Hamiltonian with RECP

Generalized RECP When core electrons of a heavy-atom molecule do not play an active role, the effective Hamiltonian with RECP can be presented in the form... 

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

The most common way of including relativistic effects in a calculation is by using relativisticly parameterized effective core potentials (RECP). These core potentials are included in the calculation as an additional term in the Hamiltonian. Core potentials must be used with the valence basis set that was created for use with that particular core potential. Core potentials are created by htting a potential function to the electron density distribution from an accurate relativistic calculation for the atom. A calculation using core potentials does not have any relativistic terms, but the effect of relativity on the core electrons is included. 

The first two-step calculations of the P,T-odd spin-rotational Hamiltonian parameters were performed for the PbF radical about 20 years ago [26, 27], with a semiempirical accounting for the spin-orbit interaction. Before, only nonrelativistic SCF calculation of the TIF molecule using the relativistic scaling was carried out [86, 87] here the P,T-odd values were underestimated by almost a factor of three as compared to the later relativistic Dirac-Fock calculations. The latter were first performed only in 1997 by Laerdahl et al. [88] and by Parpia [89]. The next two-step calculation, for PbF and HgF molecules [90], was carried out with the spin-orbit RECP part taken into account using the method suggested in [91]. 

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which... 

There is no need to explicitly include terms for direct relativistic effects, such as the dependence of mass on velocity, which are important only in the core region, in the valence-electron Hamiltonian. These terms are included as a consequence of using the Diiac-Fock wave functions. Thus, the Hamiltonian for the valence electrons is composed of the nonrelativistic Hamiltoman for the valence electrons plus the RECPs, which include the effects of the core electrons as well as the relativistic effects on the valence electrons in the core region [37]. The RECPs thns represent, for the valence electrons, the dynamical effects of relativity from the core region, the repulsion of the core electrons, the spin-orbit interaction with the nucleus, the spin-orbit interaction with the core electrons, and an approximation to the spin-orbit interaction between the valence electrons [38], which has usually been found to be quite stnaU, especially for heavier element systems [39-41]. The REP operators can be written as a summation of spin-independent potential and the spin-orbit operator, as written below, and the readers are referred to reference [39] for details. 

For many atoms and molecules, especially small open-shell systems of high symmetry, it is necessary to include spin-orbit interaction to achieve even qualitative agreement with experiment. For large systems with low symmetry or closed shells, the effect is less important because spin—orbit interaction is quenched, and these systems can therefore usually be described with a one-component method. In some cases this can also be achieved in a perturbation formalism at little additional cost. Few computer program systems have been developed for treating spin-orbit interactions at the all-electron level with a transformed Hamiltonian. In a recent review,the method and results from such calculations were discussed. Calculations including spin-orbit interactions at the RECP level have been carried out for many years.We will not discuss results, but it is clear that this will be an important method for large systems. 

The simple perturbative treatment of spin-free terms from the Pauh Hamiltonian does reasonably well for the two light members of the series, but less so for AuH. As this approximation is extremely easy to program for SCF calculations and requires almost no extra computational effort, it appears as an attractive qualitative approach to relativity in medium heavy species. Only the Dirac-Coulomb (DC) results in table 22.3 account for spin-orbit interaction. The closeness of DKH and RECP results to the DC values indicate that spin-orbit effects are of minor importance, something we would expect in closed-shell molecules, where the bonding is dominated by s orbitals. Under these conditions, the two approximate spin-free methods can compete with the full DC operator. The agreement between the results from these three schemes also indicate that the discrepancy between the calculated and experimental values is due to insufficient description of the correlation. This observation is in line with the common experience that MP2 calculations on transition-metal compounds frequently yield somewhat short bond lengths. 

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