Central-field potential

The stationary states of a one-body Dirac operator h, (2), in the presence of a central-field potential have the spatial form... 

The quantum number k takes positive and negative integer values, the appropriate electron mass parameter is given by /i, the energy parameter is W, and the central field potential energy is... 

The coefficients C/n are the parameters to be determined. The set of functions Xp is known as the basis set and often consists of atom-centred functions obtained from the solution of the Schrodinger equation with some central field potential. 

These statements hold not only for atoms, but for any system with a central field potential subject to some additional potential U. 

We begin by observing that the standard treatment of the normal unconfined hydrogen atom (UHA), solves Schrodinger s equation for a model Hamiltonian with a central-field potential (we use the usual atomic units), specifically... 

The derivation of the working equations of the CG method starts with the radial Dirac equation for one particle in a central-field potential V(r), e.g. the local potential approximation to the DHF equations... 

In this chapter we have seen how all angular dependencies can be integrated out analytically because of the spherical symmetry of the central field potential of an atom. We are now left with the task to determine the yet unknown radial functions, for which we derived the self-consistent field equations based upon the variational minimax principle. We now address the numerical solution of these equations. The mean-field potential in the set of coupled SCF equations is the reason why we cannot solve them analytically. Hence, the radial functions need to be approximated in some way in... 

H and L - wiU commute if F in // is independent of 0 and (p (central field potential). H and Lz will commute if F is independent of (p, even if it is dependent on d. This is the case for any linear systena. Therefore, Mi continues to be a sharp quantity for linear molecules hfce H2 or C2H2, but total angular momentum value does not, because L does not commute with H, so L is not a good quantum number. This is why the main term symbol for a linear molecule is based on Mi, whereas the main term symbol for an atom is based on L. 

Interpreted in terms of the symmetrical form of the periodic table (Fig. 3), the quantum numbers that define the radial distances of r = n a specify the nodal surfaces of spherical waves that define the electronic shell structure. Knowing the number of electrons in each shell, the density at the crests of the spherical waves that represent periodic shells, i.e., at 1.5,3, etc. (a), can be calculated. This density distribution, shown in Fig. 7, decreases exponentially with Z and, like the TF central-field potential, is valid for all atoms and also requires characteristic scale factors to generate the density functions for specific atoms. The Bohr-Schrodinger... 

Show how the spin-orbit coupling terms in the Dirac-Pauli Hamiltonian (11.2.17) reduce, for a particle with the central-field potential-energy function V r), to the form Hst=/(r)S L, where /(r) = (gj8VcV)r dV/dr. Verify that for an electron in the presence of a nucleus of charge Z e this reduces further to give (in the absence of a magnetic field) the first term in (11.3.13). 

Since R i is a function of r only, it depends on the central field potential U(ri). A solution to this wave function, shown in Equation 1.7, is approximated and depends on the form of the central field. 

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