\section{Geometry Optimization and Vibrational Frequency Analysis} \label{opt}

\PSIthree\ is capable of carrying out geometry optimizations (minimization
only, at present) for a variety of molecular structures using either analytic
and numerical energy gradients.  

When analytic gradients are available (see Table \ref{table:methods}),
\PSIthree\ will automatically generate and use so-called redundant internal
coordinates for carrying out the optimization.  These simple stretch, bend,
torsion, and linear bend coordinates may be read from the intco.dat file
or determined completely by distance criteria using the input geometry, and
written by to the intco.dat file.  By default, optimization is performed in
redundant internal coordinates regardless of whether cartesian or z-matrix
coordinates were given in the input.  However, optimizations cannot be
perfomed if any dummy atoms are used in the specification of the input
geometry.  

For methods for which only energies are available, \PSIthree\ will use
symmetry-adapted delocalized internal coordinates to generate geometrical
displacements for computing finite-difference gradients. The simple
coordinates can be linearly combined by hand or automatically.  The goal
is to form 3N-6(5) symmetry-adapted internal coordinates.  The automated
delocalized coordinates may work for low-symmetry molecules without
linear angles, but has not been extensively tested.  For both analytic-
and finite-difference-gradient optimizaiton methods, Hessian updates are
performed using the BFGS method.

In addition, simple internal coordinates may be generated from the z-matrix
given in the input.  The user may denote (using a dollar sign) which
coordinates should not be optimized.  However, this release of \PSIthree\
does not have a general mechanism to rigorously enforce geometrical
constraints, so the results should be examined carefully.

\PSIthree\ is also capable of computing harmonic vibrational frequencies
for a number of different methods using either analytic energy first or
second derivatives.  (At present, only RHF-SCF analytic second derivatives
are available.)  For those methods for which analytic gradients have been
coded (see Table \ref{table:methods}), \PSIthree\ will generate displaced
geometries along symmetrized, delocalized internal coordinates, compute
the appropriate first derivatives, and use finite-difference methods to
compute the Hessian.

For finite-difference procedures for vibrational frequency calculations,
the user should keep in mind that, for geometric displacements along
non-totally-symmetric coordinates, by definition, the molecular point
group symmetry will decrease.  As a result, any \PSIthree\ keywords in
the input file which depend on the number of irreducible representations
will be incorrect.  For example, if the user must specify the {\tt DOCC}
keyword in order to obtain the correct MO occupations for their system,
computation of non-symmetric vibrational modes via finite-differences
will necessarily fail.  The developers are working to further automate
the computation of vibrational frequencies such that the correlation
of irreducible representations between point groups can be handled
correctly.  This feature will be available in a future release of the
package.

The following keywords are pertinent for geometry optimizations and
vibrational frequency analyses:
\begin{description}
\item[JOBTYPE = string]\mbox{}\\
This keyword (described earlier in this manual) must be set to
{\tt OPT} for geometry optimizations and {\tt FREQ} for frequency analyses.
\item[DERTYPE = string]\mbox{}\\
This keyword (also described earlier) must be set to {\tt NONE} is only
energies are available for the chosen method and {\tt FIRST} if analytic
gradients are available.
\item[BFGS\_USE\_LAST = integer]\mbox{}\\ This keyword is used to specify
the number of gradient step for the BFGS update of the Hessian.  The default
is six.  
\end{description}
