Page 214 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

FSGO HARTREE-FOCK INSTABILITIES OF HYDROGEN 197

where is the Hamiltonian of the isolated molecule and is the z component of the
dipole moment operator (in a.u.)

with the summation over nuclei and the i summation over electrons. stands for the
nuclear charges, while z denotes the z-coordinate.

In the RHF case with the doubly occupied orbital one obtains the mean value of the
energy:

Thus, we observe that when applying a finite electric field to a molecule, in addition to
the instability with respect to the internuclear distance R, one obtains an instability
connected to a change of the electric field strentgh F, as intuitively explained previously.
In a more general way, the possible instabilities can be rationalized as follows: at some
fixed values of the internuclear distance R and the FSGO exponent a the energy may be
viewed as afunction, , of the FSGO orbital position only. In view of the cylindrical
symmetry of the problem, the position is determined by the z coordinate of the FSGO
center, _ The formula for thefunction is:

where is a constant equal to:

which contains the mean value of the kinetic energy, the electron-electron and the nuclear
repulsions, as well as the nuclear dipole moment interaction with the electric field. As one
can see, if the field is positive, the energy goes to when

When expressing the nuclear attraction integrals in the FSGO basis, one has explicitly:

where Fo is the standard error function: