Electronic spinors

Finally, the eight-component wave function ip p, En) (four ordinary electron spinor indices, and two extra indices corresponding to the two-component... 

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

Vector representations correspond to integral values of the angular momentum quantum numberj and therefore to systems with an even number of electrons. Spinor representations correspond to systems with half-integral j and therefore to systems with an odd number of electrons. Note that T is the complex conjugate of T. 

We are now in a position to investigate the effects of 3/30, 9/90 and 9/9y on the electron spin functions. When the electron spins are quantised in the molecule-fixed axis system, we see that each component of the 2 -rank spinor is an implicit function of transformation matrix (2.99). The total spinor f(S) may be expressed as a product of one-electron spinors,... 

As the standard ansatz few the many-electron wave function is an antisymmetrized product of one-electron spinors o) it may be written as a Slater determinant or in the language of second quantization (see, for example, Helgaker et al. 2000) as... 

In the case of finite-basis sets, which are used for the representation of the one-electron spinors, the basis sets for the small component must be restricted such as to maintain kinetic balance (Stanton and Havriliak 1984), which means in terms of the rearranged second equation in the matrix equations (2.4) that... 

In general, the many-electron wave function is expressed in terms of antisymmetrized products of one-electron functions and the clamped-nucleus approximation as well as the central-field and equivalence restriction for the orbitals is used. Thus the one-electron spinor takes the form... 

The most frequently used ansatz for the representation of molecular one-electron spinors is a basis expansion into Gauss-type spinors (where we have adopted the notation used in Quiney et al. (1998b))... 

The programs described so far use basis-set expansions for the one-electron spinors. The fully numerical approach, which is still a challenging task for general molecules in nonrelativistic theory (Andrae 2001), has also been tested for Dirac-Fock calculations on diatomics (DtisterhOft etal. 1994,1998 Kullie etal. 1999 Sundholm 1987,1994 Sundholm et al. 1987 v. Kopylow and Kolb 1998 v. Kopylow et al. 1998 Yang et al. 1992). The finite-element method (FEM) was tested for Dirac-Fock and Kohn—Sham calculations by Kolb and co-workers in the 1990s. However, this approach has not yet been developed into a general method for systems with more than two atoms only test systems, namely few-electron linear molecules at some fixed intemuclear distance, have been studied with the FEM. Nonetheless, these numerical techniques are able to calculate the Dirac-Fock limit and thus yield reference data for comparisons with more approximate basis-set approaches. The limits of the numerical techniques are at hand ... 

All possible determinants with 3 electrons in 6 one-electron spinors can be depicted as paths in a graph (Fig. 1). An occupied spinor P is represented by a diagonal arc connecting vertex (P-1, n-1 to vertex P,n) while unoccupied spinors are represented by vertical arcs. The determinant 246 is thus represented by the thick line in the graph. 

Figure 13. Valence spinors of the Db atom in the 6d 7s ground state configuration from average-level all-electron (AE, dashed lines) multiconfiguration Dirac-Hartree-Fock calculations and corresponding valence-only calculations using a relativistic energy-consistent 13-valence-electron pseudopotential (PP, solid lines). A logarithmic scale for the distance r from the (point) nucleus is us in order to resolve the nodal structure of the all-electron spinors. The innermost parts have been truncated.

One of the first molecular DHF EFG studies was done by Visscher et al. [162] for the hydrogen halides at the MP2, CCSD and CCSD(T) level. Here picture-change errors are absent and instead of the PCNQM model the common expression for the EFG operator can be applied. In a relativistic single-determinantal formalism the molecular electronic spinor i(r) consists of four components... 

In order to calculate expectation values for a wave function of the structure given in Eq. (8.100), it is convenient to introduce orthonormality constraints on the one-electron spinors... 

The unboundedness of the one-electron Dirac operator prohibits the application of the variational principle and we are in desperate need of a solution to this problem. In the 1980s, many authors discussed the issue of how a basis set expansion of the one-electron spinors affects the variational stability (we come back to this particular issue in chapter 10) and how this is related to the need to choose projection operators as discussed in section 8.2.3.2. 

The derivation has been general so far, i.e., we have derived the most general form of the self-consistent field equations that one may use to optimize a set of one-electron spinors for truncated Cl expansions in the MCSCF approach. 

To conclude, we have now derived the equations to determine the one-electron spinors. The next two chapters will focus on how they can be solved. The explicit solution depends on the system xmder consideration. For atoms, numerical techniques were derived early on, and we will demonstrate how... 

Note that here the summation over operators of identical particles became a summation over one-electron spinors upon resolution of the many-electron integrals, as discussed earlier in this chapter. As one would demand, in the simple case of a single electron, where Y — ip r), Eq. (8.204) reduces to the one-particle continuity equation of section 5.2.3,... 

To summarize, the electron density and the current density can be expressed in terms of one-electron spinors, and consequently we may benefit from all that has already been elaborated in section 8.1 about the interaction of two... 

The integral for the second coordinate set is of the same form. Finally, we obtain for the general Breit matrix element over one-electron spinors [201]... 

In order to describe the one-electron spinors of molecules that enter the Slater determinants to approximate the total electronic wave function it is natural to be inspired by the fact that molecules are composed of atoms. It should be noted that this at first sight obvious fact is not obvious at all as we describe only the ingredients, i.e., the elementary particles (electrons and atomic nuclei) and not individual atoms of a molecule. To define what an atom in a molecule shall be is not a trivial issue in view of the continuous total electron distribution p r) of a molecule. Nonetheless, we may utilize what we have... 

The choice and generation of basis sets has been addressed by many authors [190,192,528,554-563]. While we consider here only the basic principles of basis-set construction, we should note that this is a delicate issue as it determines the accuracy of a calculation. Therefore, we refer the reader to the references just given and to the review in Ref. [564]. In Ref. [559] it is stressed that the selection of the number of basis functions used for the representation of a shell riiKi should not be made on the grounds of the nonrelativistic shell classification nj/j but on the natural basis of j quantum numbers resulting in basis sets of similar size for, e.g., Si/2 and pi/2 shells, while the p /2 basis may be chosen to be smaller. As a consequence, if, for instance, pi/2 and p /2 shells are treated on the tijli footing, the number of contracted basis functions may be doubled (at least in principle). The ansatz which has been used most frequently for the representation of molecular one-electron spinors is a basis expansion into Gauss-type spinors. 

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