Towards the explicit computation of Bohm velocities associated to N-electron wave-functions with arbitrary spin-orientations

TL;DR

This paper extends a Bohm trajectory model to handle N-electron wave-functions with arbitrary spin orientations, simplifying the computation by approximating the total wave function as a product of spin-up and spin-down components, and validates this approach numerically.

Contribution

It introduces a new approximation method for computing Bohm velocities in multi-electron systems with arbitrary spins, reducing computational complexity.

Findings

The approximation accurately reproduces Bohm velocities in various scenarios.

The method significantly reduces computational complexity for multi-electron spin systems.

Numerical validation confirms the approximation's validity across many cases.

Abstract

The direct solution of the many-particle Schr\"odinger equation is computationally inaccessible for more than very few electrons. In order to surpass this limitation, one of the authors [X. Oriols, Phys. Rev. Lett. 2007, 98 (066803)] has recently proposed a new model to study electron-electron correlations from Bohm trajectories associated to time-dependent wave-packets solutions of pseudo single-particle Schr\"odinger equations. In the aforementioned paper only the orbital exchange interaction is considered assuming that all electrons have the same spin orientation. Then, the many-particle wave function is a complex Slater determinant of the single-particle wave-packets. In the present work the previous formalism is extended to study many-particle wave functions where the electrons have different spin orientations.The main difficulty to treat N different electron spin orientations with time-dependent wave-packets is that one must study all the possible N!N! products of permutations among spin states. To overcome this computationally inaccessible problem, in this article the total wave function is treated as a separated product of two many-particle wave functions, the first with spin up and the second with spin down. In order to numerically justify this approximation, the Bohm velocity in different antisymmetric total wave-function scenarios is computed. The computational results confirms the accurate validity of our approximation under a large number of cases.