per moment expanded state. [49,50] F I G U R E 1 Principal illustration of the response density of the water molecule (right) due to a perturbing water molecule (left). The potential originated from the left water molecule can be expanded at the responding (right) water molecule within a few basis functions…”