Barrier imaginary frequency

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

Except for reactions with low barriers (i.e. <10 kcal/mol at 7 = 300 K), or at high temperatures, the quantity /k T is large, and the last series can be neglected. The tunnel correction is then given completely in terms of the magnitude of the imaginary frequency. For small values of the first term may be Taylor expanded to give... 

Note that the imaginary frequency associated with such a barrier is zoo. For a bound mode with frequency 0)vli, an analogous approximation would be... 

All of the theoretical methods that allow for geometry optimization predict the bowl-shaped conformation of 8 to predominate over the planar alternative. However, the calculated energy differences between the two conformations differ drastically and span the range of 0.8 to 84 kcal/mol. Moreover, the calculated energy difference between the plants and bowl-shaped conformers is quite important since it represents the barrier for bowl-to-bowl inversion if (a) planar corannulene represents the transition state (TS) for the process, and (b) the entropic term in AG is negligible. Condition (a) seems to be fulfilled, since vibrational frequency calculations give one and only one imaginary frequency for the planar structure... 

FIGURE 12. Transition structure for the isomerization of peroxynitrous acid to nitric acid optimized at the B3LYP/6-311- -G(d,p) level of theory. Classical reaction barrier, TS total energy and imaginary frequency (vectors represented by the arrows) are 40.8 kcalmoL (with respect to ds-GS HO—ONO), —280.86143 au and 690i cm, respectively... 

FIGURE 30. B3LYP/6-31 +G(d,p)-optimized transition stmctures for the oxidation of trimethylamine by hydrogen peroxide (a), with one water molecule (b), and two water molecules (c). The values of the imaginary frequency and the barriers [A(E+ZPVE) ] are at the B3LYP/6-31+G(d,p) level. The barriers given in parentheses are at the B3LYP/6-31+G(d) level ... 

The optimized structure with symmetry, shown in Fig. 6(a), has one imaginary mode. The vibrational mode with an imaginary frequency is also depicted in Fig. 6(b). The imaginary mode will move the Dlh structure to C2h one. Hence the D2h symmetry structure is a saddle point between the stable C2 , structures. The energy barrier AE between the C2h minima is 0.2 meV. It should be noted that AE... 

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