Vibrational wavefunction

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]).

The last factor, the square of the overlap integral between the initial and final vibrational wavefunctions, is called the Franck-Condon factor for this transition. 

In the Bom-Oppenlieimer approxunation the vibronic wavefrmction is a product of an electronic wavefimction and a vibrational wavefunction, and its syimnetry is the direct product of the synuuetries of the two components. We have just discussed the synuuetries of the electronic states. We now consider the syimnetry of a vibrational state. In the hanuonic approximation vibrations are described as independent motions along nonual modes Q- and the total vibrational wavefrmction is a product of frmctions, one wavefunction for each nonual mode ... 

Using the results of Problem 11 of this chapter to illustrate, the sudden ionization ofN2 in its v=0 vibrational state to generate N2 produces a vibrational wavefunction... 

This example relates to the well known Franck-Condon principal of spectroscopy in which squares of overlaps between the initial electronic state s vibrational wavefunction and the final electronic state s vibrational wavefunctions allow one to estimate the probabilities ofpopulating various final-state vibrational levels. 

The normalized vibrational wavefunctions are given by the general expression... 

Vibrational wavefunctions for the states u = 0 and u = 1 are shown in Figures 1.4 and 1.5. For the sake of illustration, I have taken numerical values appropriate to The x-axis legend variable is Note that the... 

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

One of the most familiar uses of dipole derivatives is the calculation of infrared intensities. To relate the intensity of a transition between states with vibrational wavefunctions i/r and jfyi it is necessary to evaluate the transition dipole moment... 

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... 

Appendix Normal Coordinates, Vibrational Wavefunctions, and Spectral Activities. 339... 

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

As described above, the ground state vibrational wavefunction is totally symmetric for most common molecules. Therefore, the product, -(1)0 must at least contain a totally symmetric component. The direct product of two irreducible representations contains the totally symmetric representation only if the two irreducible representations are identical. Therefore, transitions can occur from a symmetrical initial state only to those states that have the same symmetry properties as the transition operator, 0. 

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. 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

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 TD wavefunction satisfying the Schrodinger equation ih d/dt) F(f) = // (/,) can be expanded in a basis set whose elements are the product of the translational basis of R, vibrational wavefunctions for r, r2, and the body-fixed (BF) total angular momentum eigenfunctions as41... 

Fig. 1. The lowest singlet (So) and first excited singlet (Si) surfaces of two hypothetical molecules. Vibrational wavefunctions for one and three vibrational levels, respectively, are indicated. Top part of a indicates schematically the time development of the nuclear geometry probability distribution after initial excitation...

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 the quantum mechanical description (in continuation of Box 2.2), the wavefunction can be described by the product of an electronic wavefunction VP and a vibrational wavefunction / (the rotational contribution can be neglected), so that the probability of transition between an initial state defined by ViXa and a final state defined by TQ/b is proportional to electron coordinates, this expression can be rewritten as the product of two terms < f i M vP2> 2 Franck-Condon factor. Qualitatively, the transition occurs from the lowest vibrational state of the ground state to the vibrational state of the excited state that it most resembles in terms of vibrational wavefunction. 

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... 

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