Augmented Lagrangian method for coupled-cluster

TL;DR

The paper tackles the core challenge of convergence in single-reference coupled-cluster theory by reframing CC as a constrained optimization problem and introducing an augmented Lagrangian (alm-CC) approach. By coupling energy minimization with CC equations and using an exponential CC ansatz, alm-CC stabilizes convergence, reduces susceptibility to unphysical stationary points, and maintains computational cost similar to conventional CC methods. Numerical results on representative systems show alm-CC reproduces CCSD where CC works well and delivers significantly more robust global convergence in harder regimes, including multiple-root scenarios. The work provides a practical optimization-based alternative to root-finding in CC, with strong theoretical underpinnings (commuting and nilpotent cluster operators) and detailed implementation guidance, underscoring its potential for broader applicability in quantum chemistry.

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.

Figures (8)

• Figure 1: Comparison of the energy error from quasi-Newton CCSD and the outer iterations of alm-CCSD.
• Figure 2: Schematic depiction of the different dissociation processes. The parameter $R = 1.4$ a.u. which is close to the equilibrium of H$_2$.
• Figure 3: Comparison of energies obtained with alm-ccsd and conventional quasi-newton CCSD. In panels (b) and (c) we included the dissociation energy of the respective H$_4$ molecules as a solid blue line, showing the size consistency of the method.
• Figure 4: Depiction of the H$_4$ model undergoing a symmetric disturbance on a circle modeled by the angle $\Theta$.
• Figure 5: Energy progress from quasi Newton using a random initialization ${\bf t}_i + {\bf t}_p$ with $\Vert {\bf t}_p \Vert/\Vert {\bf t}_i \Vert = 0.1$ for $i = 0, \ldots, 3$.

Theorems & Definitions (4)

• Proposition 1
• Proposition 2
• proof
• proof