Vibrational structure
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص484-485
2025-12-07
36
Vibrational structure
To account for the vibrational structure in electronic spectra of molecules (Fig. 14.6), we apply the Franck–Condon principle:
Because the nuclei are so much more massive than the electrons, an electronic transition takes place very much faster than the nuclei can respond. As a result of the transition, electron density is rapidly built up in new regions of the molecule and removed from others. The initially stationary nuclei suddenly experience a new force field, to which they respond by beginning to vibrate and (in classical terms) swing backwards and forwards from their original separation (which was maintained during the rapid electronic excitation). The stationary equilibrium separation of the nuclei in the initial electronic state therefore becomes a stationary turning point in the final electronic state (Fig. 14.7). The quantum mechanical version of the Franck–Condon principle refines this pic ture. Before the absorption, the molecule is in the lowest vibrational state of its lowest electronic state (Fig. 14.8); the most probable location of the nuclei is at their equilibrium separation, Re. The electronic transition is most likely to take place when the nuclei have this separation. When the transition occurs, the molecule is excited to the state represented by the upper curve. According to the Franck–Condon principle, the nuclear framework remains constant during this excitation, so we may imagine the transition as being up the vertical line in Fig. 14.7. The vertical line is the origin of the expression vertical transition, which is used to denote an electronic transition that occurs without change of nuclear geometry. The vertical transition cuts through several vibrational levels of the upper electronic state. The level marked * is the one in which the nuclei are most probably at the same initial separation Re (because the vibrational wavefunction has maximum amplitude there), so this vibrational state is the most probable state for the termination of the transition. However, it is not the only accessible vibrational state because several nearby states have an appreciable probability of the nuclei being at the separation Re. Therefore, transitions occur to all the vibrational states in this region, but most intensely to the state with a vibrational wavefunction that peaks most strongly near Re. The vibrational structure of the spectrum depends on the relative horizontal position of the two potential energy curves, and a long vibrational progression, a lot of vibrational structure, is stimulated if the upper potential energy curve is appreciably displaced horizontally from the lower. The upper curve is usually displaced to greater equilibrium bond lengths because electronically excited states usually have more antibonding character than electronic ground states. The separation of the vibrational lines of an electronic absorption spectrum depends on the vibrational energies of the upper electronic state. Hence, electronic absorption spectra may be used to assess the force fields and dissociation energies of electronically excited molecules (for example, by using a Birge–Sponer plot, as in Problem 14.2).
Fig. 14.6 The electronic spectra of some molecules show significant vibrational structure. Shown here is the ultraviolet spectrum of gaseous SO2 at 298 K. As explained in the text, the sharp lines in this spectrum are due to transitions from a lower electronic state to different vibrational levels of a higher electronic state.
Fig. 14.8 In the quantum mechanical version of the Franck–Condon principle, the molecule undergoes a transition to the upper vibrational state that most closely resembles the vibrational wavefunction of the vibrational ground state of the lower electronic state. The two wavefunctions shown here have the greatest overlap integral of all the vibrational states of the upper electronic state and hence are most closely similar.
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