Stationary point computation

Because of the nature of the computations involved, firequency calculations are valid only at stationary points on the potential energy surface. Thus, frequency calculations must be performed on optimized structures. For this reason, it is necessary to run a geometry optimization prior to doing a frequency calculation. The most convenient way of ensuring this is to include both Opt and Freq in the route section of the job, which requests a geometry optimization followed immediately by a firequency calculation. Alternatively, you can give an optimized geometry as the molecule specification section for a stand-alone frequency job. 

Verifying that the stationary point is a transition structure, and computing its zero-point energy. 

Many problems in computational chemistry can be formulated as an optimization of a multidimensional function/ Optimization is a general term for finding stationary points of a function, i.e. points where tlie first derivative is zero. In the majority of cases the desired stationary point is a minimum, i.e. all the second derivatives should be positive. In some cases the desired point is a first-order saddle point, i.e. the second derivative is negative in one, and positive in all other, directions. Some examples ... 

Tlie function to be optimized, and its derivative(s), are calculated with a finite precision, which depends on the computational implementation. A stationary point can therefore not be located exactly, the gradient can only be reduced to a certain value. Below this value the numerical inaccuracies due to the finite precision will swamp the true functional behaviour. In practice the optimization is considered converged if the gradient is reduced below a suitable cut-off value. It should be noted that this in some cases may lead to problems, as a function with a very flat surface may meet the criteria without containing a stationary point. 

There have a number of computational studies of hypothetical RMMR species [10-13, 40, 411. The simplest compounds are the hydrides HMMH. Some calculated structural parameters and energies of the linear and trans-bent metal-metal bonded forms of the hydrides are given in Table 1. It can be seen that in each case the frans-bent structure is lower in energy than the linear configuration. However, these structures represent stationary points on the potential energy surface, and are not the most stable forms. There also exist mono-bridged, vinylidene or doubly bridged isomers as shown in Fig. 2... 

Table 13-11. Computed energies [kcal/mol] of stationary points for the activation of H2 by FeO+ (Dn relative to separated FeO+ (6S+) + H2).

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

If predk = 0, then no changes Ax within the rectangular trust region (8.58) can reduce PI below the value P1(0, x ). Then x is called a stationary point of the nonsmooth function P, that is, the condition predk = 0 is analogous to the condition V/(x ) = 0 for smooth functions. If predk = 0, the PSLP algorithm stops. Otherwise predk > 0, so we can compute the ratio of actual to predicted reduction. 

The computer, going through a multidimensional search [see problem 3, Chem. Eng. Set, 45, 595-614 (1990)] came up with the arrangement of Fig. P10.6, which the authors claim is a LOCAL optimum, or a STATIONARY POINT. We are not interested in LOCAL optima, if such things exist. We are interested in finding the GLOBAL optimum. So with this in mind,... 

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