Adaptive algorithm for electronic structure calculations using reduction of Gaussian mixtures

We present a new adaptive method for electronic structure calculations based on novel fast algorithms for reduction of multivariate mixtures. In our calculations, spatial orbitals are maintained as Gaussian mixtures whose terms are selected in the process of solving equations. Using a fixed basis leads to the so-called basis error since orbitals may not lie entirely within the linear span of the basis. To avoid such an error, multiresolution bases are used in adaptive algorithms so that basis functions are selected from a fixed collection of functions, large enough to approximate solutions within any user-selected accuracy. Our new method achieves adaptivity without using a multiresolution basis. Instead, as part of an iteration to solve nonlinear equations, our algorithm selects the ‘best’ subset of linearly independent terms of a Gaussian mixture from a collection that is much larger than any possible basis since the locations and shapes of the Gaussian terms are not fixed in advance. Approximating an orbital within a given accuracy, our algorithm yields significantly fewer terms than methods using multiresolution bases. We demonstrate our approach by solving the Hartree–Fock equations for two diatomic molecules, HeH + and LiH, matching the accuracy previously obtained using multiwavelet bases.