The `aoforce` module calculates analytically harmonic vibrational
frequencies within the HF- or (RI)DFT-methods for closed-shell- and
spin-unrestricted open-shell-systems. Broken occupation numbers would
lead to results without any physical meaning. Note, that RI is only
used partially, which means that the resulting Hessian is only a (very
good) approximation to exact second derivatives of the RIDFT-energy
expression. Apart from a standard force constant calculation which
predicts all (symmetry allowed and forbidden) vibrational transitions,
it is also possible to specify certain irreps for which the
calculation has to be done exclusively or to select only a small
number of lowest eigenvalues (and eigenvectors) that are generated at
reduced computational cost.

Furthermore, the `Numforce` script allows the calculation of second
derivatives for all methods for which a program for analytic gradients
is available in `TURBOMOLE`, i.e. the main use of this script is the prediction of
vibrational spectra at the MP2 level and for excited states
using RI-CC2 or TDDFT.

If force constant calculations result in imaginary frequencies, molecular
distortions along these normal modes should lower the energy. To distort
the molecule, use the interactive module `vibration`, output of
the new coordinates is done to the general input file on `$newcoord`.

Vibrational frequencies also enable calculation of the molecular
partition function and thus prediction of thermodynamic functions at
temperatures other than 0 K and finite pressure (within the assumption
of an ideal gas and no coupling between degrees of freedom). These
functions can be obtained with the interactive module
`Freeh`, results are printed to standard I/O.

- Prerequisites
- Limitations
- Analysis of Normal Modes in Terms of Internal Coordinates
- Calculation of Raman Spectra
- Vibrational frequencies with fixed atoms using NumForce