Selection rules diatomic molecules

We now consider radiative transitions foi which both v and J change, but the electronic state does not these transitions give the vibration-rotation spectra of diatomic molecules. The selection rules for these transitions were found in Section 4.4 to be ( 2 states only)... 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection rules are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

In this section, we discuss all-electron TDDFT MPI calculations of homonuclear (N2 and F2) and heteronuclear (BF, HF, and CO) diatomic molecules [32,55]. In Figure 3.3, we present the HHG power spectra (Equation 3.11) of the N2 and CO molecules. An important difference between the N2 and CO spectra is that the latter contain even and odd harmonics. Generation of even harmonics is forbidden in systems with inversion symmetry, such as atoms and homonuclear diatomic molecules. This selection rule does not apply to the heteronuclear molecules with no inversion center (CO, BF, and HF). From Figure 3.3, one can see that, in general, HHG is more efficient... 

FIGURE 14t14 Imposition of an electric field can complicate an otherwise simple spectrum. For a diatomic molecule, the selection rule AM = 0, 1 now dictates the possible transitions. 

One of the consequences of this selection rule concerns forbidden electronic transitions. They caimot occur unless accompanied by a change in vibrational quantum number for some antisynnnetric vibration. Forbidden electronic transitions are not observed in diatomic molecules (unless by magnetic dipole or other interactions) because their only vibration is totally synnnetric they have no antisymmetric vibrations to make the transitions allowed. 

The theory of molecular symmetry provides a satisfying and unifying thread which extends throughout spectroscopy and valence theory. Although it is possible to understand atoms and diatomic molecules without this theory, when it comes to understanding, say, spectroscopic selection rules in polyatomic molecules, molecular symmetry presents a small barrier which must be surmounted. However, for those not needing to progress so far this chapter may be bypassed without too much hindrance. 

In a diatomic or linear polyatomic molecule rotational Raman scattering obeys the selection rule... 

The rotational selection rule for vibration-rotation Raman transitions in diatomic molecules is... 

For atoms, electronic states may be classified and selection rules specified entirely by use of the quantum numbers L, S and J. In diatomic molecules the quantum numbers A, S and Q are not quite sufficient. We must also use one (for heteronuclear) or two (for homonuclear) symmetry properties of the electronic wave function ij/. ... 

In the case of atoms, deriving states from configurations, in the Russell-Saunders approximation (Section 7.1.2.3), simply involved juggling with the available quantum numbers. In diatomic molecules we have seen already that some symmetry properties must be included, in addition to the available quantum numbers, in a discussion of selection rules. 

In the case of atoms (Section 7.1) a sufficient number of quantum numbers is available for us to be able to express electronic selection rules entirely in terms of these quantum numbers. For diatomic molecules (Section 7.2.3) we require, in addition to the quantum numbers available, one or, for homonuclear diatomics, two symmetry properties (-F, — and g, u) of the electronic wave function to obtain selection rules. 

For the orbital parts of the electronic wave functions of two electronic states the selection rules depend entirely on symmetry properties. [In fact, the electronic selection rules can also be obtained, from symmetry arguments only, for diatomic molecules and atoms, using the (or and Kf point groups, respectively but it is more... 

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. 

There are, however, certain selection rules for electric dipole transitions which considerably reduce the number of possible transitions. They are extensively discussed and proved in reference and for diatomic molecules consist essentially of the following three rules ... 

Upon absorption of light of an appropriate wavelength, a diatomic molecule can undergo an electronic transition, along with simultaneous vibrational and rotational transitions. In this case, there is no restriction on Au. That is, the selection rule Av = +1 valid for purely vibrational and vibrational-rotational transitions no longer applies thus numerous vibrational transitions can occur. If the molecule is at room temperature, it will normally be in its lower state, v" = 0 hence transitions corresponding to v" = 0 to v = 0,... 

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.)... 

We previously found the selection rule A7 = 1 for a 2 diatomic-molecule vibration-rotation or pure-rotation transition. The rule (4.138) forbids A/ = 1 for homonuclear diatomics this gives us no new information as far as vibration-rotation spectra are concerned, since the absence of a dipole moment insures the absence of a vibration-rotation or pure-rotation spectrum, anyway. 

The selection rule (4.138) differs from previously discussed selection rules in that it holds well for nonradiative transitions, as well as for radiative transitions. In deriving (4.138), we made no reference to the operator d, beyond the statement that it did not involve the nuclear spin coordinates. For any time-dependent perturbation that does not involve nuclear spin, the selection rule (4.138) will hold. Thus molecular collisions will not cause nonradiative transitions between symmetric and antisymmetric rotational levels of a homonuclear diatomic molecule. If we somehow start with all the molecules in symmetric levels, the collisions will not populate the antisymmetric levels. 

The electronic quantum numbers of diatomic molecules are discussed in Sections 1.19 and 4.11. The selection rule for A can be shown to be... 

Electric-quadrupole transition, 123,127 Electromagnetic radiation, 114-117. See also Radiation, electromagnetic Electromagnetic spectrum, 115 Electronic energy, 57,64,148 Electronic spectra, 130, 296-314 of diatomics, 298-306 and molecular structure, 311 of polyatomics, 71-72, 73, 75, 306-314 selection rules for, 297-301, 306-307 Electronic structure of molecules, 56-76 Electron spectroscopy for chemical analysis (ESCA), 319-320 Electron spin resonance (ESR), 130, 366-381... 

As previously mentioned, for most diatomic molecules at room temperature, the population of excited vibrational levels is negligible. We therefore first consider transitions for which the initial vibrational level is u = 0. The selection rules (4.108) allow the transitions v = 0- l, 0— 2,... 

Consideration of the matrix elements m a n of the polarizability shows that the selection rule for a pure-rotational Raman transition of a l2 diatomic molecule is (see Wilson, Decius, and Cross)... 

For a symmetric top, the selection rules are such that we can determine only B0 [see (5.85)]. Knowledge of Ib°, the moment of inertia about a principal axis perpendicular to the symmetry axis, is not sufficient to determine the molecular structure, except for a diatomic molecule. To get added information, the microwave spectra of isotopically substituted spe-... 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

This is an identical expression to that for a diatomic or linear polyatomic molecule (Equations 5.11 and 5.12) and, as the rotational selection rule is the same, namely, AJ = 1, the transition wavenumbers or frequencies are given by... 

---