Hartree-Fock frozen core

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)...

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)... 

An appropriate potential for many cases is the frozen-core Hartree—Fock potential with the addition of a core-polarisation term. 

Scattering from alkali-metal atoms is understood as the three-body problem of two electrons interacting with an inert core. The electron—core potentials are frozen-core Hartree—Fock potentials with core polarisation being represented by a further 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. 

The lowest approximation to the removal energy is seen to be — e , where e is the eigenvalue of the frozen-core Hartree-Fock equation. It should be emphasized that the valence orbital is not treated self consistently. The orbitals of the closed-shell core are determined self-consistently, then the valence electron HF equation is solved in the frozen potential of the core. From Eq. (145) it follows that there is no first-order correction to the removal energy in the frozen-core HF potential. 

To explicitly obtain the spatial distribution of the photoelectrons for the final electronic state, we assume a frozen-core Hartree Fock description where the final ionized electronic state is represented as an antisymmetrized product of a Hartree Fock ion core wavefunction, c,+, and a photoelectron... 

To obtain the final state wavefunctions needed in evaluating the photoelectron matrix elements, we assume a frozen-core Hartree-Fock model in which the core orbitals are taken to be those of the ion for both ionized and neutral electronic states. The photoelectron orbital, i2), is... 

To obtain the photoionization amplitudes, we used our Cl wavefunction for the double-minimum state and a frozen-core Hartree-Fock (FCHF) description of the wavefunction for the ionized state. For the FCHF model the wavefunction is taken to be an antisymmetrized product of HF ion orbitals and a photoelectron orbital that is a solution of a one-electron Schrodinger equation containing the Hartree-Fock potential of the ion core. The HF wavefunction provides a very adequate description of Na over all internu-clear distances of interest. 

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. 

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. 

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. 

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. 

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...

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