Page 129 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

114 J. P. MALRIEU AND J. P. DAUDEY

3.1. BEHAVIOUR ON THE CRITICAL REGION
The dominant practice in Quantum chemistry is optimization. If the geometry optimization,
for instance through analytic gradients, leads to symmetry-broken conformations, we
publish and do not examine the departure from symmetry, the way it goes. This is a pity
since symmetry breaking is a catastrophe (in the sense of Thom's theory) and the critical
region deserves attention. There are trivial problems (the planar three-fold symmetry
conformation of is a saddle point between the two pyramidal equilibrium
conformations). Other processes appear as bifurcations ; for instance in the electron transfer
problem, the energy of the rectangular system as a function of the
intersystem distance R and of the relaxation of the intra system coordinate 6 from the mean
geometry (half-way between those of behaves as a typical bifurcation [33].
The potential surface presents a symmetrical delocalized hole at short R and the symmetrical
valley for larger R values becomes a symmetrical crest beyond a critical value Rc.
Beyond R there are two symmetry-broken valleys corresponding to
C
(where A is a molecule). We have not yet met any problem which would exhibit a
multi-stable symmetry breaking where for a certain domain, one would have a co-existence
of a symmetrical valley and two symmetry broken valleys. A bistability region has been
shown to exist in the isocele triangle, between the and the neutral F states.
It is likely that the above schematic view of the potential energy surface would be relevant
for the mixed valence Donnor Acceptor Donnor (DAD) architectures such as :

which would be symmetrical and neutral for larger values and ionic and symmetry-broken
for smaller values of R, with a possible domain of multistability

3.2. ISOMORPHISMS AND INTERFERENCES BETWEEN ELECTRONIC AND
NUCLEAR RELAXATIONS

The conformational symmetry breaking in electron transfer problem is governed by the
ratio between the nuclear relaxation energy (i.e. energy stabilization when going from the
symmetrical to the localized equilibrium geometry) and the amplitude of the
electron transfer. It is therefore governed by the same inequalities that the HF symmetry
breaking for the same problem in the symmetrical conformation, the nuclear relaxation
replacing the electronic relaxation [33J.
Of course the conformational symmetry breaking may appear or disappear depending on
the level of sophistication of the computation, which may unduly favor one term of the
crucial (relaxation/resonance) ratio. As an example we would like to mention an open
controversy on the geometry of Does the surface present a double well

or a single symmetric well for a linear structure with a
delocalized hole ? A simular interference between electronic and geometrical symmetry
breaking occured for the allyl radical [46]. It may concerns the weak resonance between
two excitations [47]. It seems necessary to insist on the possible interference
between the electronic symmetry breaking of the approximate wave function and the