Kinetic balance approximate

The various two-component theories known from the literature satisfy the kinetic balance relation only to certain degrees of accuracy and hence establish only variationally stable but not variational approaches. The simplest approximation to exact kinetic balance may be obtained in the non-relativistic limit of Eqs. (14) or (16),... 

In stationary DPT we automatically care for the correct relation between X2 and i.e. we do exactly, what is done approximately in nonperturba-tive calculations by imposing the so-called kinetic balance [75]. 

The terms etc. in (10) represent the one-body mean-field potential, which approximates the two-electron interaction in the Hamiltonian, as is the practice in SCF schemes. In the DFB equations this interaction includes the Breit term (4) in addition to the electron repulsion l/rjj. The radial functions Pn ( ) and Qn/c( ) may be obtained by mmierical integration [20,21] or by expansion in a basis (for more details see recent reviews [22,23]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [24,25], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [26,27]. In the nonrelativistic limit (c oo), the small component is related to the large component by [24]... 

The approximate relation ( "kinetic balance ) between the laige and small components of the bispinor (that holds for small v/c) may be used to eliminate the small components from Eqs. (3.57) and (3.58). We obtain... 

The kinetic balance can be used to eliminate the small components from the Dirac equation. Then, the assumption c = 00 (non-relativistic approximation) leads to the Schrodinger equation for a single particle. 

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. 

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. 

The consequence of not satisfying this condition was observed in early basis set calculations (Rosicky and Mark 1979, Mark and Rosicky 1980, Mark et al. 1980). It can be shown (Dyall et al. 1984) that the kinetic energy is a maximum when the kinetic balance relation is satisfied, and therefore any approximation that does not fully span the space defined by (11.19) must lower the energy. This is indeed what was observed. 

---