Optimal Slater-determinant approximation of fermionic wave functions

TL;DR

This paper analytically investigates the best Slater-determinant approximation of fermionic wave functions, revealing structural properties and optimal constructions, especially in the case of three fermions in six orbitals.

Contribution

It provides new analytical insights and lemmas for finding optimal Slater-determinant approximations, including explicit constructions for specific fermionic states.

Findings

Optimal Slater approximation built from natural orbitals for three fermions in six orbitals.

Usefulness of Hadamard inequality in approximation problems.

Structural properties of fermionic wave functions and their approximations.

Abstract

We study the optimal Slater-determinant approximation of an $N$-fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the optimal Slater approximation of some special target wave functions.