Scaled ZORA

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... 

In order to test the various approximations of the Coulomb matrix, all electron basis set and numerical scalar scaled ZORA calculations have been performed on the xenon and radon atom. The numerical results have been taken from a previous publication [7], where it should be noted that the scalar orbital energies presented here are calculated by averaging, over occupation numbers, of the two component (i.e. spin orbit split) results. Tables (1) and (2) give the orbital energies for the numerical (s.o. averaged) and basis set calculations for the various Coulomb matrix approximations. The results from table... 

The ZORA formalism can be extended by a simple scaling procedure for the one-electron energies, thus yielding improved agreement with the fully relativistic Dirac procedure, for the core orbitals in particular. We use this scaled ZORA approach in this study (70). 

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... 

The fact that the Rayleigh quotient for the ZORA wave fiinction yields the Dirac eigenvalue logically leads into another, even simpler, approach than lORA to the improvement of ZORA, and one that preceded lORA the scaled ZORA method (van Lenthe et al. 1994). It is the renormalization terms in the lORA Hamiltonian that present the main difficulty. But for any given function, renormalization may be achieved by a simple scaling. We therefore make the approximations... 

The scaled ZORA energies for a one-electron system can easily be evaluated. We already know that the integral in the metric is equal to —E / 2mc +E ). The... 

The second term can be thought of as an effective kinetic energy operator that goes to the non-relativistic one when V 0. Proper renormalization gives the Infinite Order Regular Approximation (lORA) [17], often approximated by scaled ZORA [16], which improves on ZORA. 

Within the DFT framework, the total scaled ZORA energy can be written as... 

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... 

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