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Dissociation of H2
Do HF calculations for different
values of the H-H internuclear
distance (this distance is fixed
since we are in the BornOppenheimer picture).
Fig. 3.5: RHF potential curve for STO-3G
(ζ = 1.24) H2 compared with the
accurate results of Kolos and
Wolniewicz.
The minimal basis RHF calculation does not go to the limit of two hydrogen
atoms as the H-H distance goes to infinity. This totally incorrect behavior is
not specific for H2: if one stretches any bond for which the correct products
of dissociation must be represented by open-shell wavefunctions, then
closed-shell RHF will go badly wrong.
For H2, the products of dissociation are two localized H atoms; that is, one
electron is localized near one of the protons and the other electron is localized
near the other, distant, proton. In RHF, however, both electrons are forced to
occupy the same spatial molecular orbital. Therefore, independent of the bond
length, both electrons are described by exactly the same spatial wavefunction
and have the same probability distribution function in 3-dimensional space.
Such a description is inappropriate for two separated H atoms. Closed-shell RUF,
which restricts electrons to occupy molecular orbitals in pairs, cannot,
therefore, properly describe dissociation unless the products of dissociation are
both closed-shells.
The problem can be examined analytically: the
dissociated H2 molecule keeps a 12 11 | 11 
term because, since both electrons occupy the same
spatial orbital, there remains even at infinity some
electron-electron repulsion.

RUF
UHF gives a correct picture of the dissociation of H2
Fig. 3.19: 6-31G** potential
energy curves for H2
Does full CI get the correct dissociation picture for H2 ?
Yes: the full CI energy in minimal basis H2 is E 0  E corr  2h11  J11    2  K122 
1/ 2
In the limit
R ,  0 and Ecorr K
12
and the –K12 term exactly
cancels the J11 term.
Fig. 4.3: 6-31G** potential
energy curves for H2
CI and the size-consistency problem
Suppose you want to compute the energy difference
between reactants and products: A  B  C
where A, B are smaller than C (for example H + H  H2)
To get a reasonable answer, we need a method which works “equally
well”
 for molecules with different numbers of electrons. One simple
criterion for this is that the energy difference between (H+H) and (H2)
should be zero in the limit of infinite separation (non-interacting limit).
Such a method is called “size-consistent”.
HF is size-consistent if the monomers have a
closed shell. Full CI is also size-consistent
(since it is an exact theory).
However, for all but the smallest molecules, even with a minimal basis
set, full CI is computationally impractical. With a one-electron basis of
moderate size, there are so many possible configurations that the full
CI matrix becomes impossibly large (greater than 109 x 109).
CI and the size-consistency problem
To obtain a computationally viable scheme we must truncate the full CI
matrix (the CI expansion for the wave function). A systematic procedure
for doing this is to consider only those configurations which differ from the
HF ground state by no more than m spin orbitals.
For example, m = 4 gives single, double, triple,
and quadruple excitations in the trial function.
m = 2 gives SDCI: singly + doubly excited CI.
This choice is commonly used.
HOWEVER truncated CI is not size-consistent !
CI and the size-consistency problem
Truncated CI (e.g. SDCI) is not size-consistent !
Why? Very simple:
A + A  A2
SDCI of each of the A monomers contains double excitations
within the monomer. BUT the (non-interacting) dimer cannot
have both monomers doubly-excited, because this would
correspond to a quadruple excitation of A2. Hence the dimer
wavefunction does not have enough flexibility to yield twice
the monomer energy. (see page 264 of Szabo)
Are there schemes to compute the correlation energy
which are size-consistent? YES: Møller-Plesset (e.g. MP2)
perturbation theory
Møller-Plesset perturbation theory
(chapter 6 of Szabo)
Very useful, the “method of choice” if you need correlation.
Example: calculation
of pi-pi interactions
(dispersion) requires
electron correlation