Polynomial Approximations of Electronic Wave Functions

TL;DR

This paper develops an algebraic framework for non-linear quantum chemistry, introducing new algebraic structures related to Clifford algebras, and demonstrates polynomial approximations of wave functions with practical examples.

Contribution

It introduces a purely algebraic approach to quantum chemistry methods, connecting algebraic structures to wave function approximations and quotient Schrödinger equations.

Findings

Algebras similar to Clifford algebra are central to quantum chemistry methods.

Polynomial approximations can effectively model configuration interaction wave functions.

Quotient algebras lead to coupled cluster type equations.

Abstract

This work completes the construction of purely algebraic version of the theory of non-linear quantum chemistry methods. It is shown that at the heart of these methods there lie certain algebras close in their definition to the well-known Clifford algebra but quite different in their properties. The most important for quantum chemistry property of these algebras is the following : for a fixed number of electrons the corresponding sector of the Fock space becomes a commutative algebra and its ideals are determined by the order of excitations from the Hartree-Fock reference state. Quotients of this algebra can also be endowed with commutative algebra structures and quotient Schr{\"o}dinger equations are exactly the couple cluster type equations. Possible computer implementation of multiplication in the aforementioned algebras is described. Quality of different polynomial approximations of configuration interaction wave functions is illustrated with concrete examples. Embedding of algebras of infinitely separated subsystems in algebra of the united system is discussed