Relativistic methods singularities

Wigner-Seitz-type cells. The singularities caused by the cusps of relativistic wave functions at the nuclear sites are eliminated by suitable transformations of the sample points, which leads to an improved numerical representation of the wave functions (Bastug et al. 1995). With this method, a total of approximately 1400 sample points is needed to achieve a relative accuracy of 10 8 in calculations for diatomic molecules. 

This implies that when the ZORA ansatz is employed, the small components approach zero at the nuclei. The singular behaviour encountered in quasi-relativistic approaches based on kinetic-energy balance condition is therefore avoided. The ZORA method will be discussed in more detail in the next Section, and now we just note that the ZORA ansatz has a couple of desirable properties that shall be taken into consideration when the general ansatz function is constructed. 

In the vast field of ionic crystals doped with impurities, many interesting properties are related to large manifolds of excited electronic states well localised in a small, singular portion of the material, made of the impurity and some neighbour atoms, usually called a cluster, which is under the effect of the rest of the host. Relativistic molecular ab initio methods of the Quantum Chemistry like those described in Sections 2.1.1 and 2.1.2 are, in consequence, applicable to the cluster (or pseudomolecule) when the impurities are heavy elements, provided that the embedding effects of the rest of the host are properly... 

One seemingly sensible approach to the relativistic electronic structure theory is to employ perturbation theory. This has the apparent advantage of representing supposedly small relativistic effects as corrections to a familiar non-relativistic problem. In Appendix 4 of Methods of molecular quantum mechanics, we find the terms which arise in the reduction of the Dirac-Coulomb-Breit operator to Breit-Pauli form by use of the Foldy-Wouthuysen transformation, broken into electronic, nuclear, and electron-nuclear effects. FVom a purely aesthetic point of view, this approach immediately looks rather unattractive because of the proliferation of terms at the first order of perturbation theory. To make matters worse, many of the terms listed are singular, and it is presumably the variational divergences introduced by these operators which are referred to in [2]. Worse still, higher-order terms in the Foldy-Wouthuysen transformation used in this way yield a mathematically invalid expansion. 

In Sect. 1.4, we will demonstrate the validity of the method by analysing the relativistic Kepler problem by computing the perihelion motion of the planet Mercury, followed by Sect. 1.5, displaying the explicit connection between the Schwarzschild singularity and Gddel s theorem. The final conclusion summarises the modus operandi and its subsequent consequences. 

The relative merits of these low-order summation methods have been compared using several other S expansions in addition to the two considered here and the conclusions are as follows [11] If the residue a 2 is known exactly then hybrid Fade summation is recommended if a 2 is not known then at very low order weighted truncation generedly appears to be the method of choice, but beyond about third order the hybrid Fade summation again seems to be best. An additional advantage of weighted truncation and hybrid Fade summation is the fact that both of these methods can be applied to singularities that are more complicated than poles. For example, relativistic problems... 

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