Basis sets Kinetically balanced

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... 

The presence of the momentum operator means that the small component basis set must contain functions which are derivatives of the large basis set. The use of kinetic balance ensures that the relativistie solution smoothly reduees to the non-relativistic wave function as c is increased. 

Instead of a two-component equation as in the non-relativistic case, for fully relativistic calculations one has to solve a four-component equation. Conceptually, fully relativistic calculations are no more complicated than non-relativistic calculations, hut they are computationally demanding, in particular, for correlated molecular relativistic calculations. Unless taken care of at the outset, spurious solutions can occur in variational four-component relativistic calculations. In practice, this problem is handled by employing kinetically balanced basis sets. The kinetic balance relation is... 

The radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [22,23]. 

An application of the variational principle to an unbounded from below Dirac-Coulomb eigenvalue problem, requires imposing upon the trial function certain conditions. Among these the most important are the symmetry properties, the asymptotic behaviour and the relations between the large and the small components of the wavefunction related to the so called kinetic balance [1,2,3]. In practical calculations an exact fulfilment of these conditions may be difficult or even impossible. Therefore a number of minimax principles [4-7] have been formulated in order to allow for some less restricted choice of the trial functions. There exist in the literature many either purely intuitive or derived from computational experience, rules which are commonly used as a guidance in generating basis sets for variational relativistic calculations. 

Kinetic balance, of basis sets, 214 Kirkwood model, solvation, 395 Kirkwood-Westheimer model, solvation, 395... 

The presence of the momentum operator means that the small component basis set must contain functions which are derivatives of the large basis set. The use of kinetic balance ensures that the relativistic solution smoothly reduces to the non-relativistic wave function as c is increased.--------------------------------------------------------------------... 

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). 

In the case of finite-basis sets, which are used for the representation of the one-electron spinors, the basis sets for the small component must be restricted such as to maintain kinetic balance (Stanton and Havriliak 1984), which means in terms of the rearranged second equation in the matrix equations (2.4) that... 

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... 

If a variational calculation is to make any sense, the small component basis must be such that it has a chance to fulfill this relation. This requirement is what is referred to as the principle of kinetic balance between the large and the small component basis set. The simplest way to achieve this is to ensure that each large component basis function has a corresponding function in the small component basis fulfilling the relation above, and we must then have... 

The kinetic balance requirement in this form is quite simple to implement, but its application to Gaussian basis sets calls for some further comments. These are most easily demonstrated on Cartesian GTOs. If we use a scalar basis as described above, the main effect of the a p operator will be to differentiate the basis function. For a px GTO, we get... 

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