Molecules transition selection rules

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

As before, when pf i(Rg) (or dpfj/dRa) lies along the molecular axis of a linear molecule, the transition is denoted a and k = 0 applies when this vector lies perpendicular to the axis it is called n and k = 1 pertains. The resultant linear-molecule rotational selection rules are the same as in the vibration-rotation case ... 

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)... 

First approximation theory leads to certain wave mechanical selection rules on the basis of which a radiative electronic transition may be classified as allowed (high probability) or forbidden (vanishingly low probability). Some forbidden transitions are indeed too weak to observe easily but in actual practice with polyatomic molecules the selection rules often break down sufficiently to permit reasonably strong absorption processes to occur. The following kinds of transition are forbidden... 

The probability of the radiationless transition leading to predissociation is governed by the Franck Condon principle and by a group of selection rules first given by Kronig (1930). One reason why predissociation is so widespread is that, especially for non-linear molecules, the selection rules are extremely permissive. A striking example of a forbidden predissociation from the linear 2H (2I I) Renner state of the HCO radical has been described by Ramsay (1959). 

Figure 8.1. Rotation inversion energy levels and allowed transitions of the free NH3 molecule. The selection rules are A7 =...

The symmetry of the molecules provide selection rules for the electronic transitions in the dipole approximation. A mirror plane perpendicular to a direction requires the wavefunction un to be either symmetric un ( )=u (- ) or antisymmetric un ( )=- (- ) to this mirror plane as the electron density... 

From such considerations the symmetry species of each wavefunction associated with an energy level is determined, and these are indicated at the right in Fig. 1. It is important to realize that this symmetry label is the correct one for the true wavefunction, even though it is deduced from an approximate harmonic-oscillator model. This is significant because transition selection rules based on symmetry are exact whereas, for example, the usual harmonic-oscillator constraint that An = 1 is only approximate for real molecules. 

One of the most significant implications of the result is that an absorption spectrum measured with intense white light may be significantly different from the spectrum that would be observed using tunable monochromatic radiation. In particular, there should be a decrease in the apparent width of many lines in any absorption spectrum measured with broadband radiation. This is because, for any sample transition of frequency coq, photons of appreciably off-resonant frequency (oiq + fi) can be cooperatively absorbed and result in the excitation of two separate molecules, provided selection rules permit. In fact the Lorentzian linewidth of the concerted absorption process is readily shown to be approximately 0.64 x the ordinary absorption linewidth, if the probe radiation is assumed to be of nearly constant intensity in the frequency region of interest. Nonetheless, the observed linewidth would not be reduced to quite this extent, because of the additional and invariably stronger response associated with normal single-photon absorption. 

If a molecule changes from one rational state to another, the energy difference between the two states is made up by the emission or absorption of a quantum of radiation. For transitions between rotational states of linear molecules, the selection rule requires that AJ = 1. The energy difference between these neighboring states is... 

The situation with asymmetric top rotors is rather more complex because of the wide range of transition selection rules followed, although similar comments apply. The more intense absorption lines will tend to occur at higher frequencies although the relative intensities of lines may vary because of symmetry considerations. Symmetric top spectra occur in clumps centred around 2B J + 1) and so from the quantitative analysis viewpoint differ hardly at all from those of linear molecules. Asymmetric top spectra are more scattered and it is rather easier to choose an accessible and discrete line from them. 

Fig. 6.6a can be understood with the help of Eq. (6.28). which shows us a model of the phenomena taking place. At room temperature, most of the molecules (Boltzmann law) are in their ground electronic and vibrational states k = 0, v = 0). IR quanta are unable to change quantum number k, but they have sufficient eneigy to change v and 7 quantum numbers. Fig. 6.6a shows what in fact has been recorded. From the transition selection rules (see above), we have An — 1 — 0=1 and either the transitions of the kind AJ = (7 + 1) — 7 = +l (what is known as the R branch, right side of the spectrum) or of the kind AJ = 7— (7 + 1) = —1 (the P branch, left side). 

The SFG technique probes the second-order nonhnear hyperpolarizability tensor this tensor includes the Raman and IR susceptibihty, which requires that the molecular vibrational modes are both Raman and IR active. Since Raman- and IR-dipole moment transition selection rules for molecules with a center of symmetry indicate that a vibrational mode is either Raman or IR active but not both, only molecules in a non-centrosymmetric environment on the surface interact with the electric fields molecules in the isotropic bulk phase show inversion symmetry where the third rank hyperpolarizability tensor goes to zero [25-27]. 

A brief outline of basic information on the energy levels in atoms and molecules, as well as photon transitions/selection-rules (Chapter 2) a short... 

Not only do the experimental vibrational predissociation lifetimes require interpretation, so do the increasingly sophisticated theoretical calculations whose results often fall out of a web of coupled differential equations or the convoluted algebra of quantum mechanics. In order to offer a qualitative overview of dynamical processes in van der Waals molecules, we shall introduce a selection rule which can provide insight into possible relaxation channels of vibrationally excited molecules. This selection rule concerns the change in a quantum number, Anj., which is to remain small for efficient vibrational predissociation processes. It bears a close analogy to the selection rules of optical spectroscopy which require small changes in quantum numbers Au, AJ, AS, etc. for efficient transitions between molecular states. Let us review the origin of the vibrational predissociation selection rule which has been developed in more detail elsewhere. ... 

The Raman transition selection rules are available the same way as the electric dipole selection rules, but the transition moment operator has the symmetry of the second order firnctions x, y, z, xy, yz, and xz. If we think of the Raman transition represented in Fig. 6.17 as a dual process—absorption and then emission—then this makes sense the probability of the Raman transition depends on the transition moment for reaching the virtual state (when the incident photon hits the molecule)... 

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. 

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. 

Most stable polyatomic molecules whose absorption intensities are easily studied have filled-shell, totally synuuetric, singlet ground states. For absorption spectra starting from the ground state the electronic selection rules become simple transitions are allowed to excited singlet states having synuuetries the same as one of the coordinate axes, v, y or z. Other transitions should be relatively weak. 

The selection rule for vibronic states is then straightforward. It is obtained by exactly the same procedure as described above for the electronic selection rules. In particular, the lowest vibrational level of the ground electronic state of most stable polyatomic molecules will be totally synnnetric. Transitions originating in that vibronic level must go to an excited state vibronic level whose synnnetry is the same as one of the coordinates, v, y, or z. 

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. 

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. 

The electric dipole selection rule for a hannonic oscillator is Av = 1. Because real molecules are not hannonic, transitions with Av > 1 are weakly allowed, with Av = 2 being more allowed than Av = 3 and so on. There are other selection niles for quadnipole and magnetic dipole transitions, but those transitions are six to eight orders of magnitude weaker than electric dipole transitions, and we will therefore not concern ourselves with them. 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

The selection rules for AK depend on the nature of the vibrational transition, in particular, on the component of itrans along the molecule-fixed axes. For the second 3-j symbol to not vanish, one must have... 

In a symmetric top molecule such as NH3, if the transition dipole lies along the molecule s symmetry axis, only k = 0 contributes. Such vibrations preserve the molecule s symmetry relative to this symmetry axis (e.g. the totally symmetric N-H stretching mode in NH3). The additional selection rule AK = 0... 

When applied to linear polyatomic molecules, these same selection rules result if the vibration is of a symmetry (i.e., has k = 0). If, on the other hand, the transition is of n symmetry (i.e., has k = 1), so the transition dipole lies perpendicular to the molecule s axis, one obtains ... 

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... 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

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

As we proceed to molecules of higher symmetry the vibrational selection rules become more restrictive. A glance at the character table for the point group (Table A.41 in Appendix A) together with Equation (6.56) shows that, for regular tetrahedral molecules such as CH4, the only type of allowed infrared vibrational transition is... 

Electronic transitions mostly involve interaction between the molecule and the electric component of the electromagnetic radiation (Section 2.1). The selection rules are, therefore. 

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