Relativistic calculation

This chapter provides only a brief discussion of relativistic calculations. Currently, there is a small body of references on these calculations in the computational chemistry literature, with relativistic core potentials comprising the largest percentage of that work. However, the topic is important both because it is essential for very heavy elements and such calculations can be expected to become more prevalent if the trend of increasing accuracy continues. 

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. 

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. 

The most difficult part of relativistic calculations is that a large amount of CPU time is necessary. This makes the problem more difficult because even non-relativistic calculations on elements with many electrons are CPU-intensive. The following lists relativistic calculations in order of increasing reliability and thus increasing CPU time requirements ... 

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. 

Relativistic effects should always be included in these calculations. Particularly common is the use of core potentials. If core potentials are not included, then another form of relativistic calculation must be used. Relativistic effects are discussed in more detail in Chapter 33. 

Results from fully relativistic calculations are scarce, and there is no clear consensus on which effects are the most important. The Breit (Gaunt) term is believed to be small, and many relativistie calculations neglect this term, or include it as a perturbational term... 

RELATIVISTIC CALCULATIONS OF PHOTOEMISSION-AND LEED-INTENSITIES FOR ORDERED AND DISORDERED ALLOYS APPLICATION TO CusPt and Cu Pta... 

For both alloy systems the theoretical results for p obtained in a fully relativistic are found in very satisfying agreement with the corresponding experimental data. In addition to these calculations a second set of calculations has been done making use of the two-current model. This means the partial resistivities p have been calculated by performing scalar relativistic calculations for every spin subsystem separately. As can be seen, the resulting total isotropic resistivity p is reasonably close to the fully relativistic result. Furthermore, one notes that the relative deviation of both sets of theoretical data is more pronounced for Co2,Pdi 2, than for Co2,Pti 2,. This has to be... 

The calculation of the magnetic anisotropy of non-cubic materials requires an expansion up to 1 /c . Except in the case of fully relativistic calculations, the expansion is never carried out consistently and only the spin-orbit perturbation is calculated to second order (or to infinite order), without taking account of the other terms of the expansion. In this section, we shall follow Gesztesy et al. (1984) and Grigore et al. (1989) to calculate the terms H3 and H. Hz will be found zero and H4 will give us terms that must be added to the second order spin-orbit calculation to obtain a consistent semi-relativistic expansion. 

All calculations are scalar relativistic calculations using the Douglas-Kroll Hamiltonian except for the CC calculations for the neutral atoms Ag and Au, where QCISD(T) within the pseudopotential approach was used [99], CCSD(T) results for Ag and Au are from Sadlej and co-workers, and Cu and Cu from our own work, using an uncontracted (21sl9plld6f4g) basis set for Cu [6,102] and a full active orbital space. 

Kello, V. and Sadlej, A.J. (1996) Standardized basis sets for high-level-correlated relativistic calculations of atomic and molecular electric properties in the spin-averaged Douglas-Kroll (nopair) approximation 1. Groups Ib and 11b. Theoretica Chimica Acta, 94, 93-104. 

Strdmberg, D. and Wahlgren, U. (1990) First-order relativistic calculations on Au2 and Hg2 ". Chemical Physics Letters, 169, 109-115. 

Malkin, L, Malkina, O.L. and Malkin, V.G. (2002) Relativistic calculations of electric field gradients using the Douglas—Kroll method. Chemical Physics Letters, 361, 231-236. 

Lo, J.M.H. and Mobukowski, M. (2007) Relativistic calculations on the ground and excited states of AgH and AuH in cylindrical harmonic confinement Theoretical Chemistry Accounts, 118, 607-622. 

Philipsen, P.H.T. and Baerends, E.J. (2000) Relativistic calculations to assess the ability of the generalized gradient approximation to reproduce trends in cohesive properties of solids. Physical Review B - Condensed Matter, 61, 1773-1778. 

Figure 4. Conical parameters on the relativistic seam, (a) g (open circles), h1 (open squares), h1 (open triangles). The nonrelativistic quantities, g (filled circle) and h(nr) (filled square) (b) sw, w — x (circles), y (squares), z (triangles). Filled (open) markers from nonrelativistic (relativistic) calculations. (c) Magnitude of the invariant / = g x hr h1 as a function of R(H2—H3).

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

An expression for 8 in terms of the source and absorber nonrelativistic s electron densities at the origin, s(0) and a(0), respectively, can be obtained by considering the electrostatic interaction between the s electrons and a nucleus with a uniform charge density. A relativistic calculation yields (7) ... 

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