Energy level selection rules

These discoveries could hardly have been brought about without the extensive knowledge of energy levels, selection rules, transition probabilities and spectra assembled by laboratory spectroscopy for a very wide range of molecules. Microwave spectroscopy in the centimeter- and millimeter-wave regions is a well-established research field for further information the reader is referred to the books by Gordy and Cook (1970), and Townes and Schawlow (1955), and to a recent review on millimeter-wave spectroscopy by Winnewisser et al. (1972). Reviews on interstellar molecules are given by Snyder (1972) and Rank etal. (1971). 

Here, AE is larger than (hcOm), the highest phonon energy of the matrix, which is the condition to be considered for a multiphonon process. Such exponential laws are found also for molecules and for deep centers in semiconductors. Except in a few cases where AE is of the order of the highest vibrational energy, no selection rule is found with respect to the set of quantum numbers of the levels. 

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

Before presenting the quantum mechanical description of a hannonic oscillator and selection rules, it is worthwhile presenting the energy level expressions that the reader is probably already familiar with. A vibrational mode v, witii an equilibrium frequency of (in wavenumbers) has energy levels (also in... 

The transition between levels coupled by the oscillating magnetic field B corresponds to the absorption of the energy required to reorient the electron magnetic moment in a magnetic field. EPR measurements are a study of the transitions between electronic Zeeman levels with A = 1 (the selection rule for EPR). 

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. 

Figure 6.7(a) illustrates the rotational energy levels associated with two vibrational levels u (upper) and il (lower) between which a vibrational transition is allowed by the Au = 1 selection rule. The rotational selection rule governing transitions between the two stacks of levels is... 

The transitions between energy levels in an AX spin system are shown in Fig. 1.44. There are four single-quantum transitions (these are the normal transitions A, A, Xi, and X2 in which changes in quantum number of 1 occur), one double-quantum transition 1% between the aa and j8 8 states involving a change in quantum number of 2, and a zero-quantum transition 1% between the a)3 and fia states in which no change in quantum number occurs. The double-quantum and zero-quantum transitions are not allowed as excitation processes under the quantum mechanical selection rules, but their involvement may be considered in relaxation processes. 

Atomic spectra are much simpler than the corresponding molecular spectra, because there are no vibrational and rotational states. Moreover, spectral transitions in absorption or emission are not possible between all the numerous energy levels of an atom, but only according to selection rules. As a result, emission spectra are rather simple, with up to a few hundred lines. For example, absorption and emission spectra for sodium consist of some 40 peaks for elements with several outer electrons, absorption spectra may be much more complex and consist of hundreds of peaks. 

A further technique exists for the determination of triplet energy levels. This technique, called electron impact spectroscopy, involves the use of inelastic scattering of low-energy electrons by collision with molecules. The inelastic collisions of the electrons with the molecules result in transfer of the electron energy to the molecule and the consequent excitation of the latter. Unlike electronic excitation by photons, excitation by electron impact is subject to no spin selection rule. Thus transitions that are spin and/or orbitally forbidden for photon excitation are totally allowed for electron impact excitation. 

We have discussed the transition moment (the quantum mechanical control of the strength of a transition or the rate of transition) and the selection rules but there is a further factor to consider. The transition between two levels up or down requires either the lower or the upper level to be populated. If there are no atoms or molecules present in the two states then the transition cannot occur. The population of energy levels within atoms or molecules is controlled by the Boltzmann Law when in local thermal equilibrium ... 

The ro-vibronic spectrum of molecules and the electronic transitions in atoms are only part of the whole story of transitions used in astronomy. Whenever there is a separation between energy levels within a particular target atom or molecule there is always a photon energy that corresponds to this energy separation and hence a probability of a transition. Astronomy has an additional advantage in that selection rules never completely forbid a transition, they just make it very unlikely. In the laboratory the transition has to occur during the timescale of the experiment, whereas in space the transition has to have occurred within the last 15 Gyr and as such can be almost forbidden. Astronomers have identified exotic transitions deep within molecules or atoms to assist in their identification and we are going to look at some of the important ones, the first of which is the maser. 

As noted above, not all possible transitions between energy levels are theoretically allowed. Each energy level is uniquely characterized by a set of quantum numbers. The integer used to define the energy level in the above discussion (1,2, 3, etc.) is called the principal quantum number, n. The sub-levels described by the letters (s, p, d, /, etc.) are associated with the second quantum number, given the symbol /, with l = 1 synonymous with s, 2 =p, etc. The multiplicity of levels associated with each sub-level (i.e., the number of horizontal lines for each orbital in Figure Al.l) is defined by a third quantum number mh which has values 0, 1... 1. Thus, -orbitals only have one sub-level, p-orbitals have three (with m/ values 0 and d= 1), d-orbitals have five, etc. The selection rules can... 

Because many physical systems possess certain types of symmetry, its adaptation has become an important issue in theoretical studies of molecules. For example, symmetry facilitates the assignment of energy levels and determines selection rules in optical transitions. In direct diagonalization, symmetry adaptation, often performed on a symmetrized basis, significantly reduces the numerical costs in diagonalizing the Hamiltonian matrix because the resulting block-diagonal structure of the Hamiltonian matrix allows for the separate... 

Light emitted from a mercury lamp is caused by electronic transitions from higher-energy-level atomic orbitals to lower-energy-level atomic orbitals. The electronic transitions are subject to certain constraints known as selection rules ... 

Transitions between energy levels in organic molecules are subject to certain constraints, referred to as selection rules. 

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