Electronic structure computations relativistic effects

LANL2DZ PM3) was used to optimize structures, to reduce computational time and to reduce the memory requirements for the calculation. The use of B3LYP with ECPs facilitated the incorporation of both electron correlation and relativistic effects. 

One way to reduce the computational cost of DFT (or WFT) calculations is to recognize that the core electrons of an atom have only an indirect influence on the atom chemistry. It thus makes sense to look for ways to precompute the atomic cores, essentially factoring them out of the larger electronic structure problem. The simplest way to do this is to freeze the core electrons, or to not allow their density to vary from that of a reference atom. This frozen core approach is generally more computationally efficient. One class of frozen core methods is the pseudopotential (PP) approach. The pseudopotential replaces the core electrons with an effective atom-centered potential that represents their influence on valence electrons and allows relativistic effects important to the core electrons to be incorporated. The advent of ultrasoft pseudopotentials (US-PPs) [18] enabled the explosion in supercell DFT calculations we have seen over the last 15 years. The projector-augmented wave (PAW) [19] is a less empirical and more accurate and transferable approach to partitioning the relativistic core and valence electrons and is also widely used today. Both the PP and PAW approaches require careful parameterizations of each atom type. 

The other relativistic effect entirely neglected so far is the spin-orbit coupling. For systems in nondegenerate states, the only first-order contribution to TAE comes from the fine structures in the corresponding atoms. Their effects can trivially be obtained from the observed electronic spectra, and hence the computational cost of this correction is fundamentally zero. 

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. 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

If several electronically excited states are relevant for describing the photodissociation then one or more of the Rydberg orbitals of the molecule must be included in the (CAS) [13], As the number of orbitals and electrons increases in the CAS, the computational time increases dramatically. In order to obtain accurate potential energy surfaces for the excited electronic states, one must include diffuse functions in the basis set [4], For heavier atoms, a relativistic effective core potential (ECP) can be used to treat the scalar relativistic effects. The ECP basis sets have been developed by several research groups [15,16] and have been implemented in most of the standard electronic structure programs. 

Massively parallel (multiple instruction, multiple data) computers with tens or hundreds of processors are not readily accessible to the majority of quantum chemists at the present time. However the cost of currently available hypercube machines with tens of processors (each with about the power of a VAX) is comparable to that of superminis but with up to a hundred times the power. For applications of the type discussed above the performance of a machine with as few as 32 or 64 processors would be comparable to (or perhaps even exceed) that of a single processor supercomputer. Although computer requirements currently limit QMC applications (even with effective potentials) the proliferation of inexpensive massively parallel machines could conceivably make the application of relativistic effective potentials with C C quite competitive with more conventional electronic structure techniques. 

The radial expansion of valence orbitals introduces substantial alterations in the chemical bonding and valence spectroscopic properties of heavy-element systems. " Therefore to perform computational calculations on heavy elements such as tin, it is mandatory to include relativistic effects to determine electronic structure and to ensure quantitative agreement with experimental data. ... 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

The Dirac equation with four spinor components demands large computational efforts to solve. Relativistic effects in electronic structure calculations are therefore usually considered by means of approximate one- or two-component equations. The approximate relativistic (also called quasi-relativistic) Hamiltonians consist of the nonrelativistic Hamiltonian augmented with additional... 

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