Four-component relativistic calculation

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. 

From the above discussion it becomes clear that in order to eliminate the spin-orbit interaction in four-component relativistic calculations of magnetic properties one must delete the quaternion imaginary parts from the regular Fock matrix and not from other quantities appearing in the response function (35). It is also possible to delete all spin interactions from magnetic properties, but this requires the use of the Sternheim approximation [57,73], that is calculating the diamagnetic contribution as an expectation value. 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

Instead of a two-component equation as in the non-relativistic case, for fully relativistic calculations one has to solve a four-component equation. Conceptually, fully relativistic calculations are no more complicated than non-relativistic calculations, hut they are computationally demanding, in particular, for correlated molecular relativistic calculations. Unless taken care of at the outset, spurious solutions can occur in variational four-component relativistic calculations. In practice, this problem is handled by employing kinetically balanced basis sets. The kinetic balance relation is... 

In general, the computational requirements for full four-component relativistic calculations on molecules are so severe that cheaper alternatives must be explored. 

The field of relativistic electronic structure theory is generally not part of theoretical chemistry education, and is therefore not covered in most quantum chemistry textbooks. This is due to the fact that only in the last two decades have we learned about the importance of relativistic effects in the chemistry of heavy and super-heavy elements. Developments in computer hardware together with sophisticated computer algorithms make it now possible to perform four-component relativistic calculations for larger molecules. Two-component and scalar all-electron relativistic schemes are also becoming part of standard ab-initio and density functional program packages for molecules and the solid state. The second volume of this two-part book series is therefore devoted to applications in this area of quantum chemistry and physics of atoms, molecules and the solid state. Part 1 was devoted to fundamental aspects of relativistic electronic structure theory. Both books are in honour of Pekka Pyykko on his 60 birthday - one of the pioneers in the area of relativistic quantum chemistry. 

Aucar et al demonstrated, by means of a four-component relativistic calculation, that the origin of the diamagnetic contribution to any magnetic molecular property is due to contributions from positronic spinors in calculating the response of the system. Several approximations for the calculation of the DSO term were also investigated. As example, the DSO term for the chalcogen hydrides, XH2 (X = O, S, Se and Te), were calculated. 

Q. Sun, W. Liu, W. Kutzelnigg. Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations. Theor. Chem. Acc., 129 (2011) 423-436. 

H. J. A. Jensen, J. Oddershede. Full four-component relativistic calculations of NMR shielding and indirect spin-spin coupling tensors in hydrogen halides. J. Comp. Chem., 20(12) (1999) 1262-1273. 

G. A. Aucar, T. Saue, L. Visscher, H. J. A. Jensen. On the origin and contribution of the diamagnetic term in four-component relativistic calculations of magnetic properties. /. Chem. Phys., 110(13) (1999) 6208-6218. 

With the change in sign of k, we see that a 2pi/2 function must be balanced by a linear combination of a U1/2 function and a 3 i/2 function. As mentioned above, functions with n > + 1 (such as the 3s) are not normally used in nonrelativistic basis sets, because linear combinations of n = + 1 functions cover the same space. However, we see here that in four-component relativistic calculations the n = I + 3 functions are important for kinetic balance and must be included in the... 

There exists a wide selection of exponents for Gaussian basis sets for nonrelativistic calculations, although most of these are for lighter elements which for most purposes do not require a relativistic treatment. For four-component relativistic calculations, nonrelativistic basis sets can be used for lighter atoms, but as the relativistic effects of orbital contraction and spin-orbit splitting increase in importance, these nonrelativistic basis sets become inadequate. In some measure the orbital contraction for inner orbitals is counteracted by the use of a finite nucleus, which tends to push out the inner parts of the spinors. A major concern is the 2/ i/2 space (Matsuoka and Okada 1989) due to the 5-character of the small component at least two extra functions relative to the nonrelativistic basis are needed for the 6/ block to reduce the error in the energy to 0.5 h-... 

Krivdin and co-workers have performed four-component relativistic calculations of V(Se,C) spin-spin coupling constants in a series of 13 selenium heterocycles and their parent open-chain selenides. The authors concluded that relativistic effects play an essential role in the selenium-carbon coupling mechanism and could result in a contribution of as much as 15-25% of the total values of the V (Se,C). In the overall... 

H. Tatewaki and Y. Watanabe, Necessity of including the negative energy space in four-component relativistic calculations for accurate solutions, Chem. Phys. 389, 58-63 (2011). 

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