Hartree-Fock Numerical

Becke A D 1983 Numerical Hartree-Fock-Slater calculations on diatomic molecules J. Chem. Phys. 76 6037 5 Case D A 1982 Electronic structure calculation using the Xa method Ann. 

There is some small print to the derivation the orbitals must not change during the ionization process. In other words, the orbitals for the cation produced must be the same as the orbitals for the parent molecule. Koopmans (1934) derived the result for an exact HF wavefunction in the numerical Hartree-Fock sense. It turns out that the result is also valid for wavefunctions calculated using the LCAO version of HF theory. 

Structure calculation in which the only surviving residue is the relativistic correction to the energy. So, although the total electronic energies fluctuate by 0.1 a.u. in this study, and the results for the largest basis sets are at least 0.005 hartree above the numerical Hartree-Fock limit in all cases, the fluctuation in the relativistic corrections, Er, are significantly less than 10 hartree, which is more than sufficiently accurate for the present study. 

Many computations were made to classify the various types of behavior that no(p) can have in atoms and atomic ions [210,211,225-229], but some confusion persisted because different authors obtained different results in a few problematic cases. Purely numerical Hartree-Fock calculations, free from basis set artifacts, were then used to establish that the ground-state momenmm densities of all the atoms and their ions can be classihed into just three types [230,231] as illustrated in Figure 5.5. 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

Numerical Hartree-Fock calculations of [87], on the other hand, convincingly show that our results in real space are the same as those of the orbital space model [Eq. (4.32)] and that we are thus justified to write... 

Applications based on the even-tempered prescription (1) have shown that it can lead to atomic and diatomic Hartree-Fock ground state energies of an accuracy approaching that achieved in numerical Hartree-Fock calculations [4] It is conjectured that a comparable accuracy can be achieved for small polyatomic molecules [12], [13] by constructing basis sets according to the prescription established for diatomic molecules. Similar procedures can... 

The radial functions f (r) will be different for different atoms. Only for the hydrogen atom is the exact analytical form of the i2((r) s known. For other atoms the f (r) s will be approximate and their form will depend on the method used to find them. They might be analytical functions (e.g. Slater orbitals) or tabulated sets of numbers (e.g. numerical Hartree-Fock orbitals). 

The reader should note that no transformation operator 0A can alter the radial function J r) of an orbital and consequently the symmetry properties of the AOs are completely defined by the angular functions, Y O, ). Since these angular functions are the same in all one-electron product function approximations, the orbitals in all these approximations (Slater orbitals, numerical Hartree-Fock orbitals,... 

If comparison with experiment is not appropriate, what should be used for reference values Clearly, the desirable thing would be to eliminate as many sources of error as possible. For instance, if we wish to establish the reliability of an SCF treatment in a given one-particle basis, we could use numerical Hartree-Fock results for diatomic molecules (that is, essentially complete basis results) as benchmarks. Any difference between the finite basis and numerical results would presumably be due to inadequacies in the former, as otherwise the same approximations are made in both treatments. It is crucial to understand that this approach gives much more informa-... 

Here af and cf for the cases n = l + 1 are found from the variational principle requiring the minimum of the non-relativistic energy, whereas cf (n > l + 1) - form the orthogonality conditions for wave functions. More complex, but more accurate, are the analytical approximations of numerical Hartree-Fock wave functions, presented as the sums of Slater type radial orbitals (28.31), namely... 

The resulting atom polarizabilities fitted to experiment are summarized in Table 3-2 together with numerical Hartree-Fock results for the free atoms [112],... 

Application of the Hartree-Fock Method. - Since numerical Hartree-Fock programs dealing with complex numbers are available in many research groups, it seemed natural to apply this scheme also to the scaled Bom-Oppenheimer Hamiltonian (4.15). As a consequence, some numerical results were obtained before the theory was developed, and - as we have emphasized in the Introduction - some features seemed rather astonishing. 

A landmark in atomic theory was provided by the work of Layzer,9 who pointed out that regularities in the properties of atomic ions,10 which were hard to relate via numerical Hartree-Fock (HF) studies, could be understood via the so-called 1/Z expansion. Layzer9 showed that the total non-relativistic energy of an atomic ion could be expanded as... 

Table I. Numerical Hartree-Fock Excitation Energies ...

In order to construct atomic pseudopotentials, one needs to formulate some requirements on a pseudoorbital which is to be an eigenfunction to the atomic pseudopotential equation with the same eigenvalue as in the all-electron case. We assume that outermost part of a pseudoorbital coincides with the numerical Hartree-Fock-Slater (HFS) function as close to the nucleus as possible. The pseudoorbital should be nodeless and normalized. 

For small highly symmetric systems, like atoms and diatomic molecules, the Hartree-Fock equations may be solved by mapping the orbitals on a set of grid points. These are referred to as numerical Hartree-Fock methods. However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known functions. Any type of basis function may in principle be used expo ... 

The calculation of Mitroy started by calculating the Hartree—Fock approximation to the ground state 3s where we denote the states by the orbitals of the two active electrons in the configuration with the largest coefficient, in addition to the symmetry notation. The calculation used the analytic method with the basis set of Clementi and Roetti (1974) augmented by further Slater-type orbitals in order to give flexibility for the description of unoccupied orbitals. The total energy calculated by this method was —199.614 61, which should be compared with the result of a numerical Hartree—Fock calculation, —199.614 64. 

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. 

This recipe may be the only practical option, for example, when bond functions are used on the A B axis, or when the numerical Hartree-Fock method is employed. 

Cromer DT, Mann JB (1968) X-ray scattering factors computed from numerical Hartree-Fock wave functions. Acta Crystallogr, A-Cryst Phys Diffr Theor Gen Crystallogr 24 321... 

E.A. McCullough, Numerical Hartree-Fock methods for diatomic molecules A partial wave expansion approach. Comp. Phys. Rep. 4 (1986) 265. 

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. 

---