Optimal basis set for ab-initio calculations of energy levels in tunneling structures, using the covariance matrix of the wave functions

TL;DR

This paper introduces a statistical method using the covariance matrix of wave functions to determine an optimal basis set, significantly improving the efficiency and accuracy of energy level calculations in tunneling structures.

Contribution

It presents a novel covariance matrix-based approach to optimize basis sets for ab-initio calculations, reducing computational cost while maintaining precision.

Findings

Eigenvalues decrease rapidly, allowing basis set reduction.

Method improves speed and accuracy of SCF calculations.

Encouraging results with analytical solutions demonstrate effectiveness.

Abstract

The paper proposes a method to obtain the optimal basis set for solving the self consistent field (SCF) equations for large atomic systems in order to calculate the energy barriers in tunneling structures, with higher accuracy and speed. Taking into account the stochastic-like nature of the samples of all the involved wave functions for many body problems, a statistical optimization is made by considering the covariance matrix of these samples. An eigenvalues system is obtained and solved for the optimal basis set and by inspecting the rapidly decreasing eigenvalues one may seriously reduce the necessary number of vectors that insures an imposed precision. This leads to a potentially significant improvement in the speed of the SCF calculations and accuracy, as the statistical properties of a large number of wave functions in an large spatial domain may be considered. The eigenvalue problem has to be solved only few times, so that the amount of time added may be much smaller that the overall iterating SCF calculations. A simple implementation of the method is presented for a situation where the analytical solution is known, and the results are encouraging.