#  Why is spin-spin interaction not so important in many electron atoms?
Why can we get reasonably correct multiplet splittings just using the spin-orbit
coupling? One factor must be that spin-orbit ~Z^2<1/r^3> (Z electrons, nuclear
charge Ze) while spin-spin ~Z(Z-1)/2 <1/|r_i-r_j|^3>. In a many electron atom,
in the Thomas -Fermi approximation,  <1/|r_i-r_j|> = 2/7 <1/r>, as can be seen
from your hw results since the lhs is proportional to E_ee and the rhs is
proportional to E_en. If we naively cube this and multiply by the factor of
1/2 from (Z(Z-1)/2 ~ Z^2/2) we get 1/2 * (2/7)^3 = 0.01. So this might be it.
Also, we should consider what results from the orientation averaging,
since the dipole-dipole potential for two parallel spins is positive when the
the line joining the electrons is perpendicular to the spins and negative when the
joining line is parallel to the spins....

# He_2: Some tables list it, but books say it doesn't exist. What gives?
Maybe it is a metastable molecule consisting of, say, the metastable orthohelium
ground state bound to the parahelium ground state, or to another orthohelium ground
state. I'm totally unsure here. Somebody call a chemist!

# Justifying the Born-Oppie approx: Baym's discussion seems bogus, where he
asserts, to justify dropping the terms with derivatives on the electron wf, that
(size of nuclear wf)*grad (electron wf) ~ electron wf. First of all, if this were true,
it would not be good enough, since it would yield a term of order (m/M)^1/2 times
the electronic energy level spacing. Baym says this would be negligible, however
this is exactly the order of magnitude of the nuclear kinetic energy term, so it does
not seem negligible! However, in fact the correct eqn would be
(size of nuclear wf)*grad (electron wf) ~
[(size of nuclear wf)/(size of electron wf)] electron wf
~ (m/M)^1/4 electron wf,
so there is in fact a suppression factor of order (m/M)^1/4 ~ 10^-1...

# Discussed bonding again, and the pairing mechanism, valency, spatially
directed orbitals, and  applied this to the C_2 and O_2 molecules.
Also discussed hybridization and applied this to the Li_2 molecule.
This is all in Baym. The only place I differed was to not claim to understand
why the energy is always minimized by maximizing the overlap if a bonding
orbital is available. The explicit example was C_2: each C atom has 2, 1p electons.
A pair in the |p_z> states form a bond, then the remaining two form, say,
a |p_x> bound, according to Baym. The question is this: if the reason large
overlap is important is just because both electrons want to be in the same
deep part of the potential, then why does it make a difference if they take the
state |p_x>|p_x> rather than |p_x>p_y>? It must be the exchange energy.
But both Slater and ... say this exchange energy explanation of the covalent
bond is not correct.  ... gives the reference ... If you look this up, please
tell me what is said there.

# Labeling of diatomic molecule electron configurations by quantum numbers:

-rotational symmetry of the potential abt the z-axis: L_z is a good quantum number.
-reflection symmetry in the planecontaining the nuclei: L_z = +m and -m have the same energy.
-label states by |L_z| = "Lambda" = 0,1,2,... = "Sigma", "Pi", "Delta", ,,,
-a  nondegenerate state with L_z=0 must also be an eigenstate of the reflection
through the plane containing the nuclei, with eigenvalue +1 or -1. This is indicated
by the notation Sigma^+ or Sigma^-, with + or - as a right superscript.
-The total spin of the molecule is indicated by the multiplicity 2S+1 attached as a left-superscript.
- For a homonuclear diatomic molecule, there is the additional symmetry of reflection in the mid-plane.
Eigenstates of this with eigenvalue +1 or -1 are denoted by the right subscript g (gerade) or
u (ungerade) respectively.

Examples: C_2 is (1,Sigma,+,g), O_2 is (3,Sigma,-,g), NO is (2,Pi).

# Gave a tip about solving the problem of the van der Waals interaction V between two
H atoms in the n=2 state: use the fact that the van der Waals potential commutes with
L_z, and use the fact that x_1,2,  y_1,2,  z_1,2 are all odd under parity, and use the
fact that V is the zero component of a spin-2 tensor operator: it is the zz component
of  the tracefree part of  r^i r^j. We saw a few weeks ago that symmetric tracefree tensors
are spin-2 tensor ops (they transform irreducibly and there are 5 = 2*2 + 1 independent
components...

# hope to finish discussing hybridization with the example of the water molecule.

# start on time-dep. pert. theory