Selection rules for the harmonic oscillator

Consider a diatomic molecule that has a dipole moment due to effective charges — q and — q separated by a distance r then = qr wQ rewrite this as 

taking r in the direction of the field, the dipole transition-moment integrals have the form, 

Since m n, the first integral vanishes because J/ and ij/ are orthogonal. The normalized harmonic oscillator functions are given by Eq. (21.42), 

To evaluate this integral we use the recurrence relation Eq. (21.49) that is, This brings to the form 

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

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... 

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 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. 

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. 

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... 

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

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. 

Evaluation of the transition moment for the harmonic oscillator and for the two-particle rigid rotor gives the selection rules Au = 1 and A/ = 1 stated in Sections 4.3 and 6.4. 

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. 

The maximum of the ground-state probability density for the harmonic oscillator corresponds indeed just to the equilibrium geometry. This is why the selection rules work at all (although in an approximate way). 

Infrared Selection Rules and Intensities for the Harmonic Oscil... 

The selection rule states that, in the quantum mechanical model, a molecnle may only absorb (or anit) the hght of an energy eqnal to the spacing between two levels. For the harmonic oscillator (dipole involving two atoms), the transitions can only occur from one level to the next higher (or lower) level. 

In our treatment of IR selection rules (Appendix 6) we have written wavefunctions for the harmonic oscillator without reference to the electronic state of the molecule. In fact, all the detail of the electronic states is assumed to be contained within the spring constant for the bond. To characterize the molecule fully we would need to take into account the nuclear and electronic coordinates when defining the potential energy. Rotational and translational degrees of freedom could also be included, adding more coordinates to describe the molecular motion of the system. However, we will only consider the internal structure of molecules, and so these additional factors will be left to one side. 

Selection rules for spectroscopic experiments are derived from time-dependent perturbation theory. Transitions are allowed if the integral of the perturbing Hamiltonian and initial and final stationary states is nonzero. For the harmonic oscillator, the allowed transitions are those for which the n quantum number changes by 1, provided that there exists a dipole that varies linearly with the separation distance. (N2 has a zero dipole because of its symmetry, and there is no linear variation with distance.) For the rigid rotator, the selection rules are that the allowed transitions are a change of 1 in the / quantum number, provided that there is a nonzero permanent dipole, po-... 

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. 

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. 

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. 

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 ... 

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... 

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

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... 

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