## Chapter 13

Calculation of Vibrational Frequencies and Vibrational Spectra

Calculation of second derivatives of total energies leads to the molecular Hessian, which
enables prediction of vibrational frequencies and infrared spectra (within the harmonic
approximation) as well as the application of improved algorithms for geometry
optimization and transition state search.

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.