Levy-Leblond

The limit c —can now be taken provided that (1) V is everywhere non-singular, which is true for finite nuclei [42] but not point nuclei, and that (2) E < c which is true for the (shifted) positive-energy solutions only. With this procedure all relativistic effects are eliminated and one obtains the four-component non-relativistic Levy-Leblond equation [34,43]... 

These operators may be contrasted with the property operators obtained at the non-relativistic level. The Levy-Leblond equation given in (29) is consistent with the rigorous non-relativistic limit in that only scalar potentials are included. We now follow the standard, pragmatic approach and consider the operator form... 

Table 4. The isotropic indirect spin-spin coupling constant of calculated at various levels of theory. LL refers to the Levy-Leblond Hamiltonian, std refers to a full relativistic calculation using restricted (RKB) or unrestricted (UKB) kinetic balance, spf refers to calculations based on a spin-free relativistic Hamiltonian. Columns F, G and whether quaternion imaginary parts are deleted (0) or not (1) from the regular Fock matrix F prior to one-index transformation, from the two-electron Fock matrix G...

Table 5. The NMR shielding constant and shielding polarizabilities of the xenon atom calculated at the Hartree-Fock level using the Drrac-Coulomb Hamiltonian (SR + SO), its spin-free version (SR) as well as the non-relativistic Levy-Leblond Hamiltonian. The shielding constant is given in ppm and shielding polarizabilities in ppm/(au field2) (1 a.u. field = 5.14220642X 10" V...

This non-relativistic equation in terms of four-component spinors has been studied in detail by Levy-Leblond [44,45], who has shown that it results automatically from a study of the irreducible representations of the Gahlei group and that it gives a correct description of spin. It is easy to see that in the absence of an external magnetic field, equation (63) is equivalent to the Schrodinger equation in the sense that after elimination of the small component ... 

Earlier we mentioned briefly that the electron spin is perfectly consistent with the non-relativistic four-component Levy-Leblond theory [44,45]. The EC type interaction does not manifest in Dirac or Levy-Leblond theory. We shall show that on reducing the four-component Levy-Leblond equation into a two-component form the EC contribution arises naturally. A non-relativistic electron in an electromagnetic radiation field is described by the Levy-Leblond equation given by... 

The variational Dirac-Coulomb and the corresponding Levy-Leblond problems, in which the large and the small components are treated independently, are analyzed. Close similarities between these two variational problems are emphasized. Several examples in which the so called strong minimax principle is violated are discussed. 

Using Eq. (2) instead, we obtain the analogous expression for the Levy-Leblond energy ... 

As one can see, for each set of parameters Z, k, L and d, one can select a in such a way that Eqs. (21) and (22) are fulfilled. Then, the resulting expectation values correspond to the variational minima and are equal to the appropriate exact eigenvalues of either Dirac or Schrodinger (or rather Levy-Leblond) Hamiltonian. However the corresponding functions do not fulfil the pertinent eigenvalue equations they are not eigenfunctions of these Hamiltonians. This example demonstrates that the value of the variational energy cannot be taken as a measure of the quality of the wavefunction, unless the appropriate relation between the components of the wavefunction is fulfilled [2]. 

In this case the ground state energy in both the Dirac and Levy-Leblond case depends upon four nonlinear parameters ... 

Fig. 1. The ground state energies of a Z = 90 hydrogen-like atom obtainedfrom the Dirac (D) andfrom the Levy-Leblond (L) equations as functions of the nonlinear parameters. In the upper-rowfgures (Dl and LI) the a (abscissa) and (3 (ordinate) dependence ofE is displayed when L and S are set equal to the values corresponding to the exact solutions. In the lower-row figures (D2 and L2 ) the L (abscissa) and S (ordinate) dependence of E is displayed when a and 5 are set equal to the exact value. The arrows show directions of the gradient their length is proportional to the value of the gradient The solid line crossing the saddle corresponds to the functions andLl) or S = Sj (L) D2 and L2. For the definitions of thesefunctions see text.

Fig. 4. The same as in figures 1 and 3, but the values of L and S are not equal to the ones of the exact solutions. In the case of the Dirac equation L = S = i and in the case of the Levy-Leblond equation L = S = y have been taken. The saddlepoint coordinates (tXg, Pq are equal to (114,99) in the Dirac case and (66,84) in the Levy-Leblond case. The cross-sections of the energy surface by the planes a = p (broad solid line), and a = Uq (broken line) are plotted versus P, while the cross-sections by theplane P = Po (broken line with dots) and by the surface P = Pmax (thin solid line) areplotted versus a, in D2 (Dirac) and in L2 (Levy-Leblond). The scale o/P is shown in the horizontal axes. The scale of a has been chosen so that the curve for which P = Po und the onefor which a = Oq match at the saddle point The scale of a may be obtained by adding Oq - Po to the values o/P displayed in the axes of D2 and L2.

It has been noticed by Levy-Leblond [8] that the non-relativistic Schrddinger equation may be written as... 

The Dirac and the Levy-Leblond equations establish relationships between the large and the small components of the wavefunctions. If these relationships are to be fulfdled by the functions derived from a variational procedure, the basis sets for the large and for the small components have to be constructed accordingly. In particular, the relation... 

If the kinetic balance condition (5) is fulfilled then the spectrum of the Levy-Leblond (and Schrodinger) equation is bounded from below. Then, in each case there exists the lowest value of E referred to as the ground state. In effect, this equation may be solved using the variational principle without any restrictions. On the contrary, the spectrum of the Dirac equation is unbounded from below. It contains the negative ( positronic ) continuum. Therefore the variational principle applied unconditionally would lead to the so called variational collapse [2,3,7]. The variational collapse maybe avoided by properly selecting the trial functions so that they fulfil the boundary conditions specific for the bound-state solutions [1]. 

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