Kinetic balance condition

If the kinetic balance condition (5) is fulfilled then the spectrum of the L6vy-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]. 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

Quiney 1988). The small component s radial function has been fixed according to the kinetic balance condition (Stanton and Havriliak 1984), which has its origin in the coupled nature of Dirac s first-order differential equations and is introduced to keep the method variationally stable. The index A denotes the coordinates of the nucleus s centre RA of atom A, to which the basis function is attached, i.e. rA = r — RA. As an alternative, Cartesian Gaussians,... 

The correct nonrelativistic limit as far as the basis set is concerned is obtained for uncontracted basis sets, which obey the strict kinetic balance condition and where the same exponents are used for spinors to the same nonrelativistic angular momentum quantum number for examples, see Parpia and Mohanty (1995) and also Parpia et al. (1992a) and Laaksonen et al. (1988). The situation becomes more complicated for correlated methods, since usually many relativistic configuration state functions (CSFs) have to be used to represent the nonrelativistic CSF analogue. This has been discussed for LS and j j coupled atomic CSFs (Kim et al. 1998). 

The form of the large-component primitive set rjjf" is chosen from large-component spinors obtained by analytical solution of the one-electron Dirac equation. The small-component set is derived so that it satisfies the accurate and rigorous kinetic balance condition versus rjjf",... 

As an example, one of more well-known constraints on the basis functions is the so-called kinetic balance condition [60, 61]. Specifically, most of the finite basis functions do not form complete basis sets in the Hilbert space. If the large- and small-component radial wave functions are expanded in terms of one of these orthonormal basis sets ip such that P r) = and Q r) = Ylj then the operator identity (cr p) [Pg.168]

The requirement that the wave function should be stationary with respect to a variation in the orbitals, results in an equation that is formally the same as in non-relativistic theory, FC = SCe (eq. (3.51)). Flowever, the presence of solutions for the positronic states means that the desired solution is no longer the global minimum (Figure 8.1), and care must be taken that the procedure does not lead to variational collapse. The choice of basis set is an essential component in preventing this. Since practical calculations necessarily use basis sets that are far from complete, the large and small component basis sets must be properly balanced. The large component corresponds to the normal non-relativistic wave function, and has similar basis set requirements. The small component basis set is chosen to obey the kinetic balance condition, which follows from (8.15). 

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. 

Inserting the kinetic-balance condition Eq. (5.137) into the remaining upper component of Eq. (5.135),... 

For basis-set expansion techniques it turned out to be decisive to fulfill the kinetic-balance condition for the basis functions (see again chapter 10 for details and references), whereas fully numerical four-component calculations had already been carried out around 1970 without encountering variational collapse. In numerical approaches it is possible to search for optimized spinors in the vicinity of the nonrelativistic solution with a given number of nodes and associated orbital energy as we shall see in chapter 9. 

From Eqs. (5.136) and (5.137) in chapter 5 we know that the kinetic balance condition relates the small (lower) and large (upper) 2-spinors. In the stationary case, in which the time-dependence of the Dirac equation drops out and the total energy E if a one-electron state comes in as a separation constant according to (ih) d/df — e (cf. Eqs. (6.5)-(6.7)), we may write Eq. (5.136) as... 

The kinetic balance condition is a consequence of the 2x2 superstructure of the one-electron Hamiltonian and must therefore be fulfilled by any basis set expansion of the molecular spinors. 

Again, the small component s radial function has been fixed according to the kinetic balance condition. 

The modified Dirac equation can now be viewed from two different perspectives. The first perspective is the fact that the approximate kinetic balance condition of Eq. (5.137) has been exploited. The normalized elimination procedure then results in energy eigenvalues which deviate only in the order c from the correct Dirac eigenvalues, whereas the standard un-normalized elimination techniques are only correct up to the order c. In addition, the NESC method is free from the singularities which plague the un-normalized methods and can be simplified systematically by a sequence of approximations to reduce the computational cost [562,720,721]. From a second perspective, Eq. (14.1) defines an ansatz for the small component, which, as such, is not approximate. Hence, Eq. (14.4) can be considered an exact starting point for numerical approaches that aim at an efficient and accurate solution of the four-component SCF equations (without carrying out the elimination steps). We will discuss this second option in more detail in the next section. 

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition,

[Pg.533]

The main bottleneck of four-component calculations has its origin in the existence of a small component and in the kinetic balance condition that generates a very large basis set for this small component, which contributes only a little to expectation values for moderately large nuclear charge numbers Z. Of course, the size of this effect changes for super-heavy atoms in molecules. 

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