# Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian

## Abstract

Coupled cluster and symmetry projected Hartree-Fock are two central paradigms in electronic structure theory. However, they are very different. Single reference coupled cluster is highly successful for treating weakly correlated systems, but fails under strong correlation unless one sacrifices good quantum numbers and works with broken-symmetry wave functions, which is unphysical for finite systems. Symmetry projection is effective for the treatment of strong correlation at the mean-field level through multireference non-orthogonal configuration interaction wavefunctions, but unlike coupled cluster, it is neither size extensive nor ideal for treating dynamic correlation. We here examine different scenarios for merging these two dissimilar theories. We carry out this exercise over the integrable Lipkin model Hamiltonian, which despite its simplicity, encompasses non-trivial physics for degenerate systems and can be solved via diagonalization for a very large number of particles. We show how symmetry projection and coupled cluster doubles individually fail in different correlation limits, whereas models that merge these two theories are highly successful over the entire phase diagram. Despite the simplicity of the Lipkin Hamiltonian, the lessons learned in this work will be useful for building an ab initio symmetry projected coupled cluster theory that we expect to be accurate in the weakly andmore »

- Authors:

- Rice Univ., Houston, TX (United States)
- Consejo Superior de Investigaciones Cientificas (CSIC), Madrid (Spain). Inst. de Estructura de la Materia

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1416505

- Report Number(s):
- LLNL-SR-744060

- DOE Contract Number:
- AC52-07NA27344

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; 74 ATOMIC AND MOLECULAR PHYSICS

### Citation Formats

```
Wahlen-Strothman, J. M., Henderson, T. H., Hermes, M. R., Degroote, M., Qiu, Y., Zhao, J., Dukelsky, J., and Scuseria, G. E.
```*Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian*. United States: N. p., 2018.
Web. doi:10.2172/1416505.

```
Wahlen-Strothman, J. M., Henderson, T. H., Hermes, M. R., Degroote, M., Qiu, Y., Zhao, J., Dukelsky, J., & Scuseria, G. E.
```*Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian*. United States. doi:10.2172/1416505.

```
Wahlen-Strothman, J. M., Henderson, T. H., Hermes, M. R., Degroote, M., Qiu, Y., Zhao, J., Dukelsky, J., and Scuseria, G. E. Wed .
"Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian". United States.
doi:10.2172/1416505. https://www.osti.gov/servlets/purl/1416505.
```

```
@article{osti_1416505,
```

title = {Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian},

author = {Wahlen-Strothman, J. M. and Henderson, T. H. and Hermes, M. R. and Degroote, M. and Qiu, Y. and Zhao, J. and Dukelsky, J. and Scuseria, G. E.},

abstractNote = {Coupled cluster and symmetry projected Hartree-Fock are two central paradigms in electronic structure theory. However, they are very different. Single reference coupled cluster is highly successful for treating weakly correlated systems, but fails under strong correlation unless one sacrifices good quantum numbers and works with broken-symmetry wave functions, which is unphysical for finite systems. Symmetry projection is effective for the treatment of strong correlation at the mean-field level through multireference non-orthogonal configuration interaction wavefunctions, but unlike coupled cluster, it is neither size extensive nor ideal for treating dynamic correlation. We here examine different scenarios for merging these two dissimilar theories. We carry out this exercise over the integrable Lipkin model Hamiltonian, which despite its simplicity, encompasses non-trivial physics for degenerate systems and can be solved via diagonalization for a very large number of particles. We show how symmetry projection and coupled cluster doubles individually fail in different correlation limits, whereas models that merge these two theories are highly successful over the entire phase diagram. Despite the simplicity of the Lipkin Hamiltonian, the lessons learned in this work will be useful for building an ab initio symmetry projected coupled cluster theory that we expect to be accurate in the weakly and strongly correlated limits, as well as the recoupling regime.},

doi = {10.2172/1416505},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Wed Jan 03 00:00:00 EST 2018},

month = {Wed Jan 03 00:00:00 EST 2018}

}