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## General Information

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 [30]. 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) [31].

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 [32].

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:

• the energy change between two optimization cycles drops below the value given by threchange (default: 10-6 a.u.),
• the maximum displacement element drops below the value given by thrmax\-displ (default: 10-3 a.u.),
• the maximum gradient element drops below the value given by thrmaxgrad (default: 10-3 a.u.),
• the root mean square of the displacement elements drops below the value given by thrrmsdispl (default: 5⋅10-4 a.u.),
• the root mean square of the gradient elements drops below the value given by
thrrmsgrad (defaul:t 5⋅10-4 a.u.).

The default values for the convergence criteria can be changed using the stp menu of define. The necessary keywords are described in Section 18.2.16 below.

For structure optimization of minima with statpt as relaxation program just use:

jobex &
TS optimizations are performed by the jobex invokation:
jobex -trans &

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