Vibrational frequencies with fixed atoms using

The `NumForce` script provides with the option `-frznuclei`
a possibility to do a vibrational analysis with fixed atoms.
The atoms for which the cartesian coordinates should frozen have to be marked
in `$coord` with a "f" behind the atom type.
The frozen coordinates will be skipped during the numerical evaluation
of the force constant matrix; instead all off-diagonal elements of the
force constant matrix which refer to one or two frozen coordinates will be set
to zero, while the diagonale elements for the frozen coordinates will be set
to an arbitrarly chosen large value.

This feature is mainly intended to allow for a vibrational analysis in embbeded cluster calculations e.g. for defects in ionic crystals. The vibrational analysis uses a kind of ``frozen phonon'' approximation which corresponds to setting the masses of the fixed atoms to infinity, i.e. decoupling the fixed atoms mechanically from the ``mechanically active'' subsystem. The resulting vibrational frequencies will thus only provide good approximations to the true (harmonic) frequencies for such modes for which the mechanical coupling to the embedding environment is negligible. In particular the frequencies of stretch modes which involve bonds between the ``mechanically active'' subsystem and atoms with frozen coordinates will be strongly affected by this approximation.

Note:

- The
`-frznuclei`is not compatible with the polyhedral difference algorithm. It can only be used with central differences which should be enforced with the`-central`option. - If the option
`-frznuclei`is switched on, the program assumes that the constraints enforced by fixing coordinates remove the six external degrees of freedom for on overall rotation or translation of the system and therefore the hessian matrix is not projected onto the subspace of internal coordinates. Fixing the coordinates of only one or two atoms might does lead to some artifical small, but non-zero frequencies. - Zero-point vibrational energies calculated with the
`-frznuclei`option are only meaningful for comparison of systems with the same mechanically active atoms and similar embedding, as the contributions from the frozen coordinates are not included.