Lowering of the complexity of quantum chemistry methods by choice of representation

TL;DR

This paper demonstrates that by choosing different representations, the computational complexity of key quantum chemistry methods can be significantly reduced without approximations, with some methods scaling as low as rd of their original complexity.

Contribution

It shows how real-space, momentum-space, and time-dependent representations can lower the complexity of quantum chemistry calculations without approximations.

Findings

Hartree-Fock scales as rd of original complexity

Second-order Mf8ller-Plesset scales as rd of original complexity

Coupled cluster doubles scales as th of original complexity

Abstract

The complexity of the standard hierarchy of quantum chemistry methods is not invariant to the choice of representation. This work explores how the scaling of common quantum chemistry methods can be reduced using real-space, momentum-space, and time-dependent intermediate representations without introducing approximations. We find the scalings of exact Gaussian basis Hartree--Fock theory, second-order M{\o}ller-Plesset perturbation theory, and coupled cluster theory (specifically, linearized coupled cluster doubles and the distinguishable cluster approximation with doubles) to be $\mathcal{O}(N^3)$, $\mathcal{O}(N^3)$, and $\mathcal{O}(N^5)$ respectively, where $N$ denotes system size. These scalings are not asymptotic and hold over all ranges of $N$.