Effective core potential methods

In the present chapter we have described the Effective Core Potential method, the... 

The QMC method is ideally suited for mixed systems because electron-positron correlation, which is difficult to treat with Cl methods, is automatically treated correctly. Systems of up to a bit more than ten leptons are routinely treated. Effective core potential methods can be used to extend the method to larger systems. Expectation values of local operators for the distribution k 2 are calculated by straightforward sampling procedures, but nonlocal operators, such as those for the annihilation rate, are problematic and are under active investigation [12],... 

O. Gropen, The Relativistic Effective Core Potential Method, in Methods in Computational Chemistry, Vol. 2 Relativistic Effects in Atoms and Molecules, S. Wilson (Ed), Plenum Press, New York, 1988. 

The relativistic effective core potential method is reviewed. The basic assumptions of the model potential and pseudopotential variants are discussed and the reliability of both approaches in electronic structure calculations for heavy element systems is demonstrated for selected examples. 

Effective Core Potential methods are classified in two families, according to their basic grounds. On the one hand, the Pseudopotential methods (PP) rely on an orbital transformation called the pseudoorbital transformation and they are ultimately related to the Phillips-Kleinman equation [2]. On the other hand, the Model Potential methods (MP) do not rely on any pseudoorbital transformation and they are ultimately related to the Huzinaga-Cantu equation [3,4]. The Ab Initio Model Potential method (AIMP) belongs to the latter family and it has as a... 

In principle, it should also be possible to add a semi-loced potential to the non-relativistic all-electron Hamiltonian to eirrive at a quasi-rela-tivistic all-electron method. One such suggestion has been made by Delley [76], but the resulting method has only been tested for valence properties, which could also have been obtained by valence-only methods. Effective core potential methods have the advantage of a reduced computational effort (compared to all-electron methods) and are a valuable tool as long as one is aware of the limited domain of valence-only methods. Properties for which density variations in the atomic core are important should not be calculated this way. Examples are the electric field gradient at the nucleus or the nuclear magnetic shielding. 

U. Wahlgren, The Effective Core Potential Method. In Lecture Notes in... 

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. 

An ab initio effective core potential method derived from the relativistic all-electron Dirac-Fock solution of the atom, which we call the relativistic effective core potential (RECP) method, has been widely used by several investigators to study the electronic structure of polyatomics including the lanthanide- and actinide-containing molecules. This RECP method was formulated by Christiansen et al. (1979). It differs from the conventional Phillips-Kleinman method in the representation of the nodeless pseudo-orbital in the inner region. The one-electron valence equation in an effective potential of the core electron can be written as... 

The main difficulty here is to clearly separate effects that can hardly be separated, namely relativistic and electron-correlation effects. Nevertheless, pioneering studies of this effect date back to the mid 1970s [1140]. Four-component methods have been employed to determine the contribution which is solely due to relativity [1141]. The four-component approach, for which Dirac-Hartree-Fock and — to also account for correlation effects — relativistic MP2 calculations have been utilized, confirms results first obtained with relativistic effective core potential methods [1142,1143]. It has been found [1141] that between 10% and 30% of the lanthanide contraction and 40% to 50% of the actinide contraction are caused by relativity in monohydrides, trihydrides, and monofluorides of La, Lu and Ac, Lr, respectively. 

The large number of all-electron calculations on tetrels and pnictogens provide perhaps the best opportunity to compare all-electron and ECP results. The few results given in this section are, however, typical. Modern effective core potential methods give the same degree of accuracy in terms of predicted properties as more traditional all-electron methods, at the same level of theory and using comparable valence basis sets. 

Cundari, T. R., 8c Stevens, W. J. (1993). Effective core potential methods for the lanthanides. The Journal of Chemical Physics, 98, 5555-5565. 

Dyall K. Formal analysis of effective core potential methods. J Chem Inf Comput Sci. 2001 41 30-37. 

A. V. Titov and N. S. Mosyagin, The generalized relativistic effective core potential method Theory and calculations, Russ. J. Phys. Chem., 74, S376-S387 (2(X)0). 

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