BSHF: A program to solve the Hartree–Fock equations for arbitrary central potentials using a B-spline basis

Abstract BSHF solves the Hartree–Fock equations in a B-spline basis for atoms, negatively charged ions, and systems of N electrons in arbitrary central potentials. In the B-spline basis the Hartree–Fock integro-differential equations are reduced to a computationally simpler eigenvalue problem. As well as solving this for the ground-state electronic structure self-consistently, the program can calculate discrete and/or continuum excited states of an additional electron or positron in the field of the frozen-target N -electron ground state. It thus provides an effectively complete orthonormal basis that can be used for higher-order many-body theory calculations. Robust and efficient convergence in the self-consistent iterations is achieved by a number of strategies, including by gradually increasing the strength of the electron–electron interaction by scaling the electron charge from a reduced value to its true value. The functionality and operation of the program is described in a tutorial style example. Program summary Program Title: BSHF Program Files doi: http://dx.doi.org/10.17632/fj3y6c58dy.1 Code Ocean Capsule: https://doi.org/10.24433/CO.1226817.v2 Licensing provisions: GPLv3 Programming language: Fortran 90. External routines/libraries: LAPACK. Nature of problem: Self-consistent solution of electronic structure for atoms and electrons in arbitrary central potentials in the Hartree–Fock approximation. Solution method: A B-spline basis is employed that transforms the Hartree–Fock integro-differential equations to a computationally simpler eigenvalue problem. The eigenvalue problem is solved iteratively until self-consistency is achieved. Unusual or notable features: 1. Robust and efficient convergence in the self-consistent iterations is achieved by gradually increasing the value of the electron charge from a reduced value to its true value, i.e., increasing the strength of the electron–electron interaction. In this way all orbitals are calculated simultaneously at every iteration of the self-consistency loop, i.e., without the need to successively fill the occupied shells from the core to valence (as most other Hartree–Fock codes require for convergence). 2. In addition to atoms and negative ions, the program solves the Hartree–Fock equations for systems of N electrons confined in an arbitrary central potential specified by the user: thus the program can be used to e.g., calculate the structure of electrons confined in harmonic potentials, which are known to approximate the electron gas. 3. In addition to calculating the ground state of the N -electron system, the program calculates the discrete and/or continuum excited states for an additional electron or positron in the field of the ‘frozen’ N -electron target. It thus provides an orthonormal basis that can be used as input for many-body theory calculations. Restrictions: The program solves non-relativistic Hartree–Fock equations for spherically symmetric systems only (in modelling systems other than atoms, e.g., harmonically confined electron gas or jellium-type models of clusters, relativistic effects can often be negligible and a non-relativistic treatment is perfectly suitable). For open-shell atoms the central-field approximation is used, i.e., the potential is angularly averaged.