# Restricted Hartree Fock using complex-valued orbitals: A long-known but neglected tool in electronic structure theory

## Abstract

Restricted Hartree Fock using complex-valued orbitals (cRHF) is studied. We introduce an orbital pairing theorem, with which we obtain a concise connection between cRHF and real-valued RHF, and use it to uncover the close relationship between cRHF, unrestricted Hartree Fock, and generalized valence bond perfect pairing. This enables an intuition for cRHF, contrasting with the generally unintuitive nature of complex orbitals. We also describe an efficient computer implementation of cRHF and its corresponding stability analysis. By applying cRHF to the Be + H{sub 2} insertion reaction, a Woodward-Hoffmann violating reaction, and a symmetry-driven conical intersection, we demonstrate in genuine molecular systems that cRHF is capable of removing certain potential energy surface singularities that plague real-valued RHF and related methods. This complements earlier work that showed this capability in a model system. We also describe how cRHF is the preferred RHF method for certain radicaloid systems like singlet oxygen and antiaromatic molecules. For singlet O{sub 2}, we show that standard methods fail even at the equilibrium geometry. An implication of this work is that, regardless of their individual efficacies, cRHF solutions to the HF equations are fairly commonplace.

- Authors:

- Department of Chemistry, University of California, Berkeley, California 94720, USA and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720 (United States)

- Publication Date:

- OSTI Identifier:
- 22415819

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 142; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; CHEMICAL BONDS; ELECTRONIC STRUCTURE; EQUILIBRIUM; HARTREE-FOCK METHOD; HYDROGEN; IMPLEMENTATION; MATHEMATICAL SOLUTIONS; MOLECULES; OXYGEN; POTENTIAL ENERGY; SINGULARITY; SURFACES; SYMMETRY; VALENCE

### Citation Formats

```
Small, David W., Sundstrom, Eric J., and Head-Gordon, Martin.
```*Restricted Hartree Fock using complex-valued orbitals: A long-known but neglected tool in electronic structure theory*. United States: N. p., 2015.
Web. doi:10.1063/1.4905120.

```
Small, David W., Sundstrom, Eric J., & Head-Gordon, Martin.
```*Restricted Hartree Fock using complex-valued orbitals: A long-known but neglected tool in electronic structure theory*. United States. doi:10.1063/1.4905120.

```
Small, David W., Sundstrom, Eric J., and Head-Gordon, Martin. Wed .
"Restricted Hartree Fock using complex-valued orbitals: A long-known but neglected tool in electronic structure theory". United States.
doi:10.1063/1.4905120.
```

```
@article{osti_22415819,
```

title = {Restricted Hartree Fock using complex-valued orbitals: A long-known but neglected tool in electronic structure theory},

author = {Small, David W. and Sundstrom, Eric J. and Head-Gordon, Martin},

abstractNote = {Restricted Hartree Fock using complex-valued orbitals (cRHF) is studied. We introduce an orbital pairing theorem, with which we obtain a concise connection between cRHF and real-valued RHF, and use it to uncover the close relationship between cRHF, unrestricted Hartree Fock, and generalized valence bond perfect pairing. This enables an intuition for cRHF, contrasting with the generally unintuitive nature of complex orbitals. We also describe an efficient computer implementation of cRHF and its corresponding stability analysis. By applying cRHF to the Be + H{sub 2} insertion reaction, a Woodward-Hoffmann violating reaction, and a symmetry-driven conical intersection, we demonstrate in genuine molecular systems that cRHF is capable of removing certain potential energy surface singularities that plague real-valued RHF and related methods. This complements earlier work that showed this capability in a model system. We also describe how cRHF is the preferred RHF method for certain radicaloid systems like singlet oxygen and antiaromatic molecules. For singlet O{sub 2}, we show that standard methods fail even at the equilibrium geometry. An implication of this work is that, regardless of their individual efficacies, cRHF solutions to the HF equations are fairly commonplace.},

doi = {10.1063/1.4905120},

journal = {Journal of Chemical Physics},

number = 2,

volume = 142,

place = {United States},

year = {Wed Jan 14 00:00:00 EST 2015},

month = {Wed Jan 14 00:00:00 EST 2015}

}