#  Molecules: separation of energy scales due to small ratio of electron mass m to nuclear mass M,
m/M ~ 10^-3 - 10^-5.  Treated in Baym, Ch, 21.

# E_elec : E_vib : E_rot :: 1 : (m/M)^1/2 : m/M

# Born-Oppenheimer or adiabatic approximation:
Basic idea: solve for electron configurations at fixed nuclear positions, then
the resulting energy as a function of the nuclear positions serves as an effective potential for the
nuclear motion. To lowest order in m/M, mixing of electronic configurations does not occur, so we
get approximate eigenfunctions of the form: Psi(r,R) = N(R) e(r,R), where e(r,R) is an electronic
wavefunction of the electron positions r and the fixed nuclear positions R
(H - T_N)e(r,R) = E(R)e(r,R),
where H is the total hamiltonian and T_N is the nuclear kinetic energy, and
(T_N + E(R))N(R) = EN(R).
What has been neglected here are terms with at least one derivative of e(r,R) with respect to R.
These terms are suppressed by (m/M)^1/2.

# Most commonly we are interested in the case where e(r,R) is the ground state for the electrons
at fixed nuclear positions. Then the nuclear energy levels correspond to translational, rotational,
or vibrational excitations of the ground state.

# Upper and lower bounds on second order energy shift of ground state:
E^(2) = Sum'_i <0 | V | i >< i | V| 0>/(E_0 - E_i), where the prime means i=0 is
omitted.
Every term is negative.
A lower bound to |E^(2)| is just the abs. value of the first term.
The absolute value of the denominator is smallest for the first excited
state, hence an upper bound is obtained by taking this denominator out
of the sum:

|E^(2)| < (E_1-E_0)^-1 Sum'_i  <0 | V | i >< i | V| 0>

           = (E_1-E_0)^-1 [ <0|V^2|0> - <0|V|0>^2 ].

The last step follows since Sum' |i><i| = Sum_i |i><i| - |0><0| = 1 - |0><0|.

# Estimate of zero point oscillation amplitude in molecule: (m/M)^1/4 a.
This is typically of order 1/10 a.
 
# Molecular bonding calculations: followed Baym.

# Total wf for homonuclear diatomic molecule:
Psi(r,R) = Phi(R1,R2) phi(r, |R1-R2| ),
with
Phi(R1,R2) = Phi_cm (R1+R2) Phi_vib( |R1-R2| ) Phi_rot(R1-R2) chi_spin(s_1,s_2).
The nuclei are either bosons or fermions, so the total  nuclear wf must be either
symmetric or antisymmetric under simultaneous interchange of space and spin
coords. The cm and vib parts are symm, so it comes down to the rot and spin parts.
Under interchange the argument of the rot part changes sign, which amounts
to a parity transfn, so is even or odd for even or odd orbital angular momenta.
The symmetry of the spin part must be appropriately matched.

# Note the above is an approximation. For one thing, the factorization of the
total state is only valid in the Born-Oppenheimer approx. For another, the
vibrational and rotational states do not completely decouple.

# Quantum numbers of diatomic molecules. Confusion prevailed. Will straighten
this out next Wednesday.