Variational calculations Breit-Pauli Hamiltonian

In molecular property calculations the same mutual interplay of electron correlation, relativity and perturbation operators (e.g. external fields) occurs. For light until medium atoms relativistic contributions were often accounted for by perturbation theory facilitating quasirela-tivistic approximations to the Dirac-Hamiltonian [114-117]. It is well-known that operators like the Breit-Pauli Hamiltonian are plagued by essential singularities and therefore are not to be used in variational procedures. It can therefore be expected that for heavier elements per-turbational inclusion of relativity will eventually become inadequate and that one has to start from a scheme where relativitiy is included from the beginning. Nevertheless very efficient approximations to the Dirac equation in two-component form exist and will be discussed further below in combination with their relevance for EFG calculations. In order to calculate the different contributions to a first-order property as the EFG, Kello and Sadlej devised a multiple perturbation scheme [118] in which a first-order property is expanded as... 

Abstract Variational methods can determine a wide range of atomic properties for bound states of simple as well as complex atomic systems. Even for relatively light atoms, relativistic effects may be important. In this chapter we review systematic, large-scale variational procedures that include relativistic effects through either the Breit-Pauli Hamiltonian or the Dirac-Coulomb-Breit Hamiltonian but where correlation is the main source of uncertainty. Correlation is included in a series of calculations of increasing size for which results can be monitored and accuracy estimated. Examples are presented and further developments mentioned. 

Hay and Wadt (1985a, b) have published ECPs which are in form identical to the averaged RECPs of Christiansen, Ermler and co-workers. However, there are differences. First, the Hay-Wadt potentials are derived from the Cowan-GriflSn adaptation of the Breit-Pauli Hamiltonian into a variational computation of the atomic wave-function. From these solutions the ECPs are generated. It should be noted that the spin-orbit coupling is not included in the Hay-Wadt ECPs. Consequently, molecular calculations done using these ECPs would not include spin-orbit coupling. 

The singularity that may arise when using the Breit-Pauli Hamiltonian can cause instability in variational calculations and it is desirable that such singularity be avoided formally. Douglas and KrolE showed that the singularity... 

The Breit-Pauli (BP) approximation [140] is obtained truncating the Taylor expansion of the Foldy-Wouthuysen (FW) transformed Dirac Hamiltonian [141] up to the (p/mc) term. The BP equation has the well-known mass-velocity, Darwin, and spin-orbit operators. Although the BP equation gives reasonable results in the first-order perturbation calculation, it cannot be used in the variational treatment. 

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