Wavefunctions spectroscopy selection rules

This result is tremendously useful, it not only leads to selection rules for vibrational spectroscopy but also, as was the case with electronic wavefunctions (see 8-2), allows us to predict from inspection of the character table the degeneracies and symmetries which are allowed for the fundamental vibrational wavefunctions of any particular molecule. 

Electrons, protons and neutrons and all other particles that have 5 = are known as fermions. Other particles are restricted to 5 = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fermions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection rules. It can be shown that the spin quantum number S associated with an even number of fermions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fermions, respectively, so the wavefunction symmetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number therefore behave like individual bosons and those with odd atomic number as fermions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

Section B 1.1.2 provides a brief summary of experimental methods and instrumentation, including definitions of some of the standard measured spectroscopic quantities. Section B 1.1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefunctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, the Franck-Condon principle and selection rules are also discussed briefly. In the final section, B 1.1.4, a few applications are mentioned to particular aspects of electronic spectroscopy. 

In the main text we introduced the selection rules for IR spectroscopy via the transition dipole moment integral. This appendix gives a little more detail on the origin of the selection rules, with explicit formulae for the vibrational wavefunctions. This also allows a more complete explanation of the observation that absorption due to transitions involving neighbouring levels (e.g. n = 0 to n = 1) are more easily observed than overtones which involve transitions to higher levels in the ladder of vibrational states. 

The one-center expressions given above have been routinely used to analyze the different molecular core electron spectroscopies, and they comply with the notion that different core hole derived spectra map the same set of final states differently according to the localization character of wavefunctions and orbitals, and they thus supply local information of symmetries and densities. Although these local selection rules were stated several decades ago, they have survived as a... 

The selection rules in infrared spectroscopy arise from the integrals of the initial and final eigenstate wavefunctions with the interacting Hamiltonian in Equation 6-81. 

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