- Clearly, the number of basis functions M we use must be at least as
large as the number of occupied orbitals N.
- If M > N, in addition to the N occupied orbitals, the SCF procedure will
generate M-N virtual (i.e., unoccupied) MOs.
- Koopmans' Theorem: the orbital energies
of the occupied orbitals correspond to ionization potentials; the orbital
energies
of any virtual orbitals correspond to electron affinities.
- These Koopmans' theorem estimates are the best we can get (variationally)
without reoptimizing the MOs for the ion state.
- Does not hold for excitation energies:
is not the best estimate of the energy for exciting from orbital i to a.

- Ions take us in the direction of
*open-shell*systems: most generally, systems with unpaired electrons, e.g., H2O+, O2. - Want to extend our closed-shell Hartree-Fock method to open shells.
- Simplest approach is not to worry about pairing electrons up in orbitals at
all:
where each

*spin-orbital*may have or spin, e.g., Li atomwhere presumably and will look like a 1s orbital and like a 2s.

- Recall that spin-orbitals of different spin are orthogonal anyway: There is no
requirement here that
and
have the same space part.

- Since the terms in the Hamiltonian operator have no spin dependence,
there can be no terms in the energy involving orbitals of different spin (e.g.,
no hii).
- Fock operator (and the rest of the matrices) block into two parts, one for
spin-orbitals and one for
:
where

- The disadvantage of this approach is that no restrictions are applied to retain
symmetry properties of the wave function. Comes out
*contaminated*with other spin states. But it is cheap.

- What's wrong with (what seems to be) the simple idea of using, e.g.,
the naive open-shell extension to a closed-shell determinant?

- The closed-shell determinant is invariant to mixing the occupied orbitals among
themselves. Or an virtuals among themselves, but not occupieds with virtuals.
- The unrestricted Hartree-Fock determinant.is also invariant to mixing.
- But the
*restricted*case given above is not invariant to mixing occupieds among themselves! Because the spin-orbitals do not appear in the determinant, and if they are mixed into the other spin-orbitals they will change the wave function. - This makes the optimization of the energy much harder: have to account for
mixing between doubly and singly occupied MOs. But there is no spin
contamination in the wave function.

- Consider the molecule H2. As the bond length increases, the wave
function goes to a product 1sA1sB of the H atom wave functions on centres A and
B. Antisymmetrizing and pairing up the spins gives
- The Hartree-Fock wave function for H2 is just
where

and

- So at long distance, where S=0
- So at long distances in H2, Hartree-Fock does not give the product of two H
atom wave functions. Rather, it gives 50% (H+H), and 50%
(H
^{+}+H^{-}). "Covalent" and "ionic" terms. - Evidently, the ionic term should disappear at long distances.
- If we proceed backwards, we discover that (at long distances)
where

is the antibonding virtual orbital combination of the atomic 1s.

- As the bond forms, the variation principle tells us we can only benefit from
generalizing
to
where cu and cu can be optimized together with the MOs. This is the multiconfigurational Hartree-Fock or multiconfigurational SCF (MCSCF) method.

- At long distances,
and
,
while near equilibrium we can expect
and
.

- In the earliest days, people hoped to get away with a single STO for
each AO occupied in the atoms forming the molecule, e.g., 1s, 2s, 2px, 2py, 2pz
for B-Ne.
- Such a
*minimal*or single-zeta (SZ) basis does not work well, because it is not flexible enough to describe how the atomic orbitals deform in the molecule. - Next step, use two STOs to describe each orbital:
*double-zeta*(DZ) basis. Or even more (triple-zeta, etc.) - Molecules are lower symmetry than atoms -- orbitals empty in the atoms can
contribute, like
*d*functions in first-row molecules, or*p*functions on hydrogen!*Polarization*functions -- DZP basis, etc.

- For present purposes, we can assume that the STO designations can be
used to identify basis sets of the same quality as the STO sets.
- Thus an SZ set is not one GTO per AO (which would be useless), but enough GTOs
to describe each AO as well as one STO, etc.
- Vast number of basis sets of varying quality exist. Basis sets that give good
energies and structures do not always give good properties.
- Common sets: STO-3G (a SZ set), 4-31G, 6-31G (both DZ valence). From
GAUSSIAN. Then DZ or DZP (Dunning), and larger sets (TZP or TZ2P) etc.
- Polarization functions are very important (even though they increase the
expense) -- NH3 is planar without
*d*functions.

- We have already seen that the Hartree-Fock approach is unreliable when
we are breaking even the simplest chemical bond. This is because other
*configurations*become nearly degenerate in energy with Hartree-Fock. - There is a more subtle problem. Hartree-Fock views each electron as
interacting with an averaged potential generated by the others. But the
Hamiltonian actually contains the Coulomb repulsion between pairs of electrons
r
^{-1}. - Our knowledge of singularities in H tells us there will be cusp behavior in the
exact wave function as rij --> 0 that cancels this. In fact
So the exact wave function must involve at least terms linear in rij.

- The effect is that electrons will avoid each other
*more*than the Hartree-Fock approximation would suggest.*Electron correlation.*Correlation energy