Quasi-spinors

Note that the projection operator P = (1 a, vF)/2 projects out the particle state, wF, and the anti-particle state, ip- (or more preciselyip-), from the Dirac spinor field k. The quasi-quarks in a patch carries the residual momentum l1 and is given as... 

As a characteristic feature, both the gap functions have nodes at poles (9 = 0,7r) and take the maximal values at the vicinity of equator (9 = 7t/2), keeping the relation, A > A+. This feature is very similar to 3P pairing in liquid 3He or nuclear matter [17, 18] actually we can see our pairing function Eq. (39) to exhibit an effective P wave nature by a genuine relativistic effect by the Dirac spinors. Accordingly the quasi-particle distribution is diffused (see Fig. 3)... 

It can be formulated alternatively in terms of 4-component- or 2-component spinors, so it is directly related to both a fully relativistic theory and a quasi-relativistic theory. 

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... 

Despite recent implementations of an efficient algorithm for the four-component relativistic approach, the DC(B) equation with the four-component spinors composed of the large (upper) and small (lower) components stiU demands severe computational efforts to solve, and its applications to molecules are currently limited to small- to medium-sized systems. As an alternative approach, several two-component quasi-relativistic approximations have been proposed and applied to chemically interesting systems containing heavy elements, instead of explicitly solving the four-component relativistic equation. 

The Douglas-Kroll (DK) approach [153] can decouple the large and small components of the Dirac spinors in the presence of an external potential by repeating several unitary transformations. The DK transformation is a variant of the FW transformation [141] and adopts the external potential Vg t an expansion parameter instead of the speed of light, c, in the FW transformation. The DK transformation correct to second order in the external potential (DK2) has been extensively studied by Hess and co-workers [154], and has become one of the most familiar quasi-relativistic approaches. Recently, we have proposed the higher order DK method and applied the third-order DK (DK3) method to several systems containing heavy elements. 

Similarly as in Section 1.2, one starts from atomic AE reference calculations at the independent-particle level (some kind of quasi-relativistic HF or fully relativistic DHF). The first step now in setting up pseudopotentials consists in a smoothing procedure for valence orbitals/spinors ( pseudo-orbital transformation ). In the DHF case, to be specific, the radial part ( )/ of the large component of the energetically lowest valence spinors for each //-combination is transformed according to... 

The equivalence of the lOTC method to the four-component Dirac approach has been documented by calculations of spin orbital energies in several papers [18,20, 63]. The unitary transformation does not affect the energy eigenspectrum, though it reduces the four-component bi-spinors to two-component spinor solutions. Due to this fact the two-component methods are frequently addressed as being quasi-relativistic and it is assumed that some information is lost. It can be demonstrated [22] that the two-component lOTC wave function which is the upper component of the unitarly transformed four-component Dirac spinor I ... 

It should be remembered, of course, that Lie groups have an independent existence apart from their role in the theory of f electrons in the lanthanides. Some of the bizarre properties that turn up in the f shell might well derive from isoscalar factors that receive a ready explanation in another context. An example of this is provided by the vanishing of the spin-orbit interaction when it is set between F and G states belonging to the irreducible representation (21) of G2. In the f shell, (21) is merely a 64-dimensional representation of no special interest. However, for mixed configurations of p and h electrons, it fits exactly into the spinor representations (iiiHi 2) of SO(14) with dimensions 2 (Judd 1970). These are the analogs of the spinor representations of eq. (130), and (21) describes the quasiparticle basis of the configurations (p-t-h)". The spinor representations of SO(3) and (Hiii) of SO(ll) provide the quasi-particle bases for the p and h shells respectively and their SO(3) structures, namely S 1,2 and 512+ 912+ is/2> when coupled, must yield the L structure of (21) of G2, namely D-l-F-l-G-l-H-t-K-t-L. In this context, the F and G terms of (21) are associated with the different irreducible representations Sj,2 and S9/2. It is this property,... 

H. Moriyama, H. Tatewaki, Y. Watanabe, and H. Nakano, Molecular spinors suitable for four-component relativistic correlation calculations studies of LaF and LaF using multiconfigurational quasi-degenerate perturbation theory, Int. J. Quantum Chem. 109, 1898 - 1904 (2009). 

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