Energy scaled ZORA

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 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... 

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] . 

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