The Quasi-Harmonic Approximation

This representation of the Hamiltonian is called the quasi-harmonic (QH) approximation and should not be confused with the quadratic expansion of the correct energy in the harmonic approximation discussed in the preceding section. The quasi-harmonic PD becomes 

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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... 

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. 

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]. 

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]. 

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. 

Computation of vibrational frequencies for crystalline phases can be carried out with various methods. Perhaps the most common is the to use the quasi-harmonic approximation in lattice dynamics calculations (see Parker, this volume). Some excellent examples of this type of study are Cohen et al. 1987, Hemley et al. (1989), Wolf and Bukowinski (1987), and Chaplin et al. (1998). In general, however, such calculations serve as a validation of the modeling technique rather than as a method to interpret frequencies. Vibrational modes in crystalline solids are readily assigned because the structure is known from X-ray diffraction studies. In fact, isochemical crystalline solids are used frequently to help interpret spectra of glasses (e.g., McMillan 1984). 

At low temperatures, if most of the anharmonic effects are due to lattice expansion, the quasi-harmonic approximation can be successfully applied. However, if the average displacement of the atoms is so large that the potential energy cannot be approximated by quadratic terms anymore, the approximation fails. In such cases, we can use a classical simulation method such as molecular dynamics to sample the phase space and calculate observables using these samples. We should note that this is strictly valid only in case of high temperatures, where Tmd Tqm-... 

The low frequency Raman spectrum can provide information on the density of states g(co) in the low frequency region. In the quasi-harmonic approximation the specific heat is given by... 

Compared to the HF approach, DFT gives a better description of the crystallographic parameters. However, whilst the DFT calculations tend to overestimate the peroxide bond length, HF calculations underestimate it with a similar error. The advantages of using DFT are obvious upon comparison of calculated and experimental values of the energies of vibrational transitions on the peroxy calculated here in the T-point in the quasi-harmonic approximation (see Table 4.2). 

Systematic underestimation of the high-temperature Cp, cai T) data is a shortcoming of the quasi-harmonic approximation used to describe Ciat(T ). Westrum (1979) analyzed methods for including the contribution of crystal thermal expansion. It follows from the representation of Cd(T) in the form of the dependence... 

At the first stage, the Cexs T) contribution was subtracted, if necessary, from the low-temperature heat capacity values. The set of Qat(T) data obtained in this way was approximated by the method of least squares using Eq. (6) that describes the contribution of fhe lattice heat capacity in the quasi-harmonic approximation. As a result, we calculated the values of the 0D/ Ei/ E2/ E3/ and oe variable parameter. We then determined the analytic dependence of changes in fhese paramefers on the molar volume V. This allowed us to estimate the 0d/ ei/ e2/ e3/ and oc parameters for unstudied compounds. 

In Chap.5, anharmonic effects are considered. After an illustration of anharmonicity with the help of the diatomic molecule, we derive the free energy of the anharmonic linear chain and discuss the equation of state and the specific heat. The quasi-harmonic approximation" worked out in detail for the linear chain is then applied to three-dimensional crystals to obtain the equation of state and thermal expansion. The self-consistent harmonic approximation" is the basis for treating the effects of strong anharmonicity. At the end of this chapter we give a qualitative discussion of the response... 

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