Operators core potential

The equations to be fulfilled by momentum space orbitals contain convolution integrals which give rise to momentum orbitals ( )(p-q) shifted in momentum space. The so-called form factor F and the interaction terms Wij defined in terms of current momentum coordinates are the momentum space counterparts of the core potentials and Coulomb and/or exchange operators in position space. The nuclear field potential transfers a momentum to electron i, while the interelectronic interaction produces a momentum transfer between each pair of electrons in turn. Nevertheless, the total momentum of the whole molecule remains invariant thanks to the contribution of the nuclear momenta [7]. 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

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. 

The use of Effective Core Potential operators reduces the computational problem in three ways the primitive basis set can be reduced, the contracted basis set can be reduced and the occupied orbital space can be reduced. The reduction of the occupied orbital space is almost inconsequential in molecular calculations, since it neither affects the number of integrals nor the size of the matrices which has to be diagonalized. The reduction of the primitive basis set is of course more important, but since the integral evaluation time is in general not the bottleneck in molecular calculations, this reduction is still of limited importance. There are some cases where the size of the primitive basis set indeed is important, e.g. in direct SCF procedures. The size of the contracted basis set is very important, however. The bottleneck in normal SCF or Cl calculations is the disc storage and/or the iteration time. Both the disc storage and the iteration time depend strongly on the number of contracted functions. 

Matrix elements for the valence functions were taken with the effective core potential the coulomb and exchange terms were handled exactly, numerically, without any parameterization and a Phillips-Kleinman projection operator term was also used. Spin-orbit coupling effects amongst the valence orbitals were treated semi-empirically using the operator... 

The accurate parameterization of the effective core potential has shown that the reduction of the pseudopotential to the form of a one-particle operator is adequate. The scaling of the two-body potentials by the use of an operator65... 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

Configuration Interaction Using the Graphical Unitary Group Approach and Relativistic Core Potential and Spin—Orbit Operators. 

The method discussed here for the inclusion of relativistic effects in molecular electronic structure calculations is grounded in the Dirac-Fock approximation for atomic wave functions (29). The premise is that the major relativistic effects of the Dirac Hamiltonian are manifested in the core region, involving the core electrons, and that these effects propagate to the valence electrons. In addition, there are direct relativistic effects on valence electrons penetrating into the core region. Insofar as this is true, the valence electrons can be treated using a nonrelativistic Hamiltonian to which is added an operator, the relativistic effective core potential (REP). The REP formally, incorporates relativistic effects due to core electrons and to interactions of valence electrons with core electrons in an internally consistent way. 

Fig. 6. Average relativistic effective core potential and relativistic effective core potential energy curves for two states of Bi2. HF, Hartree-Fock GVB(pp), eight-configuration perfect-pairing generalized valence bond FVCI, full-valence Cl based on the GVB(pp) wave functions FV7R, full-valence Cl plus single and double promotions to virtual MOs relative to seven-dominant configurations. (The FVCI and FV7R calculations include the REP-based spin-orbit operator.)...

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],... 

Calculations are performed for both nickel and cobalt clusters. When the metal atoms are in the atomic d s state (see below) the atoms are treated computationally as one-electron systems by the use of Effective Core Potentials (ECP s). This is a crude but qualitatively correct description and treats the core including the 3d shell bv potentials and projection operators. Similarly for atoms in the d s state a two-electron ECP is used. 

A remark should be made here with respect to the generation and adjustment of the widely used effective core potentials (ECP, or pseudopotentials) [85] in standard non-relativistic quantum chemical calculations for atoms and molecules. The ECP, which is an effective one-electron operator, allows one to avoid the explicit treatment of the atomic cores (valence-only calculations) and, more important in the present context, to include easily the major scalar relativistic effects in a formally non-relativistic approach. In general, the parameters entering the expression for the ECP are adjusted to data obtained from numerical atomic reference calculations. For heavy and superheavy elements, these reference calculations should be performed not with the PNC, but with a finite nucleus model instead [86]. The reader is referred to e.g. [87-89], where the two-parameter Fermi-type model was used in the adjustment of energy-conserving pseudopotentials. 

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