shift up in energy. We see that the atomic shells fill up in the order 1s,2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s,
4d, 5p, 6s, 4f, 5d, 6p. The effect of screening increasing the energy of higherℓstates is clear. Its no
wonder that the periodic table is not completely periodic.
A set of guidelines, known as Hund’s rules, help us determine the quantum numbers for the ground
states of atoms. The hydrogenic shells fill up giving well definedj= 0 states for the closed shells.
As we addvalence electronswe follow Hund’s rules to determine the ground state. We get a great
simplification by treating nearly closed shells as a closed shell plus positively charged, spin^12 holes.
For example, if an atom is two electrons short of a closed shell, we treat it as a closed shell plus two
positive holes.)
- Couple the valence electrons (or holes) to givemaximum total spin.
- Now choose the state of maximumℓ(subject to the Pauli principle. The Pauli principle rather
than the rule, often determines everything here.) - If the shell is more than half full, pick the highest total angular momentum statej=ℓ+s
otherwise pick the lowestj=|ℓ−s|.
1.34 Molecules
We can study simple molecules (See section 27) to understand the physical phenomena of molecules
in general. Thesimplest molecule we can work with is theH+ 2 ion. It has two nuclei (A and
B) sharing one electron (1).
H 0 =p^2 e
2 m−
e^2
r 1 A−
e^2
r 1 B+
e^2
RABRABis the distance between the two nuclei. We calculate the ground state energy using the Hydrogen
states as a basis.
The lowest energy wavefunction can be thought of as a (anti)symmetric linear combination of an
electron in the ground state near nucleus A and the ground state near nucleus B
ψ±(
~r,R~)
=C±(R) [ψA±ψB]whereψA=
√
1
πa^30 e−r 1 A/a (^0) is g.s. around nucleus A.ψAandψBare not orthogonal; there is
overlap. Thesymmetric (bonding) state has a large probability for the electron to be
found between nuclei. The antisymmetric (antibonding) state has a small probability there, and
hence, a much larger energy. Remember, this symmetry is that of the wavefunction of one electron
around the two nuclei.
TheH 2 molecule is also simpleand its energy can be computed with the help of the previous
calculation. The space symmetric state will be the ground state.
〈ψ|H|ψ〉= 2EH+
2
(RAB)−
e^2
RAB+
〈
ψ∣
∣∣
∣
e^2
r 12∣
∣∣
∣ψ〉
The molecule canvibratein the potential created when the shared electron binds the atomsto-
gether, giving rise to a harmonic oscillator energy spectrum.