Anharmonicity

For a simple diatomic molecule, such as CO itself, the desired harmonic or mechanical frequency co may be found by fitting the energy levels to an expression 

If the effects of anharmonicity on a given vibration are serious, even more serious are the effects on the separation between different vibrations in the same molecule. Equation (17) must be replaced by the generalised form 

Unfortunately, while the data of Table 1 demonstrate the seriousness of the problem, they do not go very far to solve it. Firstly, there is the trivial observational point that extension of the treatment to other systems is liable to be extremely difficult. Carbonyl ternaries are weak, and could be masked by C—H binaries thus molecules with ligands containing CH bonds cannot be examined at all, and the study of other molecules is restricted to their solutions in carbon 

Secondly, and most seriously, the validity even of the harmonic frequencies of Table 1 may be questioned 45). The observed binary and ternary bonds are all of symmetry class T(in thehexacarbonyls) or 41 or (in the case of Mn(CO)5Br), and these symmetry classes are repeated several times both in the fundamental and in the ternary region. Thus we have satisfied the conditions for Fermi resonance. Of course, to show that Fermi resonance is symmetry-allowed is not the same as showing that it occurs, but there is every reason to suspect it in the present case. The physical origin of anharmonicity lies in the existence of direct or crossed cubic and quartic terms in the potential energy expression ). 

The number of such independent terms in a metal hexacarbonyl is 13 (10 if we discard quartic terms containing the distortion of some CO group raised to an odd power), in addition to the three harmonic force and interaction constants. Thus the number of physical quantities exceeds the number of parameters that may, with the available data, be fitted to Eq. (18). There is the further possibihty that the observed frequencies are distorted by interaction with solvent in a way that is not adequately compensated for by Eq. (18). The classical amplitude of a triply excited oscillator is greater than that for one that is only singly excited, and so jostling of solvent and solute molecules, and variability and asymmetry in the solvent sheath, may become important. This may explain the observation that binary and more especially ternary i.r. bands are considerably broader than are fundamentals in the same solvents. 

Bouteiller et al. have devised a means of treating the anharmonicities of H-bonded complexes by expanding the potential energy function as a fourth-order polynomial, fit to the ab initio calculations. These investigators use two degrees of freedom, corresponding to the X—H distance, r, and the intermolecular separation between X and Z, R. 

The frequencies that arise from this treatment are listed for the three complexes in Table 3.17. Note first that the anharmonic frequencies of v. (XH in Table 3.17) are quite a bit smaller than the harmonic values in the upper part of the table, differing by about 400 cm .  

Nonetheless, there is a strong parallel between these v, frequencies and the coefficients in Table 3.16, with either the harmonic or anharmonic treatments. There is little such parallel in the stretches (X--N) and the a j coefficients, providing a warning against using simple force constants to estimate the frequency of this intermolecular mode. It is interesting, however, that the harmonic and anharmonic values of the X N stretch are quite similar. 

With regard to various levels of theory, it is significant that inclusion of correlation (values in parentheses for ClH-NHj) lowers the frequency but has the opposite effect of increasing v. It is also worth noting that the SCF and Cl frequencies can differ by as much as 300 cm, harmonic or anharmonic. Indeed, anharmonicity and electron correlation effects are additive here the anharmonic Cl v frequency is smaller than the SCF harmonic value by a full 639 cm.  

The Bouteiller approach to anharmonicity also permits extraction of energies of excitation to vibrational levels beyond the first excitation. The various progressions are reported in Table 3.19, arising from the correlated potentials. Proceeding down each column, the spacing between successive overtones decreases as the quantum number rises, resulting from the mechanical anharmonicity. 

We have seen already, in the context of frozen phonons, that anharmonic terms appear in the direct approach simultaneously 

It turns out that the longitudinal [100] vibrations in Ge belong to the first category one finds F = F = 0 at all atomic sites K, but a perceptible cubic anharmoni ity makes F (u ) deviate from linearity  

The transverse [100] vibrations in Ge illustrj.te the other manifestation of anharmonicity. The displacement u = (0,u,u) 

The most complicated anharmonic terms are found in the transverse [111] configuration, where both forms mentioned above appear at the same time forces are not parallel to the displacement, and th ir variation is not linear. Two calculations were performed with u = +0.004 (-2,1,1) a and the calculated forces were projected on the 3 perpendicular directions (-2,1,1), (111) and (0,-l,l). Whereas all projections on (0,-1,1) are zero, the longitudinal (111) component of the force at the first-neighbor site ( =+l) is, with the above ul, as much as 46 % of the transverse (-2,1,1) one at the sites =-l, 2, the non-parallel components are still respectively 13 % and 20 % of the parallel ones. For obtaining the harmonic force constants, only the (-2,1,1) projection of the force was retained - and (in contrast to the transverse [100] case) a noticeable anharmonic behavior was still found k determined with displacements +u and -u were averaged, in order o eliminate the cubic contributions - because they still differed considerably for k j the cubic term lu in g(5.4.1) represents 7 Z of the ha onic one. It was verified by a calculation with larger u that the influence of quartic anharmonicity is negligible. 

The force constants summarized in Tab. 5.1 represent the harmonic terms of eq. (5.4.1). It is clear that the anharmonic contributions, whose presence has been considered as an unpleasant perturbation to be eliminated, carry in themselves a considerable amount of valuable information, which still awaits exploration. 

As an introduction, the chapter begins with the anharmonic diatomic molecule. Then we study the thermal properties (free energy, equation of state, thermal expansion and specific heat) of the classical anharnx)nic linear chain. Two important concepts are introduced the Gvuneisen pavametev and the quasiharmonic approximation. In this approximation, the temperature dependence of the force constants and phonon frequencies is only due to the 

1 In the literature, different definitions of the quasiharmonic approximation are used. Here, we adopt the definition of LEIBFRIED and LUDWIG [5.54] 

The setf-Qonsistent hccmonio approximation (SCHA) developed in Sect.5.4 is a method for crystals with strong anharmonicity such as the rare-gas solids. The SCHA has also been applied to the study of soft modes in ferroelectric phase transitions and to the melting process. 

In the last section, we introduce the response fuTiotion for an anharmonic crystal. The response function depends on frequenjcy widths and shifts which are evaluated on the basis of perturbation theory. This is the pseudoharmonio approximation which includes not only the effects of thermal expansion (as in the quasi harmonic approximation), but also the effects of phonon-phonon interactions. Both the widths (damping) and shifts depend on temperature and on the frequency with which the crystal is probed . 

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Pesiherbe G H and Hase W L 1996 Statistical anharmonic unimolecular rate constants for the... 

Troe J 1995 Simplified models for anharmonic numbers and densities of vibrational states. I. Application to NO2 and Chem. Phys. 190 381-92... 

Song K and Hase W L 1999 Fitting classical microcanonical unimolecular rate constants to a modified RRK expression anharmonic and variational effects J. Chem. Phys. 110 6198-207... 

Wolf R J and Hase W L 1980 Quasiperiodic trajectories for a multidimensional anharmonic classical Hamiltonian excited above the unimolecular threshold J. Chem. Phys. 73 3779-90... 

Treanor C E, Rich J W and Rehm R G 1968 Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions J. Chem. Phys. 48 1798-807... 

Hollenstein H, Marquardt R, Quack M and Suhm M A 1994 Dipole moment function and equilibrium structure of methane In an analytical, anharmonic nine-dimenslonal potential surface related to experimental rotational constants and transition moments by quantum Monte Carlo calculations J. Chem. Phys. 101 3588-602... 

Marquardt R and Quack M 1989 Infrared-multlphoton excitation and wave packet motion of the harmonic and anharmonic oscillators exact solutions and quasiresonant approximation J. Chem. Phys. 90 6320-7... 

After transforming to Cartesian coordinates, the position and velocities must be corrected for anharmonicities in the potential surface so that the desired energy is obtained. This procedure can be used, for example, to include the effects of zero-point energy into a classical calculation. 

Eqs. (D.5)-(D.7). However, when perturbations occur due to anharmonicity, the wave functions in Eqs. (D.11)-(D.13) will provide the conect zeroth-order ones. The quantum numbers and v h are therefore not physically significant, while V2 arid or V2 and I2 = m, are. It should also be pointed out that the degeneracy in the vibrational levels will be split due to anharmonicity [28]. 

The influence of solvent can be incorporated in an implicit fashion to yield so-called langevin modes. Although NMA has been applied to allosteric proteins previously, the predictive power of normal mode analysis is intrinsically limited to the regime of fast structural fluctuations. Slow conformational transitions are dominantly found in the regime of anharmonic protein motion. 

Hayward et al. 1994] Hayward, S., Kitao, A., Go, N. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Prot. Sci. 3 (1994) 936-943 [Head-Gordon and Brooks 1991] Head-Gordon, T., Brooks, C.L. Virtual rigid body dynamics. Biopol. 31 (1991) 77-100... 

Steven Hayward, Akio Kitao, and Nobuhiro Go. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Physica Scripta, 3 936-943, 1994. 

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

R. M. Levy, O. de la Luz Rojas, and R. A. Friesner. Quasi-harmonic method for calculating vibrational spectra from classical simulations on multidimensional anharmonic potential surfaces. J. Phys. Chem., 88 4233-4238, 1984. 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Harmonic analysis (normal modes) at given temperature and curvature gives complete time behavior of the system in the harmonic limit [1, 2, 3]. Although the harmonic model may be incomplete because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it serves as a first approximation for which the theory is highly developed. This model is also useful in SISM which uses harmonic analysis. 

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