# Analysis of the Extended Coupled-Cluster Method in Quantum Chemistry

## TL;DR

This paper develops the mathematical foundation of the extended coupled-cluster method in quantum chemistry, proving existence and uniqueness of solutions and analyzing its properties using a bivariational principle.

## Contribution

It provides the first rigorous mathematical analysis of the extended coupled-cluster method, including existence, uniqueness, and error bounds, using a generalized variational principle.

## Key findings

- Proved existence and uniqueness of solutions in infinite-dimensional and discretized spaces.
- Established a quadratic energy error bound in the discretized case.
- Compared the extended method to standard coupled-cluster analysis, highlighting the utility of the bivariational principle.

## Abstract

The mathematical foundation of the so-called extended coupled-cluster method for the solution of the many-fermion Schr\"odinger equation is here developed. We prove an existence and uniqueness result, both in the full infinite-dimensional amplitude space as well as for discretized versions of it. The extended coupled-cluster method is formulated as a critical point of an energy function using a generalization of the Rayleigh-Ritz principle: the bivariational principle. This gives a quadratic bound for the energy error in the discretized case. The existence and uniqueness results are proved using a type of monotonicity property for the flipped gradient of the energy function. Comparisons to the analysis of the standard coupled-cluster method is made, and it is argued that the bivariational principle is a useful tool, both for studying coupled-cluster type methods, and for developing new computational schemes in general.