Contributions to normal modes

These examples make it clear that we cannot expect to find characteristic bands for the CO and skeletal deformation modes in the polypeptide chain. These coordinates contribute to normal modes in ways specific to the structures in question. In fact, this means that this spectral region should be particularly sensitive to conformation, and that normal-mode analysis should provide a useful approach to the determination of structural differences. 

CONTRIBUTIONS OF PARTICULAR INTERNAL COORDINATES TO NORMAL MODES... 

In this contribution the concept of instantaneous normal modes is applied to three molecular liquid systems, carbon monoxide at 80 K and carbon disulphide at ambient temperature and two different densities. The systems were chosen in this way because pairs of them show similarities either in structural or in dynamical properties. The systems and their simulation are described in the following section. Subsequently two different types of molecular coordinates are used cis input to normal mode calculations, external, i.e. translational and rotational coordinates, and internal, i.e. vibrational coordinates of strongly infrared active modes, respectively. The normal mode spectra are related quantitatively to molecular properties and to those of liquid structure and dynamics. Finally a synthesis of both calculations is attempted on qualitative grounds aiming at the treatment of vibrational dephcising effects. 

Figure 20. Contribution of normal modes to rms fluctuations (A) as a function of frequency selected atoms and the radius of gyration are included.

Figure 1 The normal modes of motion for the three stretch modes of DCCH. (Adapted from T. A. Holme and R. D. Levine, Chem. Phys. Lett. 150 393 (1988).) The displacement of the atoms in each mode is shown by an arrow. Note that while all atoms contribute to all modes, the respective contributions do vary and the v, mode is almost a localized CH stretch. For recent studies of the overtone spectroscopy of HCCH and its isotopomers see J. Lievin, M. Abbouti Temsamani, P. Gaspard, and M. Herman, Chem. Phys. Lett. 190 419 (1995) M. J. Bramley, S. Carter, N. C. Handy, and 1. M. Mills, J. Mol. Spectrosc. 160 181 (1993) B. C. Smith and J. S. Winn, J. Chem. Phys. 89 4638 (1988) K. Yamanouchi, N. Ikeda, S. Tuschiya, D. M. Jonas, J. K. Lundberg, G. W. Abramson, and R. W. Field, J. Chem. Phys. 95 6330 (1991).)...

There are no explicit criteria that help to define a suitable amplitude // needed to describe the contribution of internal modes to normal modes and, then, to judge on the quality of this definition. However, there are properties that are implicitly assumed to be associated with amplitudes / . These can be formulated in the following way [20] ... 

Note from Fig. 5.1 that Za and zb are opposite in sign (the liquid model used here gives limfc o[ A(fc), ajB(fc)] = [—0.11, 0.99] when the norm of the eigenvector is normalized to unity). Thus the atoms A and B contribute to this mode with the out-of-phase fashion in terms of Eq. (5.146). Since the magnitude of Za gauges the efficiency of the atom a for participating in the orientational motion, the optical mode is evidently related to the rotational motion of the molecules. This fact is also obvious by noting that Eq. (5.148) depends on the moment of inertia of the molecule. 

Equations (15) and (16) define the so-called effective conjugation coordinate of ECC theory. The important conclusion that can be derived from Eqs. (11) and (17) is that only normal modes that contain an oscillation of the dimerization amplitude ( R vibration) can have relevant Raman cross sections. Then the two relevant lines of the Raman spectrum of polyenes are necessarily assigned to normal modes that involve a large contribution by the fl oscillation (FI modes in the ECC theory). More precisely, the treatment of the dynamical problem of polyenes in terms of ECC theory assigns the two strong Raman lines to two different combinations (in-phase and out-of-phase) of the R oscillation with C—H wagging vibration. 

How is a set of normal modes represented Normal modes describe positions of the beads of a polymer chain. The position of a polymer chain containing N + l monomers (or beads) may be written as a 3A + 3-dimensional vector, namely the chain conformation vector = (xq,yo>zo,x, zn)- This vector lists the position coordinates of the polymer s -I-1 beads, (vo, yo, zo) being the coordinates of the hrst bead of the polymer. Each normal mode may similarly be written as a iN + 3-dimension vector whose coordinates represent the extent to which each monomeric coordinate contributes to the mode. Normal modes, treated as vectors, are orthogonal to each other. 

Harmonic analysis (normal modes) at given temperature and curvature gives complete time behavior of the system in the harmonic limit [1, 2, 3]. Although the harmonic model may be incomplete because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it serves as a first approximation for which the theory is highly developed. This model is also useful in SISM which uses harmonic analysis. 

Two degrees of freedom lead to two modes of motion. These two modes of motion, synchronous and antisynchronous, are the normal modes of motion for this system. If only synchronous motion is excited, the antisynchronous mode will never contribute to the motion. The same is true for the pure antisynchronous mode (Fig. 5-2b) there will never be a synchronous conPibution. Under these conditions, but only under these conditions, energy does not pass from one mass to the other. 

One of the main attractions of normal mode analysis is that the results are easily visualized. One can sort the modes in tenns of their contributions to the total MSF and concentrate on only those with the largest contributions. Each individual mode can be visualized as a collective motion that is certainly easier to interpret than the welter of information generated by a molecular dynamics trajectory. Figure 4 shows the first two normal modes of human lysozyme analyzed for their dynamic domains and hinge axes, showing how clean the results can sometimes be. However, recent analytical tools for molecular dynamics trajectories, such as the principal component analysis or essential dynamics method [25,62-64], promise also to provide equally clean, and perhaps more realistic, visualizations. That said, molecular dynamics is also limited in that many of the functional motions in biological molecules occur in time scales well beyond what is currently possible to simulate. 

The entropy difference A5tot between the HS and the LS states of an iron(II) SCO complex is the driving force for thermally induced spin transition [97], About one quarter of AStot is due to the multiplicity of the HS state, whereas the remaining three quarters are due to a shift of vibrational frequencies upon SCO. The part that arises from the spin multiplicity can easily be calculated. However, the vibrational contribution AS ib is less readily accessible, either experimentally or theoretically, because the vibrational spectrum of a SCO complex, such as [Fe(phen)2(NCS)2] (with 147 normal modes for the free molecule) is rather complex. Therefore, a reasonably complete assignment of modes can be achieved only by a combination of complementary spectroscopic techniques in conjunction with appropriate calculations. 

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