Page 117 - Bruno Linder Elementary Physical Chemistry

August 18, 2010 11:36 9in x 6in b985-ch09 Elementary Physical Chemistry

102 Elementary Physical Chemistry

9.8. Structure, Transitions and Selection Rules

We have already mentioned that in a hydrogenic atom, the electron can have
three quantum numbers n, l,and m l . We must now also add m s .Thus,

n =1, 2,...
l =0, 1,... ,n − 1
m l =0, +1, −1, +2, −2,...
m s =+1/2, −1/2

Note that l cannot exceed n − 1, although it may be smaller.
2
2
Each energy level is n -fold degenerate, meaning that there are n states
which have the same energy. This rule is true only for hydrogenic atoms
and does not apply to other atoms.

9.9. Many-Electron Atoms
The Schr¨odinger equation can only be solved exactly for hydrogenic atoms,
yielding exact analytic expressions for the wave-functions. In all other cases,
solutions are approximate.
As a first approximation, one can think of the wave-function of the
atom as the product of the wave-functions of the individual electrons, i.e.

Ψ atom = ψ(1)ψ(2)ψ(3)... (9.10)
where ψ(1) is the orbital of electron 1, ψ(2) the orbital of electron 2, etc. but
with nuclear charge that is modified by the presence of all other electrons.
This effective nuclear charge, Z eff , is the charge of the nucleus shielded by
the other electrons. Thus, the nuclear charge an electron “sees” is not the
actual charge, Z e (e being the absolute value of an electronic charge) but
Z e−σ = Z eff ,where σ is a shielding constant that can be approximated.
The Z eff values are different for s, p, d, etc. orbitals. For example, the
s electron has greater penetration to the nucleus than the p electron; the
p electron has greater penetration than the d electron, etc. But there are
exceptions; for example, 4s precedes 3d.

9.10. Pauli Exclusion Principle
This Principle states that no more than two electrons can occupy the same
orbital. Actually, the Pauli Exclusion Principle really states that in an atom