Explicit Relativistic Calculations

The WB approach was used to generate both model potentials as well as pseudopotentials. The DKH method was applied in explicit relativistic calculations together with model potentials and also to provide molecular AE results for calibration studies with various valence-only schemes (cf. below). 

There are also ways to perform relativistic calculations explicitly. Many of these methods are plagued by numerical inconsistencies, which make them applicable only to a select set of chemical systems. At the expense of time-consuming numerical integrations, it is possible to do four component calculations. These calculations take about 100 times as much CPU time as nonrelativistic Hartree-Fock calculations. Such calculations are fairly rare in the literature. 

Relativistic effects are significant for the heavier metals. The method of choice is nearly always relativistically derived effective core potentials. Explicit spin-orbit terms can be included in ah initio calculations, but are seldom used because of the amount of computational effort necessary. Relativistic calculations are discussed in greater detail in Chapter 33. 

Bioinorganic systems often contain heavy elements that need to be treated with an explicit relativistic method. It is now possible to carry out an explicit relativistic electronic structure calculation on the fly (152). The scalar-relativistic Douglas - Kroll - Hess method was implemented by us recently in the BOMD simulation framework (152). To use the relativistic densities in a non-relativistic gradient calculations turned out to be a valid approximation of relativistic gradients. An excellent agreement between optimized structures and geometries obtained from numerical gradients was observed with an error smaller than 0.02 pm. 

In summary, conventional relativistic ECP s provide an efficient mean to calculate molecular properties up to and including the third row transition elements in cases where the spin-orbit coupling is weak. ECP s can also be used together with explicit relativistic no-pair operators. Such ECP s are somewhat more precise at at the atomic level, but of essentially the same quality as conventional relativistic ECP s in molecular applications. It should also be possible to combine the ECP formalism with full Fock-Dirac methods, but this has yet not been done. 

For d = 3, the solution of Eq. (7) is the familiar Coulomb wave function in 3 dimensions. For d 3 this is not true. This represents an additional problem since the non-relativistic calculations have to be done without an explicit knowledge of the wave function. Fortunately, cancelation of all divergences can be ensured on the operator level using the Schrodinger equation in d-dimensions. Once divergences are canceled, the limit d —> 3 can be taken and a non-trivial (but now finite) matrix elements can be easily computed. 

In the calculations based on effective potentials the core electrons are replaced by an effective potential that is fitted to the solution of atomic relativistic calculations and only valence electrons are explicitly handled in the quantum chemical calculation. This approach is in line with the chemist s view that mainly valence electrons of an element determine its chemical behaviour. Several libraries of relativistic Effective Core Potentials (ECP) using the frozen-core approximation with associated optimised valence basis sets are available nowadays to perform efficient electronic structure calculations on large molecular systems. Among them the pseudo-potential methods [13-20] handling valence node less pseudo-orbitals and the model potentials such as AIMP (ab initio Model Potential) [21-24] dealing with node-showing valence orbitals are very popular for transition metal calculations. This economical method is very efficient for the study of electronic spectroscopy in transition metal complexes [25, 26], especially in third-row transition metal complexes. 

Another option that reduces the number of functions, particularly when heavy atoms are involved, is the replacement of inner shell electrons by effective (or pseudo) potentials. Such procedures have been incorporated into many ab initio program systems including ACES II. Since the core electrons are not explicitly considered, effective potentials can drastically reduce the computational effort demanded by the integral evaluation. However, because the step is an inexpensive part of a correlated calculation, the role of effective potentials in correlated calculations is less important, due to the fact that dropping orbitals is tantamount to excluding them via effective potentials. An exception occurs when relativistic effects are important, as they would be in a description of heavy atom systems. Most such chemically relevant effects are due to inner shell elearons their important physical effects, like expanding the Pt valence shell, can be introduced via effective potentials that are extracted from Dirac-Fock or other relativistic calculations on atoms. ° Similarly, some effeaive potentials introduce some spin-orbital effects as well. Thus, besides simplifying the computation, effective potential calculations could include important physical effects absent from the ordinary nonrelativistic methods routinely applied. 

In a similar manner, explicitly correlated calculation of the l/r,ic expectation value leads to an absolute shielding constant for the helium atom of 59.9367794 ppm (Drake 2006) (the most accurate value obtained in a recent study including relativistic, quantum-electrodynamic, and nuclear mass effects is cne = 59.96743 ppm (Rudzifiski et al. 2009)). For all the rare-gas atoms the... 

In this paper, we present another application of the semi-relativistic expansion by evaluating the relativistic corrections to the energy up to 1/c and l/c". This gives us explicit correction terms to the usual calculation of anisotropy energy in magnetic systems. 

The purpose of this paper was to explain the new semi-relativistic expansion and to show how it can be used to carry our explicit calculations in physical cases. The results are simple. 

For all results in this paper, spin-orbit coupling corrections have been added to open-shell calculations from a compendium given elsewhere I0) we note that this consistent treatment sometimes differs from the original methods employed by other workers, e.g., standard G3 calculations include spin-orbit contributions only for atoms. In the SAC and MCCM calculations presented here, core correlation energy and relativistic effects are not explicitly included but are implicit in the parameters (i.e., we use parameters called versions 2s and 3s in the notation of previous papers 11,16,18)). 

As the MSFT (38) additivity model predicts only very weak core correlation and scalar relativistic contributions to the proton affinity, we have not attempted their explicit (and very expensive) calculation. 

Atomic units will be used throughout. The explicit density functionals representing the different contributions to the energy from the different terms of the hamiltonian are found performing expectation values taking Slater determinants of local plane waves as in the standard Fermi gas model. Those representing the first relativistic corrections are calculated in the Appendix. 

Relativistic Quantum Defect Orbital (RQDO) calculations, with and without explicit account for core-valence correlation, have been performed on several electronic transitions in halogen atoms, for which transition probability data are particularly scarce. For the atomic species iodine, we supply the only available oscillator strengths at the moment. In our calculations of /-values we have followed either the LS or I coupling schemes. 

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