Atomic Kinetic Balance

L. PISANI, J.-M. ANDRE and E. CLEMENTI, Study of relativistic effects in atoms and molecules by the kinetically balanced LCAO approach. J. Chem. Educ., 70, 894 (1993). 

Any realistic description of molecules containing heavy atoms has to take into account relativistic effects (13,41). Attempts to use the algebraic approach to solve the Dirac-Hartree-Fock (DHF) equations are now well advanced (42-45). The difficulties encountered axe much greater than in the nonrelativistic case since the basis sets used have to be larger and have to fulfil the kinetic balance criterion to guarantee the proper description of the large and small components of the molecular orbitals (46-49). 

Quiney 1988). The small component s radial function has been fixed according to the kinetic balance condition (Stanton and Havriliak 1984), which has its origin in the coupled nature of Dirac s first-order differential equations and is introduced to keep the method variationally stable. The index A denotes the coordinates of the nucleus s centre RA of atom A, to which the basis function is attached, i.e. rA = r — RA. As an alternative, Cartesian Gaussians,... 

The correct nonrelativistic limit as far as the basis set is concerned is obtained for uncontracted basis sets, which obey the strict kinetic balance condition and where the same exponents are used for spinors to the same nonrelativistic angular momentum quantum number for examples, see Parpia and Mohanty (1995) and also Parpia et al. (1992a) and Laaksonen et al. (1988). The situation becomes more complicated for correlated methods, since usually many relativistic configuration state functions (CSFs) have to be used to represent the nonrelativistic CSF analogue. This has been discussed for LS and j j coupled atomic CSFs (Kim et al. 1998). 

The nature of basis sets suitable for 4-component relativistic calculations is described. The solutions to the Dirac equation for the hydrogen atom yield the fundamental properties that such basis functions must satisfy. One requirement is that the basis sets for the large and small component be kinetically balanced, and the consequences of this are discussed. Schemes for the optimization of basis sets and choice of symmetry and shell structure is discussed, as well as the advantages offered the use of family sets for scalar basis sets. Special considerations are also required for the description of correlation and polarization in these calculations. Finally the applicability of finite basis sets in actual applications is discussed... 

While SCF basis sets are the starting point for molecular calculations, it is necessary to augment the basis sets that describe the atomic orbitals in order to describe polarization of the atomic orbitals in the molecule and to describe electron correlation. These two requirements have some overlap, and the functions used for electron correlation are often adequate for molecule formation. Both of these effects usually involve basis functions with higher angular momentum than in the atomic valence orbitals. This presents a challenge in relativistic calculations because the requirements of kinetic balance for the small component adds one further unit of angular momentum to the basis for any function added to the large component for polarization or correlation. 

Some other versions of the DFT method like the Beijing Density Functional method (BDF) (see the chapter of C. van Wuellen in this issue) were also used for small compounds of the heaviest elements like 111 and 114 [115-117]. There, four-component numerical atomic spinors obtained by finite-difference atomic calculations are used for cores, while basis sets for valence spinors are a combination of numerical atomic spinors and kinetically balanced Slater-type functions. The non-relativistic GGA for F is used there. 

This depends on the kinetic balance between the surface reaction of the adsorbed hydrocarbon with oxygen species and the further dissociation of the hydrocarbon into adsorbed carbon atoms which can be dissolved in the nickel crystal. The kinetic balance is illustrated in a simplified mechanism in Fig. 4. This differs slightly from the mechanism presented earlier (4). 

The ratio diss/f is about 0.3-0.5 therefore, the direct dependence of the dissociation rate coefficient on electron temperature is not significant. Assuming that atom formation is due to the dissociation of molecules and atom losses are due to vacuum pumping with speed Sp (the loading effect is discussed next), the kinetic balance equation for atomic oxygen density o can be presented as... 

The main bottleneck of four-component calculations has its origin in the existence of a small component and in the kinetic balance condition that generates a very large basis set for this small component, which contributes only a little to expectation values for moderately large nuclear charge numbers Z. Of course, the size of this effect changes for super-heavy atoms in molecules. 

The formation of whisker carbon cannot be tolerated in a tubular reformer. The important question is whether or not carbon is formed, and not the rate at which it may be formed. In terms of the growth mechanism, it means to extend the induction period (to in Equation 5.1) to infinity. This is achieved by keeping the steady-state activity of carbon smaller than one (refer to Section 5.2.4). The carbon formation depends on the kinetic balance between the surface reaction of the adsorbed hydrocarbon with oxygen species and the further dissociation of the hydrocarbon into adsorbed carbon atoms, which can nucleate to whisker carbon. However, this approach is complex and there is a need for simple guidelines using simple thermodynamic calculations. 

O. Matsuoka, M. Klobukowski, and S. Huzinaga, Chem. Phys. Lett., 113, 395 (1985). Kinetically Balanced Calculations on Relativistic Many-Electron Atoms. 

When it comes to contracted basis sets, kinetic balance strictly applied to the contracted large component can lead to problems. While it would be possible to apply the kinetic balance relation to derive a small-component basis from a set of large-component contracted basis functions, this procedure has been shown to be unsuitable in practice (Visscher et al. 1991). The best approach for generating contracted basis sets for relativistic four-component calculations has been to start with an uncontracted large-component basis, and to construct a small-component basis from this basis using kinetic balance. This set is then used in an uncontracted DHF calculation for the atom in question, yielding large- and small-component atomic functions that are kinetically... 

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