Internal frequencies

The second triad element was determined from international frequencies for the basic initiating events. One fault tree for sphere rupture was constructed and evaluated using the fault tree program FTW (1991). 

In Eq. (77), x = h(o/2kg T is the reduced internal frequency, q = EJhoi the reduced solvent reorganization energy, p = hElha> the reduced electronic energy gap and / (z) the modified Bessel function of order m. The quantity S is a coupling parameter which defines the contribution of the change in the internal normal mode ... 

The Kieffer model correctly predicts the systematic change of the reduced partition functions of various minerals with structure, as indicated by Taylor and Epstein (1962). For anhydrous sihcates, the decrease in the sequence framework-chain-orthosilicate reflects the decreasing frequency of antisymmetric Si-O stretching modes. The internal frequencies of the carbonate ion give a high reduced partition function at all T. The value for rutile is low because of the low frequencies of the Ti-0 modes (Kieffer, 1982). 

TABLE 20. Adiabatic internal frequencies of cyclopropane and some simple hydrocarbons from Reference 163 ... 

M. E. Tobar et al., Joint Meeting of the European Frequency and Time Forum and the ieee International Frequency Control Symposium 573, 1999... 

R. W. Cernosek, SJ. Martin, A.R. Hillman and H.L. Bandey, 1997 IEEE International Frequency Control Symposium (Orlando, FI, May 28-30, 1997). 

L. Mitchell, M. A. Proc. of the 1993 IEEE International Frequency Control Symposium IEEE New York, 603-608 (1993). 

By modifying the classical theory and using a quantum-mechanical model for the critically activated complex it is possible to extend the detailed theory [Eqs. (XI.8.3) and (XI.8.3a)] in such a way as to include consideration of structural changes on the internal frequencies. In Eq. (XI.8.3) V represents a weighted average of the internal fre(i[uencies of a species that has the configuration of the transition state and represents the equilibrium constant for the equilibrium between this transition state and normal molecules. and respectively represent the standard... 

The transition-state theory itself as given in Eqs. (XI.8.4) and (XI.8.4a) differs from the previous theories in its attempt to eliminate the uncertainty inherent in the frequency factor v by the replacement with the universal frequency factor kT/h, We can, for example, derive Eq. (XI.8.4a) from Eq. (XI.8.3a) if we are willing to say that the partition function Q= = for the transition-state complex can be factored into a product of partition functions (including the internal vibrations) and that one of the internal frequencies, let us call it Vej corresponds to the motion over the top of the barrier. If then this frequency Ve kT/h so that the corresponding vibrational partition function qe can be expanded and approximated as = kT/hve and finally that v — vej we obtain Eq. (XI.8.4a), where refers to the partition function for the transition-state molecule from which one degree of vibration has been factored. X, and are then... 

In recent years, theoretical calculations of the molecular vibrations that characterize solid polymers have become sophisticated and complete enough to permit detailed comparison with measured spectra. These calculations include the internal frequencies of the basic repeat unit, skeletal modes involving cooperative or phased vibrations of an entire chain, and interchain vibrations. The skeletal modes are determined by intrachain forces and chain conformation interchain modes depend upon crystal symmetry and spacing and upon interchain forces (usually much weaker than intrachain forces). 

Berkowitz and Wahl9 have reviewed the experimental and theoretical estimates of the dissociation energy of molecular fluorine. The Raman and far-i.r. spectra of crystalline F2 show that in this state the element resembles 02 more closely than it does the other halogens.10 The intermolecular forces, in particular, are extremely weak, as exemplified by the small shifts of the internal frequencies from their gas-phase values, the absence of observable factor-group splitting of the fundamental and overtones, and the low value of the external (lattice) vibrations. 

We assume a parametric region where the internal frequencies of the oscillator and of the spin yield the fastest time scales,... 

Conners Advantage (Frequency Precision). The dispersive instrument depends on calibration (polystyrene at 1601 cm 0, and the ability of gears and levers to move slits and gratings in a reproducable fashion. The FTIR carries its own internal frequency standard, usually a He-Ne gas laser, that serves as the master timing clock, tracking mirror movement and frequency calibration to a precision and accuracy of better than 0.01 wavenumbers (cm ). 

R.A. McGill, T.E. Mlsna and R. Mowery, in IEEE International Frequency Control Symposium, 1998, pp. 630. 

Once the internal force constant kn (see Eq. 45) and the internal mass mn (see Eq. 54) have been derived, the internal frequency wn is given by Eq. (55) ... 

Clearly, the best correlation pattern complying exactly with the expected Lorentzian form is obtained in the case of the AvAF amplitudes in connection with a comparison of frequencies with adiabatic internal frequencies (Og. Adiabatic internal modes, the amplitude definition of Eq. (64) and the force constant matrix f as a suitable metric for comparison provide the right ingredients for a physically well-founded CNM analysis. 

Figure 3. Frequency uncertainty test for benzocyclobutadiene according to FiF/6-31G(d,p) calculations. The correlation diagrams correspond to correlations between normalized amplitudes and frequency differences = cOp - with cOp being a normal mode frequency of a molecular fragment (t)p and 0) being a normal mode frequency. Amplitudes AvAF, AvPF, CvAF, CvPF are employed in connection with adiabatic internal frequencies CO3 and c-vector frequencies using a nonredundant set of internal coordinates (a - d) or a strongly redundant set (e - h). In all cases, points that have A/ = 0 for all tests within a given row of diagrams are removed.

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... 

In this situation, an attractive alternative is provided by the adiabatic internal frequencies. For example, the McKean relationship between C-H stretching frequencies and equilibrium C-H bond lengths can easily be reproduced with the help of adiabatic CH stretching frequencies calculated at the HF/6-31G(d,p) level of theory as is shown in Figure 7. The r coefficient obtained is 0.998, which is clearly better than the r coefficient for the correlation of the experimental frequencies (0.991, Figure 6). Eq. (71) gives the relationship between C-H equilibrium bond lengths and internal frequencies, which can be used... 

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 13 gives the correlation between dissociation energy [48] and adiabatic internal frequency for a number of CH bonds. It can be expressed by Eq. (81)... 

Altindal A., Patel R., Zhou R., Josse R, Ozturk Z. Z., and Bekaroglu O., Soluble dode-cylsulfanylphthalocyanines as sensitive coatings for chemical sensors in gas phase. Proceedings of the 1998 IEEE International Frequency Control Symposium, pp. 676-684, Pasadena, CA, USA, 27-29 May 1998. 

Josse F., Bender F., Cernosek R. W., and Zinszer K., Guided SH-SAW sensors for liquid-phase detection, presented at IEEE International Frequency Control Symposium and PDA Exhibition, June 6-8, 2001, Seattle, Washington, USA. 

Morgan D. R, History of SAW devices, presented at IEEE International Frequency control symposium. May 27-29, 1998, Ritz-Carlton Hotel, Pasadena, California. 

Based on "Resonant Piezoelectric Devices as Physical and Biochemical Sensors , F. Josse, and R. W. Cemosek, jHesented at the 2002 IEEE International Frequency Control Meeting as a Symposium Tutorial, New Orleans, LA. 

R. Lucklum, P. Hauptmann, Thin Film Shear Modulus Determination with Quartz Crystal Resonators A Review, IEEE International Frequency Control Symposium, (2001) 408-17. 

Prestage, J.D. Tjoelker, R.L. Dick, G.J. Maleki, L. Progress report on the improved linear ion trap physics package. Proc. of the 1995 IEEE International Frequency Control Symposium, San Francisco, May 31-June 03 1995, 82-85. 

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