Stationary points nature

As with the distillation ROMs, the profiles lying outside the MET may not be physically achievable, but the relevance of this global map is veiy important, and will be highlighted in subsequent sections. Note that it is also possible to identify stable, unstable, and saddle nodes, and each of these stationary points nature and location provide insight into the behavior of the curves (refer to Section 2.5.2). 

To identify the nature of stationary points on the potential energy surface. 

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. 

Another use of frequency calculations is to determine the nature of a stationary point found by a geometry optimization. As we ve noted, geometry optimizations converge to a structure on the potential energy surface where the forces on the system are essentially zero. The final structure may correspond to a minimum on the potential energy surface, or it may represent a saddle point, which is a minimum with respect to some directions on the surface and a maximum in one or more others. First order saddle points—which are a maximum in exactly one direction and a minimum in all other orthogonal directions—correspond to transition state structures linking two minima. 

Figure 4. CASSCF(8,8)/6-31G" optimized geometries of stationary points in the ring expansion of singlet phenylcarbene ( A -la).57 Bond lengths in angstroms, and bond angles in degrees. The ball-and-stick pictures are profiles that indicate the nonplanar nature of these species.

If both first and second derivatives vanish at the stationary point, then further analysis is required to evaluate the nature of the function. For functions of a single variable, take successively higher derivatives and evaluate them at the stationary point. Continue this procedure until one of the higher derivatives is not zero (the nth one) hence,/ (jc ),/"(jc ),. . ., /(w-1)(jc ) all vanish. Two cases must be analyzed ... 

For general nonlinear objective functions, it is usually difficult to ascertain the nature of the stationary points without detailed examination of each point. 

However, this particular experimental design only covered values of x3 up to 1.68 consequently, the saddle point is only predicted by the model and not exhibited by the data. This is the reason the lack-of-fit tests of Section IV indicated neither model 3 nor model 4 of Table XVI could be rejected as inadequately representing the data. As is apparent, additional data must be taken in the vicinity of the stationary point to confirm this predicted nature of the surface and hence to allow rejection of certain models. This region of experimentation (or beyond) is also required by the parameter estimation and model discrimination designs of Section VII. 

Table 1 NBO charges of the stationary points (black natural charge, red change to the previous stationary point B3LYP/6-31++G(d,p)...

Similarly, dihydrogen-bonded complexes of LiH with a variety of proton donors (e.g., HF, HCl, H2O, H2S, and NH3) have been studied by Kulkami [3] and Kulkami and Srivastava [4]. Some details of these studies are very interesting and show a dramatic dependence of dihydrogen bonding on the natme of proton donors and the level of theory. All the possible structures for these systems have been optimized at the HF/6-31++G(d,p) and MP2/6-31-H-G(d,p) levels, and the nature of stationary points has been examined by calculating their vibrational frequencies at the MP2/6-31++G(d,p) level. 

Also as already noted above, taking advantage of molecular symmetry can provide very large savings in time. However, structures optimized under the constraints of symmetry should always be checked by computation of force constants to verify tlierr nature as stationary points on die full PES. Additionally, it is typically worthwhile to verify that open-shell wave functions obtained for symmetric molecules are stable with respect to orbital changes that would generate other electronic states. 

Obviously, if as t —> oo the stationary solution dnj( )/d = 0 exists, indeed the asymptotic solution rij(oo) of (2.1.1) is one of the solutions n(- of the set (2.1.14). Here we have an example of a simple but very important case of a stable stationary solution. Other stationary points cannot be ascribed to the asymptotic solutions, i.e., n nj(oo), but they are also important for the qualitative treatment of the set of equations. Note that striving of the solutions for stationary values is not the only way of chemical system behaviour as —> oo another example is concentration oscillations [4, 7, 16]. Their appearance in a set (2.1.2) depends essentially on a nature of... 

Table 4.2 The location and nature of turning points, stationary points and points of inflection are given by the first, second and, where appropriate, third and fourth derivatives...

Interestingly, in the last row of Table 4.2 we see that a turning point may exist for which f 2 x) = 0. In such cases,/(3)(x) = 0, and the nature of the turning point is determined by the sign of the fourth derivative. An example of a function for which this latter condition applies is y=f x) (x - l)4. If there is any doubt over the nature of a stationary point, especially if the second derivative vanishes, it is always helpful to sketch the function ... 

A stationary point could of course be characterized just from the number of negative force constants, but the mass-weighting requires much less time than calculating the force constants, and the frequencies themselves are often wanted anyway, for example for comparison with experiment. In practice one usually checks the nature of a stationary point by calculating the frequencies and seeing how many imaginary frequencies are present a minimum has none, a transition state one, and a hilltop more than one. If one is seeking a particular transition state the criteria to be satisfied are ... 

Free energy second derivatives are mainly used to analyse the nature of stationary points on the PES, and to compute harmonic force constants and vibrational frequencies to perform such calculations in solution, one needs analytical expressions for Qa second derivatives with respect to nuclear displacements (the alternative of using numerical differentiation of gradients is far too much expensive except for very small molecules). 

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