Selection rules factor

Often it is possible to resolve vibrational structure of electronic transitions. In this section we will briefly review the symmetry selection rules and other factors controlling the intensity of individual vibronic bands. 

The synnnetry selection rules discussed above tell us whether a particular vibronic transition is allowed or forbidden, but they give no mfonnation about the intensity of allowed bands. That is detennined by equation (Bl.1.9) for absorption or (Bl.1.13) for emission. That usually means by the Franck-Condon principle if only the zero-order tenn in equation (B 1.1.7) is needed. So we take note of some general principles for Franck-Condon factors (FCFs). 

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

The matrix elements (60) represent effective operators that still have to act on the functions of nuclear coordinates. The factors exp( 2iAx) determine the selection rules for the matrix elements involving the nuclear basis functions. 

These results provide so-called "selection rules" because they limit the L and M values of the final rotational state, given the L, M values of the initial rotational state. In the figure shown below, the L = L + 1 absorption spectrum of NO at 120 °K is given. The intensities of the various peaks are related to the populations of the lower-energy rotational states which are, in turn, proportional to (2 L + 1) exp(- L (L +1) h /STi IkT). Also included in the intensities are so-called line strength factors that are proportional to the squares of the quantities ... 

Observed on the wing of the CS2 bending mode. Occurs in violation of the selection rules of the point group Dsd but is IR active under the Csi factor group of the crystal. Could also be a combination vibration or caused by the CS2 impurity which was present in the sample (see text)... 

We point out one more selection rule for R. Let Aq — Ai + A2, A2 = Al >0. Choose R in such a way that the two-layer scheme possesses the factorized operator... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

These selection rules are affected by molecular vibrations, since vibrations distort the symmetry of a molecule in both electronic states. Therefore, an otherwise forbidden transition may be (weakly) allowed. An example is found in the lowest singlet-singlet absorption in benzene at 260 nm. Finally, the Franck-Condon principle restricts the nature of allowed transitions. A large number of calculated Franck-Condon factors are now available for diatomic molecules. 

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 ... 

In more complicated cases, the derivation of selection rules from symmetry requires more formal application of group theory. The fundamental problem is to derive the symmetry properties of a product from the symmetry properties of the factors. For only if the product contains a totally symmetric component can the matrix element have a non-zero value. 

The second factor is at the origin of the so called monopole selection rule. Symmetry requirements impose that the two W s must correspond to the same irreducible representation in order for the overlap integral not to vanish. 

The observed oscillator strength for any given transition is seldom found to be unity. A certain degree of forbiddenness is introduced for each of the factors which cause deviation from the selection rules. The oscillator strength of a given transition can be conveniently expressed as the product of factors which reduce the value from that of a completely allowed transition ... 

Figure 10.6—Vibrational-rotational bands of carbon monoxide (P = 1000 Pa). The various lines illustrate the principle of the selection rules (see Fig. 10.7). In this case, AV = +1 and AJ = 1. Branch R can be seen on the left-hand side of the spectrum while branch P is on the right. The distance between the rotational bands allows the moment of inertia I of the molecule to be calculated. I is not constant due to the anharmonicity factor.

For C22H2, the nuclear spin of C12 is zero and contributes a factor of 1 to the nuclear statistical weights. The statistical weights are therefore the same as in H2. For the ground vibronic state, the even J levels are s and have nuclear statistical weight 1, corresponding to the one possible ns the odd J levels are a and have nuclear statistical weight 3. The usual selection rule (4.138) holds for collisions as well as radiative transitions, and we have ortho and para acetylene. The two forms have not been separated. 

The integral < vib vib) maY vanish because of symmetry considerations. For example, the C02 normal mode v3 in Fig. 6.2 has eigenvalue — 1 for the inversion operation. Hence (Section 6.4), the v3 factor in the vibrational wave function is an even or odd function of the normal coordinate Q3, depending on whether v3 is even or odd. For a change of 1,3,5,... in the vibrational quantum number v3, the functions p vib and p"ib have different parities and their product is an odd function, so that ( ibl vib) vanishes. Thus we have the selection rule Ac3 = 0,2,4,... for electronic transitions in... 

A number of techniques have been used previously for the study of state-selected ion-molecule reactions. In particular, the use of resonance-enhanced multiphoton ionization (REMPI) [21] and threshold photoelectron photoion coincidence (TPEPICO) [22] has allowed the detailed study of effects of vibrational state selection of ions on reaction cross sections. Neither of these methods, however, are intrinsically capable of complete selection of the rotational states of the molecular ions. The TPEPICO technique or related methods do not have sufficient electron energy resolution to achieve this, while REMPI methods are dependent on the selection rules for angular momentum transfer when a well-selected intermediate rotational state is ionized in the most favorable cases only a partial selection of a few ionic rotational states is achieved [23], There can also be problems in REMPI state-selective experiments with vibrational contamination, because the vibrational selectivity is dependent on a combination of energetic restrictions and Franck-Condon factors. 

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