Uniform electron gases: III. Low-density gases on three-dimensional spheres

Abstract

By combining variational Monte Carlo (VMC) and complete-basis-set limit Hartree-Fock (HF) calculations, we have obtained near-exact correlation energies for low-density same-spin electrons on a three-dimensional sphere (3-sphere), i.e.~the surface of a four-dimensional ball. In the VMC calculations, we compare the efficacies of two types of one-electron basis functions for these strongly correlated systems, and analyze the energy convergence with respect to the quality of the Jastrow factor. The HF calculations employ spherical Gaussian functions (SGFs) which are the curved-space analogs of cartesian Gaussian functions. At low densities, the electrons become relatively localized into Wigner crystals, and the natural SGF centers are found by solving the Thomson problem (i.e. the minimum-energy arrangement of $n$ point charges) on the 3-sphere for various values of $n$. We have found 11 special values of $n$ whose Thomson sites are equivalent. Three of these are the vertices of four-dimensional Platonic solids --- the hyper-tetrahedron ($n=5$), the hyper-octahedron ($n=8$) and the 24-cell ($n=24$) --- and a fourth is a highly symmetric structure ($n=13$) which has not previously been reported. By calculating the harmonic frequencies of the electrons around their equilibrium positions, we also find the first-order vibrational corrections to the Thomson energy.

6 pages, 2 figures, accepted for publication in the Journal of Chemical Physics

References in corpus (13)

• Continuum variational and diffusion quantum Monte Carlo calculations
• Full optimization of Jastrow-Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules
• Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals
• Two electrons on a hypersphere: a quasi-exactly solvable model
• Ground state of two electrons on a sphere
• Unrestricted Hartree-Fock theory of Wigner crystals
• The Hartree-Fock ground state of the three-dimensional electron gas
• Correlation energy of two electrons in the high-density limit
• Uniform electron gases
• Chemistry in One Dimension
• Ground state of two electrons on concentric spheres
• Nodal surfaces and interdimensional degeneracies
• Basis functions for electronic structure calculations on spheres

Cited by in corpus (5)

• Coulomb and Riesz gases: The known and the unknown
• A weight-dependent local correlation density-functional approximation for ensembles
• Exchange functionals based on finite uniform electron gases
• Two and three electrons on a sphere: A generalized Thomson problem
• Transient Uniform Electron Gases