Page 128 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

QUANTUM CHEMISTRY IN FRONT OF SYMMETRY BREAKINGS 113

2.8. THE DIFFICULTY TO RESTORE SYMMETRY
The major drawback of symmetry-broken solutions is the difficulty to exploit them at a
higher level of accuracy. There are three possible attitudes (besides simply refusing
symmetry-breaking).

One attitude would consist in restoring symmetry by a symmetric superposition of the
degenerate and linearly independent but non orthogonal symmetry-broken solutions,
considering the gerade and ungerade combinations of the solutions in the
electron transfer problem, or of the solutions in the bond breaking. Due
to the non orthogonality the calculation of the overlap and of the hamiltonian matrix
elements between these solutions is rather difficult (although it is routinely done in GVB
programs). This is the first drawback. The second one is that in some cases this
combination will not satisfy all symmetry requirements. For instance if one combines spin-
polarized UHF solutions the result has no reason to be a spin eigenfunction. Finally one
does not see how to go simply beyond this step to treat later on the dynamical correlation
effects.

A second attitude consists in projecting the symmetry-broken solution on to the appropriate
symmetry-adapted subspace. The exact or approximate projected HF methods have been
the subject of an important litterature but the cost of the projections is non negligible
compared to a CI and they do not compare efficiently with the traditional avenue which
consists in respecting the symmetry from the beginning and performing CI.
The third attitude consists in performing the CI from one symmetry-broken HF solution,
using the corresponding MOs. The idea is that if one goes sufficiently close to Full CI
(which is independent of the choice of the MOs), the symmetry breaking of the intermediate
step will be unimportant. Usual CI codes are written assuming the equivalence between
and MOs and cannot be used for UHF solutions, but they might be exploited for singlet-
type symmetry breakings, in order to study the convergence of the symmetry. Unrestricted
Moller-Plesset (UMP) perturbative expansions have been written to the 4th order
(essentially for the study of doublet or triplet states in [39] and the convergence of UMPn
expansion for an UHF solutions in single bond breaking appears to be fantastically
poor, as shown by several authors [40–42]. The reasons for that poor convergence, i.e. of
that failure to restore symmetry, have been analysed in details [43] and are twofold. The
first one is due to the lack of meaning of the energy denominators in that problem, (a defect
which disappears if one uses an Epstein Nesbet zeroth-order Hamiltonian). The other one
is the strong coupling between the doubly excited and the singly excited determinants in
UHF SB solution, which only plays a role from the 4th order in energy and slows the
perturbation convergence. So far the HF symmetry-broken solutions appear as deserving to
be searched and analysed, since they tell us very instructive stories about the physical
trends acting on the electronic population, but they do not appear as a shortcut towards the
exact solution.

3. Symmetry breaking of the nuclear conformation
There is not much to say about this well accepted phenomenon. We would simply like to
stress on two peculiar aspects.