Core-valence interaction

The core electrons are replaced by a gaussian expansion which reproduces electrostatic and exchange core-valence interactions. 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

This has enabled these authors to derive a computational means for obtaining the effective core-valence interaction ((7eore) as a local one-electron operator.)... 

Having described the variety of ways in which the core-valence interaction may be parameterized it is clear that we should examine how they perform in actual calculations. Generally the advances in the complexity of the parameterization have produced commensurate improvements in the accuracy of the results. However, by introducing a large number of parameters the simplicity of the core-valence concept is lost and, in practice, the fitting of the parameters themselves can be expensive in terms of computer time, although they only need to be obtained once for each atom. 

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

Once a nodeless orbital has been generated the one-electron atomic Fock equation is easily inverted to produce a (radially) local operator, the EP, which represents the core-valence interactions (22,23). 

The expression for the anisotropic part of hyperfine coupling involves an integral over the spatial distribution of the unpaired electron, which is relatively easy to compute accurately even at a relatively low level of theory. The contact term, however, includes a delta-function that chips out the wave function amplitude at the nucleus point. The latter is quite difficult to compute both because standard Gaussian basis sets do not reproduce the wavefunction cusp at the nucleus point and because additional flexibility has to be introduced into the core part of the basis to account for the now essential core valence interaction. " ... 

It seems worthwhile emphasizing that, apart from the fact that they lead to the spurious xc-tail, the nonlinear contributions to the core-valence interaction seem to be less important for the exact Ex than for the LDA. This is illustrated in Table 4.9, which lists atomic excitation energies associated with the transfer of an electron from one spin channel into the other. While in the case of the LDA, NLCCs are required for the accurate reproduction of these transfer energies, linear unscreening is sufficient for the exact Ex. 

Here a pseudo-potential is used to represent the core-valence interaction. Thereby the SCF procedure is reduced in scale to encompass only the valence electrons. The development of the method is described in an application to the uranium atom [60]. The procedure is as follows. First a set of valence pseudoorbitals is formed from a linear combination of atomic orbitals, with coefficients... 

For core-valence interaction, a non-relativistic or quasi-relativistic cf. below) Hartree-Fock (HF) description of the core naturally leads to a one-electron potential... 

Griffin calculation in LS-coupling) and doing a similar averaging for the equations (8) and (9) can readily be evaluated. The steps leading to equations (8) and (9) have relevance not only for the relativistic operators but also for the core-valence interaction, equation (5) the first two terms in equation (5) are local operators anyway (i.e. multiplicative potentials), c Kc can be written as an /-dependent operator, for an atom as in equations (7) and (8), and the orbitals necessary for building up Jc and Kc can be taken from an LS Cowan-Griffin AE calculation as above. 

When working with the final model potentials of equations (10) and (11) in molecular applications, we are interested in valence properties and, in fact, do not need to include core orbitals in our wavefunction any more. This is so, since core-valence interaction has been accounted for implicitly in equation (10), and the sum of energies of individual cores with fixed orbitals is a constant, anyway. However, in order to get an aufbau principle for the valence wavefunction, core orbitals > have to be shifted from energies c to energies above those of the (occupied) valence ones, and this level-shift is done by... 

DFT-Based Pseudopotentials. - The model potentials and shape-consistent pseudopotentials as introduced in the previous two sections can be characterized by a Hartree-Fock/Dirac-Hartree-Fock modelling of core-valence interactions and relativistic effects. Now, Hartree-Fock has never been popular in solid-state theory - the method of choice always was density-functional theory (DFT). With the advent of gradient-corrected exchange-correlation functionals, DFT has found a wide application also in molecular physics and quantum chemistry. The question seems natural, therefore Why not base pseudopotentials on DFT rather than HF theory ... 

Although this does not enter into the discussion of correlation effects, we point out the role of higher-order relativistic effects, such as the Breit interaction, on the spin-orbit splitting, which are not explicitly included. For the neutral Pb atom, the Breit interaction estimated by a four-component all-electron calculation using first order perturbation theory lowers the SCF spin-orbit splitting by 166 cm thus compensating partially the increase due to core-core and core-valence interactions [60]. 

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. 

Therefore, when parameterized against a DK relativistic atomic reference, the local potentials not only approximates the core-valence interaction, but also all the differences between the two kinetic operators, which are non-local operators. The non-locality of the DK hamiltonian is hidden in the K (Eq.8.22) and A (Eq. 8.23) operators and those operators show up in almost every term of the expansion of the DK hamiltonian [17,156]. Thus, the non-local to local approximation of the Version I model core potential is more severe. Based on our experience, this error is... 

The quality of an atomic pseudopotential is measured by its transferability, that is, its performance in describing a given atom in different valence-electron configurations (for example, isolated, but in different electronic configurations, in a molecule, a cluster or a solid). This property strongly depends on the way the core-valence interaction is... 

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