Selection rules, for the harmonic

STATIC ELECTRIC-DIPOLE SELECTION RULES FOR THE HARMONIC OSCILLATOR... 

Selection Rules for all Laue Classes. The selection rules for the harmonic coefficients are derived from the invariance of the pole distribution to the operations of the crystal and sample Laue groups. The invariance conditions are applied to every function t/ (h, y) from Equation (36), as the terms of different / in this equation are independent. If we compare Equations (38) and (39) with (37) we observe that they have an identical structure. On the other side the sample and the crystal coordinate systems were similarly defined. As a consequence the selection rules for the coefficients of", flf", and respectively y) ", resulting from the sample symmetry must be identical with the selection rules for the coefficients A P and resulting from the crystal symmetry, if the sample and the crystal Laue groups are the same. The exception is the case of cylindrical sample symmetry that has no correspondence with the crystal symmetry. In this case, only the coefficients af and y l are different from zero, if they are not forbidden by the crystal symmetry. 

The selection rule for the harmonic oscillator, Eq. (25.55), requires that An = 1. Under the influence of a light beam the harmonic oscillator makes transitions only to states immediately above and below its original state. The existence of selection rules simplifies the interpretation of spectra enormously. 

According to the selection rule for the harmonic oscillator, any transitions corresponding to An = 1 are allowed (Sec. I-2). Under ordinary conditions, however, only the fundamentah that originate in the transition from u = 0 to u = 1 in the electronic ground state can be observed because of the Maxwell-Boltzmann distribution law. In addition to the selection rule for the harmonic oscillator, another restriction results from the symmetry of the molecule (Sec. 1-9). Thus the number of allowed transitions in polyatomic molecules is greatly reduced. The overtones and combination bands of these fundamentals are forbidden by the selection rule of the harmonic oscillator. However, they are weakly observed in the spectrum because of the anharmonicity of the vibration... 

Vibrational Selection Rules for the Harmonic OsdUator. If the higher terms in (7) are neglected, and if the vibrational wave function f/y is assumed to be strictly of the form described earlier in this chapter, that is, a product of harmonic oscillator functions, then the selection rules for vibrational transitions are very restrictive indeed. The integral for fix becomes... 

VIII. Time-Dependent Perturbations Radiation Theory Time-Dependent Perturbations, 107. The Wave Equation for a System of Charged Particles under the Influence of an External Electric or Magnetic Field, 108. Induced Emission and Absorption of Radiation, 110. The Einstein Transition Probabilities, 114. Selection Rules for the Hydrogen Atom, 116. Selection Rules for the Harmonic Oscillator, 117. Polarizability Rayleigh and Raman Scattering, 118. 

The vibrational selection rule for the harmonic oscillator, Au = 1, applies to polyatomic molecules just as it did to diatomic molecules. Vibrational energy can, therefore, change in units of hcoi/ln. Transitions in which one of the three normal modes of energy changes by Au = - -1 (for example Ui = 0 1, U2 = U3 = 0 or 1 = 1) 2 = 3, i>3 = 2 3) result from absorption of a photon having one of three fundamental frequencies of the molecule. In the actual case, anharmonicities also allow transitions with Au, = 2, 3,... so that, for example, weak absorption also occurs at 2coi, 3(Ui, etc. and at coi + coj, 2vibrational transitions often play major roles in planetary spectroscopy. 

The selection rules for the QM harmonic oscillator pennit transitions only for An = 1 (see Section 14.5). As Eq. (9.47) indicates diat the energy separation between any two adjacent levels is always hm, the predicted frequency for die = 0 to n = 1 absorption (or indeed any allowed absorption) is simply v = o). So, in order to predict die stretching frequency within the harmonic oscillator equation, all diat is needed is the second derivative of the energy with respect to bond stretching computed at die equilibrium geometry, i.e., k. The importance of k has led to considerable effort to derive analytical expressions for second derivatives, and they are now available for HF, MP2, DFT, QCISD, CCSD, MCSCF and select other levels of theory, although they can be quite expensive at some of the more highly correlated levels of theoiy. 

Allowed transitions in the harmonic approximation are those for which the vibrational quantum number changes by one unit. Overtones - that is, the absorption of light at a whole number times the fundamental frequency - would not be possible. A general selection rule for the absorption of a photon is that the dipole... 

Let us consider the case of the harmonic approximation (see Equation 4.15). Because of its mathematical properties, the quantum mechanical solution yields that the selection rule for the IR and Raman transitions is [12]... 

The dominant term for vibrational transitions is, of course, the second, which gives the primary selection rule for a harmonic oscillator of Ac = 1. The overtone transitions Av = 2, 3, etc., are very much weaker because of the rapid convergence of (6.325). 

Let a molecule have two vibrational modes with frequencies uia and uii,. If the second-order resonance condition 2uia uif, is fulfilled, then the huif, transition in the infrared spectrum can split if the interaction is allowed by symmetry of molecule into two lines of comparable intensity. The second line cannot be explained as a result of the interaction of light with the vibrational a mode because the transition with excitation of two hwa quanta is forbidden due to the well-known n — n 1 selection rule for a harmonic oscillator. Fermi explained (7) this experimental observation as a result of a nonlinear resonance interaction of two vibrational modes with each other. Since that time the notion of Fermi resonance has been generalized to processes with participation of different types of quanta (e.g. + iv2 u>3, lo + iv2 — UJ3 — 0J4, and so on) and to elec-... 

The electric dipole selection rule for a harmonic oscillator is Av = 1. Because real molecules are not harmonic, transitions with Av > 1 are weakly allowed, with Av = 2 being more allowed than Av = 3 and so on. There are other selection rules for quadrupole and magnetic dipole transitions, but those transitions are six to eight orders of magnitude weaker than electric dipole transitions, and we will therefore not concern ourselves with them. 

As stated in Sec. 1-2, some overtones and combination bands are observed weakly because the actual vibrations are not harmonic and some of them are allowed by symmetry selection rules. For the symmetry selection rules of these nonfundamental vibrations, see Refs. 3, 7, and 8. 

Selection rules for the high-order harmonic generation spectra 404... 

SELECTION RULES FOR THE HIGH-ORDER HARMONIC GENERATION SPECTRA... 

Deriving the selection rules for the IR and Raman spectra, we assumed that the equivalent atoms can differ only by the sign of the deviation from the equilibrium position, but its absolute value is the same. This is how it would be for ahamionic oscillator. An anharmonicity introduces, therefore, another reason why a (harmonically) forbidden transition will have a non-negligible intensity. 

Finally, let us consider the relationship between the vibrational motions and the infrared (IR) absorption spectra. The IR spectra show the frequencies corresponding to the energy gaps in the transitions between vibrational eigenstates, with the peak intensities proportional to the transition moments. The transitions between vibrational states have rules called selection principles for the harmonic oscillator, transitions take place for the eigenstate pairs with A = 1. This selection principle comes from the fact that the transition moment, which is proportional to the transition dipole moment. 

Using an identity from Appendix F, derive the selection rule for a harmonic oscillator, An = 0, 1. 

For a harmonic oscillator, the selection rule for the Raman effect is the following. Let us assume that the harmonic oscillator is originally in the state a with quantum number n. Then the matrix element (ci R y) will be different from zero only if the state j has the quantum number nil. Similarly, if state h has the quantum number nij 0 R ) will be different from zero only if state y has the quantum number mil. Both matrix elements will be simultaneously different from zero only if m = n or m = n 2, so that we may conclude that the selection rule for Raman scattering by a harmonic oscillator is An = 0, 2. The first possibility corresponds to scattering of light of the incident frequency v the second corresponds to scattering of light of frequency v db 2vg, where is the fundamental frequency of the harmonic oscillator. 

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