Relativistic methods framework

In this chapter, we therefore consider whether it is possible to eliminate spin-orbit coupling from four-component relativistic calculations. This is a situation quite different from that of more approximate relativistic methods where a considerable effort is required for the inclusion of spin-orbit coupling. We have previously shown that it is indeed possible to eliminate spin-orbit coupling from the calculation of spectroscopic constants [12,13]. In this chapter, we consider the extension of the previous result to the calculation of second-order electric and magnetic properties, i.e., linear response functions. Although the central question of this article may seem somewhat technical, it will be seen that its consideration throws considerable light on the fundamental interactions in molecular systems. We will even claim that four-component relativistic theory is the optimal framework for the understanding of such interactions since they are inherently relativistic. 

Bioinorganic systems often contain heavy elements that need to be treated with an explicit relativistic method. It is now possible to carry out an explicit relativistic electronic structure calculation on the fly (152). The scalar-relativistic Douglas - Kroll - Hess method was implemented by us recently in the BOMD simulation framework (152). To use the relativistic densities in a non-relativistic gradient calculations turned out to be a valid approximation of relativistic gradients. An excellent agreement between optimized structures and geometries obtained from numerical gradients was observed with an error smaller than 0.02 pm. 

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

The Douglas-Kroll approach to relativistic electronic structure theory in the framework of density functional theory was reviewed focussing on recent method developments and illustrative applications which demonstrate the capabilities of this approach. Compared to other relativistic methods, which often are only applied to small molecules for demonstration purposes, the DK approach has been used in a variety of fields. Besides the very popular pseudopotential approach, which accounts for relativistic effects by means of a potential replacing the core electrons, until now the scalar relativistic variant of the second-... 

S. Komorovsy, M. Repisky, O. L. Malkina, V. G. Malkin, 1. Malkin, M. Kaupp. A fully relativistic method for calculation of nudear magnetic shielding tensors with a restricted magnetically balanced basis in the framework of the matrix Dirac-Kohn-Sham equation. /. Chem. Phys., 128 (2008) 104101. 

Our ambition is to provide a modern introduction to the field of relativistic quantum chemistry, aimed at the advanced student and the practicing nonspecialist researcher. The material has been divided into five parts. Parts I and II provide the necessary background from classical physics, relativistic quantum mechanics, and group theory. Part III covers the application of these principles to fully relativistic methods for quantum chemistry within a four-component framework. Part IV deals with the main... 

In this respect, the single-configurational Hartree-Fock method looks more promising and universal when combined with accounting for the relativistic effects in the framework of the Breit operator and for correlation effects by the superposition-of-configurations or by some other method (e.g. by solving the multi-configurational Hartree-Fock-Jucys equations (29.8), (29.9)). 

This expression excludes self-interaction. There have been a number of attempts to include into the Hartree-Fock equations the main terms of relativistic and correlation effects, however without great success, because the appropriate equations become much more complex. For a large variety of atoms and ions both these effects are fairly small. Therefore, they can be easily accounted for as corrections in the framework of first-order perturbation theory. Having in mind the constantly growing possibilities of computers, the Hartree-Fock self-consistent field method in various... 

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. 

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

In summary, the recommended method for the inclusion of spin-orbit coupling and other relativistic effects for molecules containing heavy elements, considering computational complexity and accuracy factors, is one based on ab initio REPs. Future developments should include more extensive spin-orbit and Cl procedures in the A -S framework, the development and implementation of Cl computational codes in to-co coupling, the incorporation of reliable methods for the treatment of core-valence polarization and correlation effects, and selected benchmark all-electron calculations. 

The 1998 Nobel prize for chemistry was awarded to two scientists whose principal contribution was to devise methods that brought approximate quantum theory calculations for the medium-sized molecules within the realms of practicality. The suite of programs that the modern chemist has available for calculating molecular structures is extensive and sophisticated. But, in practice, a compromise always has to be made in terms of the computational effort versus the level of approximation, and some issues of approximation cannot be avoided within the framework of the suite. One such example is the assumption of stationary nuclei another is the problem of relativistic velocity effects, which become significant for the electrons of elements heavier than about iron (that is, the heavier two thirds of the elements). The time-independent Schrbdinger equation is based on Newtonian rather than relativistic mechanics. 

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