Valence-region energy, equations

Now, the most obvious merit of this equation is that it expresses valence-region energies in terms of the total (electronic and nuclear) potential energies of the individual nuclei Z. Contrasting with Eq. (9), V is not any longer required. Applying Eq. (4), we can reformulate (10) in a more compact manner ... 

Essentially two lines are followed for the derivation of pseudopotentials or effective-core potentials creating nodeless pseudo-orbitals. One approach uses as reference data the shape of the valence orbitals or spinors in the spatial valence region and their corresponding one-particle energies. In the spatial core region the one-particle functions are smoothed according to special prescriptions and the radial Fock equation is solved for the potential, which is then usually fitted by linear combinations of Gaussians times powers of r. 

In many cases we are not interested in the full ionization spectrum, but only in the first few IPs for the outer valence electrons. The method which takes into account only the outer valence region is known as the OVGF method. It was shown by Cederbaum et al." that in the OVGF method G(IP) can be used in a diagonal form. Diagonal approximations in the self-energy matrix in the ab initio framework have proven successful for outer valence lEs and EAs of closed shell molecules. By taking this into account, equation (9) can be rewritten as... 

The Pauli approximation may be used in conjunction with this method by neglecting the small component spinors Q) of the Dirac equation, leading to RECPs expressed in terms of two-component spinors. The use of a nonrelativistic kinetic energy operator for the valence region, and two-component spinors leads to Hartree-Fock-like expressions for the pseudoorbitals. Note that the V s (effective potentials) in this expression are not the same for pseudo-orbitals of different symmetry. Thus the RECPs are expressed as products of angular projectors and radial functions. In the Dirac-Fock approximation, the orbitals with different total j quantum numbers, but which have the same / values are not degenerate, and thus the potentials derived from the Dirac-Fock calculations would be y-dependent. Consequently, the RECPs can be expressed in terms of the /y-dependent radial potentials by equa-... 

A truncation of the expansion (3.5) defines the zero- and first-order regular approximation (ZORA, FORA) (van Lenthe et al. 1993). A particular noteworthy feature of ZORA is that even in the zeroth order there is an efficient relativistic correction for the region close to the nucleus, where the main relativistic effects come from. Excellent agreement of orbital energies and other valence-shell properties with the results from the Dirac equation is obtained in this zero-order approximation, in particular in the scaled ZORA variant (van Lenthe et al. 1994), which takes the renormalization to the transformed large component approximately into account, using... 

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