## Chapter 14

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.