Selection rules for transitions

The vanishing integral rule is not only usefi.il in detemiining the nonvanishing elements of the Hamiltonian matrix H. Another important application is the derivation o selection rules for transitions between molecular states. For example, the hrtensity of an electric dipole transition from a state with wavefimction "f o a... 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

The methyl iodide molecule is studied using microwave (pure rotational) spectroscopy. The following integral governs the rotational selection rules for transitions labeled J, M, K... 

To investigate the spectra of diatomic molecules, we need the selection rules for radiative transitions. We now investigate the electric-dipole selection rules for transitions between vibration-rotation levels belonging to the same 2 electronic state. (Transitions in which the electronic state changes will be considered in Chapter 7.)... 

Conditions of non-zero values of the submatrix elements of the electron transition operators define the selection rules for radiation. The latter coincide with those non-zero conditions for the quantity Q. On the other hand, the selection rules for Q are defined by the conditions of polygons for 3nj-coefficients, in terms of which they are expressed. The requirement (24.21) must also be kept in mind. The selection rules for transitions (25.8)-(25.17) are summarized in Table 25.1. In all cases the selection rules JJ k, hhk and l2 +12 + k is even number are valid. The table contains only those polygons which have the quantum numbers of both configurations, because only in such a case do these conditions serve as the selection rules for radiation. If a certain quantum number has no restrictions from this point of view, this means that it does not form a polygon with quantum numbers of the other configuration. Such quantities are placed in curly brackets. 

These rules show that the G<- G transition, in contrast with the others, is purely rotational. In the coordinate system shown in Figure 8.20, the transition states for the cis and trans paths of interconversion have symmetry axes and C2y and relate to the symmetry groups and C2h, respectively. The different symmetries of the transition states results from the fact that the same permutation relates to different symmetry operations in C2v and C2h. For example, (ab)(14)(28)(36) is equivalent to inversion in C2h, while in it corresponds to the reflection in the axy plane. The symmetry of the reaction path does not affect the symmetry of states with even Ka (and Ka = 0). However, the selection rules for transitions Ka = 1 0 are different for cis and trans paths. The classifica-... 

Fig. 2. Schematic fine-structure energy scheme for % state of Cr3+ in LiCAF. Selection rules for transitions are indicated.

Linear molecules belong to either the D h (with an inversion centre) or the (without an inversion centre) point group. Using the vibrational selection rule in Equation (6.56) and the D h (Table A.37 in Appendix A) or (Table A. 16 in Appendix A) character table we can see that the vibrational selection rules for transitions from the zero-point level (LjJ in Di0o/l, A1 in ) allow transitions of the type... 

As in Section 5.2.4 on rotational spectra of asymmetric rotors, we do not treat this important group of molecules in any detail, so far as their rotational motion is concerned, because of the great complexity of their rotational energy levels. Nevertheless, however complex the stack associated with the v = 0 level, there is a very similar stack associated with each excited vibrational level. The selection rules for transitions between the rotational stacks of the vibrational levels are also complex but include... 

If the Schrodinger equation (Section 1.3) is solved using this potential, the resulting eigenvalues are expressed as a function of a quantum number, n. Selection rules for transitions are An = 1 for IR and An = 2 for Raman. The values of a and b must be chosen so that calculated IR and Raman fequencies agree with those observed. As an example, consider 1,3-disilacyclobutane ... 

Spectroscopy is concerned with the observation of transitions between stationary states of a system, with the accompanying absorption or emission of electromagnetic radiation. In this section we consider the theory of transition probabilities, using time-dependent perturbation theory, and the selection rules for transitions, particularly those relevant for rotational spectroscopy. 

In most of the examples described in this book, the rotational angular momentum is coupled to other angular momenta within the molecule, and the selection rules for transitions are more complicated than for the simplest example described above. Spherical tensor methods, however, offer a powerftd way of determining selection rules and transition intensities. Let us consider, as an example, rotational transitions in a good case (a) molecule. The perturbation due to the oscillating electric component of the electromagnetic radiation, interacting with the permanent electric dipole moment of the molecule, is represented by the operator... 

The selection rules for transitions between the Is (Ai) state and the stress-split sublevels can be deduced from group theory when one knows the IRs under which the components of the dipole moment (actually, x, y, and z) transform. As an example, Fig. 8.3 shows the polarization features of the allowed transitions from the Is (Ai) state for F// [100] in silicon. The splitting under stress of the Is (T2) donor level in silicon is the same as those of the npo and np-1-1 levels shown in this figure. For F// [100], the highest- and lowest-energy... 

The symmetry properties also enable us to establish the selection rules for transitions between states. For the group C2 , transitions between and U2 states and between bi and 2 states are forbidden. All others are allowed. Thus the transition between Ai is... 

The selection rule for transitions between J levels is therefore... 

The transition AMj = 0 is polarized along the z axis, while those with AZIZ = 1 are right and left circularly polarized in the xy plane. These are the selection rules for transitions between states that separate in a magnetic held. 

Our aim is to examine selection rules for transitions from a ground state to excited states of different multiplicity. We will assume the ground state to be a singlet state but this also mixes with higher triplets... 

The propensity rules are based on the observation that, to the extent that states of given total symmetry are built upon a single MO, transitions between these states are governed by selection rules for transitions between the corresponding MO. In this way the propensity rules arise naturally out of the theory and are not dependent upon empirical deductions. Their simplicity is partly due to the structure of the operators responsible for non-radiative (auto-ionization) a nd radiative decay. 

Figure 1 MCD mechanism for a molecule with 7=1/2 in the ground and excited states, (a) derivative shaped /I-term and its components when kT >g(5H and (b) absorption shaped C-term and its component when kT gf3H. The selection rules for transitions of rep and Icp light between the degenerate ground and excited states split by a magnetic field is shown at the bottom.

The orbital selection rules for transitions between individual crystal field levels are displayed in terms of the point group irreps of the initial and final state wavefunctiOTis, and for a forced ED transition between 4 Fj 41 F[. 

The excess photon energy of the order up to a few thousand wavenumbers will end up partly as internal energy of the ion. The resulting vibrational populations depend on the amount of excess energy, the selection rules for transitions from the intermediate vibronic states and the corresponding Franck-Condon factors. 

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