Dirac radial

The RQDO radial, scalar, equation derives from a non-unitary decoupling of Dirac s second order radial equation. The analytical solutions, RQDO orbitals, are linear combinations of the large and small components of Dirac radial function [6,7] ... 

It is occasionally argued that the missing nodes in the Dirac radial density explain how electrons get across the nodes [129] as there are no longer any nodes. However, one must keep in mind that this is a somewhat artificial question as time has been eliminated from the stationary Dirac and Schrodinger equations. The question remains why the density dramatically... 

We now turn to the discretization of the coupled first-order differential equations as they occur in the solution of the Dirac radial equation for atoms (see chapter 9). While the Numerov scheme is well established for second-order differential equations — and, hence, for the solution of the radial Schrodinger equation for atoms — this is not the case for first-order differential equations. Indeed, it was long believed that the Numerov scheme cannot be used at all in this context [1165]. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

The radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], 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 [22,23]. 

The relativistic version (RQDO) of the quantum defect orbital formalism has been employed to obtain the wavefunctions required to calculate the radial transition integral. The relativistic quantum defect orbitals corresponding to a state characterized by its experimental energy are the analytical solutions of the quasirelativistic second-order Dirac-like equation [8]... 

Eq. (36) may also be expressed as a system of two first-order equations, i.e. as the radial Dirac equation in the representation of Biedenharn. Let us rewrite the radial Dirac-Pauli equation (18) with V = —Zjr in the form... 

Figure 1. The radial parts of the large component of the 6pi/2 bispinor and the corresponding pseudospinor obtained in equivalent Dirac-Fock and 21-electron GRECP/SCF calculations for the state averaged over the relativistic 65 /26 1/2 configuration of thallium. Their difference is multiplied by 1000. The GRECP is generated for the state averaged over the nonrelativistic 6s 6p 6d configuration.

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