Hartree-Fock algorithm

The most famous case concerns the symmetry breaking in the Hartree-Fock approximation. The phenomenon appeared on elementary problems, such as H2, when the so-called unrestricted Hartree-Fock algorithms were tried. The unrestricted Hartree-Fock formalism, using different orbitals for a and p electrons, was first proposed by G. Berthier [5] in 1954 (and immediately after J.A. Pople [6] ) for problems where the number of a andp electrons were different. This formulation takes the freedom to deviate from the constraints of being an eigenfunction. 

The development and efficient implementation of a parallel direct SCF Hartree-Fock algorithm, with gradients and random phase approximation solutions, are described by Feyereisen and Kendall, who discussed details of the structure of the parallel version of DISCO. Preliminary results for calculations using the Intel-Delta parallel computer system were reported. The data showed that the algorithms were efficiently parallelized and that throughput of a one-processor Cray X-MP was reached with about 16 nodes on the Intel-Delta. The data also indicated that sequential code, which was not a bottleneck on traditional supercomputers, became time-critical on parallel computers. 

All the early work was concerned with atoms, with Sir William Hartree regarded as the father of the technique. His son, Douglas R. Hartree, published the definitive book, The Calculation of Atomic Structures, in 1957, and in this he derived the atomic HF equations and described numerical algorithms for their solution. Charlotte Froese Fischer was a research student working under the guidance of D. R. Hartree, and she published her own definitive book. The Hartree—Fock Method for Atoms A Numerical Approach in 1977. The Appendix lists a number of freely available atomie structure programs. Most of these can be obtained from the Computer Physics Communications Program Library. 

The SCF-MI algorithm, recently extended to compute analytic gradients and second derivatives [18,41], furnishes the Hartree Fock wavefunction for the interacting molecules and also provides automatic geometry optimisation and vibrational analysis in the harmonic approximation for the supersystems. The Ml strategy has been implemented into GAMESS-US package [42]. 

The results obtained with this algorithm were very good except when the indices ijr corresponded to occupied orbitals in a Hartree-Fock state and ksp to unoccupied ones (and vice versa) that is, when the 3-RDM element corresponded to the expectation value of a three-body elemental excitation. 

This equation has at least two exact solutions. Thus, both the set of RDM s obtained in a Hartree-Fock HF) calculation and that obtained from a FCI one fulfill exactly this effective one-body equation. Unfortunately, the iterative method sketched above converged to the HF solution in all the cases tested. This may be due to the fact that in our algorithm, the correlation effects are estimated through a renormalization procedure, which may not be sufBciently accurate in the first order case. To improve this aspect is one of the motivations of our present line of work. 

Eq.(8) is the starting point for a direct variational approach to Density Functional Theory, proposed by Teter, Payne and Allan [23,24], and called band-by-band (or state-by-state) conjugate-gradient (CG) algorithm. By contrast, Eqs.(10-12) have been in use since many years. They parallel the well-known SCF approach to the Hartree-Fock approximation. In the spirit of Teter, Payne and Allan, a variational approach to the treatment of perturbations within DFT is now presented. 

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