- 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
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.
Open Shell SCF
- 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
where each spin-orbital may have
spin, e.g., Li atom
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
have the same space part.
Unrestricted Hartree-Fock Theory
- 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.,
- Fock operator (and the rest of the matrices) block into two parts, one for
spin-orbitals and one for
- 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.
Restricted Hartree-Fock Theory
Breakdown of Hartree-Fock Model
- 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
- 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)
is the antibonding virtual orbital combination of the atomic 1s.
- As the bond forms, the variation principle tells us we can only benefit from
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,
while near equilibrium we can expect
- 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
- 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
- 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.
GTO Basis Sets
- 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.
The Hartree-Fock Model Revisited