Stationary points are places on the potential energy surface (PES) with a zero gradient, i.e. zero first derivatives of the energy with respect to atomic coordinates. Two types of stationary points are of special importance to chemists. These are minima (reactants, products, intermediates) and first-order saddle points (transition states).
The two types of stationary points can be characterized by the curvature of the PES at these points. At a minimum the Hessian matrix (second derivatives of energy with respect to atomic coordinates) is positive definite, that is the curvature is positive in all directions. If there is one, and only one, negative curvature, the stationary point is a transition state (TS). Because vibrational frequencies are basically the square roots of the curvatures, a minimum has all real frequencies, and a saddle point has one imaginary vibrational ``frequency''.
Structure optimizations are most effectively done by so-called quasi-Newton-Raphson methods. They require the exact gradient vector and an approximation to the Hessian matrix. The rate of convergence of the structure optimization depends on anharmonicity of the PES and of the quality of the approximation to the Hessian matrix.
The optimization procedure implemented in statpt belongs to the
family of quasi-Newton-Raphsod methods [25]. It is
based on the restricted second-order method, which employes Hessian
shift parameter in order to control the step length and direction.
This shift parameter is determined by the requirement that the step
size should be equal to the actual value of the trust radius,
tradius, and ensures that the shifted Hessian has the correct
eigenvalue structure, all positive for a minimum search, and one
negative eigenvalue for a TS search. For TS optimization there is
another way of describing the same algorithm, namely as a minimization
on the "image" potential. The latter is known as TRIM (Trust Radius
Image Minimization) [26].
For TS optimizations the TRIM method implemented in statpt tries to maximize the energy along one of the Hessian eigenvectors, while minimizing it in all other directions. Thus, one ``follows'' one particular eigenvector, hereafter called the ``transition'' vector. After computing the Hessian for your guess structure you have to identify which vector to follow. For a good TS guess this is the eigenvector with negative eigenvalue, or imaginary frequency. A good comparison of different TS optimization methods is given in [27].
Structure optimizations using statpt are controlled by the keyword
$statpt to be present in the control file. It can be
set either manually or by using the stp menu of define. The
type of stationary point optimization depends on the value of
itrvec specified as an option within $statpt. By
default itrvec is set to 0, which implies a structure
minimization. A value itrvec > 0 implies a transition state
optimization using the eigenvalue-following TRIM algorithm, where the
index of the transition vector is specified by itrvec. Note,
that statpt orders eigenvalues (and eigenvectors) of the Hessian in
ascending order, shifting six (or five in the case of linear
molecules) zero translation and rotation eigenvalues to the end.
Note: this order differs from that used for vibrational frequencies in the control file, where rotational and translational eigenvalues are not shifted.
By default a structure optimization is converged when all of the following criteria are met:
threchange (default: 10-6a.u.),
thrmax\-displ (default: 10-3a.u.),
thrmaxgrad (default: 10-3a.u.),
thrrmsdispl (default:
5⋅10-4a.u.),
thrrmsgrad (defaul:t
5⋅10-4a.u.).
The default values for the convergence criteria can be changed using the stp menu of define. The necessary keywords are described in Section 15.2.15 below.
For structure optimization of minima with statpt as relaxation program just use:
jobex &
jobex -trans &