Spinor space

Spin waves, 753 Spin wave states, 757 Spin zero particles, 498 "Spinor space, 428 Spinors, 394,428, 524 column, 524 Dirac... 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

It is easy to see that Eq. (1) is invariant with respect to a similarity transformation in the four-dimensional spinor space. If k and E are solutions of Eq. (1) then... 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

The absence of spin symmetry in the relativistic case makes the indices run over all spinor space, which is twice as wide as the non-relativistic orbital space. 

Crucial is the selection of the active spinor space that determines the model space. It should be large enough to take all important interactions between nearby states into account, but also be small enough to keep the formalism computationally tractable and numerically stable. Direct diagonalization of this model space would be equivalent to a small Cl expansion and does of course not take dynamic electron correlation into account. Correlation is included by defining the so-called wave operator U... 

The first step consists in a decoupling of electronic from positronic states by means of the not norm-conserving transformation Wa of Eq. (148). At the same step we project to electronic states, since we are only interested in these, i.e. we perform a projecting transformation that transforms from a 4-component spinor space to a 2-component spinor space. 

We apply the following notation for operator matrices in the discussion to come matrices written as M indicate real matrices in the basis of a set of spin-free (one-component) basis functions A. The notation M denotes matrices in two-component spinor space cpfi and M refers to the complete basis function space in which the 4-spinors (tpf, q> ) are expanded. We therefore write the ( two-component ) standard matrices... 

Similar arguments to these can be marshaled for the higher-order terms in (8.10). The conclusion is that the energy is invariant to rotations within the occupied spinor space and within the unoccupied spinor space. The matrix elements of k within these spaces may therefore be set to zero, and the only matrix elements to be considered are those that connect occupied and unoccupied spinors. 

The behavior of the SCF solutions is not the only issue of interest in the energetics of model potentials. Because Bk is finite, the core spinors are not moved an infinite distance from the valence space, but a finite distance. Any representation of the core spinors remains in the molecular basis set, and can contaminate a correlated calculation, if they are not removed (Klobukowski 1990). The result can be an overestimation of correlation effects. Diagonalizing B in the virtual spinor space should provide a means of recognizing and removing core-like spinors. 

---