Translation operators, vibrational modes

The fact that using excitation in the CT absorption a number of Raman bands appears in the intermolecular mode region,while the intramolecular bands practically disappear, can be understood on the basis of the dimeric model. The modulation of the transfer integral t by the intermolecular vibrations, particularly the antiphase translational modes, provides an efficient mechanism for intensity enhancement at resonance with the CT transition. No such mechanism is operative for the intramolecular vibrational modes. 

Thus, the fifth term in eq. (5.9) arises from the expectation value of the linear term of the intermolecular potential function (which is assumed to be expanded up to second order in the oscillator coordinates) and is referred to as the V-T (vibration-translation) coupling term. The V-T term mainly depends on the linear force fk, the first term of which contains the coupling term BkF which couples the reaction path to the perpendicular vibrational mode A , and a matrix element hk s) of the second quantized creation and anhilation operators. BkF is a measure of the curvature of the reaction path and determines the amount of coupling to a given vibrational degree of freedom. 

The factor-group analysis predicts a total of 98 lattice-vibration modes however, the character table for the point group Oh shows that the translation operations are of symmetry species Fiu only. One acoustic mode will also be found in this species and will be infrared-inactive. Therefore, this analysis predicts seventeen infrared-active lattice modes. In table 29.17, ten frequencies were listed for the RIG series. Twelve frequencies were listed for the RGaG series (table 29.19), and fifteen frequencies were listed for the RAG series (table 29.18). They suggest that the powder-transmission method does not yield all of the infrared-active modes. 

We present in Section 2 the formalism giving the equations for the reduced density operator and for competing instantaneous and delayed dissipation. Section 3 presents matrix equations in a form suitable for numerical work, and the details of the numerical procedure used to solve the integrodiffer-ential equations with the two types of dissipative processes. In Section 4 on applications to adsorbates, results are shown for quantum state populations versus time for the dissipative dynamics of CO/Cu(001). The fast electronic relaxation to the ground electronic state is shown first without the slow relaxation of the frustrated translation mode of CO vibrations, for comparison with previous work, and this is followed by results with both fast and slow relaxation. In Section 5 we comment on the general conclusions that can be reached in problems involving both vibrational and electronic relaxation at surfaces. 

The internal modes can be seen in Fig. 5.2(b) above 1200 cm. This region of the spectrum is very rich and again there are too many features to be explained by the two fundamentals alone. The external mean square displacement value is so big that most of the intensity arising from the internal vibrations is actually found in the phonon wings ( 2.6.5). Here the full wing is operative and includes translational and librational contributions. Indeed few of the sharp features present in this part of the spectrum are due to zero phonon transitions of the fundamentals. This is but a slightly more severe case of the analysis that will be covered in detail for benzene and we end our analysis of the pure ammonium bromide salt here. 

The operator of slow processes jf-j consists of the operator of W2 vibrational transitions between molecules of different species, the operator describing the transfer of vibrational energy into rotational and translational modes as well as the operator of chemical... 

LMOD operates in neither torsion space nor Cartesian space, nor any other user-defined search space LMOD generates its own search space LMOD determines the low-mode search directions in conformational space automatically and is appropriate for any kind of molecular systems including rings and molecular complexes. LMOD is inherently well suited for conformational searching of complexes, because the relative translations and rotations necessary to guide one molecule with respect to the other(s) appear as low-frequency vibrations on the intermolecular PES. 

The term hk k/ k arises from the expectation value of the linear terms in (8.66). These terms are called phonon VT terms because they exchange one quantum of vibrational energy when forced by the translational motion. The second derivative terms in (8.66) contain operators such as ar> and Ord y i.e., these terms create a quantum in one mode and destroy one in another. It has, however, turned out that these so-called VV terms are of minor importance for energy transfer in solids [62]. These terms may therefore be neglected. Thus if the higher-order terms are dropped we also have the simplification 0jt(r) = (Okt in eq. (8.73). 

---