Delocalized motions

Minimized structures gained from MD simulations are also often basis of normal mode analysis (NMA) [41-43]. NMA assumes that all atoms harmonically oscillate around their equilibrium points. The oscillations deflned by frequency and amplitude (normal mode) are extracted and reflect directions of internal protein motions. Given all its normal modes, the entire protein motion can be expressed as a superposition of modes. The modes vith lo vest frequency correspond to rather delocalized motions in proteins in vhich a large number of atoms oscillate in coordinated motion vith considerable amplitude. Modes vith higher frequency represent more localized motions. Linear combinations of the most relevant normal modes can be employed to depict essential protein motions. Stepwise displacement of atoms of the original structure along the modes can be applied to build up an ensemble of relevant protein conformations [44, 45]. 

The third example presented in Sect. 6 illustrates our preliminary theoretical dynamical spectra in the far-IR domain below 1,000 cm, where vibrational anharmonicities arising from mode-couplings and delocalized motions have been captured with great accuracy on a model phenylalanine neutral di-peptide, in relation to cold IR-MPD experiments. 

Boson Peak, a Signature of Delocalized Collective Motions in Glasses (Example FC as Sensor Molecule)... 

The so-called Boson peak is visible as a hump in the reduced DOS, g(E)IE (Fig. 9.39b), and is a measure of structural disorder, i.e., any deviation from the symmetry of the perfectly ordered crystal will lead to an excess vibrational contribution with respect to Debye behavior. The reduced DOS appears to be temperature-independent at low temperatures, becomes less pronounced with increasing temperature, and disappears at the glass-liquid transition. Thus, the significant part of modes constituting the Boson peak is clearly nonlocalized on FC. Instead, they represent the delocalized collective motions of the glasses with a correlation length of more than 20 A. 

Beyond the Boson peak, the reduced DOS reveals for all studied glasses a temperature-independent precisely exponential behavior, g(E)/E exp( / o) with the decay energies Eo correlating with the energies E of the Boson peak. This finding additionally supports the view that the low-energy dynamics of the glasses are indeed delocalized collective motions because local and quasilocal vibrations would be described in terms of a power law or a log-normal behavior [102]. 

In a second possibility, the polaron-like hopping model, a structural distortion of the DNA stabilizes and delocalizes the radical cation over several bases. Migration of the charge occurs by thermal motions of the DNA and its environment when bases are added to or removed from the polaron [23]. 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

These TR results demonstrate that the localized model of Ru(bpy) + is valid on the timescales of electronic motions and molecular vibrations. It is virtually certain that delocalization (via, for example, intramolecular electron transfer or dynamic Jahn-Teller effects) occurs on some longer timescale. 

Petersen et al., Petersen and Voth,i Spohr, Spohr et al., and Walbran and Kornyshev developed EVB-based models to study the effect of con-finemenf in nanometer-sized pores and fhe role of acid-functionalized polymer walls on solvation and transport of protons in PEMs. The calculations by the Voth group revealed an inhibiting effect of sulfonate ions on proton motions. The EVB model by Kornyshev, Spohr, and Walbran was specifically designed to sfudy effecfs on proton mobilify due to charge delocalization within SOg groups, side chain packing density, and fluctuations... 

There is, however, an exception in an infinite homogeneous electron gas, electrons are delocalized. Neglecting their orbital motion does not contradict quantum mechanics, and lacking localization is unproblematic when only cross... 

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