Selection rules molecules

This spectrum is called a Raman spectrum and corresponds to the vibrational or rotational changes in the molecule. The selection rules for Raman activity are different from those for i.r. activity and the two types of spectroscopy are complementary in the study of molecular structure. Modern Raman spectrometers use lasers for excitation. In the resonance Raman effect excitation at a frequency corresponding to electronic absorption causes great enhancement of the Raman spectrum. 

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. 

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. 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

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. 

Spectroscopists observed that molecules dissolved in rigid matrices gave both short-lived and long-lived emissions which were called fluorescence and phosphorescence, respectively. In 1944, Lewis and Kasha [25] proposed that molecular phosphorescence came from a triplet state and was long-lived because of the well known spin selection rule AS = 0, i.e. interactions with a light wave or with the surroundings do not readily change the spin of the electrons. 

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. 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

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 ... 

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 ... 

For non-linear molecules of the spherical or symmetric top variety, pf j(Rg) (or dpf j/dRa) may be aligned along or perdendicular to a symmetry axis of the molecule. The selection rules that result are... 

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 hydrogenolyaia of cyclopropane rings (C—C bond cleavage) has been described on p, 105. In syntheses of complex molecules reductive cleavage of alcohols, epoxides, and enol ethers of 5-keto esters are the most important examples, and some selectivity rules will be given. Primary alcohols are converted into tosylates much faster than secondary alcohols. The tosylate group is substituted by hydrogen upon treatment with LiAlH (W. Zorbach, 1961). Epoxides are also easily opened by LiAlH. The hydride ion attacks the less hindered carbon atom of the epoxide (H.B. Henhest, 1956). The reduction of sterically hindered enol ethers of 9-keto esters with lithium in ammonia leads to the a,/S-unsaturated ester and subsequently to the saturated ester in reasonable yields (R.M. Coates, 1970). Tributyltin hydride reduces halides to hydrocarbons stereoselectively in a free-radical chain reaction (L.W. Menapace, 1964) and reacts only slowly with C 0 and C—C double bonds (W.T. Brady, 1970 H.G. Kuivila, 1968). 

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... 

For a symmetric rotor molecule the selection rules for the rotational Raman spectmm are... 

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

The procedure for determining the selection rules for a particular molecule is ... 

Linear molecules belong to either the (with an inversion centre) or the (without an inversion centre) point group. Using the vibrational selection rule in Equation (6.56) and the (Table A. 3 7 in Appendix A) or (Table A. 16 in Appendix A) character table we can... 

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... 

Although, as in linear and symmetric rotor molecules, the term values are slightly modified by Coriolis forces in a degenerate (T2) state, the rotational selection rules... 

Such bands also obey the rotational selection rules in Equation (6.82) and appear similar to a U-I band of a linear molecule. 

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. 

In non-linear polyatomic molecules the process of deterioration of quantum numbers continues to such an extent that only the total electron spin quantum number S remains. The selection rule... 

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... 

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

The ir spectra acquired in this way are extremely sensitive to the orientation of the surface molecules. Molecules must have a significant component of a molecular vibration perpendicular to the surface to be sensed by coupling with the highly directional electric field. Molecules whose dipole moments are perfectly parallel to the surface caimot couple to the existing electric fields, and therefore, are ir transparent by this method. This selectivity of the approach for molecule dipole moments perpendicular as opposed to parallel to the surface is known as the surface selection rule of irras. 

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