Rotational spectra selection rules

Stoicheff investigated the pure rotational Raman spectrum of CS2. The first few lines could not be observed because of the width of the exciting line. The average values of the Stokes and anti-Stokes shifts for the first few observable lines (accurate to 0.02 cm-1) are Ap = 4.96, 5.87, 6.76, 7.64, and 8.50 cm-1, (a) Calculate the C=S bond length in carbon disulfide. (Assume centrifugal distortion is negligible. The rotational Raman selection rule for linear molecules in 2 electronic states is AJ = 0, 2.) (b) Is this an R0 or Re value (c) Predict the shift for the 7 = 0—>2 transition. 

In accordance with these considerations, the pure-rotational Raman spectrum (selection rule AJ= 2) of has every second line missing, whereas that of Na has all lines present, but those arising from even-J states axe more intense than those arising from. odd-J states (2). Yoshino and Bernstein (ll) have observed intensity alternations having statistical origins in both the pure-rotational Raman spectrum of Ha, and in the rotational fine-structure (selection rules AJ=0, 2) of the vibrational band in the Raman spectra of both and Da. 

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. 

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

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 ... 

The dipole and polarization selection rules of microwave and infrared spectroscopy place a restriction on the utility of these techniques in the study of molecular structure. However, there are complementary techniques that can be used to obtain rotational and vibrational spectrum for many other molecules as well. The most useful is Raman spectroscopy. 

Here a third selection rule applies for linear molecules, transitions corresponding to vibrations along the main axis are allowed if Aj = 1. The A/=0 transition is only allowed for vibrations perpendicular to the main axis. Note that because of this selection rule the purely vibrational transition (called Q branch) appears in the gas phase spectrum of C(X but is absent in that of CO. In both cases, two branches of rotational side bands appear (called P and R branch) (see Fig. 8.3 for gas phase CO). 

The rotation-vibration interaction of Section 4.32 produces different effects in nonlinear molecules than those discussed in the previous section. In nonlinear molecules the quantum numbers are vavhvcKJM >. The connection between the group quantum numbers Ico , co2> xi > 2 -A 3/ > and the usual quantum numbers is given by Eq. (4.85). The different effect can be traced to the different nature of the rotational spectrum. In lowest order, the spectrum of a bent molecule is given by Eq. (4.107) and Figure 4.21. The rotation-vibration interaction introduces terms with selection rules... 

In the ideal case of free Eu + ions, we first must observe that the components of the electric dipole moment, e x, y, z), belong to the irreducible representation in the full rotation group. This can be seen, for instance, from the character table of group 0 (Table 7.4), where the dipole moment operator transforms as the T representation, which corresponds to in the full rotation group (Table 7.5). Since Z)° x Z) = Z) only the Dq -> Fi transition would be allowed at electric dipole order. This is, of course, the well known selection rule A.I = 0, 1 (except for / = 0 / = 0) from quantum mechanics. Thus, the emission spectrum of free Eu + ions would consist of a single Dq Ei transition, as indicated by an arrow in Figure 7.7 and sketched in Figure 7.8. 

Figure 3.3 shows some of these possible transitions for HCI. Those with A7 = +1 are known as the R branch and occur at the high-energy side of the hypothetical transition At = 1, A7 = 0 (this is not allowed because of the selection rule, A7 = +1). Those with A7 = — 1 on the low-frequency side of the hypothetical transition form the P branch. Figure 3.4 shows the absorption spectrum of HCI at room temperature, with the rotational transitions responsible for each line. The relative intensities of the lines reflect the relative populations of the absorbing rotational levels the peaks are doublets due to the separate absorptions of the two chlorine isotopes, that is, H35C1 and H37C1, which have different reduced masses and hence values of the rotational constant B. 

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.

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. 

Symmetric tops with no dipole moment have no microwave spectrum. For example, planar symmetric-top molecules have a C axis and a ak symmetry plane such molecules cannot have a dipole moment. Thus benzene has no microwave spectrum. For a symmetric top with a permanent electric dipole moment, the selection rules for pure-rotation transitions are... 

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

Figure 9.24 shows part of the laser Stark spectrum of the bent triatomic molecule FNO obtained with a CO infrared laser operating at 1837.430 cm-1. All the transitions shown are Stark components of the qP1 8) rotational line of the I,1, vibrational transition, where v, is the N-F stretching vibration. The rotational symbolism is that for a symmetric rotor (to which FNO approximates) for which q implies that AK = 0, P implies that AJ = — 1 and the numbers indicate that K" = 7 and J" = 8 (see Section 6.2.4.2). In an electric field each J level is split into (J + 1) components (see Section 5.2.3), each specified by its value of Mj. The selection rule when the radiation is polarized perpendicular to the field (as here) is AMj = 1. Eight of the resulting Stark components are shown. 

Figure 9.37 shows the first CRD spectrum to be obtained and is that of the v = I 0 vibronic band in the b-X electronic band system of 02. The length of the cavity was 1 m and it was filled with air at atmospheric pressure containing about 20% oxygen. The rotational selection rules for such an unusual type of transition allows a P branch, an R branch and two Q branches these are labelled rQ and PQ. (In general, pre-superscripts. .. 0, p, q, r, s... 

In practice values of B are also often quoted in cm-1. For the simple rigid rotor the rotational quantum number J takes integral values, J = 0, 1, 2, etc. The rotational energy levels therefore have energies 0, 2B, 6B, 12B, etc. Elsewhere in this book we will describe the theory of electric dipole transition probabilities and will show that for a diatomic molecule possessing a permanent electric dipole moment, transitions between the rotational levels obey the simple selection rule A J = 1. The rotational spectrum of the simple rigid rotor therefore consists of a series of equidistant absorption lines with frequencies 2B, 4B, 6B, etc. 

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