A Fortran 90 program to solve the Hartree–Fock equations for interacting spin- [formula omitted] fermions confined in harmonic potentials

Abstract

A set of weakly interacting spin- 1 2 Fermions, confined by a harmonic oscillator potential, and interacting with each other via a contact potential, is a model system which closely represents the physics of a dilute gas of two-component fermionic atoms confined in a magneto-optic trap. In the present work, our aim is to present a Fortran 90 computer program which, using a basis set expansion technique, solves the Hartree–Fock (HF) equations for spin- 1 2 Fermions confined by a three-dimensional harmonic oscillator potential, and interacting with each other via pair-wise delta-function potentials. Additionally, the program can also account for those anharmonic potentials which can be expressed as a polynomial in the position operators x, y, and z. Both the restricted-HF (RHF), and the unrestricted-HF (UHF) equations can be solved for a given number of Fermions, with either repulsive or attractive interactions among them. The option of UHF solutions for such systems also allows us to study possible magnetic properties of the physics of two-component confined atomic Fermi gases, with imbalanced populations. Using our code we also demonstrate that such a system exhibits shell structure, and follows Hund's rule. Program summary Program title: trap.x Catalogue identifier: AEBB_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBB_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 17 750 No. of bytes in distributed program, including test data, etc.: 205 138 Distribution format: tar.gz Programming language: mostly Fortran 90 Computer: PCs—SUN, HP Alpha, IBM Operating system: Linux, Solaris, Tru64, AIX Classification: 7.7 Nature of problem: The simplest description of a spin 1 2 ; trapped system at the mean field level is given by the Hartree–Fock method. This program presents an efficient approach to solving these equations. Additionally, this program can solve for time-independent Gross–Pitaevskii and Hartree–Fock equations for bosonic atoms confined in a harmonic trap. Thus the combined program can handle mean-field equations for both the Fermi and the Bose particles. Solution method: The solutions of the Hartree–Fock equation corresponding to the Fermi systems in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Hartree–Fock equations which comprise a set of nonlinear integro-differential equations, are transformed into a matrix eigenvalue problem. Thereby, solutions are obtained in a self-consistent manner, using methods of computational linear algebra. Running time: The run times of example jobs are from a few seconds to a few minutes. For jobs involving very large basis sets, the run time can extend into hours.