Large component basis

The molecular orbitals in the nonrelativistic and one-component calculations and the large component in the Dirac-Fock functions were spanned in the Cd s Ap9d)l[9slp6d basis of [63] and the H (5s 2p)/[35 l/>] set [61]. Contraction coefficients were taken from corresponding atomic SCF calculations. The basis for the small components in the Dirac-Fock calculations is derived by the MOLFDIR program from the large-component basis. The basis set superposition error is corrected by the counterpoise method [64]. The Breit interaction was found to have a very small effect and is therefore not included in the results. 

In developing relativistic basis sets, it is only natural to focus on the large component which clearly accounts for most of the electron density. In particular for ligh.ter elements, one would expect this to be very close to the non-relativistic wavefunction. But the relation above tells us that the small component basis is dependent on the large component basis, in particular using the expansion in eq. 10 we must have that... 

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... 

In chapter 10, we have already discussed how the size of the small-component basis set can be made equal to that of the large-component basis set by absorbing the kinetic-balance operator into the one-electron Hamiltonian. In this chapter, we have elaborated on this by introducing a pseudo-large component that has led to the modified Dirac equation. 

The role of kinetic balance in these bounds is also critical. If the small-component basis set is not related to the large-component basis set by kinetic balance, the large positive term that involves U is reduced toward zero, and the eigenvalues can approach those of the large-component potential as expressed in (11.25). 

It should also be borne in mind that the large-component basis is not totally independent of the small-component basis, and that the size of the total basis set will be determined by the kinetic balance requirement. If we use the relation derived previously in (11.19),... 

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 large-component basis must be slightly larger than for the nonrelativistic calculation of the same quality. (This applies also to 2-spinor basis sets.)... 

The principle of kinetic balance requires that the small component basis contains roughly 2.5 times as many functions as the large-component basis. 

In a contracted scalar basis set, the core spin-orbit splitting must be accounted for, and the calculations become a little more difficult to do directly. The worst case for basis set size would be a basis set in which all functions are contracted, for example, a generally contracted basis set or an atomic natural orbital (ANO) basis set. Then, the large-component basis set would double in size, apart from the functions. The small-component basis set, however, would not change, provided the contractions were properly restricted for instance, the component was eliminated from the d functions for the small component of the p3/2 contracted functions. 

Allowing for a 40% increase in the large-component basis size, the (LL LL) set would increase by a factor of 4 and the LL SS) set by a factor of 2, giving 4-1-25-1-39 = 68 as the overall factor, which is a little more than for the 2-spinor basis. If the increase is more like 70%, the (LL LL) set would increase by a factor of 9 and the LL SS) set by a factor of 3, giving 9-1-39-1- 39 = 87. The 2-spinor method therefore has a size advantage in a contracted basis. If the small-component contractions were not properly restricted, however, the integral count in these two cases increases uniformly by a factor of 4 or of 9, which is much larger than in the nonrelativistic case. 

The expression in square brackets is the matrix of 2mc — V evaluated over the set of basis functions ((a p)/2mc)x/i, which is essentially the small-component basis set. Likewise, T can be regarded as the integral over (or p) between the small- and large-component basis set. This makes it clear that the denominator in ZORA must be evaluated over the small-component basis functions. 

Levy-Leblond Hamiltonian as the unperturbed operator. It turns out that when so-called kinetic balance is obeyed (roughly speaking, when the basis set describing the small component is represented by functions derived from the large-component basis functioas xl by xs = the same results... 

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