Harmonic oscillator selection rules

In deriving the relations between infrared absorption intensities and dipole moment derivatives we have restricted the treatment to transitions involving a change of a single 

The harmonic oscillator selection rules for vibrational transition can be evaluated using the expression of the dipole moment as a power series with respect to normal coordinates. On the basis of expressions (1.25) - (1.27) tiie dipole moment matrix element for a transition between vibrational states V and V may be written as 

The harmonic approximation restricts the dipole moment expansion to the constant and linear terms. Thus, the selection rule associated with the approximation of electrical hannonicity, states that transitions involving a change by 1 of just one of the vibrational quantum numbers of the linear harmonic oscillator functions defining the vibrational states of molecules are only allowed. Aside from this, for a transition to take place at least one of tiie Cartesian components of the (dp/dQk)o derivatives should differ from 

The third term in expression (l.Sl) considers transitions associated with a change in a vibrational quantum number by 2. It governs the intensities of the usually weak overtone bands. The possibility of observing such transitions is determined by the fact that vibrations in real molecules are not strictly harmonic, both with respect to potential energy and dipole moment functions. When one quantum number is changed by 2, the linear term has also a finite, usually small, contribution to die matrix element 

The fourth term is associated with the intensities of the weak combination bands (vibrational sum or difference bands). These bands are due to transitions involving changes by 1 of two vibrational quantum numbers. 

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As for a diatomic molecule, the general harmonic oscillator selection rule for infrared and Raman vibrational transitions is... 

There is one additional property of the overtones that might reveal the anharmonic character of y, the absorption coefficient. The appearance of overtones in the IR spectrum, in violation of the harmonic oscillator selection rules, is attributed to electrical and mechanical anharmonicity. If the H bond caused additional anharmonicity, the intensity of the harmonics should increeise with H bond formation. 

As seen in Eqs. 1.216 and 1.217, both A and B terms involve summation over v, which is the vibrational quantum number at the electronic excited state. Since the harmonic oscillator selection rule (Au = =t 1) does not hold for a large u, overtones and combination bands may appear under resonance conditions. In fact, series of these nonfundamental vibrations have been observed in the case of A-term resonance (Sec. 1.23). 

In vibrational Raman scattering, which is the primary technique of interest in studies of proton conductors, the Born approximation is invoked to write the state i > as the product of an electronic wave function, e>, and a vibrational wave function, d >, i.e. i> = k> o >. The subscript m on the vibrational wave function designates the mth normal vibrational mode. Usually the transition j k occurs between the vibrational level v in the ground electronic state g and the vibrational level v also in the ground electronic state, so the transition jy - k> may be written Gy m + f>- Here the harmonic oscillator selection rule =... 

Similar symmetiy restrictioas also apply for overtone and combination bands. As already discussed, these transitions are not allowed under the harmonic oscillator selection rules. It should be pointed out that even if a given transition is not forbidden under both symmetry and harmonic oscillator selection rules, it may have a very low intensity. This will be determined by the particular form of the vibration and die electronic structure of the molecule. The assignment of a given band to infiared active or forbidden transition is, therefore, not always a straightforward task. 

Experimental. The vibrational spectrum of an ideal harmonic oscillator would consist of one line at frequency v corresponding to A = hv, where A is the distance between levels on the vertical energy axis in Fig. 10-la. In the harmonic oscillator, AE is the same for a transition from one energy level to an adjacent level. A selection rule An = 1, where n is the vibrational quantum number, requires that the transition be to an adjacent level. 

The selection rules illustrated above are general, as they depend only on the symmetry properties of the functions involved. However, more limiting, selection rules depend on the form of the wavefunctions involved. A relatively simple example of the development of specific selection rules is provided by the harmonic oscillator. The solution of this problem in quantum mechanics,... 

The general form of the energy of the harmonic oscillator indicates that the vibrational energy levels are equally spaced. Due to the vector character of the dipole transition operator, the transition between vibronic energy levels is allowed only if the following selection rule is satisfied ... 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

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. 

The derivation above may be generalized to wave functions other than electronic ones. By evaluation of transition dipole matrix elements for rigid-rotor and harmonic-oscillator rotational and vibrational wave functions, respectively, one arrives at the well-known selection rules in those systems that absorptions and emissions can only occur to adjacent levels, as previously noted in Chapter 9. Of course, simplifications in the derivations lead to many forbidden transitions being observable in the laboratory as weakly allowed, both in the electronic case and in the rotational and vibrational cases. 

An alternative but not so general selection rule (it is restricted to the harmonic oscillator approximation) is that jVoI> / v5i dr is zero if dfiJdQ (evaluated for the equilibrium nuclear configuration) is zero, i.e. if there is no linear dependence of the dipole moment on the normal coordinate Q . 

What are the electric-dipole selection rules for a three-dimensional harmonic oscillator exposed to isotropic radiation ... 

Let r and be the irreducible representations to which the vibrational wave functions in (9.189) belong. According to the italicized statement following (9.187), the integral (9.189) vanishes unless T+<8>riratls contains T, . This is the basic IR selection rule. (Note that its deduction does not involve the harmonic-oscillator approximation or the approximation of... 

Since the selection rule for nonzero Qi and Pi matrix elements in the harmonic oscillator basis is Av = 1, and since the definition of a polyad is such that all pairs of states differing by only one vibrational quantum number... 

Under the assumptions of a harmonic oscillator and an interaction potential, which is linear in (r — re), only the next lowest vibrational channel n = n — 1 can be populated. The propensity rule n — n — 1 is a strict selection rule. 

If only the first derivatives in tie dipole-moment function and the second derivatives (k=j) in the potential function are retained, the strict harmonic-oscillator-linear-dipole-moment approximation, the selection rules are strict ... 

According to quantum mechanics, only those transitions involving Ad = 1 are allowed for a harmonic oscillator. If the vibration is anhar-monic, however, transitions involving Au = 2, 3,. .. (overtones) are also weakly allowed by selection rules. Among many Au = 1 transitions, that of u = 0 <-> 1 (fundamental) appears most strongly both in IR and Raman spectra. This is expected from the Maxwell-Boltzmann distribution law, which states that the population ratio of the u = 1 and u = 0 states is given by... 

Overtones and combination tones are forbidden in the harmonic oscillator approximation. Mechanical anharmonicity is one factor that might give them intensity through violating the Av = I selection rule. There is another possible cause for this, however, which might operate even when the oscillator is perfectly harmonic. It is electrical anharmonicity. 

STATIC ELECTRIC-DIPOLE SELECTION RULES FORTHE HARMONIC OSCILLATOR... 

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). 

Selection Rules. The harmonic-oscillator, rigid-rotor selection mles are Av = 1 and AJ = 1 that is, infrared emission or absorption can occur only when these allowed transitions take place. For an anharmonic diatomic molecule, the A7 = 1 selection mle is still valid, but weak transitions corresponding to An = 2, 3, etc. (overtones) can now be observed. Since we are interested in the most intense absorption band (the fundamental ), we are concerned with transitions from various J levels of the vibrational ground... 

From such considerations the symmetry species of each wavefunction associated with an energy level is determined, and these are indicated at the right in Fig. 1. It is important to realize that this symmetry label is the correct one for the true wavefunction, even though it is deduced from an approximate harmonic-oscillator model. This is significant because transition selection rules based on symmetry are exact whereas, for example, the usual harmonic-oscillator constraint that An = 1 is only approximate for real molecules. 

It follows that the selection rule for electric-dipole transitions [cf. Eq (4.70)] in a harmonic oscillator is An = 1. 

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