Components of the spinor

In Eq. (168), the first, magnetic-field term admixes different components of the spinors both in the continuity equation and in the Hamilton-Jacobi equation. However, with the z axis chosen as the direction of H, the magnetic-field temi does not contain phases and does not mix component amplitudes. Therefore, there is no contribution from this term in the continuity equations and no amplitude mixing in the Hamilton-Jacobi equations. The second, electric-field term is nondiagonal between the large and small spinor components, which fact reduces its magnitude by a further small factor of 0 particle velocityjc). This term is therefore of the same small order 0(l/c ), as those terms in the second line in Eqs. (164) and (166) that refer to the upper components. 

In most cases of interest, the spin density is collinear that is, the direction of the magnetization density m(7) is the same over the space occupied by the system it is customary to identify this as the -direction. The Hamiltonian is then diagonal if the external potential is diagonal, which allows one to decouple the spin f and spin J, components of the spinors and to obtain two... 

The components of the spinor e+, e define standing waves for fermions and anti-fermions respectively in each case by a combination of two spherical waves that respectively, converges to and diverges from r = 0, e.g. ... 

Here subscripts correspond to the covariant components of the spinor, which are related to the contravariant components (superscripts) through the metric spinor... 

This relation between the components of the spinor ensures that states below -2meC are omitted (otherwise ihd/dt E would not be small compared to the rest energy). This approximation will turn out to be very important in the relativistic many-electron theory so that a few side remarks might be useful already at this early stage. Eq. (5.137) will become important in chapter 10 as the so-called kinetic-balance condition (in the explicit presence of external vector potentials also called magnetic balance). It shows that the lower component of the spinor Y is by a factor of 1/c smaller than Y (for small linear momenta), which is the reason why Y is also called the large component and Y the small component. In the limit c oo, the small component vanishes. 

A formal solution is the so-called minimax principle [354], which states that the problem of variational collapse is avoided by determining the minimum of the electronic energy with respect to the large component of the spinor, while guaranteeing a maximum of the energy with respect to the small component. How such a saddle point may look has been shown by Schwarz and Wechsel-Trakowski [217]. The minimax principle has also been discussed in great detail for the complicated two-electron problem [355] (see also Refs. [356,357]). 

Of course, what has just been stated for the one-electron Dirac Hamiltonian is also valid for the general one-electron operator in Eq. (11.1). However, the coupling of upper and lower components of the spinor is solely brought about by the off-diagonal ctr p operators of the free-partide Dirac one-electron Hamiltonian and kinetic energy operator, respectively. We shall later see that the occurrence of any sort of potential V will pose some difficulties when it comes to the determination of an explicit form of the unitary transformation U. A universal solution to this problem will be provided in chapter 12 in form of Douglas-Kroll-Hess theory. 

For a relativistic Hamiltonian with inversion symmetry, it can be shown that the large and small components of the spinor have different parity. We do this by writing the spinor, 1 , as a column vector of large and small components... 

We start by splitting each of the four spatial components of the spinor into a real and an imaginary part ... 

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). 

As we can see, the FW two-component wave function is not the large component of the Dirac spinor, but it is related to it by an expression involving X. Consider a similarity transformation based on U parameterized as... 

As seen from equation (50), the ESC Hamiltonian is energy dependent and Hermitian. For a fixed value of E, the ESC Hamiltonian can be diagonalized and the resulting solutions, in principle, form a complete orthonormal set. The eigenfunctions of are identical to the large component of the Dirac spinor. When Z — 0, equations (38) and (44) give us the similarity transformed Hamiltonian... 

The orbital angular momentum quantum numbers, = I and corresponding, respectively, to the large and to the small components of the Dirac spinor are equal to... 

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,... 

The l/REP(r), U ARKP(r), and terms At/f EP(r) in 11s0 of Eqs. (23), (31), and (34) or Eq. (6), respectively, are derived in the form of numerical functions consistent with the large components of Dirac spinors as calculated using the Dirac-Fock program of Desclaux (27). These operators have been used in their numerical form in applications to diatomic systems where basis sets of Slater-type functions are employed (39,42,43). It is often more convenient to represent the operators as expansions in exponential or Gaussian functions (32). The general form of an expansion involving M terms is... 

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