Frequencies - adiabatic

Adiabatic frequencies of 1 are compared in Table 20 with those of some other hydro-carbons The adiabatic CC frequency is about 80 and the adiabatic CH stretching frequency about 100 cm larger than the corresponding values for cyclohexane. Compared to ethene, adiabatic CH stretching frequencies are almost identical, which is in line with the high dissociation energy of the CH bond of 1 (see Section V. E). The same observation has been made by McKean using isolated CH frequencies obtained by appropriate deuteration of 1 ... 

In addition to the adiabatic frequency for the motion perpendicular to the RPO one can also determine the stability frequency or what is known as a characteristic eigenvalue (61)... 

The calculated adiabatic frequencies reveal that the CO stretching modes increase by 70 cm i upon geminal F-substitution while the uncoupled OO stretching mode frequency decreases by just 44 cm. Compared to oxirane (adiabatic CO stretching to 1130 cm- ), the adiabatic CO stretching frequency of 1 (1121 cm". Table 3) is normal while it is considerably increased for 2 (1189 cm" ). 

Sym exp. Difluorodioxirane (2) CCSD(T),sc. Characterization Dioxirane (1) CCSD(T) Characterization Adiabatic Frequencies 2 1 ... 

The adiabatic internal frequencies calculated for the three benzynes are listed in Table 10 together with the associated internal coordinates. They have to be compared with the corresponding adiabatic frequencies of benzene obtained at the HF/6-31G(d,p) level of theory CC 1406, HC 3348, CCC 997, HCC in-plane 1441, CCCC 653, and HCCC out-of-plane 969 cm T... 

Figure 7 confirms the McKean relationship [30] and, furthermore, suggests that calculated adiabatic internal frequencies are as useful or even more useful as the measured "isolated" C-H stretching frequencies. However, the real advantage of adiabatic frequencies will become obvious if one attempts to set up McKean relationships also for other bonds. 

Ab initio frequencies of normal vibrational modes and, by this also, adiabatic frequencies suffer from the harmonic approximation used in the calculation. Even when applying efficient scaling procedures, there is no guarantee that ab initio frequencies accurately reproduce the exact fundamental frequencies of the experiment. Therefore, one has to ask whether the adiabatic internal frequencies might not be much more meaningful if they would be based on experimental frequencies rather than frequencies calculated within the harmonic approximation. 

Figure 11. Comparison of directly calculated adiabatic frequencies (HF/6-31G(d,p) calculations) and adiabatic frequencies derived from experimental vibrational frequencies (see text).

Once adiabatic modes are known either from calculations or experimental data, adiabatic frequencies can be used to characterize chemical bonds. For example, it is easy to verify a McKean relationship [30] between adiabatic CH or CC stretching frequencies and the corresponding bond lengths (Section 11). It can... 

There is a close connection between the adiabatic frequency and the stability frequency (cf. Eq, 19) of a periodic orbit. Since at the pods the time average of is exactly d Em/du one finds that the adiabatic stability frequency is just the first order Magnus approximation to the exact stability frequency.From a practical point of view it is actually easier to compute the stability frequency. Finding the adiabatic frequency implies actual construction of the (u,v) coordinate system. 

The vibrationally adiabatic approximation is coordinate dependent. However one may formulate the quantal adiabatic approximation using the coordinate system defined by pods. One can then show that up to terms of order that if at Uq there exists a pods v/ith action (n+l/2)h, and energy (0 ) then also quantally Uq is an adiabatic barrier or well of the n-th vibrationally adiabatic potential energy surface.Furthermore, the quantal adiabatic frequency for motion perpendicular to the pods is excellently approximated by the adiabatic frequency of the pods. Finally, one can show, that to order fi, all quantal nonadiabatic coupling elements are identically zero at Uq In other words, one should expect that just as in the classical case, the coordinate system defined by the pods is also quantally, the optimal coordinate system for the vibrationally adiabatic approximation. One should also expect that the semiclassical barrier and well energies and frequencies computed via the pods are an excellent approximation to the quantal energies and frequencies., ... 

Here we assume that the final state /> in Eq. (1) belongs to the electronic ground state d). If we take all vibrations to be harmonic with adiabatic frequencies co, the energies corresponding to these vibronic states are... 

F. Vewinger, J. Appel, E. Figueroa, And A. I. Lvovsky. Adiabatic frequency conversion of optical information in atomic vapor. Optics Letters 2007 Oct 1 32(19) 2771-2773. 

FIGURE 7.4 The effective frequency 0(r) in the integrand of Equation (6.170) for the two transversal modes y = l(n3 = 1), 3(ni = 1) in HO2 radical. Solid lines show the results of numerical solutions of Equations (6.191) and (6.193). Dotted lines represent the adiabatic frequencies. (Taken from Reference [45] with permission.)... 

Adiabatic internal modes can be defined for equilibrium points on the PES as well as for all points along the RP where in the latter case it is required that the harmonic part of the energy in equation (28b) is minimized with regard to displacements in the 3K — L — 1 )-dimensional vibrational space while relaxing all internal parameters but one. In this way, generalized adiabatic modes aifs) together with the corresponding adiabatic frequencies a>k s) and force constants kkis) are calculated where k denotes a particular internal coordinate Rk. 

The analysis of adiabatic force constants and adiabatic frequencies associated with the internal coordinates that describe the reaction complex provides direct insight into how the reaction complex changes along s. 

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