This program finds virtually exact solutions of the Hartree-Fock and density functional theory equations for diatomic molecules and atoms. The quality of a solution depends on grid size and arithmetic precision used. The lowest energy eigenstates of a given irreducible representation and spin can be obtained. The DFT support is facilitated via the libxc library (see -l|L options of the ./x2dhfctl script). The lxcctl script is provided to help test the support of the x2dhf program for the libxc functionals.

The program can also used to obtain the ground and excited states of one-electron systems with the (smoothed) Coulomb, Green-Sellin-Zachor, or Kramers-Hennenberger potentials, as well the superposition of atomic potentials.

Single particle two-dimensional numerical functions (orbitals) are used to construct an anti-symmetric many-electron wave function of the restricted open-shell Hartree-Fock model. The orbitals are obtained by solving the Hartree-Fock equations in the form of the coupled two-dimensional second-order (elliptic) partial differential equations (PDE). The Coulomb and exchange potentials are obtained as solutions of the corresponding Poisson equations. The PDEs are discretized by the 8th-order central difference stencil on a two-dimensional grid, while a Newton-Cotes quadrature rule is used to evalaute integrals. The resulting large and sparse system of linear equations is solved by the (multi-colour) successive overrelaxation method ((MC)SOR). The self-consistent-field iterations are interwoven with the (MC)SOR ones and orbital energies and normalisation factors are used to monitor the convergence. The accuracy of solutions depends mainly on the grid-size and the system under consideration.

See the following articles for the detailed description of the program and examples of its usage and accuracy:

• L. Laaksonen, P. Pyykkö, and D. Sundholm, Fully numerical Hartree-Fock methods for molecules, Comp. Phys. Reports 4 (1986) 313-344. http://doi.org/10.1016/0167-7977(86)90021-3

• L. Laaksonen, D. Sundholm, and P. Pyykkö, in "Scientific Computing in Finland", Eds. K. Kankaala and R. Nieminen, Research Report R1/89, Centre for Scientific Computing, (1989) p. 183.

• P. Pyykkö, in Numerical Determination of the Electronic Structure of Atoms, Diatomic and Polyatomic Molecules (NATO ASI Series C271) Eds. M. Defranceschi and J. Delhalle, (1989) p. 161. http://doi.org/10.1007/978-94-009-2329-4

• J. Kobus, Finite-difference versus finite-element methods, Chem. Phys. Lett. 202 (1993) 7-12. http://doi.org/10.1016/0009-2614(93)85342-L

• J. Kobus, Vectorizable algorithm for the (multicolour) successive overrelaxation method, Comput. Phys. Commun. 78 (1994) 247-255. http://doi.org/10.1016/0010-4655(94)90003-5

• J. Kobus, L. Laaksonen, D. Sundholm, A numerical Hartree-Fock program for diatomic molecules, Comp. Phys. Commun. 98 (1996) 346-358. http://doi.org/10.1016/0010-4655(96)00098-7

• J.Kobus, D.Moncrieff, and S.Wilson, A comparison of the electric moments obtained from finite basis set and finite difference {Hartree-Fock} calculations for diatomic molecules, Phys. Rev. A 62 (2000), 062503/1--9

• J.Kobus, D.Moncrieff, and S.Wilson, Comparison of the polarizabilities and hyperpolarizabilities obtained from finite basis set and finite field, finite difference {Hartree-Fock} calculations for diatomic molecules, J. Phys. B: At. Mol. Opt. Phys. 34 (2001) 5127-5143.

• J. Kobus, Numerical Hartree-Fock methods for diatomic molecules, Handbook of Molecular Physics and Quantum Chemistry (Chichester), ed. S. Wilson (Wiley, 2002)

• J. Kobus, Hartree-Fock limit values of multipole moments, polarizabilities and hyperpolarizabilities for atoms and diatomic molecules, Comp. Lett. 3 (2007) 71-113. http://doi.org/10.1163/157404007782913408

• J. Kobus, A finite difference Hartree-Fock program for atoms and diatomic molecules, Comp. Phys. Commun. 184 (2013) 799-811. http://dx.doi.org/10.1016/j.cpc.2012.09.033

• J.Kobus, Hartree-Fock limit values of multipole moments, polarizabilities and hyperpolarizabilities for atoms and diatomic molecules, Phys. Rev. A 91 (2015) 022501, URL: http://link.aps.org/doi/10.1103/PhysRevA.91.022501, DOI: https://doi.org/10.1103/PhysRevA.91.022501

Density functionals are evaluated with the libxc library, which has been described in

• S. Lehtola et al, SoftwareX 2018, 7, 1–5, DOI: https://doi.org/10.1016/j.softx.2017.11.002

The programming language used is Fortran 90 and C (if the multi-threaded version employing pthreads is requested). Therefore Fortran 90 and C compilers together with CMake are needed to compile and build the program. This can be done by running ./x2dhfctl -b. See ./x2dhfctl -h and INSTALL for more details.

./doc/users-guide.pdf contains the description of the program's input data and its usage.

test-sets/ directory contains the dozens of specific examples and the testctl script should be used to list and run them (try 'testctl -h').

./lda_orbitals and ./hf_orbitals directories contain LDA and HF orbitals for a number of atomic systems. These orbitals can be used to start the SCF process. The data were generated by means of the HelFEM program and a finite-difference HF program for atoms (qrhf), respectively,

./bin directory contains the x2dhf executable(s) and several BASH and Perl scripts to facilitate the usage of the program (run source .x2dhfrc to adjust the PATH variable accordingly). First of all try xhf -h and testctl -h. See also elpropctl -h, lxcctl -h and pecctl -h.

To give the program a try, execute the following commands: source .x2dhfrc cd tests testctl run h/set-01

Jacek Kobus

2024-07-23