Dirac spinors four-component

In the relativistic method, the Dirac-Fock four-component spinors are partitioned into core and valence spinors. The many-electron relativistic wavefunction for a single configuration can be thus expressed as... 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

In this paper, for functions (pi r) we shall use the four-component spinors r) being solutions of the Dirac equation... 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

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... 

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

To construct the Dirac-Fock equations, it is assumed that the wave function for an atom having N electrons may be expressed as an antisymmetrized product of four-component Dirac spinors of the form shown in Eq. (9). For cases where a single antisymmetrized product is an eigenfunction of the total angular momentum operator J2, the JV-electron atomic wave function may be written... 

On the other hand, in the pioneering DHF and post-DHF program package MOLFDIR [3] and the well-developed four-component relativistic program package DIRAC [4], the molecular four-component spinors are expanded into decoupled... 

The time-dependent Schrodinger equation (2.43) presents a serious problem from the point of view of relativity theory it does not treat space and time in a symmetric way, because second-order derivatives of the wavefunction with respect to spatial coordinates are accompanied by a first-order derivative with respect to time. One way out, as actually proposed by Schrodinger and later known as the Klein-Gordon equation, would be to have also second-order derivatives with respect to time. However, that would lead to a total probability for the particle under consideration which would be a function of time, and to a variation of the number of particles of the universe (which, at the time, was completely unacceptable). In 1928, Dirac sought the solution for this problem, by accepting first-order derivation in the case of time and forcing the spatial derivatives to also be first order. This requires the wavefunction to have four components (functions of the spatial coordinates alone), often called a four-component spinor . 

These atomic orbitals are four-component Dirac spinors. The symmetriza-tion coefficients are obtained by the use of group theoretical projection operators [21]... 

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