Molecular Normalized

Vatta M, Stetson SJ, Jimenez S, et al. Molecular normalization of dystrophin in the failing left and right ventricle of patients treated with either pulsatile or continuous flow-type ventricular assist devices. J Am Coll Cardiol. Mar3 2004 43(5) 811-817. 

Each molecular vibration factor in Equation (3) is a type of molecular time correlation function for the internal vibrational dynamics. In the harmonic approximation, i) and f) would reduce to the harmonic vibrational eigenstates and the qj would be the actual molecular normal modes. Then one has the simplification... 

A striking feature in the vibronic structure of the low temperature emission spectrum of ruthenocene is a repetitive pattern of clusters of bands. The pattern consists of the main peak and three side-peaks which are separated from the main peak by about 47 cm , 65 cm , and 112 cm . The first two side peaks are often not well resolved. The separation between the bands within a cluster is less than the energy of any normal mode. A possible explanation for the side-peaks could be phonon wings on the main 333 cm progression. However, the spectra obtained from organic glasses contain the same repetitive pattern. Thus the structure must arise from molecular normal modes and not from crystal lattice modes. 

Figure 13.16 Model geometry of intercalated Mg ions, optimised at the B3LYP/DZ level, (a) perspective view along the molecular normal, (h) view along a crystal plane. This geometry was used for the calculation of the Huang-Rhys factors visualised in Figure 13.15.

The diametrically opposite treatment to the phonon description in the previous sections is to regard the impurity and its immediate neighbours in the solid as an isolated unit, or a molecule. The vibrational motion of the atoms in such a molecule is represented by the molecular normal modes, with amplitudes Qr7 (having dimensions as in (2.5) of length x square root of the reduced mass M) and their-mo-menta-conjugates Pr7 (T is an irreducible representation of the molecular point group and 7 its component). The electron-vibration coupling enters the Hamiltonian as... 

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

These rules are quite obviously what one would expect on the basis of a first-order time-dependent perturbation theory approach to colhsional energy transfer in which the interaction potential is expanded only to first order in the molecular normal coordinates. There will certainly be many situations where such a simple picture is expected to fail. One may ask why these simple rules are even qualitatively successful in many cases. The answer may be that V-V energy-transfer processes which have been observed to date using laser techniques, have almost always been faster than V-T/R relaxation or they could not have been detected. (Among the obvious exceptions to this statement are and SO where laser... 

The global molecular normalization naturally remains valid. For a neutral molecule A with N electrons, this means that... 

As pointed out in Section IIB, it is possible to approach the lattice dynamics problem of a molecular crystal by choosing the cartesian displacement coordinates of the atoms as dynamical variables (Pawley, 1967). In this case, all vibrational degrees of freedom of the system are included, i.e., translational and librational lattice modes (external modes) as well as intramolecular vibrations perturbed by the solid (internal modes). It is then obviously necessary to include all intermolecular and intramolecular interactions in the potential function O. For the intramolecular part a force field derived from a molecular normal coordinate analysis is used. The force constants in such a case are calculated from the measured vibrational frequencies, The intermolecular part of O is usually expressed as a sum of terms, each representing the interaction between a pair of atoms on different molecules, as discussed in Section IIA. 

Here G(f2) is a molecular normalized distribution with respect to the solid angle which defines the average over all the possible orientations. Substituting the expression for the induced electric dipole jlind, one gets ... 

When a minimum is reached, the resulting equilibrium structure can be compared with experimental spectroscopic or electron diffraction and X-ray diffraction results. Isomerization energies can be calculated as differences between the strain energies of the isomers. From the second derivatives of the potential at the minimum, which must all be positive if the state is a true minimum, one can recalculate the molecular normal vibration frequencies as predicted by the molecular mechanics run, and from them, by a simple statistical mechanics calculation, the vibrational contribution to the... 

Neglecting the effect of the surrounding dielectric environment, the total reorganization energy A and its component along the molecular normal modes can be obtained from a simple series of computations on isolated molecules. Coropceanu et al. provided a valuable example of these computations having evaluated the local electron... 

Let us use the carbon dioxide molecule as an example. It should have 3N - 5 or four modes of vibration. We can represent each mode by the "direction" of the associated normal coordinate, that is, the directions in which each of the atoms move in the course of the vibration. Recall that a normal mode of vibration is the simplest motion of a system of particles, and that for a system vibrating in one mode, all the particles move in phase and at the same frequency. They reach their maximum points of displacement at the same instant, and they pass through their equilibrium positions at the same instant. If we were able to take a "freeze-frame" view of carbon dioxide vibrating purely in one of its normal modes, the directions the atoms move at the instant the particles are at their equilibrium positions would serve to describe the nature of the vibration. We could represent these directions of motion by arrows at each atom, and in fact, this is a very common way of representing molecular normal modes. 

With this machinery in hand, we look at the general Nd problem, vhere N may be identified vith some or all molecular modes. Since the molecule in the sth electronic state generally does not share the symmetries of the Ith state, we must consider the largest subgroup common to both the s and I state conformations to classify the molecular normal modes. As we have already noted the mass-weighted normal coordinates of the I electronic state manifold and of an arbitrary secondary manifold s are linearly related... 

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