Approximation quasi-harmonic

We have presented here the quasi harmonic approximation epitomized by Eq.(51) to show one way to represent the dynamics of nuclear motions in a quantum mechanical scheme. A general solution for these equations cannot be obtained. However, a number of particular cases exist for which solutions have been worked out in the literature. 

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

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. 

With the advance of computing techniques classic LD programs have become more and more sophisticated. The PHONON program, provided from Daresbury Laboratory [69], is one such excellent example. PHONON uses the quasi-harmonic approximation and has a wide range of two body potentials embodied in the code. In addition, angular three-body bending potentials, four-body torsion potentials are also included. The program has been widely used for simulations of a variety of properties, such as dispersion curves, defects and surface phonons of crystalline and amorphous materials. 

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. 

A triple axis spectrometer ( 3.4.1) was employed in these measurements [10,11,12,13]. The vibrational calculations were performed on the primitive cell once its calculated Helmholtz free energy, F, had been minimised using the quasi-harmonic approximation. Quantum effects are present due to the lightness of both atoms [14,15]. 

Omission of dynamics. Minimization identifies the static configuration of lowest energy and there is no representation of the vibrational or other dynamical properties of the system. In formal terms, these are zero Kelvin calculations, with zero point motion omitted. It is, however, relatively straightforward to add a treatment of the vibrational properties of the system within the harmonic (or quasi-harmonic) approximations. Such methods will be discussed in Chapter 3. 

If the thermodynamic properties are calculated within the harmonic approximation, in which the normal modes of vibration are assumed to be independent and harmonic, the cell has no thermal expansion. PARAPOCS (Parker and Price, 1989) extends this to the quasi-harmonic approximation. In this method the vibrations are assumed to be harmonic but their frequencies change with volume. This provides an approach for obtaining the extrinsic anharmonicity which leads to the ability to calculate thermal expansion. 

As noted earlier, this approach assumes the quasi-harmonic approximation which includes important anharmonic effects associated with the variation of free energy with volume (extrinsic anharmonicity). It does, however, neglect intrinsic anharmonicity which becomes important at elevated temperatures. To investigate crystals at high temperatures Molecular Dynamics (MD) can be used in which intrinsic anharmonic effects are treated explicitly. This method is considered in detail in Chapter 4. 

T is thus given in the quasi-harmonic approximation by the sum of static and vibrational contributions ... 

However, the drawback of ab initio calculations is that they usually refer to the athermal limit (T = 0 K), so that pressure but not temperature effects are included in the simulation. Although in principle the ab initio molecular dynamics approach[13] is able to overcome this limitation, at the present state of the art no temperature-dependent quantum-meehanieal simulations are feasible yet for mineral systems. Thus thermal properties have to be dealt with by methods based on empirical interatomic potential functions, containing parameters to be fitted to experimental quan-tities[14,15, 16]. The computational scheme applied here to carbonates is that based on the quasi-harmonic approximation for representing the atomic motion[17]. 

Fig. 6.7. The difference between harmonic and quasi-harmonic approximations for a diatomic molecule, fa) The potential energy for the harmonic oscillator, (b) The harmonic approximation to the oscillator potential V n,r(.R) for a diatomic molecule is not realistic since at If = 0 fand at If < 0), the energy is finite, whereas it should go asymptotically to infinity when R tends to 0. tc A more realistic (quasi-harmonic) approximation is as follows the potential is harmonic up to R = 0, and for negative R, it goes to infinity. The difference between the harmonic and quasi-harmonic approximations pertains to such high energies (high oscillation amplitudes), that it is practically of negligible importance. In cases b and c, there is a range of small amplitudes where the harmonic approximation is applicable.

Let me stress once more that the problem appears when making the quasi-harmonic approximation, not in the real system we have. 

Conclusion the quasi-harmonic approximation means almost the same as the (less realistic) harmonic one. 

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