A B-spline Hartree–Fock program

Abstract

A B-spline version of a Hartree–Fock program is described. The usual differential equations are replaced by systems of non-linear equations and generalized eigenvalue problems of the form ( H a − ε a a B ) P a = 0 , where a designates the orbital. When orbital a is required to be orthogonal to a fixed orbital, this form assumes that a projection operator has been applied to eliminate the Lagrange multiplier. When two orthogonal orbitals are both varied, the energy must also be stationary with respect to orthogonal transformations. At such a stationary point, the matrix of Lagrange multipliers, ε a b = ( P b | H a | P a ) , is symmetric and the off-diagonal Lagrange multipliers may again be eliminated through projection operators. For multiply occupied shells, convergence problems are avoided by the use of a single-orbital Newton–Raphson method. A self-consistent field procedure based on these two possibilities exhibits excellent convergence. A Newton–Raphson method for updating all orbitals simultaneously has better numerical properties and a more rapid rate of convergence but requires more computer processing time. Both ground and excited states may be computed using a default universal grid. Output from a calculation for Al 3 s 2 3 p P 2 shows the improvement in accuracy that can be achieved by mapping results from low-order splines on a coarse grid to splines of higher order onto a refined grid. The program distribution contains output from additional test cases. Program summary Program title: SPHF version 1.00 Catalogue identifier: AEIJ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIJ_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 13 925 No. of bytes in distributed program, including test data, etc.: 714 254 Distribution format: tar.gz Programming language: Fortran 95 Computer: Any system with a Fortran 95 compiler. Tested on Intel Xeon CPU X5355, 2.66 GHz Operating system: Any system with a Fortran 95 compiler Classification: 2.1 External routines: LAPACK ( http://www.netlib.org/lapack/) Nature of problem: Non-relativistic Hartree–Fock wavefunctions are determined for atoms in a bound state that may be used to predict a variety atomic properties. Solution method: The radial functions are expanded in a B-spline basis [1]. The variational principle applied to an energy functional that includes Lagrange multipliers for orthonormal constraints defines the Hartree–Fock matrix for each orbital. Orthogonal transformations symmetrize the matrix of Lagrange multipliers and projection operators eliminate the off-diagonal Lagrange multipliers to yield a generalized eigenvalue problem. For multiply occupied shells, a single-orbital Newton–Raphson (NR) method is used to speed convergence with very little extra computation effort. In a final step, all orbitals are updated simultaneously by a Newton–Raphson method to improve numerical accuracy. Restrictions: There is no restriction on calculations for the average energy of a configuration. As in the earlier HF96 program [2], only one or two open shells are allowed when results are required for a specific LS coupling. These include: 1. ( nl ) N n ′ s , where l = 0 , 1 , 2 , 3 2. ( np ) N n ′ l , where l = 0 , 1 , 2 , 3 , … 3. ( nd ) ( n ′ f ) Unusual features: Unlike HF96, the present program is a Fortran 90/95 program without the use of COMMON. It is assumed that Lapack libraries are available. Running time: For Ac 7 s 2 7 p P 2 the execution time varied from 6.9 s to 9.1 s depending on the iteration method.