Page 217 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

200 J. M. ANDRÉ ET AL.
At first sight, we are forced to solve this equation numerically, but its overall form allows
a qualitative insight into the number of solutions and their approximate values. For
example, one easily see that S represents a sum of two identical quasi-atomic (one-
dimensional) functions each centered on the corresponding hydrogen nucleus. The
functions are quite similar to Gaussian functions, but they differ by their one-
dimensionality and by a different radial dependence. Indeed, instead of the usual
exponential behaviour, one has the function that is positive, even with respect to
z, and, as it seen from the previous equation, has its maximum at while it vanishes
for large z.
This information is sufficient to analyze the qualitative behaviour of Indeed, two
limiting cases may be considered.

For one limiting case (small values of i.e., close to equilibrium distances), the
function can be easily evaluated. By putting it looks like a single one-dimensional
orbital centered in the symmetry center of the molecule. As one can see for
there is only one solution, of the equation and it corresponds obviously to a
single minimum of the energy (symmetric solution) as seen in Figure 3. For very large
values of the equation cannot be longer satisfied and does not attain negative
values. This corresponds to the energy changing monotonically qualitatively. The energy
minimum at is clearly unstable when F increases. As one sees from Figure 4 for
moderate values of F, one should observe two values satisfying the equation. One of
them corresponds to a minimum and the other to a maximum of the energy. One example
of a single minimum is given in Figure 3 for the cases, and 0.25 a.u. Thus,
there is a certain critical electric field value for which the energy curve changes
qualitatively from the one having a single minimum to that with no minimum at all. From
Figure 3, one easily see the existence of one and no minima in the curve
according to the strength of the field. For large values of F, there is no root in the S-
function and no minimum is found for the This is the case for in Figure 3.
Since the coordinate origin has been placed at the center of the molecule, the contribution
of the nuclei to the dipole moment is 0 and the total dipole moment is equal to (in
a.u.). At large values of the energy of the molecule mainly comes from the interaction
of the dipole moment with the electric field and therefore has the asymptotic form
From Figure 3, it is also seen that the energy changes linearly with for large For
small values, one has a strong influence of the nuclei (through the first term of the
equation). Also, it is easily proved that the asymptotic behaviour of the energy
comes from its linear dependence of for large values of this variable and that its
asymptotic slope is 2F.
The other limiting case corresponds to the limit of large values of In order to remain
concise, this has not been illustrated. For one finds three values of for which
The first one is _ and the remaining two are close to and
(the two positions of the nuclei). They correspond to the broken symmetry behaviour
analyzed in section 2. For very large values of no value of satisfies the equation,
and the function does not attain negative values. This means that the energy has
no extremum.