The Harmonic Approximation

The classical harmonic approximation is adequate at low enough temperatures, where most of the contribution to S. comes from the bottom part of the potential energy well (except near absolute zero, where quantum effects become important ). This approximation is expected to be less adequate at higher temperatures, where the contribution of the anharmonic wings of a localized microstate become significant. Also, the contribution of the higher frequencies should be calculated quantum mechanically. 

Hagler et al. suggested calculating the quantum mechanical (QM) entropy using the Einstein harmonic oscillators formula. For that, one calculates the vibrational frequencies which lead to 

Calculation of entropies of both types is straightforward and efficient because simulations are not required. This allows one to study the conformational stability of many localized microstates of the same molecule by comparison of their harmonic free energies. However, the method is limited only to models of macromolecules in vacuum. 

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This algorithm was improved by Chen et al. [78] to take into account the surface anhannonicity. After taking a step from Rq to R[ using the harmonic approximation, the true surface information at R) is then used to fit a (fifth-order) polynomial to fomi a better model of the surface. This polynomial model is then used in a coirector step to give the new R,. 

While it is not essential to the method, frozen Gaussians have been used in all applications to date, that is, the width is kept fixed in the equation for the phase evolution. The widths of the Gaussian functions are then a further parameter to be chosen, although it appears that the method is relatively insensitive to the choice. One possibility is to use the width taken from the harmonic approximation to the ground-state potential surface [221]. 

S(p) is thus the harmonic approximation counterpart of the function/(p). To simplify the orthography, we use symbol instead of the usual For a... 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

In the harmonic approximation, V does not involve the cross-term pj-Pe because pj- and are the symmetry coordinates. It is thus of the form... 

The vibraiimial rroqueiicics are tlenved lioin the harmonic approximation, which assiiiines that the potential surface has a quadratic form. 

The centrifugal distortion constant depends on the stifthess of the bond and it is not surprising that it can be related to the vibration wavenumber co, in the harmonic approximation (see Section 1.3.6), by... 

Many thermodynamic quantities can be calculated from the set of normal mode frequencies. In calculating these quantities, one must always be aware that the harmonic approximation may not provide an adequate physical model of a biological molecule under physiological conditions. 

This formula resembles (3.32) and, as we shall show in due course, this similarity is not accidental. Note that at n = 0 the short action 1 2 ( q) taken at the ground state energy Eq is not equal to the kink action (3.68). Since in the harmonic approximation for the well Tq = 2n/o)o, this difference should be compensated by the prefactor in (3.74), but, generally speaking, expressions (3.74) and (3.79) are not identical because eq. (3.79) uses the semiclassical approximation for the ground state, while (3.74) does not. 

P Q-) =p Q-,Q-,p), which in the harmonic approximation is described by (3.16), PhiQ-iQ-,P) exp(— CO Q1 tanh co ). Having reached the point Q, the particle is assumed to suddenly tunnel along the fast coordinate Q+ with probability A id(Q-), which is described in terms of the usual one-dimensional instanton. The rate constant comes from averaging the onedimensional tunneling rate over positions of the slow vibration mode,... 

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

Even for such a simple molecule, which 1 deliberately constrained to be lineiir and where I assumed that the harmonic approximation was applicable, the potential energy function will have cross-terms. 

Ebend is the energy required for bending an angle formed by three atoms A-B-C, where there is a bond between A and B, and between B and C. Similarly to Estr, Fbend is usually expanded as a Taylor series around a natural bond angle and terminated at second order, giving the harmonic approximation. 

Vibrational Spectra Many of the papers quoted below deal with the determination of vibrational spectra. The method of choice is B3-LYP density functional theory. In most cases, MP2 vibrational spectra are less accurate. In order to allow for a comparison between computed frequencies within the harmonic approximation and anharmonic experimental fundamentals, calculated frequencies should be scaled by an empirical factor. This procedure accounts for systematic errors and improves the results considerably. The easiest procedure is to scale all frequencies by the same factor, e.g., 0.963 for B3-LYP/6-31G computed frequencies [95JPC3093]. A more sophisticated but still pragmatic approach is the SQM method [83JA7073], in which the underlying force constants (in internal coordinates) are scaled by different scaling factors. 

The harmonic approximation can also be used to provide an estimate of the vibrational free energy, using (Refs. 1 and 6). 

The starting point for the determination of mode frequencies is the harmonic approximation. Here we assume that we begin with an equilibrium geometry and investigate the restoring forces generated for small displacements from equilibrium. 

The force constant for the vibrations, K = in the harmonic approximation is proportional to oP. Differentiating Equation 24.6 to obtain K and substituting into Equation 24.21, the result is [84,85]... 

The harmonic approximation is only valid for small deviations of the atoms from their equilibrium positions. The most obvious shortcoming of the harmonic potential is that the bond between two atoms can not break. With physically more realistic potentials, such as the Lennard-Jones or the Morse potential, the energy levels are no longer equally spaced and vibrational transitions with An > 1 are no longer forbidden. Such transitions are called overtones. The overtone of gaseous CO at 4260 cm (slightly less than 2 x 2143 = 4286 cm ) is an example. 

Next we discuss the effect of deuteratlon on low frequency modes Involving the protons> Because of the anharmonlc variation of the energy as a function of tilt angle a (Fig. 4b), the hindered rotations of H2O and D2O turn out to be qualitatively different. The first vibrational excited state of H2O Is less localized than that of D2O, because of Its larger effective mass. The oscillation frequency of the mode decreases by a factor 1.19 and the matrix elements by a factor 1.51 upon deuteratlon. Therefore, the harmonic approximation, which yields an Isotopic factor 1.4 for both the frequency and the Intensity, Is quite Inappropriate for this mode. 

In good agreement with our calculation. In Ref. 14, using the harmonic approximation, the anomalous Isotopic factor for the frequency was Interpreted as due to mixing with the hindered translation. However, as we have shown, the harmonic approximation Is Inappropriate In this case. 

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