The Vibrational Wavefunctions

Q) is the vibrational frequency of the pth oscillator. The eigenfunctions and eigenvalues of (1.46) are known and given by 

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Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

A plot of the square of the vibrational wavefunction with v = 30 is shown in Figure 1.8. 

If we further assume that the vibrational wavefunctions associated with normal mode i are the usual harmonic oscillator ones, and r = u + 1, then the integrated intensity of the infrared absorption band becomes... 

The vibrational wavefunctions may be expressed as functions of the jth normal coordinate ... 

Fig. 2 The experimentally determined potential energy V(), expressed as a wavenumber for convenience, as a function of the angle in the hydrogen-bonded complex H20- HF. The definition of

We note from Fig. 2 that the hypothetical equilibrium conformation is pyramidal, with 0e = 46(8)°, even though the geometry of the complex is effectively planar in the zero-point state (i.e. the vibrational wavefunction has C2v symmetry) because the PE barrier at the planar (0 = 0) form is low. At the time of the publication of [112] this was a critical result because it demon-... 

It is possible that the complexes benzene- -HX can be described in a similar way, but in the absence of any observed non-rigid-rotor behaviour or a vibrational satellite spectrum, it is not possible to distinguish between a strictly C6v equilibrium geometry and one of the type observed for benzene- ClF. In either case, the vibrational wavefunctions will have C6v symmetry, however. 

Fig. 5. Square of the vibrational wavefunction of NO for v = 0 and v = 15. Vibrational amplitude in v = 15 induces large changes in the electronic structure of the molecule, when one compares the molecule at its outer turning point, where it is already beginning to resemble separated atoms.

The variations in efficiency (rate) of radiationless transitions result from differences in the Franck-Condon factor, visualised by superimposing the vibrational wavefunctions, / (or /2 - the probability distributions), of the initial and final states. We will consider three cases illustrated in Figure 5.2. 

In Figure 5.16(a), the maximum of the absorption spectrum (at 0 K) corresponds to the line AB, the maximum overlap of the vibrational wavefunctions. This transition terminates in the vibrational level corresponding to point B, which is below the crossover point, X. This proces s is followed by a fast down-relaxation by multiphonon emission to the point C, from which the emission originates. Thus, the emission spectrum has its maximum at an energy corresponding to the line CD. Finally, another multiphonon emission process takes place by down-relaxation from D to the departing point A. 

The methods described above are all based on the Born-Oppenheimer approximation. Therefore, they can be used to calculate polarizabilities of diatomic molecules for a given internuclear distance R. However, if one is interested in values of the polarizability tensors, and C", for a particular vibrational state /i )), one has to average the polarizability radial functions a(R) and C(R) with the vibrational wavefunction i.e., one has to... 

For diatomic molecules the vibrational wavefunctions can be obtained numerically as solution of the one-dimensional Schrddinger equation... 

In Table 1 the predicted dipole and quadmpole polarizability tensor components ay and C,y for the vibrational states with quantum number v are given. They were calculated for all vibrational states supported by the potential energy function as expectation values of the polarizability radial functions a(R) and C(R) over the vibrational wavefunction (equation (14)). The latter were obtained from... 

Note that the potential energy V(x) rises to infinite values at sufficiently large displacement. One should expect this boundary condition to mean that the vibrational wavefunction will fall to zero amplitude at large displacement (as in the square well case, but less abruptly). One should also expect that the confining potential well would lead to quantized solutions, as is indeed the case ... 

The most likely electronic transition will occur without changes in the positions of the nuclei (e.g., little change in the distance between atoms) in the molecular entity and its environment. Such a state is known as a Franck-Condon state, and the transition is referred to as a vertical transition. In such transitions, the intensity of the vibronic transition is proportional to the square of the overlap interval between the vibrational wavefunctions of the two states. See Fluorescence Jablonski Diagram Comm, on Photochem. (1988) Pure and Appl. Chem. 60, 1055. 

Again consider the v=0 N2 ionization treated in Problem 11 of this chapter. If subsequent to ionization, the N2+ ions produced were probed to determine their internal vibrational state, a fraction of the sample equal to 2 = 0.92 would be detected in the v=0 state of the N2+ ion. For this sub-sample, the vibrational wavefunction becomes, and remains from then on,... 

Solutions for which J O require the vibrational wavefunction and energy to respond to the presence of the centrifugal potential given by -h2 J(J+l)/(2 oR2) these solutions obey the full coupled V/R equations given above. 

Figure 3.6 shows the Morse potential energy curves for two hypothetical electronic states of a diatomic molecule, the vibrational energy levels for each, and the shape of the vibrational wavefunctions (i//) within... 

Thus, the absorption to the excited electronic state depends on the electronic transition dipole moment, the Franck-Condon (EC) overlap between the vibrational wavefunctions in both electronic states and the vibrational excitation probability. Indeed, as seen from the schematic representation in Eigure 2.1b, the absorption spectrum represents the reflection of the wavefunction, but it is also dependent on the EC factors that lead to intensity alterations in the observed features. 

We then discover an extremely important fact each normal coordinate belongs to one of the irreducible representations of the point group of the molecule concerned and is a part of a basis which can be used to produce that representation. Because of their relationship with the normal coordinates, the vibrational wavefunctions associated with the fundamental vibrational energy levels also behave in the same way. We are therefore able to classify both the normal coordinates and fundamental vibrational wavefunctions according to their symmetry species and to predict from the character tables the degeneracies and symmetry types which can, in principle, exist. 

Furthermore, knowledge of the irreducible representations to which the vibrational wavefunctions belong coupled with the vanishing integral rule tells us a good deal about the infra-red and Raman spectra of the molecule under consideration. 

Classifying the vibrational energy levels means finding out to which irreducible representation of the molecular point group the vibrational wavefunction(s) associated with a given level belong. 

The dependence of the vibrational wavefunctions v j v2j2) on the rotational quantum numbers is sometimes a weak one and may in that case be suppressed, in particular in the rototranslational bands, 1 17 1 272) ItfiOt O), or short v v2). 

Returning to equation (25), evaluation of the total vibrational overlap integral, (Xj X7), is less formidable than it appears. The vibrational wavefunctions are a complete orthonormal set for which ( 1 0 )= where S is the Kronecker delta. For the vast majority of normal modes, S (and AQe) = 0. For these modes the vibrational overlap integrals become (yjy,/) = 1 if v = v, and = 0 if v v . Except for the requirement that the vibrational quantum number must... 

Here p is the density of vibrational levels of states Sj and Sf at the energy of the electronic transition E. The overlap of the electronic wavefunctions 0i5 0f and of the vibrational wavefunctions (0i 0f) are factorized according to the Born-Oppenheimer approximation just as in the case of radiative transitions. The density of vibrational levels is greater for the lower (final) state Sf... 

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