FD HF method

Due to these developments of the FD HF method even moderate computer resources are sufficient to treat diatomic systems with up to 20-25 electrons routinely. Data from Table 1 can be used to estimate the CPU time needed to solve the FD HF equations for a given system. Typically, in every scf cycle, 10 (MC)SOR iterations are performed on each orbital and potential. For example, for the AIF molecule there are in all 57 Poisson equations to be solved in each scf iteration and this require about 60 seconds on a Sun SS20 workstation. Depending on the quality of the initial guess for the orbitals and the potentials and the accuracy of the solution 5-30 hundred scf cycles might be needed. The current version of the program will soon become available (20). 

Work is under way to calibrate basis sets for the InF and TIF molecules to the same level of accuracy in order to study the effects of parity and time reversal non-conservation in the latter (26,27) and to show that the FD HF method can be applied to such systems. 

The above demonstrated possibility of obtaining numerical virtual orbitals indicate that the FD HF method can also be used as a solver of the Schrodinger equation for a one-electron diatomic system with an arbitrary potential. Thus, the scheme could be of interest to those who try to construct exchange-correlation potential functions or deal with local-scaling transformations within the functional density theory (32,33). 

This is the exact second-order equation for the large component of the DHF a orbital which together with Eq. (3) could be used in the second-order formulation of the DHF method for atoms. It is believed that the corresponding equations for diatomic molecules in the prolate spheroidal coordinates can be solved by the same technique as the one used in the FD HF method. 

Work is under way to extend the formulation of the FD HF method from diatomic molecules to the linear polyatomic ones. Following Becke (54) the idea is to divide, say, a linear triatomic molecule into two overlapping diatomic regions . The solution of the orbital and potential equations for the whole molecule would be obtained by solving the HF equations for the two diatomic subsystems alternatively, until self-consistency is achieved. 

A decade ago Laaksonen et al. published a paper giving an outline of the finite difference (FD) (or numerical) Hartree-Fock (HF) method for diatomic molecules and several examples of its application to a series of molecules (1). A summary of the FD HF calculations performed until 1987 can be found in (2). The work of Laaksonen et al. can be considered a second attempt to solve numerically the HF equations for diatomic molecules exactly. The earlier attempt was due to McCullough who in the mid 1970s tried to tackle the problem using the partial wave expansion method (3). This approach had been extended to study correlation effects, polarizabilities and hyper-fine constants and was extensively used by McCullough and his coworkers (4-6). Heinemann et al. (7-9) and Sundholm et al. (10,11) have shown that the finite element method could also be used to solve numerically the HF equations for diatomic molecules. 

For the same reasons as in the nonrelativistic case the availability of a numerical solver of the DHF equations for molecules would be very much desired. One possible way to proceed would be to deal with the DHF method cast in the form of the second-order equations instead of the system of first-order coupled equations and try to solve them by means of techniques used in the FD HF approach. The FD scheme was used by Laaksonen and Grant (50) and Sundholm (51) to solve the Dirac equation. Sundholm used the similar approach to perform Dirac-Hartree-Fock-Slater calculations for LiH, Li2, BH and CH+ systems (52,53). 

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