A novel method for the solution of the Schrödinger equation in the presence of exchange terms

Abstract

In the Hartree-Fock approximation the Pauli exclusion principle leads to a Schrödinger Eq. of an integro-differential form. We describe a new spectral noniterative method (S-IEM), previously developed for solving the Lippman-Schwinger integral equation with local potentials, which has now been extended so as to include the exchange nonlocality. We apply it to the restricted case of electron-Hydrogen scattering in which the bound electron remains in the ground state and the incident electron has zero angular momentum, and we compare the acuracy and economy of the new method to three other methods. One is a non-iterative solution (NIEM) of the integral equation as described by Sams and Kouri in 1969. Another is an iterative method introduced by Kim and Udagawa in 1990 for nuclear physics applications, which makes an expansion of the solution into an especially favorable basis obtained by a method of moments. The third one is based on the Singular Value Decomposition of the exchange term followed by iterations over the remainder. The S-IEM method turns out to be more accurate by many orders of magnitude than any of the other three methods described above for the same number of mesh points.2

The Canonical Function Method and its applications in quantum physics

“…5. Our results (crosses) fall exactly on top of the continuous curves for the singlet and the triplet obtained by Rawitscher et al [49] with the S-IEM (see Fig. 12).…”

“…Recently, Rawitscher et al [49] obtained very accurate results for the phaseshift at l = 0 for momentum k = 0.2/a 0 in the absence of any polarization potential, but with rigorous inclusion of the Fock exchange term. They obtained the phaseshifts and scattering lengths in the singlet and triplet states.…”

“…The latter has to respect Pauli exclusion principle, stating that the wave function of the incident electron be anti-symmetric with the wave functions of the electrons in the target atom or molecule. In the Hartree-Fock formulation, this requirement leads to the presence of non local terms in the resulting coupled differential equations satisfied by the radial parts of the free electron wavefunctions [49,62]. The non-local kernel of the integrodifferential type equation originating from the exchange operator is often simplified and a central LEE potential is used, or some decoupling procedure is employed [23].…”

A Spectral Integral Equation Solution of the Gross-Pitaevskii Equation

“…These functions require wave-numbers, rather than energies as input parameters. The calculations are done by means of a semi-spectral Chebyshev expansion method that gives a reliable accuracy [31][32][33][34][35][36][37][38].…”

“…In the S-IEM the full radial domain is divided into partitions, and in each partition the wave function is expanded in a series of Chebyshev polynomials, whose coefficients are calculated by solving linear equations [29,30]. The S-IEM has been applied to the solution of several physics problems [31][32][33][34][35][36][37][38][39][40], and a pedagogical description is available in [41,42].…”