Wave function relativistic coupled perturbed

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

Also in the molecular case perturbation theory helps to clarify individual contributions to the property ax cording to their order [120] but more crucial is the fact that perturbation theory becomes an inadequate means for the proper description of relativistic effects in heavy atoms. Furthermore spin-orbit effects can substantially influence the result and methods using multi-component wave functions incorporating spin-orbit coupling from the beginning are favorable. 

Gagliardi and Roos conducted a series of studies on actinide compounds. They follow a combined approach with DKH/AMFI Hamiltonians combined with CASSCF/CASPT2 for the energy calculation and an a posteriori added spin-orbit perturbation expanded in the space of nonrelativistic CSFs. This strategy aims to establish a balance of sufficiently accurate wave function and Hamiltonian approximations. Since the CASSCF wave function provides chemically reasonable but not highly accurate results (as witnessed, for instance, in the preceding section), it is combined with a quasi-relativistic Hamiltonian, namely the sc alar-relativistic DKH one-electron Hamiltonian. Additional effects — dynamic correlation and spin-orbit coupling — are then considered via perturbation theory. 

In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part is added to the Hamiltonian. The effects of the spin-orbit term on the energies and wave functions are commonly estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry [2, 3]. 

Second-order properties are often evaluated using coupled-perturbed Hartree-Fock (CPHF) theory. The CPHF wave function is essentially the first-order perturbed wave function, which, as we saw above, must include the negative-energy states. Thus, in the relativistic case, the CPHF method must include both the positive- and negative-energy states. 

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