Rotational spectroscopy mechanisms

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

Although the determination of the zero field eigenvalues and wavefunctions is described in many standard texts on rotational spectroscopy, we will briefly recall the principles and give some results for later reference. From Eq. (IV. 59 a) and IV.59b), the zero field Hamiltonian of an asymmetric top molecule is given by (from now on quantum mechanical operators will be denoted by underlining) ... 

Figure 34 shows also experimental results obtained by Schneider et using the method of Auger spectroscopy. For the system S + Ar it is seen that the RAD(cr) process provides the major contribution to the Ar L excitation. However, for Si + Ar the RAD((r) process becomes less important. In this case an additional process different from vacancy sharing has to be considered. This process, analyzed by Wille, is attributed to the lS-2ir-4or rotational coupling mechanism (Figure 31). As expected the respective cross sections from this process, labeled ROT in Figure 34, are practically the same for the systems S + Ar and Si + Ar. On the contrary, with increasing asymmetry of the system the RAD(o-) process loses importance, since the energy gap between the L shells of the collision partners increases (Figure 31). Hence, it is concluded that pure L-vacancy sharing may be studied only in rather symmetric collision systems.

In this chapter, we extend our treatment of rotation in diatomic molecules to nonlinear polyatomic molecules. A traditional motivation for treating polyatomic rotations quantum mechanically is that they form a basis for experimental determination for bond lengths and bond angles in gas-phase molecules. Microwave spectroscopy, a prolific area in chemical physics since 1946, has provided the most accurate available equilibrium geometries for many polar molecules. A background in polyatomic rotations is also a prerequisite for understanding rotational fine structure in polyatomic vibrational spectra (Chapter 6). The shapes of rotational contours (i.e., unresolved rotational fine structure) in polyatomic electronic band spectra are sensitive to the relative orientations of the principal rotational axes and the electronic transition moment (Chapter 7). Rotational contour analysis has thus provided an invaluable means of assigning symmetries to the electronic states involved in such spectra. 

Despite the fact that we treated vibrational and rotational spectroscopy first, the astute student will recognize that one of the mysteries of classical mechanics involved electronic spectroscopy. The inability to explain the (electronic) spectrum of the hydrogen atom was a major reason for the development of quantum mechanics. Yet, we have put off a detailed discussion of it until after considering rotational and vibrational spectra. 

In the present section all basic information required for understanding rotational spectroscopy is provided. We give as already assumed the Born-Oppenheimer approximation [21], which allows the separation of nuclear and electronic motion, as well as the separation of the various nuclear motions themselves (vibrational, rotational, translational). We therefore focus only on the quantum mechanics elements related to the rotational motion. 

The theory of rotational spectroscopy depends upon an understanding of the quantum mechanics of angular momentum. Useful results from the quantum theory of angular momentum are given in Table 1 and useful results in spherical tensor notation are given in Table 2. 

The rotation of a rigid, linear triatomic or polyatomic molecule is mechanically equivalent to the rotation of a rigid diatomic molecule. All are the rotations of an "infinitesimally thin rod" with two or more point masses attached. The basic analysis for the rotational spectroscopy of linear polyatomic molecules follows that of diatomic molecules however, it requires a generalization of the moment of inertia to more than two atoms. For a linear arrangement of point masses, the moment of inertia, I, about the center of mass is... 

In order to identify the dynamic nature of the macrocyclic component inside PRF-25 a label was placed on the macrocycle and monitored by SSNMR spectroscopy. The VT-SSNMR spectra clearly indicated that a dense array of soft 24C6 macrocycles were able to rapidly rotate while mechanically linked to the rigid, three-periodic framework of PRF-25 (Figure 18a). The motion was shown to be thermally driven and the rate of rotation of the macrocyclic component was estimated to be >10 MHz at temperatures >150 °C. This motion was shown to be completely reversible as cooling of the material caused reabsorption of H2O molecules and halted the free rotation. PRF-25 was the first known MOF material to show dynamic behavior of an interlocked component in the solid-state. 

Other relaxation mechanisms come into play when dipole-dipole relaxation is less efficient. The spin-rotation (SR) interaction is important for spin-5 nuclei in smaller molecules, particularly in the gas phase. The nucleus experiences magnetic fields due to the differential rotation of charge with the molecular frame, and fluctuations of these fields, as with collisions, induce relaxation. This mechanism is distinguished by increase in the rate with increase in temperature and with decrease in viscosity, in contrast to the mechanisms depending on molecular tumbling. This is because the mechanism is more effective with increased population of the higher rotational states. The rate depends on the spin-rotation coupling tensor (C) and its anisotropy (as the square) and appropriate moments of inertia (/), values of which may be available from rotational spectroscopy. 

Two important conclusions can be drawn from the solution obtained. One concerns the quantization of the rigid rotator energy the other describes the properties of angular momentum in quantum mechanics. We will use the first one later in Section 7.8.2 in the description of rotational spectroscopy here, we consider the properties of the angular momentums in quantum mechanics. 

If two different three-dimensional arrangements in space of the atoms in a molecule are interconvertible merely by free rotation about bonds, they are called conformationsIf they are not interconvertible, they are called configurations Configurations represent isomers that can be separated, as previously discussed in this chapter. Conformations represent conformers, which are rapidly interconvertible and are thus nonseparable. The terms conformational isomer and rotamer are sometimes used instead of conformer . A number of methods have been used to determine conformations. These include X-ray and electron diffraction, IR, Raman, UV, NMR, and microwave spectra, photoelectron spectroscopy, supersonic molecular jet spectroscopy, and optical rotatory dispersion (ORD) and CD measurements. Some of these methods are useful only for solids. It must be kept in mind that the conformation of a molecule in the solid state is not necessarily the same as in solution. Conformations can be calculated by a method called molecular mechanics (p. 178). 

There exist a series of beautiful spectroscopy experiments that have been carried out over a number of years in the Lineberger (1), Brauman (2), and Beauchamp (3) laboratories in which electronically stable negative molecular ions prepared in excited vibrational-rotational states are observed to eject their extra electron. For the anions considered in those experiments, it is unlikely that the anion and neutral-molecule potential energy surfaces undergo crossings at geometries accessed by their vibrational motions in these experiments, so it is believed that the mechanism of electron ejection must involve vibration-rotation... 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

The vibrational and rotational motions of the chemically bound constituents of matter have frequencies in the IR region. Industrial IR spectroscopy is concerned primarily with molecular vibrations, as transitions between individual rotational states can be measured only in IR spectra of small molecules in the gas phase. Rotational - vibrational transitions are analysed by quantum mechanics. To a first approximation, the vibrational frequency of a bond in the mid-IR can be treated as a simple harmonic oscillator by the following equation ... 

The conformation of the mixed p agonist/5 antagonist H-Tyr-c[-D-Orn-2-Nal-D-Pro-Gly-] in comparison to that of H-Tyr-c[-D-Orn-Phe-D-Pro-Gly-] was studied in DMSO-d6 by NMR spectroscopy and by molecular mechanics calculations [62,64]. Neither peptide showed nuclear Overhauser effects between C H protons or chemical exchange cross peaks in spectra obtained by total correlation and rotating frame Overhauser enhance-... 

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