Hartree-Fock calculations for

Hartree-Fock Calculations for Atoms and Slater s Rules... 

Pisani C and R Dovesi 1980. Exact-Exchange Hartree-Fock Calculations for Periodic Systems. I. Illustration of the Method. International Journal of Quantum Chemistry XVII 501-516. 

Fig. 5.4 Changes in the iron orbital with the number of d-electrons as predicted by Hartree-Fock calculations for the configuration (N = 1—7). (Taken from [19])...

In view of the impracticability of the Hartree-Fock calculation for common molecules, the LCAO MO spatial functions may be used in place of Hartree-Fock ones. The MO s a and b are given by... 

Suhai128 investigated water dimer and an infinite chain of hydrogen-bonded water molecules by means of the DFT and post-Hartree-Fock calculations. For the infinite system, the DFT(BLYP), MP2, and MP4 binding energies were within 0.2 kcal/mol, whereas the corresponding interatomic distances were within 0.04 A. A similar agreement was reported for water dimer. 

Table 1. Perchlorate Hartree-Fock calculations for somewhat distorted tetrahedral conformations...

There is far less reported experience for ab initio studies of electronically excited states than for ground states. Matrix Hartree-Fock calculations for excited states cannot be considered routine. Often the same basis set is used for both the ground and excited state even though as long ago as 1958 Shull and Lowdin [17] demonstrated... 

It has been demonstrated that for the excited states of the atoms He, Li and Be considered in the present work, a simple optimization of the a and [3 parameters for each size of basis set leads to a sequence of even-tempered basis sets capable of supporting high accuracy in Hartree-Fock calculations for excited state energies of atoms. Furthermore, optimization of the a and f3 parameters for the smallest basis set in a sequence, M = 6 in the present study, followed by application of the recursion formulae (40) and (41) represents a good compromise which undoubtedly proved useful in case where full optimization of these parameters for each size of basis set is computationally demanding. 

One-center expansion was first applied to whole molecules by Desclaux Pyykko in relativistic and nonrelativistic Hartree-Fock calculations for the series CH4 to PbH4 [81] and then in the Dirac-Fock calculations of CuH, AgH and AuH [82] and other molecules [83]. A large bond length contraction due to the relativistic effects was estimated. However, the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in the entire coordinate space. It is notable that the RFCP and one-center expansion approaches were considered earlier as alternatives to each other [84, 85]. 

Fig. 5. Splitting of the five 4f75d crystal-field states of Tb3+ in UYF4 (taken from van Pieterson et al., 2002b). The parameter A represents the f-d interactions as explained in the text. On the left (A = 0) the parameters for these interactions are set to zero and on the right (A = 1) they have the values from Hartree-Fock calculations for the free...

For the particular case of H2, very accurate calculations including correlation effects have been carried out for the potential curves of ground and some excited states of H2 by Kolos and Wolniewicz [42-44],and by Davidson [45-47], Extended Hartree-Fock calculations for the ground state [48] and double configuration SCF calculations for some excited states of H2 [49] have been carried out by Wahl and co-workers. For Hj a very rigorous calculation has been carried out by Bates and co-workers [50]. 

Using atomic densities calculated from tabulated atomic wave functions, the summation was found [214] to produce results equivalent to the most elaborate molecular Hartree-Fock calculations for a series of small molecules, at a fraction of the computing expense. Surface areas and volumes computed by the two methods were found virtually identical. The promolecule calculation therefore has an obvious advantage in the exploration of surface electron densities, surface areas and molecular volumes of macromolecules for the analysis of molecular recognition. 

The numerical values for these quantities have been extracted and summarized in Table V. These results did not surprise us, since they were predicted by ionic model calculations (19) as well as one ab initio Hartree-Fock calculation for lithium fluoride (20) (a subsequent one is also shown in Table V) which treated both monomer and dimer. However, the trend is opposite to that observed with metal and noble gas dimers, whose I.P. s are lower than the corresponding monomers. It is simply a consequence of the relative bonding strengths of the two units in the neutral and ionic forms. Alakll halide dimers are more stable as neutrals metal and noble gas dimers are generally more stable as ions. 

Potential curves for H2 and (plotted from tabulated data in Ref, 255). The dissociation energies (indicated by arrows) are determined from Hartree-Fock calculations for H2, H, F2 and F for F2 the calculated dissociation energy has an incorrect sign (see text). 

These are chosen to span the space required to describe the occupied states. The choice can be optimised to lower the total energy E, but this is a laborious process requiring a complete Hartree—Fock calculation for each variation. An extensive investigation of this has resulted in the tables of Clementi and Roetti (1974) for the low-lying states of neutral atoms, positive ions and isoelectronic series of ions up to Z=54. These eigenvectors have been very sensitively verified by the (e,2e) reaction (chapter 11) and form an excellent start for structure calculations. 

Christiansen, P.A. and McCullough, E.A. (1977) Numerical Hartree-Fock calculations for N2, FH and CO comparison with optimized LCAO results. J. Chem. Phys., 67, 1877-1882. 

We have also added a method of calculating improved virtual orbitals. Our use of this procedure for N electron excited state virtual orbitals (8l) in the framework of the SCF calculation of the N-l electron problem closely resembles those proposed by Huzinaga (82). We have also investigated Huzinagafs recent method for improved virtual orbitals in the extended basis function space (83) This is also a useful procedure where there are convergence problems for the Hartree-Fock calculations for the N-electron occupied space of the excited states. This should also be helpful in optimizing virtual orbitals to use them in perturbation theory expressions. 

The Hartree-Fock calculation for ( ) yields a stabilization energy with respect to the separated atoms which is nearly identical to the GVB-PP calculation, and taking the appropriate correction into... 

The use of systematic sequences of even-tempered basis sets in mar trix Dirac-Hartree-Fock calculations for the argon atom ground state. 

Some matrix Hartree-Fock calculations for the water ground state... 

The AIMP method in its present form starts from a quasirelativistic all-electron Hartree-Fock calculation for the atom under consideration in a suitable electronic state and approximates the operators on the left-hand side of Equation (3.10) for an atomic core X as described in the following. 

Fig. 5.12. Nonrelativistic Hartree-Fock calculations for Ba+, showing the bi-modal behaviour of the nf orbitals resulting from centrifugal barrier effects (after J.-P. Connerade and M.W.D. Mansfield [212]).

This suggests a very simple way to include a substantial part of the correlations. One may simply perform a Hartree-Fock calculation for the ground and for the excited states, calculate the length and velocity forms of the cross section and take the geometric mean of the two. This is referred to as the HFU approximation if the continuum states are unrelaxed, and as the HFR if the Hartree-Fock geometric mean is calculated using relaxed final states. 

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