Regular approximations

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

In contrast to eq. (8.13), the factor Ej 2mc — V) is always much smaller than 1. K may now be expanded in powers of Ej(2mc — V), analogously to eq. (8.22). Keeping only the zero-order term (i.e. setting K = 1) gives the Zero-Order Regular Approximation (ZORA) method. ... 

Note that in this case the spin-orbit coupling is included already in zero order. Including the first-order term from an expansion of K defines the Eirst-Order Regular Approximation (FORA) method. 

In order to find approximate solutions of the equations for Ci t) and gi,..j t) one can use regular approximate methods of statistical physics, such as the mean-field approximation (MFA) and the cluster variation method (CVM), as well as its simplified version, the cluster field method (CFM) . In both MFA and CFM, the equations for c (<) are separated from those for gi..g t) and take the form... 

Wolff, S. K., Ziegler, T., van Lenthe, E., Baerends, E. J., 1999, Density Functional Calculations of Nuclear Magnetic Shieldings Using the Zeroth-Order Regular Approximation (ZORA) for Relativistic Effects ZORA Nuclear Magnetic Resonance , J. Chem. Phys., 110, 7689. 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

In this paper we use a regular approximation of the Dirac Fock formalism known as... 

Equation (1) is obtained by using an expansion in E/ 2c - Vc) on the Dirac Fock equation. This expansion is valid even for a singular Coulombic potential near the nucleus, hence the name regular approximation. This is in contrast with the Pauli method, which uses an expansion in (E — V)I2(. Everything is written in terms of the two component ZORA orbitals, instead of using the large and small component Dirac spinors. This is an extra approximation with respect to the original formalism. 

Within the DFT framework, we apply two different approaches to deal with relativistic effects, the so-called quasi-relativistic (QR) method (73) and the more modem "Zeroth Order Regular Approximation for Relativistic Effects" (ZORA) (14-16). The QR method is also known as the Pauli approach. 

It seems natural to suppose that the tetragonal distortion of the tri-anion results from the Jahn-Teller effect. In order to study the problem more thoroughly we undertook recently the DFT calculations of this cluster as well as of several other hexanuclear rhenium chalcohalide clusters. The technical details of these calculations can be found in the original publication [8]. Here we only want to note that the introduction of relativistic corrections for Re atoms is crucial for the correct reproduction of the geometry of clusters. In our calculations, this was done by the zero order regular approximation (ZORA) Hamiltonian [9] within ADF 2000.02 package [10]. 

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