Coulomb pseudo-potential

The nonlocal part of the pseudo potential requires the calculation of overlap integrals between the basis functions and the projectors. As the projectors can be written themselves as Cartesian Gaussian functions the same formulas as for the basis set overlap integrals can be used. The error function part of the local pseudo potential and the core potential can be written as a special case of a Coulomb integral... 

In some applications, Va is simply the Coulomb attraction between the bare nucleus and the electrons, Va r) = —Z jr, where Za is the nuclear charge. In other cases the use of the Coulomb potential renders the calculation unfeasible, and one has to resort to pseudo-potentials (see Sect. 6.3.1). 

A Gaussian screened Coulomb potential B Cartesian force constants for ions in channels C Matrix elements for nearly free ions D Matrix elements in pseudo-Bloch functions... 

We have seen earlier that it is possible to expand almost any smooth function which goes to zero at infinity in terms of Gaussian functions, so that the natural first choice for an expansion of the core potential is a linear combination of Gaussians. We have seen how to generate the explicit numerical forms of the pseudo, Coulomb and exchange potential available from atomic calculations so that we may use both these forms and the Gaussian expansion method to guide our choice. 

Extensive introductions to the effective core potential method may be found in Ref. [8-19]. The theoretical foundation of ECP is the so-called Phillips-Kleinman transformation proposed in 1959 [20] and later generalized by Weeks and Rice [21]. In this method, for each valence orbital (pv there is a pseudo-valence orbital Xv that contains components from the core orbitals and the strong orthogonality constraint is realized by applying the projection operator on both the valence hamiltonian and pseudo-valence wave function (pseudo-valence orbitals). In the generalized Phillips-Kleinman formalism [21], the effect of the projection operator can be absorbed in the valence Pock operator and the core-valence interaction (Coulomb and exchange) plus the effect of the projection operator forms the core potential in ECP method. 

Some flexibility remains in the selection of terms to be included in Eq. 9.36. An assortment of ECPs are possible within this framework because the form of the pseudo-orbital within the cutoff radius is not completely defined. So-called soft ECPs have been designed so that wi cancels the Coulomb singularity at the nucleus [133, 135]. This is valuable for QMC calculations because their efficiency is sensitive to rapid changes of the potential. Several sets of soft ECPs have been designed specifically for QMC so that Gaussian basis function can be used in QMC calculations without special consideration of the electron-nucleus cusp conditions [136, 137]. 

In Eq. (167) the atomic pseudopotential is a one-electron operator which takes into account the interaction of a valence electron with the core. At large distance from the nuclei, E tends to the Coulomb potential — z/r where z is the net charge of the core of the atom. The valence pseudo-Fock operator takes the form... 

Figure 2.5. Schematic representation of the construction of the pseudo-wavefunction

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