L spinors

L-spinors, [87], are related to Dirac hydrogenic functions in much the same way as Sturmian functions [93,94] are related to Schrodinger hydrogenic functions. The radial parts are given by [87, III]... 

S-spinors. It is unfortunate that L-spinors are virtually useless for treating... 

G-spinors are appropriate for distributed charge nuclear models, and are much the most convenient for relativistic molecular calculations. Whereas neither L-spinors nor S-spinors satisfy the matching criterion (140) for finite c (although they do in the nonrelativistic limit), G-spinors are matched according to (140) for all values of c. The radial functions can be written... 

For the relativistic case there are three analogous choices of expansion functions to those discussed above. The hydrogenic functions have their analogue in the L-spinors obtained from the solution of the Dirac-Coulomb equation [4]. Again their use is mainly restricted to analytic work in atomic calculations, due to the difficulties in evaluating the integrals [5]. The analogue of the STO is the S-spinor which may be written in the form... 

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. 

This is verified by applying U(a,A) to the left and U(a,A) l to the right of both sides of Eqs. (11-142) and (11-97), mid making use of the transformation properties of (11-184) and (11-185) of the fields. Equations (11-241) and (11-242) were to be expected. They guarantee that four-vector, respectively. We are now in a position to discuss the transformation properties of the one-particle states. Consider the one-negaton state, p,s,—e>. Upon taking the adjoint of Eq. (11-241) and multiplying by ya we obtain... 

Let T be the fundamental representation of SO(4), and be the positive half-spinor representation. Let Mi = T 0ru(D) and M2 = S +0cHom(D, W). If we choose a complex structure on in other words, a reduction of the symmetry group from SO(4) into SU(2), T could be identified with A° L Hence Mi can be identified with A° 0c End(D). More explicitly, choosing a basis for T, we could write the identification as... 

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. 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

Other G-spinor parameters depend on the component index /3. Thus - l = j + f3r]i,l2. 

Thus, the non-relativistic wave function (1.14) of an electron is a two-component spinor (tensor having half-integer rank) whereas its relativistic counterpart is already, due to the presence of large (/) and small (g) components, a four-component spinor. The choice of / in the form (1 + l — l ) is conditioned by the requirement of a standard phase system for the wave functions (see Introduction, Eq. (4)). 

By substituting the previous definition for these spinors into (56) and using (79), it follows that A is a null vector, (e.g., AjAJ = 0), which, according to (78), implies l = l2= 0. 

In order to address polarization/rotation modulation—not just static polarization/rotation—an algebra is required which can reduce the ambiguity of a static representation. Such an algebra which is associated with 2, r, and that reduces the ambiguity up to a sign ambiguity, is available in the twistor formalism [28]. In this formalism, polarization/rotation modulation can be accomodated, and a spinor, K, can be represented not only by a null direction indicated by 2, q, or C, but also a real tangent vector L indicated in Fig. 4. 

All representations except S are two-dimensional. Subscripts g and u have the usual meaning, but a superscript + or is used on S representations according to whether x(oy) = l. For L > 0, x(C2), an(f x(°v) are zero. In double groups the spinor representations depend on the total angular momentum quantum number and are labeled... 

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