Augmented Lagrangian method for coupled-cluster

TL;DR

This paper introduces an augmented Lagrangian approach to improve the convergence stability and reliability of the coupled-cluster method in quantum chemistry, addressing issues of convergence failure and unphysical solutions.

Contribution

The paper develops an augmented Lagrangian formulation for coupled-cluster, enhancing convergence stability and reducing susceptibility to local minima without increasing computational cost.

Findings

Enhanced convergence towards the ground state

Reduced occurrence of unphysical solutions

Comparable computational cost to conventional CC

Abstract

We propose to improve the convergence properties of the single-reference coupled cluster (CC) method through an augmented Lagrangian formalism. The conventional CC method changes a linear high-dimensional eigenvalue problem with exponential size into a problem of determining the roots of a nonlinear system of equations that has a manageable size. However, current numerical procedures for solving this system of equations to get the lowest eigenvalue suffer from two practical issues: First, solving the CC equations may not converge, and second, when converging, they may converge to other -- potentially unphysical -- states, which are stationary points of the CC energy expression. We show that both issues can be dealt with when a suitably defined energy is minimized in addition to solving the original CC equations. We further propose an augmented Lagrangian method for coupled cluster (alm-CC) to solve the resulting constrained optimization problem. We numerically investigate the proposed augmented Lagrangian formulation showing that the convergence towards the ground state is significantly more stable and that the optimization procedure is less susceptible to local minima. Furthermore, the computational cost of alm-CC is comparable to the conventional CC method.