Relaxation energy, harmonic approximation

In the limit of static fields, the nuclear relaxation contribution (from now on just vibrational ) to the polarizabilities can be computed in the double harmonic approximation, i.e. assuming that the expansions of both the potential energy and the electronic properties with respect to the normal coordinates can be limited to the quadratic and the linear terms, respectively (i.e. assuming both mechanical and electric harmonicity). 

Common spectroscopic techniques test small portions of the ground and/or excited state PES either around the gs minimum (IR and non-resonant Raman spectra, electronic absorption spectra.) or in the proximity of the excited state minimum (steady-state fluorescence). These spectra are then satisfactorily described in the best harmonic approximation, a local harmonic approach that approximates the PES with parabolas whose curvatures match the exact curvatures calculated at the specific position of interest [78]. Anharmonicity in this approach manifests itself with the dependence of harmonic frequencies and relaxation energies on the actual nuclear configuration [79]. Along these lines we predicted softened (hardened) vibrational frequencies for the ground (excited) state [74], amplified and p-dependent infrared and Raman intensities [68, 74], different Frank-Condon... 

Figure 20. The 5j 12j dispersed fluorescence of ultracold anthracene in a supersonic beam. The available vibrational energy is 1792 cm1. The parameters of the optically active modes are given in Table II. The top figure is the experimental spectrum.60 The bottom figure is the emission in the harmonic approximation (y6.6 = 0). The calculation clearly fails to reproduce the broad redistributed emission. The middle figure was calculated with IVR [Eqs. (131)]. Only one b ) state (the ground vibrational state f/> = 0 was used. yt.t/y, = 40. The relaxed emission was calculated in the fast modulation limit [Eq. (116a)], with f0 = 2f = 75 cm-1.61...

The contribution of each vibrational mode to X, can be obtained by expanding the potential energies of the neutral and cation states in a power series of the normal coordinates (denoted here as 2, and Q2). In the harmonic approximation, the relaxation energy X writes [1-5,17-21] ... 

The role of two-phonon processes in the relaxation of tunneling systems has been analyzed by Silbey and Trommsdorf [1990]. Unlike the model of TLS coupled linearly to a harmonic bath (2.39), bilinear coupling to phonons of the form Cijqiqja was considered. In the deformation potential approximation the coupling constant Cij is proportional to (y.cUj. There are two leading two-phonon processes with different dependence of the relaxation rate on temperature and energy gap, A = (A Two-phonon emission prevails at low temperatures, and it is... 

Z is proportional to the gas pressure, and, since Z1>0, the collision number for energy transfer, is constant for a particular transition, the actual value of fi is inversely proportional to the pressure. For convenience relaxation times are usually referred to a pressure of 1 atm. Equation (1) is an approximation, and requires modification to take into account the reversibility between quantum states 0 and 1. For example, the correct equation for vibrational relaxation of a simple harmonic oscillator of fundamental frequency, v, is... 

Equation (13.39) implies that in the bilinear coupling, the vibrational energy relaxation rate for a quantum hannonic oscillator in a quantum harmonic bath is the same as that obtained from a fully classical calculation ( a classical harmonic oscillator in a classical harmonic bath ). In contrast, the semiclassical approximation (13.27) gives an error that diverges in the limit T 0. Again, this result is specific to the bilinear coupling model and fails in models where the rate is dominated by the nonlinear part of the impurity-host interaction. 

We have seen that vibrational relaxation rates can be evaluated analytically for the simple model of a hannonic oscillator coupled linearly to a harmonic bath. Such model may represent a reasonable approximation to physical reality if the frequency of the oscillator under study, that is the mode that can be excited and monitored, is well embedded within the spectrum of bath modes. However, many processes ofinterest involve molecular vibrations whose frequencies are higherthan the solvent Debye frequency. In this case the linear coupling rate (13.35) vanishes, reflecting the fact that in a linear coupling model relaxation cannot take place in the absence of modes that can absorb the dissipated energy. The harmonic Hamiltonian... 

Stretching motion (amide I mode) was found to occur on a 1 picosecond or subpicosecond timescale [24-28], whereas vibrational relaxation of photolyzed carbon monoxide in the heme pocket can last from tens to hundreds of picoseconds in different Mb mutants [7,29,30], Starting from a basis of harmonic normal modes and including the contribution of cubic anharmonicity in the potential energy, providing nonlinear coupling, Leitner and coworkers estimated the thermal diffusivity of Mb at 300K to be 14 ps", approximately the value for water. The thermal conductivity... 

Combination and difference bands Besides overtones, anharmonicity also leads to the appearance of combination bands and difference bands in the IR spectrum of a polyatomic molecule. In the harmonic case, only one vibration may be excited at a time (the transition dipole moment integral vanishes when the excited state is given by a product of more than one Hermite polynomial corresponding to different excited vibrations). This restriction is relaxed in the anharmonie case and one photon can simultaneously excite two different fundamentals. A weak band appears at a frequency approximately equal to the sum of the fundamentals involved. (Only approximately because the final state is a new one resulting from the anharmonie perturbation to the potential energy mixing the two excited state vibrational wave functions.)... 

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