Calculating the distance from an electronic wave function to the manifold of Slater determinants through the geometry of Grassmannians

TL;DR

This paper introduces a Riemannian geometry-based algorithm to measure the distance from a correlated electronic wave function to the manifold of Slater determinants, aiding in quantifying entanglement and correlation.

Contribution

It develops an efficient geometric algorithm using Grassmannian tools to find critical Slater determinants related to a given wave function.

Findings

Algorithm converges to critical Slater determinants.

Efficient implementation with fixed orbital basis.

Applicable to various electronic wave functions.

Abstract

The set of all electronic states that can be expressed as a single Slater determinant forms a submanifold, isomorphic to the Grassmannian, of the projective Hilbert space of wave functions. We explored this fact by using tools of Riemannian geometry of Grassmannians as described by Absil et. al [Acta App. Math. 80, 199 (2004)], to propose an algorithm that converges to a Slater determinant that is critical point of the overlap function with a correlated wave function. This algorithm can be applied to quantify the entanglement or correlation of a wave function. We show that this algorithm is equivalent to the Newton method using the standard parametrization of Slater determinants by orbital rotations, but it can be more efficiently implemented because the orbital basis used to express the correlated wave function is kept fixed throughout the iterations. We present the equations of this method for a general configuration interaction wave function and for a wave function with up to double excitations over a reference determinant. Applications of this algorithm to selected electronic systems are also presented and discussed.