Frozen-core treatment

The Pauli operator of equations 2 to 5 has serious stability problems so that it should not, at least in principle, be used beyond first order perturbation theory (20). These problems are circumvented in the QR approach where the frozen core approximation (21) is used to exclude the highly relativistic core electrons from the variational treatment in molecular calculations. Thus, the core electronic density along with the respective potential are extracted from fully relativistic atomic Dirac-Slater calculations, and the core orbitals are kept frozen in subsequent molecular calculations. 

In electron correlation treatments, it is a common procedure to divide the orbital space into various subspaces orbitals with large binding energy (core), occupied orbitals with low-binding energy (valence), and unoccupied orbitals (virtual). One of the reasons for this subdivision is the possibility to freeze the core (i.e., to restrict excitations to the valence and virtual spaces). Consequently, all determinants in a configuration interaction (Cl) expansion share a set of frozen-core orbitals. For this approximation to be valid, one has to assume that excitation energies are not affected by correlation contributions of the inner shells. It is then sufficient to describe the interaction between core and valence electrons by some kind of mean-field expression. 

Table 17. Convergence of the anisotropy of the static hypermagnetizability An(0) of neon with the correlation treatment (from [39] finite field orbital-relaxed results (unless specified) for d-aug-cc-pVDZ, frozen core approximation for Is shell, numbers in atomic units.)...

The electronic many-body Hamiltonian in equation (1) is treated in the framework of the independent-electron frozen-core model. This means that there is only one active electron, whereas the other electrons are passive (no dynamic conelation is accounted for) and no relaxation occurs. In this model the electron-electron interaction is replaced by an initial-state Hartree-Fock-Slater potential [37]. This treatment is expected to be highly accurate for heavy collision systems at intermediate to high incident energies. The largest uncertainties of the independent-electron model will show up for low-Z few-electron systems, such as H -F H and H + He° or for high multiple-ionization probabilities. 

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 interelectronic energy of an electron in orbital i with two paired electrons in orbital / consists of two parts Jij for the different-spin interaction and Jy — Xy for the same-spin interaction, which together give 2 Jt] — Kij. Within the orbital i only Jn should appear but this term, due to relation (14), may be replaced by 2Ju —Ku. It is important to realize that this self-adjustment occurs only for occupied orbitals — thanks to the property (/< — K ) y)i = 0 — but not for virtual orbitals since for those the operator 2J —Kf is present, and an electron in a virtual orbital feels the full interaction of N electrons. For this reason it is often said that virtual-orbital solutions of (5) are appropriate for (N 4- l)-electron systems It would be natural to use an operator Hke (3) to obtain appropriate virtual orbitals for N-electron systems. This heis been done by Kelly in his extensive perturbation calculation of Be 29-65) by Hunt and Goddard in their calculation of the excited states of H2O >, and by Lefebvre-Brion et al. (frozen-core approximation) Goddard s method will serve to illustrate this general type of treatment. 

Full ab initio treatments for complex transition metal systems are difficult owing to the expense of accurately simulating all of the electronic states of the metal. Much of the chemistry that we are interested in, however, is localized around the valence band. The basis functions used to describe the core electronic states can thus be reduced in order to save on CPU time. The two approximations that are typically used to simplify the basis functions are the frozen core and the pseudopotential approximations. In the frozen core approximations, the electrons which reside in the core states are combined with the nuclei and frozen in the SCF. Only the valence states are optimized. The assumption here is that the chemistry predominantly takes place through interactions with the valence states. The pseudopotential approach is similar. 

Huckel (properly, Huckel) molecular orbital theory is the simplest of the semiempirical methods and it entails the most severe approximations. In Huckel theory, we take the core to be frozen so that in the Huckel treatment of ethene, only the two unbound electrons in the pz orbitals of the carbon atoms are considered. These are the electrons that will collaborate to form a n bond. The three remaining valence electrons on each carbon are already engaged in bonding to the other carbon and to two hydrogens. Most of the molecule, which consists of nuclei, nonvalence electrons on the carbons and electrons participating in the cr... 

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