Variational solution of the Schrödinger equation using plane waves in adaptive coordinates: The radial case

Abstract

A new method for solving the Schrödinger equation is proposed, based on the following details. First, a map u=u(r) from Cartesian coordinates r to a new coordinate system u is chosen. Second, the solution (orbital) psi(r) is written in terms of a function U depending on u so that psi(r)=/J(u)/(-1/2)U(u), where /J(u)/ is the Jacobian determinant of the map. Third, U is expressed as a linear combination of plane waves in the u coordinate, U(u)= sum (k)c(k)e(ik x u). Finally, the coefficients c(k) are variationally optimized to obtain the best energy, using a generalization of an algorithm originally developed for the Coulomb potential [J. M. Perez-Jorda, Phys. Rev. B 58, 1230 (1998)]. The method is tested for the radial Schrödinger equation in the hydrogen atom, resulting in micro-Hartree accuracy or better for the energy of ns and np orbitals (with n up to 5) using expansions of moderate length.

Parameterized Bases for Calculating Vibrational Spectra Directly from ab Initio Data Using Rectangular Collocation

“…Owing to the advantages of a small basis set, there are many basis optimization approaches. In electronic structure theory, plane wave basis functions can be optimized by varying the definition of the coordinate; − optimization of the basis also improves the description of electron correlation . Neural networks provide a flexible means of representing orbitals and wave functions. , …”

Fast solution of elliptic partial differential equations using linear combinations of plane waves

“…Adaptive coordinates have been employed in a number of solid-state or quantum chemical applications [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. For example, they have been used [12] for pseudopotential-free all-electron calculations on diamond.…”