The ZORA ansatz

Another method to avoid the singularities in the vicinity of the nuclei was proposed by van Lenthe et al. [26,27,35]. They suggested an method that also includes the interaction potential, V (r), in the denominator of the ansatz for the small component. This ansatz was used in the derivation of the so called zeroth-order regular approximation (ZORA) Hamiltonian. The ZORA ansatz can thus be the written as 

For atoms and molecules, the total potential, F(r), close to the nucleus is completely dominated by the nuclear-electron attraction potential, even for many-electron systems. At the nucleus, the potential is thus proportional to — 

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This implies that when the ZORA ansatz is employed, the small components approach zero at the nuclei. The singular behaviour encountered in quasi-relativistic approaches based on kinetic-energy balance condition is therefore avoided. The ZORA method will be discussed in more detail in the next Section, and now we just note that the ZORA ansatz has a couple of desirable properties that shall be taken into consideration when the general ansatz function is constructed. 

The present idea is to replace the ZORA ansatz, which already is an approximation to the energy-dependent elimination of the small component approach, with another but similar expression that relates the large and the small components. The general ansatz function should have the same shape as the ZORA function close to the nucleus. Its first derivative should also be reminiscent of that of the ZORA function. A general function f[r) that fulfills the desired asymptotic conditions for r —0 and for r -> < can for example consist of one exponential function or of a linear combination of a couple of exponential functions as... 

The simplest exponential-type function that simulates the ZORA ansatz in the vicinity of the nucleus is... 

The zeroth-order regular approximation (ZORA) Hamiltonian can be derived from the upper part of the transformed Dirac equation (20). By using the ZORA ansatz for the small component (5) and assuming that the upper and the lower components are equal, the final ZORA equation for the upper component becomes... 

The quasi-relativistic model obtained by using the ZORA ansatz in combination with a fully variational derivation is the infinite-order regular approximation (lORA) previously derived by Sadlej and Snijders [46] and by Dyall and Lenthe [47]. The lORA method has recently been implemented by Klop-per et al. [48]. The ZORA model can be obtained from the lORA equation by omitting the relativistic correction term to the metric. However, the indirect renormalization contribution is as significant as the relativistic interaction operator in the Hamiltonian. This is the reason why ZORA overestimates the... 

The ansatz used in the construction of the ERA Hamiltonian has one adjustable parameter namely the scaling factor y in the denominator of the exponent of the ansatz function in equation (9). Since 7 = 1 yields at the nucleus the same first derivative of the ansatz function with respect to r as with the ZORA ansatz, this value for 7 has been used in the calculations presented above. How-... 

The Z4 method can alternatively be justified from a scaled ZORA ansatz, as discussed by van Lenthe and Baerends [26] and Aquino et al. [13], leading to the same working equations. The relation with scaled ZORA is seen by the appearance of the term /C /(4c ) both of these equations (12.16) and (12.6). A scalar relativistic variant is obtained by replacing... 

Figure 1. The Exponential Regular Approximation (ERA with y= 1), the Zeroth-Order-Regular Approximation (ZORA), and the Kinetic-Energy Balance Condition (KEBC) ansatz functions for uranium (Z = 92). The distance R is given in bohrs.

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