Low-frequency modes

In light of tire tlieory presented above one can understand tliat tire rate of energy delivery to an acceptor site will be modified tlirough tire influence of nuclear motions on tire mutual orientations and distances between donors and acceptors. One aspect is tire fact tliat ultrafast excitation of tire donor pool can lead to collective motion in tire excited donor wavepacket on tire potential surface of tire excited electronic state. Anotlier type of collective nuclear motion, which can also contribute to such observations, relates to tire low-frequency vibrations of tire matrix stmcture in which tire chromophores are embedded, as for example a protein backbone. In tire latter case tire matrix vibration effectively causes a collective motion of tire chromophores togetlier, witliout direct involvement on tire wavepacket motions of individual cliromophores. For all such reasons, nuclear motions cannot in general be neglected. In tliis connection it is notable tliat observations in protein complexes of low-frequency modes in tlie... 

A particular advantage of the low-mode search is that it can be applied to botli cyclic ajic acyclic molecules without any need for special ring closure treatments. As the low-mod> search proceeds a series of conformations is generated which themselves can act as starting points for normal mode analysis and deformation. In a sense, the approach is a system ati( one, bounded by the number of low-frequency modes that are selected. An extension of th( technique involves searching random mixtures of the low-frequency eigenvectors using Monte Carlo procedure. 

Although the Lanczos is a fast efficient algorithm, it does not necessarily give savings in memory. To save memory a number of techniques divide the molecule into smaller parts that correspond to subspaces within which the Hessian can be expressed as a matrix of much lower order. These smaller matrices are then diagonalized. The methods described below show how one then proceeds to achieve good approximations to the true low frequency modes by combining results from subspaces of lower dimension. 

In another promising method, based on the effective Hamiltonian theory used in quantum chemistry [19], the protein is divided into blocks that comprise one or more residues. The Hessian is then projected into the subspace defined by the rigid-body motions of these blocks. The resulting low frequency modes are then perturbed by the higher... 

Coupling to these low-frequency modes (at n < 1) results in localization of the particle in one of the wells (symmetry breaking) at T = 0. This case, requiring special care, is of little importance for chemical systems. In the superohmic case at T = 0 the system reveals weakly damped coherent oscillations characterised by the damping coefficient tls (2-42) but with Aq replaced by A ft-If 1 < n < 2, then there is a cross-over from oscillations to exponential decay, in accordance with our weak-coupling predictions. In the subohmic case the system is completely localized in one of the wells at T = 0 and it exhibits exponential relaxation with the rate In k oc - hcoJksTY ". 

Next we discuss the effect of deuteratlon on low frequency modes Involving the protons> Because of the anharmonlc variation of the energy as a function of tilt angle a (Fig. 4b), the hindered rotations of H2O and D2O turn out to be qualitatively different. The first vibrational excited state of H2O Is less localized than that of D2O, because of Its larger effective mass. The oscillation frequency of the mode decreases by a factor 1.19 and the matrix elements by a factor 1.51 upon deuteratlon. Therefore, the harmonic approximation, which yields an Isotopic factor 1.4 for both the frequency and the Intensity, Is quite Inappropriate for this mode. 

The low-frequency shift and the broadening of the CO spectra at 0 ps suggest that the low-frequency modes of adsorbed CO, that is, stretching, frustrated rotation, and frustrated translation modes of Pt-CO, were thermally excited by pump pulses, as reported by Bonn et al. [82] Thus, it is concluded that the transient site migration of adsorbed CO on the Pt electrode surface was caused by a transient rise in the surface temperature of Pt induced by pump pulses. 

It can bi Ieen from eq. (18) that K. must be proportional to L7 in order for G to be intensive. Weithus have a proof of what may ngw.jbe asserted to be the theorem0 0 that K, is proportional to v /0 for an approximately cubical body. Itiis seen that this statement is not restricted to elastomers, but is valid for all crystalline and amorphous materials. It is also the case that for sufficiently low frequency modes, the spectral density g(ic)dic... 

The vibrational frequency of the special pair P and the bacteriochlorophyll monomer B have also been extracted from the analysis of the Raman profiles [39,40,42,44,51]. Small s group has extensively performed hole-burning (HB) measurements on mutant and chemically altered RCs of Rb. Sphaeroides [44,45,48-50]. Their results have revealed low-frequency modes that make important contribution to optical features such as the bandwidth of absorption line-shape, as well as to the rate constant of the ET of the RCs. 

Lee and colleagues36 have computed the vibrational frequencies of all-traws-octatetra-ene. They have found that the mean absolute percentage deviation for frequencies is 12% at the HF level, while it decreases to 4% at the MP2 level. Among the low-frequency modes, the frequencies of the in- and out-of-plane CCC skeletal bends are lower than the experimental values by 16%. When d basis functions on each carbon atom are added, the frequencies of some of the low-frequency modes approach the observed frequencies. 

Fig. 4.1. Coupling of the adsorbate low-frequency mode with substrate phonons K level of the adsorbate (a) initial quasicontinuous phonon spectrum of the substrate not perturbed by the adsorbate, bold lines designating the levels which correspond to the specified wave vector K (b) level shifts in the K subsystem caused by the coupling of the adsorbate K mode and substrate phonons (c).

Represent the Hamiltonian for low-frequency modes of the adsorbate and substrate as a sum of three terms ... 

The high-frequency mode is coupled through the anharmonicity coefficient y with the low-frequency mode which is a resonant one due to harmonic interaction with the surface reservoir. For simplicity, the last term is written in the Gaitler-London approximation (compare with more general Eq. (4.1.8) where this restriction is absent). The required GF of the high-frequency mode can be obtained from the... 

Fig. 4.2. Low-frequency modes of librational vibrations of a molecular complex with a hydrogen bond.

Generalization of the exchange dephasing model to various cases of anharmonic coupling between high-frequency and low-frequency modes, and to the case of collectivized excitations in adsorbate... 

Considering only biquadratic anharmonic coupling, the dephasing of local vibrations was treated in the special case that only high-frequency modes underwent collectivization174 and subsequently with the allowance also made for collectivized low-frequency modes.138 175 It should be emphasized that the possibility for... 

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