I have a question about the number of parameters in Full-CI.
Take HeH^+ in a minimal basis for example. If I start from Hartree-Fock, I will have two MOs (bonding and anti-bonding),
c_1 \Chi_1 + c_2 \Chi_2,
c_3 \Chi_1 + c_4 \Chi_2.
Here \Chi_1 and \Chi_2 are the AO basis. There are four parameters, c1-c4. Two normalization and one orthogonal constraints give 4-2-1=1 parameters.
Now we go to full-CI. There are three Slater Determinants (SDs) to span the exact state, (sigma)^2, (sigma) (sigma*), (sigma*)^2. Here '*' means antibonding orbital. There are three parameters for three SDs. Since there is a global phase factor without physical consequence (although we can normalize it). I have 3-1=2 parameters.
Counting HF+Full-CI, 1+2=3.
Next we turn to VB (non-orthogonal) scheme. The wavefunction can be written as spatial*spin. The spatial is symmetric, spin is antisymmetric for the singlet state. We use product of AOs to construct the spatial wavefunction, i.e. \Chi_1 \Chi_1, \Chi_1 \Chi_2 +\Chi_2 \Chi_1 , \Chi_2 \Chi_2. As before, there is one global phase factor, so 3-1=2 parameters.
Why I need 3 in MO (orthogonal basis) but 2 in VB (non-orthogonal)?
The full-CI parametrization is known to be redundant, H. Nakatsuji, J. Chem. Phys., 113, 2949-2956 (2000), is this an example?
Thank you very much in advance
with best regards