Levy-Leblond Functional

The functionals Fik i>2k) play a central role in stationary direct perturbation theory. Fq iPq) has been called the Levy-Leblond functional [23], since its stationarity condition is the LLE. For 4( 2) th name Hylleraas-Rutkowski functional has been suggested [23], since this belongs to the class of Hylleraas functionals of second-order perturbation theory, and since it has first been proposed by Rutkowski [73, 74] in a slightly different form. 

The implication of the above arguments is that we have taken the maximum of the second part of Fq and the minimum of the first part. Fq is called the Levy-Leblond functional because the solutions of the Levy-Leblond equation make this functional stationary. The Lagrange multiplier is simply the nonrelativistic 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]. 

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.

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]. 

Fig. 5. The ground state energies of a Z = 30 hydrogendike atom in the Gaussian basis obtained from the Dirac (D) andfrom the L y-Leblond (L) equations as functions of a and. The saddle point coordinates Pq) are equal to (282,168) in the Dirac case and (271,164) in the Levy-Leblondcase. The cross-sections of the energy surface by the planes a = p and a = c% are plotted versus p while the cross-sections by the plane P = Po nd by the surface p = Pmax y - versus a, in D2 (Dirac) and in L2 (Levy-Leblond). The scale of is shown in the abscissas. The scale of a has been chosen so that the curve for which P = Po the onefor which a = OCq match at the saddle point The scale of a may be obtained by adding tto - Po to the values of p displayed in the abscissas of D2 and L2. All conventions concerning the notation are the same as in figures 1 and 4.

Levy-Leblond Hamiltonian as the unperturbed operator. It turns out that when so-called kinetic balance is obeyed (roughly speaking, when the basis set describing the small component is represented by functions derived from the large-component basis functioas xl by xs = the same results... 

Fig. 5. The ground state energies of a Z 30 hydrogen-like atom in the Gaussian basis obtained from the Dirac (D) andfrom the L y-Leblond (L) equations as functions of a and. The saddle point coordinates equal to (282,168) in the Dirac case and (271,164) in the Levy-Leblondcase. The cross-sections of...

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