ZORA Hamiltonian

Quasl-relativistic Pauli Hamiltonian. ZORA approximation. 

A number of types of calculations can be performed. These include optimization of geometry, transition structure optimization, frequency calculation, and IRC calculation. It is also possible to compute electronic excited states using the TDDFT method. Solvation effects can be included using the COSMO method. Electric fields and point charges may be included in the calculation. Relativistic density functional calculations can be run using the ZORA method or the Pauli Hamiltonian. The program authors recommend using the ZORA method. 

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. 

Hamiltonians that are needed to deal with coupling constants are the ZORA-DSO,... 

ZORA Method. Very recently, we began applying a new relativistic approach, the ZORA method of van Lenthe and co-workers (14-16). The ZORA Hamiltonian is given by... 

Within the Kohn-Sham formalism of DFT, V is substituted by the Kohn-Sham potential Vks Expanding equation 7 a little, we note that spin-orbit effects are implicitly included in the ZORA Hamiltonian ... 

The older, Pauli-Hamiltonian based QR method has recently been compared to the more modem ZORA approach (10). The comparison has been done based on methane derivatives. We have summarized the results in Table II. Both the ZORA and QR methods agree well for molecules containing atoms no heavier than Cl. This shouldn t be surprising since relativistic effects are still small in such molecules. However, at least for this sample of molecules, the ZORA method is clearly superior for molecules containing heavy nuclei like Br or I. This is reflected in the mean absolute deviation between theory and experiment of 9.2 ppm (ZORA) and 15.6 ppm (QR), respectively (10). Note that some of the same systems have also been studied by other authors (35-37). 

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]. 

Unfortunately, in the presence of F(r), the unitary matrix giving the exact decoupling is not found in a closed form. A number of different approximations to the exact FW transformation have been suggested and analyzed in the literature. - With the special choice of approximations to the exact decoupling, the effective two-component ZORA Hamiltonian in the presence of electromagnetic fields is... 

Equation (4.10a) is the electrostatic potential, scalar relativistic (SR), and spin-orbit (SO) terms of the ZORA Hamiltonian in the absence of electromagnetic fields. The remaining part, (4.10b) and (4.10c), represents the hyper-fine terms due to the presence of the nuclear magnetic moments, JIa and ps. 

The nonrelativistic limit of this operator yields the paramagnetic spin-orbit (PSO) contribution of Ramsey s theory. The remaining terms in eq. (4.10b) result in the ZORA relativistic spin-orbit Hamiltonian,... 

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