^{1}and Peter M. W. Gill

^{2,a)}

### Abstract

Restricted Hartree–Fock calculations have been performed on the Fermi configurations of electrons confined within a cube. The self-consistent-field orbitals have been expanded in a basis of particle-in-a-box wave functions. The difficult one- and two-electron integrals have been reduced to a small set of canonical integrals that are calculated accurately using quadrature. The total energy and exchange energy per particle converge smoothly toward their limiting values as increases; the highest occupied molecular orbital–lowest unoccupied molecular orbital gap and Dirac coefficient converge erratically. However, the convergence in all cases is slow.

I. INTRODUCTION

II. MODEL, BASIS SET, AND SYMMETRY

III. SCF THEORY

IV. COULOMB INTEGRALS

V. RESULTS AND DISCUSSION

VI. CONCLUDING REMARKS

### Key Topics

- Dirac equation
- 11.0
- Electron correlation calculations
- 6.0
- Wave functions
- 5.0
- Statistical properties
- 4.0
- Band gap
- 3.0

## Figures

Total energy for electrons (mean density ) in a cube. RHF theory with basis functions was used.

Total energy for electrons (mean density ) in a cube. RHF theory with basis functions was used.

Fock exchange energy for electrons (mean density ) in a cube. RHF theory with basis functions was used.

Fock exchange energy for electrons (mean density ) in a cube. RHF theory with basis functions was used.

Dirac exchange energy for electrons (mean density ) in a cube. RHF theory with basis functions was used.

Dirac exchange energy for electrons (mean density ) in a cube. RHF theory with basis functions was used.

Mean square density for electrons (mean density ) in a cube. RHF theory with basis functions was used.

Mean square density for electrons (mean density ) in a cube. RHF theory with basis functions was used.

HOMO-LUMO gap for electrons (mean density ) in a cube. RHF theory with basis functions was used.

HOMO-LUMO gap for electrons (mean density ) in a cube. RHF theory with basis functions was used.

The ratio (5.7) for electrons (mean density ) in a cube. RHF theory with basis functions was used.

The ratio (5.7) for electrons (mean density ) in a cube. RHF theory with basis functions was used.

## Tables

Irreducible representations of spanned by a set of basis functions.

Irreducible representations of spanned by a set of basis functions.

Energies and symmetries of the particle-in-a-box wave functions.

Energies and symmetries of the particle-in-a-box wave functions.

Number of distinct, nonvanishing integrals with , .

Number of distinct, nonvanishing integrals with , .

A selection of low-order canonical integrals [see Eq. (4.2) for definition] to ten-decimal places.

A selection of low-order canonical integrals [see Eq. (4.2) for definition] to ten-decimal places.

Total energy , Fock exchange energy , Dirac exchange energy , and HOMO-LUMO gap for electrons and basis functions. Restricted Hartree–Fock (RHF) theory and a mean density were used in all cases.

Total energy , Fock exchange energy , Dirac exchange energy , and HOMO-LUMO gap for electrons and basis functions. Restricted Hartree–Fock (RHF) theory and a mean density were used in all cases.

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