Anharmonic interaction

This formula is derived in Appendix 3). With regard to various cubic and quartic anharmonic interactions, the quantity ft is characterized by a certain combination of these anharmonic contributions and becomes dependent on k (see Eq. (4.3.14) for a related quantity and Ref. 140). However, this dependence is insignificant compared to the k-dependence appearing in the denominators of Eqs. (4.3.32) and (4.3.34). Therefore, spectral characteristics defined by formulae (4.3.32) can with good reason be regarded as proportional to certain functions of lateral interaction parameters and of the resonance width 77 ... 

Contents Lattice Dynamics. - Symmetry. - Inter-molecular Potentials. - Anharmonic Interactions. - Two-Phonon Spectra of Molecular Crystals. -Infrared and Raman Intensities in Molecular Crystals. 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

As is expected for the B value of a linear triatomic monohydride, the rotational constant increases very slowly with o2 at low V2 [15]. Then, owing to strong 2 1 anharmonic interaction with the CP stretch (for which B is expected and observed to decrease with 03 at low 03 [15]), Bo>U2,o begins to decrease slowly and continues to decrease slowly and monotonically to V2 = 34 (Bo = 0.6663 cm-1, B34 = 0.6454 cm-1). Suddenly Bo>vz,o reverses... 

A non-perturbative theory of the multiphonon relaxation of a localized vibrational mode, caused by a high-order anharmonic interaction with the nearest atoms of the crystal lattice, is proposed. It relates the rate of the process to the time-dependent non-stationary displacement correlation function of atoms. A non-linear integral equation for this function is derived and solved numerically for 3- and 4-phonon processes. We have found that the rate exhibits a critical behavior it sharply increases near a specific (critical) value(s) of the interaction. 

To explain the idea of the method [13,14], let us consider a two-phonon decay of a highly excited local mode caused by the interaction /7irl( = QY.rf V3 A,, where V3 lI/ are the cubic anharmonicity interaction parameters, Q is the coordinate... 

Here Hint is the anharmonic interaction in the collinear-configurational approximation (see Refs. [5,7]), Vmk = v/n,7i/2displacement operators of the host atoms with respect to the atom(s)of the mode (Y.i eimeim = M- We take into account that the strongly excited mode can be considered classically, and replace its coordinate operator by Q(t) = A cos( )/f), where A is the initial amplitude of the mode. Then... 

If the anharmonic interaction is not weak, then D(t) should be found from the equation of motion, which in case k > 2 turns out to be a non-linear integral equation. To get the corresponding equation, one should start from the equation of motion for the displacement operator(s) q. From equation (5) it follows that... 

As an example, we consider the multiphonon relaxation of a local mode caused by an anharmonic interaction with a narrow phonon band. We suppose that the mode is localized on an atom and take into account two diagonal elements of the Green function which stand for the contribution of two nearest atoms of the lattice to the interaction the non-diagonal elements are usually much smaller [16] and approximate the density of states of the phonon band by the parabolic distribution... 

To sum up, we have developed a general non-perturbative method that allows one to calculate the rate of relaxation processes in conditions when perturbation theory is not applicable. Theories describing non-radiative electronic transitions and multiphonon relaxation of a local mode, caused by a high-order anharmonic interaction have been developed on the basis of this method. In the weak coupling limit the obtained results agree with the predictions of the standard perturbation theory. 

Here, HFree is the Hamiltonian of the free harmonic high frequency oscillator and Hint is that giving the anharmonic interaction between the slow and fast modes, which are, respectively, given by... 

We already noted that, for a harmonic interaction, o, = constant and (z) = z0 = Zq — (p/B) varies linearly with the applied pressure. However, for an anharmonic interaction, oa is a function of the applied pressure (or, equivalently, ofthe average thickness ofthe film (z), which differs from Zq because of the asymmetry of the distribution). The functional dependence of the pressure on the average thickness differs in the anharmonic and harmonic cases. The pressure is higher in the former case, because ofthe... 

Vo is the anharmonic interaction when the CO stretch is in the ground vibrational state, and Vi is the anharmonic interaction when the CO stretch is in its first excited vibrational state. To simplify the analysis, take Eo = 0, and transform into the interaction representation with respect to all of the qA. Furthermore, take V0 = 0, which restricts the analysis to the excited vibrational state but does not fundamentally change the nature of the results. Then, the effective Hamiltonian for the CO stretch is... 

The motion actually observed by i.r. absorption is a small part of the whole pattern, since for effective absorption one must match both frequency and wavelength in the electromagnetic and crystal osdllations. Absorption occurs essentially at the frequency of the longest waves present in the crystal These waves are damped by anharmonic interaction with the numerous short-wave modes, except at very low temperatures where interaction with ctystal ddects and surfaces takes over. In spite of the complexity of the process one has to expect a simple lineshape and a fairly simple temperature dependence for a wave propagating in one crystal direction. 

The quantum theory of vibrational relaxation in low-temperature ordered solids is well develojjed, at least for weak interactions. Starting from the harmonic solid, with known normal mode energies , the anharmonic interactions between modes are introduced as an ordered perturbation and the renormalized mode energies are calculated, usually by temperature Green s function methods, for each order of jierturbation. The calculated energy shifts j — are complex. 

Probably of more importance are the contributions of anharmonic interaction terms [Eq. (3.19)]. Since the zero-point contribution of these terms will be different for the various isotopic species, a difference in the effective potential is expected. 

If we consider only the effect of the anharmonic interaction terms up to fourth degree through first order, the following result is obtained... 

In Table 5 the features that have been found in the luminescence spectra of the (UOs Vp) centre have been assigned. The vibronic features assigned to vibronic transitions involving the vs and V4 vibrational modes have a broad band character. This is probably due to anharmonic interaction of these modes with lattice phonons occuring in the same frequency region... 

Figure 5.4 Fermi resonances energies E of the first vibrational levels of a mode q and of a mode in a C-O-H group (upper diagrams) and corresponding IR spectra (lower diagrams). The values E/hc (c, velocity of light in cm sec ) of these two modes are represented when no interaction is present (left upper diagram) and when an anharmonic interaction of the form fqqg is present (right upper drawing / is a constant). Arrows indicate transitions that appear in vibrational spectroscopy (IR or Raman).

This review focused exclusively on frequency-domain spectroscopy. There is, however, major progress in time-domain spectroscopy. Methods such as 2D-IR spectroscopy offer exciting possibilities, especially for studies of large molecules, and of molecules in condensed phases [160]. Anharmonic interactions are of the essence in 2D-IR spectroscopy. The impressive experimental development of recent years [160], and pioneering theoretical studies, such as those conducted by Mukamel and coworkers [161], contribute to our impression that this may become a major direction for ab initio spectroscopic studies. 

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