Frozen-core Hartree—Fock calculations

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. 

It is of course possible to solve the Dirac—Fock problem with a linear combination of analytic orbitals. However, owing to the rapid variation of the orbitals near the nucleus it requires an awkwardly-large basis. If an analytic representation is convenient for a reaction calculation it may be obtained by a least-squares fit to a numerical orbital. 

Hartree—Fock calculations may be performed to find sets of orbitals describing the lowest-lying states of different symmetry manifolds of an atom. It is found that each different state has a closed-shell core whose orbitals are closely independent of the state. 

A frozen-core calculation involves choosing a particular state (for example the one lowest in energy), performing a Hartree—Fock calculation to find the best orbitals, then using the orbitals of the core to generate a nonlocal potential (5.27), which is taken to represent the core in calculations of further states. 

A good example is provided by the alkali-metal atoms, which consist of one electron outside a closed-shell core in the single-configuration model. If the frozen-core approximation is valid a frozen-core calculation of the orbital occupied by one electron will give the same result as a Hartree—Fock calculation and the core orbitals will not depend on the state. 

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The n=l and 2 shells were frozen at the ground-state Hartree—Fock values. The orbital set included the 4s,3p,3d,4f and 5g natural orbitals and 3p,3d,4s,4p,4d,4f,5s,5p,5d,5f orbitals from frozen-core Hartree—Fock calculations to provide representations for states whose dominant configuration is 13s n/). This set was again augmented by extra ad hoc orbitals to increase flexibility. The full set contained 24 orbitals (6 s-type, 7 p-type, 6 d-type, 3 /-type, 2 g-type) which were all orthogonalised using the prescription for two orbitals a) and b)... 

Table 5.3. One-electron separation energies for the lower-energy states of sodium (units eV). Experimental data (EXP) are from Moore (1949). The calculations are FCHE, frozen-core Hartree—Fock and POL, frozen-core Hartree—Fock with the phenomenological core-polarisation potential (5.82)...

Essentially-complete agreement with experiment is achieved by the coupled-channels-optical calculation. We can therefore ask if scattering is so sensitive to the structure details in the calculation that it constitutes a sensitive probe for structure. The coupled-channels calculations in fig. 9.3 included the polarisation potential (5.82) in addition to the frozen-core Hartree—Fock potential. Fig. 9.4 shows that addition of the polarisation potential has a large effect on the elastic asymmetry at 1.6 eV, bringing it into agreement with experiment. However, in general the probe is not very sensitive to this level of detail. 

All the above methods are somehow based on an orbital hypothesis. In fact, in the multipolar model, the core is typically frozen to the isolated atom orbital expansion, taken from Roothan Hartree Fock calculations (or similar [80]). Although the higher multipoles are not constrained to an orbital model, the radial functions are typically taken from best single C exponents used to describe the valence orbitals of a given atom [81]. Even tighter is the link to the orbital approach in XRCW, XAO, or VOM as described above. Obviously, an orbital assumption is not at all mandatory and other methods have been developed, for example those based on the Maximum Entropy Method (MEM) [82-86] where the constraints/ restraints come from statistical considerations. 

The second step specifies which orbitals are frozen and which ones will be optimized in the VB calculation. After the file specification (three first lines), the first line shows that there are 28 basis functions. The second line specifies that 2 MOs arising from the Hartree—Fock calculation will be frozen during the VB calculation, and that among the 28 basis functions, only 26 will be kept as the basis on which the VB orbitals will be expanded. The third and fourth lines indicate, respectively, the MOs that are frozen, and the basis functions that are kept for the variational procedure. In the present case, the MOs that are frozen correspond to the Is core, and the Is basis functions are eliminated from the VB orbitals. 

Table 5.1 illustrates the frozen-core approximation for the case of sodium using a simple Slater (4.38) basis in the analytic-orbital representation. The core (Is 2s 2p ) is first calculated by Hartree—Fock for the state characterised by the 3s one-electron orbital, which we call the 3s state. The frozen-core calculation for the 3p state uses the same core orbitals and solves the 3p one-electron problem in the nonlocal potential (5.27) of the core. Comparison with the core and 3p orbitals from a 3p Hartree—Fock calculation illustrates the approximation. The overwhelming component of the 3p orbital agrees to almost five significant figures. 

In our calculations, only the valence electrons are treated at the VB level. The inactive electrons are kept in a frozen core obtained through an atomic Hartree-Fock (HF) calculation. All geometry optimizations or relaxations are also performed at the HF level. 

Table 5.1. Comparison of a frozen-core (FC) calculation for the sodium 3p orbital with a Hartree—Fock (HF) calculation of the same state. The basis column gives the parameters nt and Cw of the basis Slater orbitals (4.38). The other columns give the coefficients Ci( (5.36). The frozen core is the 3s Hartree-Fock core...

Although the frozen-core approximation underlies all ECP schemes discussed so far, both static (polarization of the core at the Hartree-Fock level) and dynamic (core-valence correlation) polarization of the core may accurately and efficiently be accounted for by a core polarization potential (CPP). The CPP approach was originally used by Meyer and co-workers (MUller etal. 1984) for all-electron calculations and adapted by the Stuttgart group (Fuentealba et al. 1982) for PP calculations. The... 

With the exception of Ligand Field Theory, where the inclusion of atomic spin-orbit coupling is easy, a complete molecular treatment of relativity is difficult although not impossible. The work of Ellis within the Density Functional Theory DVXa framework is notable in this regard [132]. At a less rigorous level, it is relatively straightforward to develop a partial relativistic treatment. The most popular approach is to modify the potential of the core electrons to mimic the potential appropriate to the relativistically treated atom. This represents a specific use of so-called Effective Core Potentials (ECPs). Using ECPs reduces the numbers of electrons to be included explicitly in the calculation and hence reduces the execution time. Relativistic ECPs within the Hartree-Fock approximation [133] are available for all three transition series. A comparable frozen core approximation [134] scheme has been adopted for... 

The prediction of molecular Rydberg spectra, as well as the analysis of the available laboratory data, has constituted a challenge on the theory. A number of ab initio calculations have been carried out on transition probabilities of Rydberg molecules, such as the frequently quoted Hartree-Fock (HF) study of H3 by King and Morokuma (15), the self-consistent-field frozen-core calculation with floating-spherical-Slater-type orbitals (FSSO) on the second-row Rydberg... 

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