Four-spinor

Introducing the moduli ai and phases <]) for the four spinor components t]ij (i = 1,2,3,4), we note the following relations (in which no summations over i are implied) ... 

It exhibits a complicated spin-dependent structure arising from the Dirac four spinor, while it is reduced to a simple form,... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

Four-spinor angular momentum labels etc. are those of the... 

The basic variational principle (3.4) is applied directly in relativistic (extended) Thomas-Fermi models [12, 21,46] in which an approximate density functional representation for the complete [/] is utilised. The mainstay of applications is, however, the KS-scheme. In order to set up this scheme one first introduces auxiliary single-particle four spinors i (x), in terms of which the exact interac-... 

The next task is to derive an alternative form, more useful in practice, of the fundamental variational equations of Section 4.2.1. The basic idea is to represent the elementary density variables of RDFT in terms of auxiliary single-particle four spinors... 

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... 

In this section I will outline the different methods that have been used and are currently used for the computation of parity violating effects in molecular systems. First one-component methods will be presented, then four-component schemes and finally two-component approaches. The term one-component shall imply herein that the orbitals employed for the zeroth-order description of the electronic wavefunction are either pure spin-up spin-orbitals or pure spin-down spin-orbitals and that the zeroth-order Hamiltonian does not cause couplings between the two different sets ( spin-free Hamiltonian). The two-component approaches use Pauli bispinors as basic objects for the description of the electronic wavefunction, while the four-component schemes employ Dirac four-spinors which contain an upper (or large) component and a lower (or small) component with each component being a Pauli bispinor. 

If the atomic wavefunctions Xiii, 9, basis sets for relativistic calculations [161] in the first part of this book)... 

In the FF approach, the extra computational expense of constructing an explicit matrix representation of the back-transformation U can be avoided. Instead, the four-spinors xJ used to compute the Coulomb interaction of the transition densities Xm Xn with the partial density ft as matrix elements of the transformed operators. In fact, the matrix elements of the potential... 

The spinor product ipl pk represents the charge density associated with the four-spinor ipk, while c aqtpk represents the g-component of the current. The matrix elements of these operators may be reduced to elementary G-spinor integrals of the form... 

In Table 2 we present the expectation values of the operator = (a x r), which determines the interaction strength of a state tpo with a homogeneous magnetic field of magnitude B. Here, each one-electron four-spinor, tpo, is determined for the Dirac-Hartree-Fock ground-state of the neon atom using BERTHA. 

Table 2 Calculations of = (c(a X r) /2) for H-like neon, using Eq. (34) and four-spinors generated by BERTHA, are compared with the analytic values...

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