Reduction of multivariate mixtures and its applications

Adaptive algorithm for electronic structure calculations using reduction of Gaussian mixtures

“…Conceptually our method is simpler and, also, less technical than either [19,31,32] or [4] and, as far as we know, is novel. Clearly, the key to enabling our approach is the reduction algorithm described and analyzed in [12]. This reduction algorithm has complexity O r 2 N + p (d) rN , where N is the initial number of terms, r is the number of skeleton terms and p (d) is the cost of computing the inner product between the terms of a Gaussian mixture (p (d) is a small constant in dimension d = 3).…”

“…We observed that, within several additional iterations, the orbital energies improved by gaining additional accurate digits. However, as pointed out in [12], the quadruple precision implementation is more than 10 times slower than the double precision version (if the same accuracy is required). We do not include these results since, by modifying the reduction algorithm, it should be possible to avoid using quadruple precision.…”

“…Our approach is novel since we do not select a basis in advance and let the iteration for solving a system of integral equations identify the necessary functions to achieve a desired accuracy. The reduction algorithm developed in [12] selects the "best" subset of linearly independent terms after each operation that generates a large number of terms. These terms are Gaussians (identified by their parameters) and, since potentials and Green's functions are also represented via Gaussian mixtures, all integrals are evaluated explicitly.…”

“…We present a new adaptive method for electronic structure calculations based on algorithms for reduction of multiresolution multivariate mixtures [12]. While we represent solutions using a linear combination of Gaussians (which has a long history in quantum chemistry), these Gaussians are not selected in advance to be used as an approximate basis but are generated in the process of solving equations.…”

“…Using Gaussian geminals to improve accuracy has a long history in quantum chemistry (see [23,2] on this topic and references therein). We note that incorporating geminal functional forms into the adaptive approach of this paper appears possible (but by no means trivial) since the reduction algorithm for multivariate mixtures discussed in [12] (and on which the approach of this paper is based) is applicable to much more general functions than those used in our two examples. We also note that a different approach to incorporate cusps and to address basis set error was recently proposed in [17] by using a range-separated DFT formulation.…”