Resonance selection rules

One has at hand, therefore, a completely general semiclassical mechanics which allows one to construct the classical-limit approximation to any quantum mechanical quantity, incorporating the complete classical dynamics with the quantum principle of superposition. As has been emphasized, and illustrated by a number of examples in this review, all quantum effects— interference, tunnelling, resonances, selection rules, diffraction laws, even quantization itself—arise from the superposition of probability amplitudes and are thus contained at least qualitatively within the semiclassical description. The semiclassical picture thus affords a broad understanding and clear insight into the nature of quantum effects in molecular dynamics. 

This spectrum is called a Raman spectrum and corresponds to the vibrational or rotational changes in the molecule. The selection rules for Raman activity are different from those for i.r. activity and the two types of spectroscopy are complementary in the study of molecular structure. Modern Raman spectrometers use lasers for excitation. In the resonance Raman effect excitation at a frequency corresponding to electronic absorption causes great enhancement of the Raman spectrum. 

The hyperpolarizability tensor is obtained in a way similar to the case of SHG. However, the selection rules for an SFG resonance at the IR frequency implies that the vibrational mode is both IR and Raman active, as the SF hyperpolarizability tensor elements involve both an IR absorption and a Raman-anti-Stokes cross-section. Conversely, the DFG hyperpolarizability tensor elements involve an IR absorption and a Raman-Stokes cross-section. The hyperpolarizability tensor elements can be written in a rather compact form involving several vibrational excitations as [117] ... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

A. Jorio, A.G.S. Filho, V.W. Brar, A.K. Swan, M.S. Unlu, B.B. Goldberg, A. Righi, J.H. Hafner, C.M. Lieber, R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Polarized resonant Raman study of isolated single-wall carbon nanotubes symmetry selection rules, dipolar and multipolar antenna effects. Phys. Rev. B 65, 121402.1-121402.4 (2002). 

Thus for an x-wave (l = 0), which corresponds to hitting the bull s eye and so usually dominates unless forbidden by selection rules or upstaged by a resonance, the total cross-section (of which the major component is normally elastic scattering) is given by... 

Unfortunately, the different selection rules that apply to resonant and normal Raman scattering were not taken into account in this spectral interpretation. In the following, it is shown that the conclusions and assignments mentioned above have to be modified on the basis of symmetry considerations as discussed by Ricchiardi et al. (41). 

In a case where the transition of an energy state is from 0 to 1 in any one of the vibrational states (vi,v2,v3,. ..), the transition is considered as fundamental and is allowed by selection rules. When a transition is from the ground state to v — 2,3,. .., and all others are zero, it is known as an overtone. Transitions from the ground state to a state for which Vj = 1 and vj = 1 simultaneously are known as combination bands. Other combinations, such as v — 1, Vj = 1, v = 1, or v, — 2, v7 — 1, etc., are also possible. In the strictest form, overtones and combinations are not allowed, however they do appear (weaker than fundamentals) due to anharmonicity or Fermi resonance. 

Infrared, Raman, microwave, and double resonance techniques turn out to offer nicely complementary tools, which usually can and have to be complemented by quantum chemical calculations. In both experiment and theory, progress over the last 10 years has been enormous. The relationship between theory and experiment is symbiotic, as the elementary systems represent benchmarks for rigorous quantum treatments of clear-cut observables. Even the simplest cases such as methanol dimer still present challenges, which can only be met by high-level electron correlation and nuclear motion approaches in many dimensions. On the experimental side, infrared spectroscopy is most powerful for the O—H stretching dynamics, whereas double resonance techniques offer selectivity and Raman scattering profits from other selection rules. A few challenges for accurate theoretical treatments in this field are listed in Table I. 

As a consequence of the quantum-mechanical selection rules for resonance energy transfer involving the spin wave functions of the donor and acceptor,... 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

Electric-quadrupole transition, 123,127 Electromagnetic radiation, 114-117. See also Radiation, electromagnetic Electromagnetic spectrum, 115 Electronic energy, 57,64,148 Electronic spectra, 130, 296-314 of diatomics, 298-306 and molecular structure, 311 of polyatomics, 71-72, 73, 75, 306-314 selection rules for, 297-301, 306-307 Electronic structure of molecules, 56-76 Electron spectroscopy for chemical analysis (ESCA), 319-320 Electron spin resonance (ESR), 130, 366-381... 

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