Scaled ZORA equation

It was previously shown [7] that one can, starting from the Dirac-Fock equation, derive the two component scaled ZORA equation... 

Using only the one component part of the operator we arrive at the scalar scaled ZORA equation... 

A truncation of the expansion (3.5) defines the zero- and first-order regular approximation (ZORA, FORA) (van Lenthe et al. 1993). A particular noteworthy feature of ZORA is that even in the zeroth order there is an efficient relativistic correction for the region close to the nucleus, where the main relativistic effects come from. Excellent agreement of orbital energies and other valence-shell properties with the results from the Dirac equation is obtained in this zero-order approximation, in particular in the scaled ZORA variant (van Lenthe et al. 1994), which takes the renormalization to the transformed large component approximately into account, using... 

To evaluate the integral, we may make use of the scaled coordinate system that was used to relate the Dirac and the ZORA equations in the previous section to show that... 

What is the relation between the lORA energies and the ZORA and Dirac energies There is a correspondence at =0 and we expect that the correspondence continues in the vicinity of this point. Unlike the ZORA equation, we cannot perform a scaling to obtain a relation with the Dirac ESC equation, and therefore we cannot obtain a direct relation with the Dirac eigenvalues. What we can do is to make use of the Rayleigh quotient for (18.37) to obtain a relation between the ZORA and lORA eigenvalues, since ZORA and lORA have the same Hamiltonian but a different metric. For an arbitrary wave function r] . 

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... 

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 renormalization terms commute with the ZORA Hamiltonian and simply give a scaling factor. The lORA equation now becomes... 

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