Rotational levels, allowed

The primary significance of microwave spectroscopy for chemistry is in determination of molecular structure. Assignment of microwave spectral lines to transitions between specific rotational levels allows determination of the rotational constants A0, B0, and C0, and the corresponding moments of inertia. The moments of inertia are dependent on the molecular bond distances and bond angles. 

While a laser beam can be used for traditional absorption spectroscopy by measuring / and 7q, the strength of laser spectroscopy lies in more specialized experiments which often do not lend themselves to such measurements. Other techniques are connnonly used to detect the absorption of light from the laser beam. A coimnon one is to observe fluorescence excited by the laser. The total fluorescence produced is nonnally proportional to the amount of light absorbed. It can be used as a measurement of concentration to detect species present in extremely small amounts. Or a measurement of the fluorescence intensity as the laser frequency is scaimed can give an absorption spectrum. This may allow much higher resolution than is easily obtained with a traditional absorption spectrometer. In other experiments the fluorescence may be dispersed and its spectrum detennined with a traditional spectrometer. In suitable cases this could be the emission from a single electronic-vibrational-rotational level of a molecule and the experimenter can study how the spectrum varies with level. 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

In Figure 7.25 are shown stacks of rotational levels associated with two electronic states between which a transition is allowed by the -F -F and, if it is a homonuclear diatomic, g u selection rules of Equations (7.70) and (7.71). The sets of levels would be similar if both were states or if the upper state were g and the lower state u The rotational term values for any X state are given by the expression encountered first in Equation (5.23), namely... 

Figure 10.7 The population of excited energy states relative to that of the ground state for the CO molecule as predicted by the Boltzmann distribution equation. Graph (a) gives the ratio for the vibrational levels while graph (b) gives the ratio for the rotational levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations. The dots represent ratios at integral values of v and 7. which are the only allowed values.

As the molecule vibrates it can also rotate and each vibrational level has associated rotational levels, each of which can be populated. A well-resolved ro - vibrational spectrum can show transitions between the lower ro-vibrational to the upper vibrational level in the laboratory and this can be performed for small molecules astronomically. The problem occurs as the size of the molecule increases and the increasing moment of inertia allows more and more levels to be present within each vibrational band, 3N — 6 vibrational bands in a nonlinear molecule rapidly becomes a big number for even reasonable size molecules and the vibrational bands become only unresolved profiles. Consider the water molecule where N = 3 so that there are three modes of vibration a rather modest number and superficially a tractable problem. Glycine, however, has 10 atoms and so 24 vibrational modes an altogether more challenging problem. Analysis of vibrational spectra is then reduced to identifying functional groups associated... 

We have now looked at the way photons are absorbed. Photons of UV and visible light cause electrons to promote between orbitals. Infrared photons have less energy, and are incapable of exciting electrons between orbitals, but they do allow excitation between quantized vibrational levels. The absorption of microwaves, which are less energetic still, effects the excitation between quantized rotational levels. 

Because of the large number of rotational levels in the upper and lower states, the overlap between the exciting laser line and the dopp-ler broadened absorption profile may be nonzero simultaneously for several transitions (u", / ) (v, f) with different vibrational quantum numbers v and rotational numbers J. This means, in other words, that the energy conservation law allows several upper levels to be populated by absorption of laser photons from different lower levels. 

From precise wavelength measurements of the fluorescence spectrum (which may be performed e. g. by interferometric methods accurate values for the molecular constants can be obtained since the wavelength differences of subsequent lines in the fluorescence progression yield the energy separation of adjacent vibrational and rotational levels as a function of v . From these spectroscopically deduced molecular constants, the internuclear distance can be calculated A special computer programm developed by Zare ) allows the potential curve to be constructed from the measured constants and, if the observed fluorescence progression... 

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. 

The rotational energy levels for a homonuclear diatomic molecule follow Eq. 8.16, but the allowed possibilities for j are different. (The rules for a symmetric linear molecule with more than two atoms are even more complicated, and beyond the scope of this discussion.) If both nuclei of the atoms in a homonuclear diatomic have an odd number of nuclear particles (protons plus neutrons), the nuclei are termed fermions if the nuclei have an even number of nuclear particles, they are called bosons. For a homonuclear diatomic molecule composed of fermions (e.g., H— H or 35C1—35C1), only even-j rotational states are allowed. (This is due to the Pauli exclusion principle.) A homonuclear diatomic molecule composed of bosons (e.g., 2D—2D or 14N—14N) can only have odd- j rotational levels. 

Any molecule with a permanent electric dipole moment can interact with an electromagnetic field and increase its rotational energy by absorbing photons. Measuring the separation between rotational levels (for example, by applying a microwave field which can cause transitions between states with different values of /) let us measure the bond length. The selection rule is A/ = +1—the rotational quantum number can only increase by one. So the allowed transition energies are... 

Let us consider some specific examples whose spectra occur in this book. A very simple case is O2 inits excited1 Ag state. The predominant nucleus, 160, has / = 0 and so is a boson. IT is also zero and there is only one nuclear spin function (IT = 0, M/r = 0) which is symmetric with respect to Pn. Thus for each value of J only one A-doublet component is allowed by the exclusion principle, namely, that which has positive parity all the rotational levels of O2 in its 1 Ag state therefore have positive parity. The other A-doublet component is missing. 

In order to understand the observed spectra it is first necessary to consider the importance of nuclear spin in determining the nature of the allowed rotational levels. Each 14N nucleus has spin 7=1, and in the homonuclear N2 system the individual spins are coupled to form a total nuclear spin IT of 2, 1 or 0. The most appropriate basis system for 3 + N2 is Hund s case (b) coupling ... 

Consideration of symmetry with respect to nuclear interchange shows, however, that there are restrictions on the allowed combinations of N and h Even values of It (0 or 2) correspond to ortho-N2, and can only combine with odd values of N. The odd value of It (1), which corresponds to para-N2, can only combine with even values of N. This association is the reverse of that which occurs in the ground state of N2 because of the u character of the excited state. The situation for the first four rotational levels (N = 0 to 3) is summarised schematically in figure 8.22 this pattern, of course, applies for all vibrational levels. 

Figure 8.22. Schematic energy level diagram for the first four rotational levels of N2 in its A 3E+ state, showing the nuclear hyperfine states which are allowed to combine with each N level. Relative vertical spacings are not drawn to scale [43].

Figure 8.45. /I-doublet and proton hyperfine splitting for the lowest rotational level in OH ( J = 3/2, 2 FT 3/2 and the allowed electric dipole transitions.

As we mentioned in the introduction to this section, it was known forty years ago from optical spectroscopy that CH is a component of interstellar gas clouds and the search was on for a spectroscopic detection of the radical at higher resolution so that the /l-doubling in the lowest rotational level (J = 1/2) could be measured or predicted accurately. This would enable the detection of CH by radio-astronomers and so allow the distribution of CH in these remote sources to be mapped out. The race was won by Evenson, Radford and Moran [48] using the then new technique of far-infrared LMR in the Boulder laboratories of the NBS (now known as NIST). They realised that there was a good near-coincidence between the water discharge laser line at 118.6 qm (84.249 cm ) and the N = 3 <- 2, J = 7/2 5/2 transition of CH in the / ) spin... 

Figure 9.24. Lower rotational levels of a2 + molecule, like the CN radical, showing the spin-doubling and the allowed electric dipole transitions between N = 2 and 1.

Figure 9.25. Zeeman splitting of the N = 1 and 2 rotational levels in the CN radical. In region 1 the rotational transition is electric dipole allowed and magnetically tunable. In region 3 the magnetically-tunable transitions are magnetic dipole electron spin transitions the electric dipole transitions are not magnetically tunable. Region 2 is intermediate between these limiting cases.

We now use the results we have obtained to calculate the energy levels in a magnetic field, determine the field values for the allowed electric dipole transitions, and compare the results with the experimental spectrum [56]. It is already clear that in the case (b) basis set we shall have to take note of the extensive mixing of different rotational levels by the AN = 2 off-diagonal matrix elements of the spin-spin interaction. In SO the spin spin parameter X is comparable with the rotational constant B0, and, as we shall see, in heavier molecules like SeO, X is so much larger than B0, because of spin orbit coupling, that a case (a) basis is more appropriate. 

Figure 11.58. Hyperfine energy levels and magnetic dipole transitions for Hj in the N = 1 rotational level (not to scale). The dashed lines indicate magnetic-dipole allowed transitions which were not observed experimentally (see text).

Recombination may also proceed via an electronically excited state if during the course of a bimolecular collision the system may transfer from the nonquantized part of the potential curve associated with one electronic state to a second state from which emission is allowed. This process is called preassociation or inverse predissociation, and the selection rules that control the probability of crossing in both directions are well known [109]. In such encounters total angular momentum must be conserved. For diatomic molecules, the system can pass only into the rotational level of the excited bound state which corresponds to the initial orbital angular momentum in the collision. 

The formation of vibrationally excited products is nearly always energetically possible in an exothermic reaction, and these products can be detected by observing either an electronic banded system in absorption or the vibration-rotation bands in emission. In principle, rotational level distributions may be determined by resolving the fine structure of these spectra, but rotational energy is redistributed at almost every collision, so that any non-Boltzmann distribution is rapidly destroyed and difficult to observe. In contrast, simple, vibrationally excited species are much more stable to gas-phase deactivation and the effects of relaxation are less difficult to eliminate or allow for. 

Fig. 6. Symmetry classification of the rotational levels in the ground state of NH3. Arrows inversion and inversion-rotation transitions allowed by selection rules discussed in Section 4.3. Numbers in parenthesis behind the species symbols spin statistical weights...

A succession of levels like those of a linear molecule can be calculated for each quantum number K, which in this case describes the quantized component of the angular momentum about the unique a-axis. K cannot exceed 7, the quantum number for the total angular momentum, i.e., K = 0, 1,... dz7. For an oblate symmetric top the rotational constant A j has to be replaced by Q ]. In relation to the case of A" = 0, other K quantum numbers allowed will thus result in lower energies Ejk, which is in contrast to the prolate top with a positive term of (A[ j - 6 ]). Evidently, all rotational levels with 0 are doubly degenerate. It should be noted that each level still possesses an M-degeneracy of (27 -f 1) as discussed in connection with the linear molecule. This is due to space quantization. 

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