Harmonic vibrations vibration coupling

Phonons are normal modes of vibration of a low-temperature solid, where the atomic motions around the equilibrium lattice can be approximated by harmonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled normal modes (phonons) if a harmonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

Figure A.4 Visualization of lattice vibrations as coupled harmonic oscillations of spheres connected by springs.

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

O. Hino, T. Kinoshita, G. K.-L. Chan, and R. J. Bartlett, Tailored coupled cluster singles and doubles method applied to calculations on molecular structure and harmonic vibrational frequencies of ozone. J. Chem. Phys. 124, 114311 (2006). 

The application of Stepanov s theory to intramolecular F bonded systems has been criticized [42], In this case the low frequency vibration described above as vXH Y is also partly constrained by a more nearly harmonic vibration involving skeletal bending motions of the rest of the molecule, and the X, H, and Y atoms are not collinear. These factors would seem to suggest that (I) the vXll Y type of vibration will be of higher frequency than in the usual case (perhaps 200-300 cm"1 rather than 100-200 cm"1) so that the sub-bands will be more widely spaced and may not be recognised as part of the rXH band (2) the motion of the H atom will have less effect on rXY and (3) H-bond bending vibrations may also couple considerably with vXH. The observation of rather smaller frequency shifts for vXR and narrower absorption bands w such cases are in reasonable agreement with this picture,... 

Messina et al. [25] test the time-dependent Hartree reduced representation with a simple two-degree-of-freedom model consisting of the h vibration coupled to a one-harmonic-oscillator bath. The objective function is a minimum-uncertainty wavepacket on the B state potential curve of I2. Figure 12, which displays a typical result, shows that this approximate representation gives a rather good account of the short-time dynamics of the system. 

For centrosymmetric complexes the intensities of the parity-forbidden d< d bands arise through vibronic interactions and consequently show substantial temperature dependence. It can be shown that for the ideal case of a si ngle harmonic vibration of frequency y coupled to the electronic system the intensity of a band should be given by105-106... 

In addition to the individual and uncorrelated particle motions, we also have collective ones. In a strict sense, the hopping of an individual vacancy is already coupled to the correlated phonon motions. Harmonic lattice vibrations are the obvious example for a collective particle motion. Fixed phase relations exist between the vibrating particles. The harmonic case can be transformed to become a one-particle problem [A. Weiss, H. Witte (1983)]. The anharmonic collective motion is much more difficult to treat theoretically. Correlated many-particle displacements, such as those which occur during phase transformations, are further non-trivial examples of collective motions. 

The nuclear function %a(R) is usually expanded in terms of a wave function describing the vibrational motion of the nuclei, and a rotational wave function [36, 37]. Analysis of the vibrational part of the wave function usually assumes that the vibrational motion is harmonic, such that a normal mode analysis can be applied [36, 38]. The breakdown of this approximation leads to vibrational coupling, commonly termed intramolecular vibrational energy redistribution, IVR. The rotational basis is usually taken as the rigid rotor basis [36, 38 -0]. This separation between vibrational and rotational motions neglects centrifugal and Coriolis coupling of rotation and vibration [36, 38—401. Next, we will write the wave packet prepared by the pump laser in terms of the zeroth-order BO basis as... 

Note that a harmonic vibration and linear coupling is insufficient to shift the frequency or cause dephasing nonlinearity must be present in either the vibration or in the coupling. 

To demonstrate the potential of two-dimensional nonresonant Raman spectroscopy to elucidate microscopic details that are lost in the ensemble averaging inherent in one-dimensional spectroscopy, we will use the Brownian oscillator model and simulate the one- and two-dimensional responses. The Brownian oscillator model provides a qualitative description for vibrational modes coupled to a harmonic bath. With the oscillators ranging continuously from overdamped to underdamped, the model has the flexibility to describe both collective intermolecular motions and well-defined intramolecular vibrations (1). The response function of a single Brownian oscillator is given as,... 

I = vibronic coupling parameter, v = (1/2jt) , k = harmonic vibrational constant). (Ref 19. Reproduced by permission of Klnwer Academic Publishers)... 

The hv and Kj parameters are the harmonic vibrational frequency and anhar-monicity constant respectively of the vibrational Hamiltonian (Hq). The Aj and A2 parameters are the first-order and second-order Jahn Teller coupling constants respectively for the Jahn-Teller part of the Hamiltonian (Hjj). 

Besler, B. H., Scuseria, G. E., Scheiner, A. C., and Schaefer, FI. F., A systematic theoretical study of harmonic vibrational frequencies The single and double excitation coupled cluster (CCSD) method, J. Chem. Phys, 89, 360-366 (1988). 

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