Beyond the Harmonic Approximation

2 Beyond the Harmonic Approximation Thus far, our analysis has been entirely predicated upon the harmonic approximation advanced in eqn (5.2). On the other hand, real crystals are characterized by nonlinear force laws and the resulting anharmonic effects are part of the story from the outset. If we confine our interest to the specific heat, we see that much may be made of the harmonic approximation. However, if we aim to press our modeling efforts further, for example, to consider the thermal expansion or the thermal conductivity, we will see that the harmonic approximation does not suffice. Let us flesh out the logic in the case of the coefficient of thermal expansion. Our first ambition will be to demonstrate that the harmonic approximation predicts 

We have seen that as a consequence of the harmonic Hamiltonian that has been set up thus far, our oscillators decouple and in cases which attempt to capture the transport of energy via heating, there is no mechanism whereby energy may be communicated from one mode to the other. This shortcoming of the model may be amended by including coupling between the modes, which as we will show below, arises naturally if we go beyond the harmonic approximation. The simplest route to visualizing the physics of this problem is to assume that our harmonic model is supplemented by anharmonic terms, such as 

This result urges us to consider the nonlinear terms which must supplement the quadratic model set forth in eqn (5.2). The coefficients are defined in analogy 

There are certain subtleties associated with the order to which the expansion of eqn (5.67) is carried and for a discussion of which we refer the perspicacious reader to chap. 25 of Ashcroft and Mermin (1976). If we maintain the use of normal coordinates, but now including the anharmonic terms which are represented via these coordinates, it is seen that the various modes are no longer independent. For the purposes of the present discussion, we see that these anharmonic terms have the effect of insuring that the equation of motion for the n mode is altered by coupling to the rest of the mode variables. This result bears formal resemblance to the Navier-Stokes equations which when written in terms of the Fourier variables yield a series of mode coupled equations. We have already noted that the physics of both thermal expansion and thermal conductivity demand the inclusion of these higher-order terms. 

Recall that the coefficient of thermal expansion is defined as 

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The potential U3 of the bending mode has been generally approximated by a harmonic potential [22,23,53,71-73]. Extension of this model should go beyond the harmonic approximation used in the description of the three vibrational modes, by substituting to the previous potentials (66), (67), and (68) a Morse-type potential [13]. 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

Note that beyond the harmonic approximation for the H-bond bridge and according to Eqs. (14) and (18), the full Hamiltonian (12) is... 

Besides, p(0), the IR absorption transition operator at time t = 0 and p(t) is this same operator at time t. Then, beyond the harmonic approximation and in view of Eq. (9) giving the expression of the transition moment operator, the ACF of the bare weak H-bond, may be written... 

Within the harmonic approximation of the transition state theory (TST), the prefactors are determined as the ratios between the product of all the eigenmode frequencies at the minimum and that of all the real eigenmode frequencies at the saddle points. This approximation is adequate for systems with slowly varying PES s, to the extent that no anharmonicities are involved. Going beyond the harmonic approximation of TST, in Ref [5], the rate prefactor probabilities are determined as... 

Carbonniere, P., Lucca, T., Pouchan, C., Rega, N., and Barone, V., Vibrational computations beyond the harmonic approximation performances of the B3LYP density functional for semirigid molecules, J. Comput. Chem., 26, 384-388 (2005). 

Barone, V., Vibrational spectra of large molecules by density functional computations beyond the harmonic approximation the case of pyrrole and furan, Chem. Phys. Lett., 383, 528-532 (2004). 

There have been a number of calculations recently on the vibrations of H5OJ beyond the harmonic approximation. As noted already, such calculations are essential due to the highly anharmonic nature of these vibrations. Attempts to go beyond the harmonic approximation have been done in reduced dimensionality [27, 56] and also with the additional vibrational adiabatic approximation [56], These calculations selected the three proton degrees of freedom and the OO-stretch as the reduced dimensionality space. While such approaches are better in... 

Recently, CHj has been the subject of ab initio integral path calculations, which include nuclear quantum effects such as zero-point motion and tunneling in full dimensionality and go beyond the harmonic approximation. These studies confirm the above results, in that while protons undergo large-amplitude pseudorotational motion and become scrambled and statistically equivalent, this motion is concerted and the situations in which the vibrating nuclear skeleton having an H2 moiety attached to a CH3 tripod are the most important contributors to the overall appearance of the cation. 

For the harmonic oscillator, the root-mean square values of Qa scale as and can also be expected to be of order unity for all quasi-rigid motions beyond the harmonic approximation. 

The evaluation of vibrational frequencies beyond the harmonic approximation, which requires the evaluation of high-order (third and fourth) energy derivatives will be treated in a further section. 

The derivation of equations for the transition moments at the anharmonic level is made difficult by the need to account for both the anharmonicity of the potential energy surface (PES) and of the property of interest. Owing to the complexity of such a treatment, various approximations have been employed, in particular, by considering independently the wave function and the property, so that different levels of theory can be apphed to each term and only one of them is treated beyond the harmonic approximation [244,245]. Following the first complete derivation by Handy and coworkers [243], Barone and Bloino adopted the alternative approach presented by Vazquez and Stanton [240] and proposed a general formulation for any property function of the normal coordinates or their associated momenta, which can be expanded in the form of a polynomial truncated at the third order. In this work, we will follow the latter approach, as applied to the infrared (IR) and Raman spectra. 

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