Complex Atoms

Complex Atoms
1. Shell Model of the Atom
Consider forming a system of several electrons electrostatically bound to an infinitely heavy nucleus of charge Ze. The electrons also electrostatically repel each other, so a Hamiltonian for the system can be written that, in atomic units, looks like

…..(1)
or



where is the position of electron i with respect to the nucleus. ri and rij are defined by the corresponding expression in the equation above. While the Hamiltonian is not complete, all additional terms can quite successfully be treated as perturbations.
Using a central potential H may be written as:

…….(2)

The Schr?dinger equation is:


where


……(3)

and

Here ui(ai) is the wave function of the ith electron having a set of quantum numbers ai. It follows then that each ui is a solution of the one-electron central force problem and must be of the form:
….(4)

which is a one-electron spin orbital. The form of the is unspecified and depends on the particular choice of central potential. So ai must represent the set of four quantum numbers:

….(5)

2.2 Angular Momentum for Complex Atoms






For two electrons, everything carries over from the coupling of two angular momenta. So,






Even for more than two electrons, one can write:





Each total J value has (2J + 1) different MJ values.
For two electrons in the configuration 2p3p, the possible values for the total orbital angular momentum are L = 0, 1, or 2 whose symbols are S, P, or D. Since the spin for each is si = 1/2, the total spin can be S = 1 or 0. These possibilities are expressed as:


an expression like:
is called a term
is called a level
is called a state
The permitted states of angular momentum that can arise from a system composed of two sources of angular momentum are given by the Clebsch–Gordan series: