#  H. A. Bethe and E. E. Salpeter: Quantum Mechanics of One- and Two- Electron Atoms.

# Improving variational method: add more parameters, e.g. series expansion. Another idea:
allow for two different effective nuclear charges Z1 and Z2, on the grounds that one electron
would be closer to the nucleus than the other, but antisymmetrize of course. Turns out Chandrasekhar
had this idea too, and it is a good enough ansatz to reveal that the H- ion has a bound state
(while the single Z' ansatz does not show this). The actual ground state energy of the H- ion
is around -0.75 eV, and there is only one bound state.
Since the unpertubed ground state is a spin singlet, it is reasonable to use a spin singlet
ansatz. In fact, if we ignore the spin-dependent fine structure terms in the Hamiltonian,
it commutes with S^2, so the perturbation of the singlet must be another singlet state
Let |Z1> be the ground state of the Coulomb potential with nuclear charge Z1. The
ansatz is then  (|Z1>|Z2> + |Z2>|Z1>)|0,0>/Sqrt[2].

# Note the doubly excited states of helium, where both electrons are excited, have higher
energy than the ground state of the helium ion He+, so there is enough energy to eject an
electron. These are called auto-ionizing states. According to Cohen-Tannoudji et al,
some doubly excited states are primarily auto-ionizing, but some actually decay first by
emission of radiation. The auto-ionizing effect is called the Auger effect (according to
Bethe & Salpeter).

# excited states: 1s2s vs. 1s2p: 2s closer to the nucleus so less screened so lower energy.
#    Symmetric and antisymmetric states:
 |+> = (|1s,2s> + |2s,1s>)|00>/Sqrt[2]
 |->  = (|1s,2s> - |2s,1s>)|1m_s>/Sqrt[2].
V_ee is diagonal in this basis for the degenerate subspace, so just evaluate the expectation values:
<+|V_ee|+> = <1s,2s|V_ee|1s,2s> + <1s,2s|V_ee|2s,1s>
<-|V_ee|->  = <1s,2s|V_ee|1s,2s>  - <1s,2s|V_ee|2s,1s>
The second term is called the exchange integral and its value the exchange energy.
It is obviously positive in this case, and physically it is clear that it must be positive
even when complex states replace 1s and 2s, since the energy must decrease when the
antisymmetric wave function is chosen since then the electrons are less often found near
each other.
[One can prove nicely in general that the exchange integral is positive using the fact that
1/|r-r'| is the operator inverse of  (-Laplacian), which is a positive operator. The physical
idea seems to rely only on the fact that 1/|r-r'| is smaller the farther apart the points r and r'
are. This suggests to me a conjecture about finite real symmetric NxN matrices:
 

MATHEMATICAL CONJECTURE: If M_ij is a real, symmetric NxN matrix, and  M_ij < M_kl  whenever |i-j| mod N > |k-l| mod N  (i.e. if i,j,k,l = 1,2,...,N are arranged on a circle the distance  from i to j is more than  that from k to l), then all the eigenvalues of M are positive.

Please let me know if you prove it!

# Discussed metastability of ground state of orthohelium (spin triplet): lifetime around 10^4 s!
Something has to flip the spin: a relativistic S.B type interaction with the quantized electromagnetic
field. (The 1s2s parahelium (spin singlet) state is also metastable, decaying with a lifetime of
19.7 ms via a two-photon electric dipole transition.)

# Excited states of helium.

# Many electrons. Hund's partially justified rules:
1) maximize S
2) maximize L
3) J=|L-S| if L-shell < half-filled, L+S if > half-filled.

# H, He, Li, Be, B fairly obvious.
C: (2p)^2. Spin state symmetric lower energy since space antiymmetric, so S=1.
Space has 1x1 = 2 + 1 + 0, which are symm, antisymm, symm respectively.
So the unique anti symm choice is 1, so L=1.
Thus J = 2,1,0. Hund's 3rd rule says J=0.

A simple way to apply Hund's rules:
Put in all the spins up into all the available states, starting with the highest m_l working down.
If you fill all the states, and still  have some spins left, put in the remaining spins down starting
with  the largest m_l. In the resulting configuration, the total m_s is
equal to S and the total m_l is equal to L.

Why does this work? You are finding the state with the top values
of L_z and S_z subject to the Pauli exclusion principle, and
giving S_z primacy over L_z. These top values must be equal to
L and S respectively.

 # In class i was asked for an example of a partially  antisymmetric-in-space fermion wave function,
to illustrate why maximizing the spin maximizes the spatial antisymmetry.  Here it is:

Let 1,2,3 denote the space and spin labels of three particles. Suppose the function f(1,2,3) is antisymmetric
in the first two slots: f(1,2,3) = - f(2,1,3).  Then we can make a totally antisymmetric wf by summing
over cyclic permutations:

psi(1,2,3) = f(1,2,3) + f(2,3,1) + f(3,1,2)

You can check that psi is antisymmetric under any pair interchange of its arguments.

Now let's apply this to three electrons in the 2p shell for example.
Let f(1,2,3) be the state

f(1,2,3) = |110> (|+-+> - |-++>)

where the first factor means L_z values m_1=1, m_2=1, m_3=0 and the second factor denotes the spin state.
This state has S=1/2 and L=2, and is antisymmetric under (12) interchange, but not under (23) or (31).
So let us sum over the cyclic perms:

psi(1,2,3) =  |110> (|+-+> - |-++>)    +   |101> (|-++> - |++->)   +    |011> (|++-> - |+-+>)

This state is totally antisymmetric, but is not antisymmetric under just space permutations.
The probability of finding all  three particles as the same space point is zero (check this)
but the prob. of finding the first two at x and the third at y  is not zero:

|xxy><xxy|psi> =  |xxy> [ <x|1><x|1><y|0> (|+-+> - |-++>)     +     <x|1><x|0><y|1> (|-++> - |+-+>)] ,

               not = 0.
 

# Note in many electron atoms the orbitals 1s,2s,2p, etc from which the configurations are
constructed are eigenfunctions of the effective central potential created by the nucleus
together with the other electrons. This postential depends on the number of electrons.
The atom convention is to label the orbitals by the angular momentum letter s,p,d,f, etc,
and by the integer n = (#nodes of radial fn) + 1 + k, so for hydrogen it would be the
principal quantum number.

# Reason for Hund's second rule: electrons more spread out. To see this, note degeneracy in m_L,
so look at m_L=L. Example Ti, (3d)^2 = 2x2 = 4 + 3 + 2 + 1 + 0. Both 3 and 1 are antisymm so possible.
Top state of 3 has (m1,m2) = (2,1) and  (1,2). Top state of 1 has (2,-1), (1,0), (0,1), (-1,2) .  The larger
m_L orbitals are the most spread out.

# LS multiplet splitting arises from spin-orbit and spin-spin interaction.Must be spin-spin
averages to a small result since it is rarely discussed for many electron atoms. [However, in
the excited states of two-electron atoms it is important according to Bethe and Salpeter. In fact,
the order of the levels 3^P_2,1,0 is different for He than for Li+, and yet different for Z>10.]

# Origin of spin-orbit interaction: electron spin feels magnetic field created by current of moving
nucleus in its rest frame. Or, moving magnetic dipole generates electric dipole moment. Better yet,
just get it from the Dirac eqn!.

# H_s.o. = Sum_i  X(r_i) . s_i,  where i = 1,2,3,... labels the particles
and X(r) = (1/2m^2c^2)(1/r)(dV_eff/dr)L.
Note this is a sum of  L_i . S_i  terms, NOT the total L dot the total S. Nevetheless, Wigner-Eckart
comes to the rescue again, since the form of the projection theorem. One finds that
in a state of definite LS the expectation value of H_s.o. is equal to some zeta(N,L,S) times the expectation
value of the total L dot the total S, where N labels the other quantum numbers.
Now 2L.S = J^2 - L^2 - S^2, so the spin-orbit Hamiltonian is diagonalized in the J,M_J basis,
and the difference in the pertubations of the J and J-1 levels is  zeta(N,L,S) J. This is the Lande interval rule.
We looked at the example of the ^5D multiplet of  Fe,  from Bethe & Jackiw, Intermediate Quantum Mechanics.
There the interval rule seems good to about 10%. Presumably the corrections come from the spin-spin
interactions and higher orders in perturbation theory. My guess is primarily from the former.

# Hund's third rule: zeta(N,L,S) turns out ot be positive for less than half-filled shells and negative for more
than half-filled shells. This can be shown in the "Hartree-Fock" approximation. See Baym.

# Hartree-Fock idea: self consistent system of nonlinear equations for orbitals in one or more Slater determinants,
where the potential seen by each electron includes a term which is created by the charge density of the
other electrons. Could solve iteratively, as linear equations, by using the potential constructed from solutions
to the nth iteration as the potential in the (n+1)st iteration.

# Thomas-Fermi: treat electrons as degenerate Fermi gas, locally, at each point. Assumes many electrons
with wavelengths much smaller than length scale over which the effective potential changes much.