Quasi-harmonic model

B. Results from Harmonic and Quasi-Harmonic Models.178... 

In this section we briefly discuss the harmonic and quasi-harmonic models that are commonly used to describe the molecular motions, i.e.,... 

Since it became clear from various observations that the librational motions of the molecules, even in the ordered a and y phases of nitrogen at low temperature, have too large amplitudes to be described correctly by (quasi-) harmonic models, we have resorted to the alternative lattice dynamics theories that were described in Section IV. Most of these theories have been developed for large-amplitude rotational oscillations, hindered or even free rotations, and remain valid when the molecular orientations become more and more localized. 

As for the quantities related directly to the quasi-harmonic model, i.e. Grtineisen parameter and thermal expansion coefficient, the only data derived from measurements are, to our knowledge, those reported in diagrams by Salje and Viswanathan[20] y = 1.80 and a= 6.5 x 10" K at room temperature. These should be compared with the values computed by the RIM (y = 2.61, a= 11.5X 10-5 K-i) and Rjjyji ( = 1.50, = 5.8 x lO ... 

SELECTION OF AN OPTIMUM QUASI-HARMONIC MODEL OF AN ORGANIC CRYSTAL. 

Fig. 4.4 Phonon dispersion relations and the vibrational density of states (VDOS) calculated for guest-free clathrate Sii36 using DFT methods at T = 0 K. At right are shown the corresponding mode Gruneisen parameters calculated using a quasi-harmonic model as a function of temperature (from [66])...

DFT calculations of the static lattice were performed within the local density approximation (LDA) using planewave basis sets and ultrasoft pseudopotentials, and the results were used to construct a force constant matrix within a large superceU model. The phonon spectrum was then evaluated as a function of temperature using quasi-harmonic models that allowed us to constmct mode Gruneisen relationships [66] (Fig. 4.4). The results indicated a dip in the V(T) relation at 80 K, that was slightly smaller than that observed for diamond-structured Si (Fig. 4.5). 

An alternative approach to the calculation of the Helmholtz free energy is the k-space quasi-harmonic model (QC-QHMK) ° " introduced in 2001 by Aluru et al. This method, still a generalization of the quasi-continuum... 

QC-QHMK Quasi-continuum iF-space quasi-harmonic model... 

The Cd, cor(T) confribufion fhaf removes femperafure limifafions of fhe quasi-harmonic model used fo defermine Ciat(T) was described at the third stage of calculafions. The analytic form of Cd, cor(T) was foimd by comparing Cp cai(T) and Cp exp(T) Cp exp(T) values were calculated from the experimenfal high-femperafure enfhalpy increments. 

Expression 11.3 provides the whole simulated spectrum, while a detailed vibrational analysis requires the unambiguous assignment of each mode contribution. Recently, a number of methods appeared in the literature aimed at the extraction of normal-mode-like analysis from ab initio dynamics [58-63]. Some of these [58-60] refer to the quasi-harmonic model introduced by Karplus [64,65] in the framework of classical molecular dynamics and individuate normal-mode directions as main components of the nuclear fluctuations in the NVE or NVT ensemble. The quasinormal model relies on the equipartition of the kinetic energy among normal modes thus problems arise when the simulation time required to obtain such a distribution is computationally too expensive, as is often the case for ab initio dynamics. Other approaches [61-63] carry out the time evolution analysis in the momenta subspace instead of the configurational space. In these approaches the basic consideration is that, at any temperature, generalized normal modes g, correspond to uncorrelated momenta such that [61]... 

The pyroelectric coefficient at constant strain, p is expressed by the polarization model, using the quasi-harmonic approximation, as... 

RO, Fig. 3d) (2) higher-frequency, smaller amplitude, quasi-harmonic oscillations (QHO, Fig. 3a) and (3) double-frequency oscillations containing variable numbers of each of the two previous types. By far the most familiar feature of the BZ reaction, the relaxation oscillations of type 1 were explained by Field, Koros, and Noyes in their pioneering study of the detailed BZ reaction mechanism.15 Much less well known experimentally are the quasiharmonic oscillations of type 2,4,6 although they are more easily analyzed mathematically. The double frequency mode, first reported by Vavilin et al., 4 has been studied also by the present author and co-workers,6 who explained the phenomenon qualitatively on the basis of the Field-Noyes models of the BZ reaction. 

Section VI. It is possible to unblock the first drawback (i), if to assume a nonrigidity of a dipole—that is, to propose a polarization model of water. This generalization roughly takes into account specific interactions in water, which govern hydrogen-bond vibrations. The latter determine the absorption R-band in the vicinity of 200 cm-1. A simple modification of the hat-curved model is described, in which a dipole moment of a water molecule is represented as a sum of the constant (p) and of a small quasi-harmonic time-varying part p(/j. 

The cosine-squared potential model was simplified in terms of the so-called stratified approximation, for which the spectral function Tcs(Z) is given in GT, p. 300 and in VIG, p. 462. We remark that the dielectric spectra calculated rigorously for the CS model agree with this approximation, while simpler quasi-harmonic approximation (GT, p. 285 VIG, p. 451) used in item A yields for p > la too narrow theoretical absorption band. 

Molecular dynamics (MD) simulations, the most suitable theoretical tool for the investigation of internal motions, can be used to explore both equilibrium properties and time-dependent phenomena. Based on both experimental and theoretical observations two models for the internal motion of proteins have been suggested. Within the framework of the first model internal motions arise from harmonic or quasi-harmonic vibrations that occur in a single multidimensional well on the potential energy surface [4,5,6,7]. The second model assumes that motions are a superposition of oscillations within a well and... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

The experimental and computational study of bacterial thioredoxin, an E. coli protein, at THz frequencies is presented. The absorption spectrum of the entire protein in water was studied numerically in the terahertz range (0.1 - 2 THz). In our work, the initial X-ray molecular structure of thioredoxin was optimized using the molecular dynamical (MD) simulations at room temperature and atmospheric pressure. The effect of a liquid content of a bacterial cell was taken into account explicitly via the simulation of water molecules using the TIP3P water model. Using atomic trajectories from the room-temperature MD simulations, thioredoxin s THz vibrational spectrum and the absorption coefficient were calculated in a quasi harmonic approximation. 

For both models, we used atomic trajectories from our room-temperature MD simulations to calculate thioredoxin s THz spectra in a quasi harmonic approximation. The absorption coefficient was calculated for different orientations of the molecule with respect to the electric field polarization. 

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