Finite nucleus component

Here, the summation extends over both negative and positive energy spinors. Pnn > ) and Qr if) are the large and small radial components and are expanded in G spinors, x il and xfi, that satisfy the boundary conditions associated with the finite nucleus [Ishikawa et al. (1997b)),... 

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-... 

The three components of the electric field gradient tensor are related by Poisson s equation, as shown earlier. However, the electrons that have a finite probability density at the nucleus, the s and p1/2 electrons, have a spherically symmetric distribution around the nucleus and as such do not contribute to E2. Thus, in the computation of E2, the Un can be related by... 

A slightly higher energy state of carbon is described by the electron configuration ls22s2p3, and from the 2s and the three 2p orbitals four equivalent mixed or hybrid orbitals are obtained. In the case of carbon in methane, and saturated carbon in most other molecules, the hybridization is described as sp3. These orbitals have some characteristics of the component atomic orbitals. Accordingly, an sp3 orbital has finite charge density at the nucleus, as does the 2s (but not the 2p) orbital also it has directionality, in common with the 2p orbitals, with lobes now of unequal size. 

The components of A contain contributions from two types of terms, an anisotropic dipole-dipole interaction and an isotropic ( contact ) interaction owing to the presence of a finite electron spin density at the nucleus. Contributions from the former may be expressed as... 

While this is by no means the only possible choice for relativistic four-component calculations, it is definitely the most efficient and convenient. It permits programmers to exploit an extensive technology that has been refined and tested through years of development of nonrelativistic methods. The usual complaint about the poor behavior of Gaussian functions close to nuclei is less severe for relativistic calculations, where nuclei of finite size are normally used and not point nuclei. As we discussed in chapter 7, Gaussians actually are particularly suitable for describing the wave function close to a nucleus of finite size. 

The source of the electric field can be an externally applied field, or it can originate in the components of the nuclear potential that are not included in the internal component of the field (that is, the nuclear potential V). Such components arise from the nonspherical nature of the nucleus, the lowest-order term of which is the quadrupole moment. The implementation of a finite-nuclear model is quite straightforward we simply expand the nuclear charge distribution in a series ... 

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