Normal-mode frequencies

Wall M R and Neuhauser D 1995 Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. 

Many thermodynamic quantities can be calculated from the set of normal mode frequencies. In calculating these quantities, one must always be aware that the harmonic approximation may not provide an adequate physical model of a biological molecule under physiological conditions. 

The vibrational frequencies are obtained from the force-field using the second derivatives of the potential energy with respect to displacements of the atoms (in a more elaborate version of the argument used in Section 8.2 for the one-dimensional chain) the calculation is analogous to the calculation of normal mode frequencies for molecules. The resulting vibrational frequencies can be compared with those... 

The vibrational spectrum of methylguanine-methylcytosine (GC) complex consists of 99 normal modes frequencies. Differently from the AT base pair, in the GC complex the normal modes of the two bases are coupled together, thus an analysis of the shift relatively to the isolated bases is extremely complicated. This stronger coupling can possibly he ascribed to the presence of three h-bonds, rather than two as in AT. However, we tentatively discuss some significant shifts. 

Diagonalization, of General Quantum Eigenvalues or Classical Normal Mode Frequencies from a Small Number of Residues or a Short-Time Segment of a Signal. I. Theory and Application to a Quantum-Dynamics Model. 

The vibrational frequencies of isotopic isotopomers obey certain combining rules (such as the Teller-Redlich product rule which states that the ratio of the products of the frequencies of two isotopic isotopomers depends only on molecular geometry and atomic masses). As a consequence not all of the 2(3N — 6) normal mode frequencies in a given isotopomer pair provide independent information. Even for a simple case like water with only three frequencies and four force constants, it is better to know the frequencies for three or more isotopic isotopomers in order to deduce values of the harmonic force constants. One of the difficulties is that the exact normal mode (harmonic) frequencies need to be determined from spectroscopic information and this process involves some uncertainty. Thus, in the end, there is no isotope independent force field that leads to perfect agreement with experimental results. One looks for a best fit of all the data. At the end of this chapter reference will be made to the extensive literature on the use of vibrational isotope effects to deduce isotope independent harmonic force constants from spectroscopic measurements. 

In Equation 4.71 the individual qvib s have been specified qVib(v ) to indicate that these partition functions depend on the normal mode frequencies. It is interesting to note that the partition function for translation, which is usually considered in terms of the problem of the particle in a three dimensional rectangular box, is, itself a product of three partition functions one for motion in the x dimension, one for y, etc. 

The second product is over the 3N—6(3N—5) normal mode frequencies of the ideal gas harmonic molecule to which Equation 4.78 applies. Thus the product over vibrations Equation 4.90 is indeed the quantum mechanical contribution to the molecular partition function for the ideal gas. 

The sum is called the partially Fourier transformed dynamical matrix, which depends only on Q, and For each wave vector Q the normal mode frequencies of the crystal can be found by setting the secular determinant equal to zero ... 

For this case, the primary change that is observable in the IR spectrum is due to changes in the vibrahonal frequencies of the probe molecule due to modificahons in bond energies. This can lead to changes in bond force constants and the normal mode frequencies of the probe molecule. In some cases, where the symmetry of the molecule is perturbed, un-allowed vibrational modes in the unperturbed molecule can be come allowed and therefore observed. A good example of this effect is with the adsorption of homonuclear diatomic molecules, such as N2 and H2 (see Section 4.5.6.8). 

Normal Mode Frequency (cm x) Atom Eigenvector (A) x Y Z Mode Type... 

There is a one to one correspondence between the imperturbed fi equencies CO, C0j j = 1,. .., N,. .. appearing in the Hamiltonian equivalent of the GLE (Eq. 3) and the normal mode frequencies. The diagonalization of the potential has been carried out exphcitly in Refs. 88,90,91. One finds that the imstable mode frequency A is the positive solution of the Kramers-Grote Hynes (KGH) equation (7). This identifies the solution of the KGH equation as a physical barrier fi-equency. 

Figure 2.7. Experimental frequencies [56] and calculated [HF/6-31G(d,p)] normal modes, frequencies (x 0.8929), and IR intensities (km/mol) of ammonia.

As usual, the Qk and w fc = 1,. .., 31V-/- 7 are the normal coordinates and the corresponding normal-mode frequencies. The kinetic energy is then... 

The effects of coupling of the DTO and RB units in not only one- but also three-dimensional arrays are discussed below and molecular weight trends illustrated. A fundamental connection between relaxation times and normal mode frequencies, shown to hold in all dimensions, allows the rapid derivation of the common viscoelastic and dielectric response functions from a knowledge of the appropriate lattice vibration spectra. It is found that the time and frequency dispersion behavior is much sharper when the oscillator elements are established in three-dimensional quasi-lattices as in the case of organic glasses. 

The partition functions Z and Za are also easily evaluated in the harmonic approximation from products of the stable normal mode frequencies at the sad-... 

This means that, in the harmonic approximation and to lowest order in h, the classical transition rate is multiplied by a factor depending only on the sums of squares of the normal mode frequencies at the saddle point and minimum ... 

The solution yields [1], beside the final set of normal modes frequencies, the final nonadiabatic form of the fermionic Hamiltonian, that is ... 

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