Rotational spectroscopy, three-dimensional

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). 

Rotational motion is spinning of the entire molecule around an axis in three-dimensional space. Figure 10 illustrates the rotational motion of a water molecule. Rotational motion occurs in liquid and gas phases of water and, to a limited extent, through defects in the solid phase (ice). Rotational motion of water molecules can be measured using NMR and dielectric spectroscopy (Belton, 1994). 

A coirplete understanding of the role of carbohydrates in biological systems requires knowledge of the distribution at equilibrium of the various conformers in aqueous solution. The conformational behavior of carbohydrates in solution can be examined from different vantage points (1,), but the most relevant approach is, no doubt, study of dilute solutions themselves. At present, high resolution NMR spectroscopy is the primary tool for determination of three-dimensional structure of oligosaccharides in solution. Optical rotation is also very sensitive to conformation (2) and there is a new, semi-enqpirical theory of optical rotation of oligosaccharides ( ). 

The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modem theory of complex atoms and ions is based. 

Much of the beauty of high-resolution molecular spectroscopy arises from the patterns formed by the fine and hyperfine structure associated with a given transition. All of this structure involves angular momentum in some sense or other and its interpretation depends heavily on the proper description of such motion. Angular momentum theory is very powerful and general. It applies equally to rotations in spin or vibrational coordinate space as to rotations in ordinary three-dimensional space. 

NMR experiments include COSY, TOCSY, Cheteronuclear NMR experiments, NOESY (nuclear overhauser enhancement spectroscopy) and ROESY (rotating frame overhauser effect spectroscopy) as well as other two- and three-dimensional methodologies (Fossen and Andersen, 2006). 

Fig. 2.4. (A) Sketch of the cryostat insert for single-molecule spectroscopy by fluorescence excitation. The focus of lens L is placed in the sample S by the magnet/coil pair M, C. (B) Scan over the inhomogeneous line (a) with a 2 GHz region expanded (b) to show isolated single-molecule absorption profiles. (C) Three-dimensional pseudo-image of single molecules of pentacene in p-terphenyl. The measured fluorescence signal (z-axis) is shown over a range of 300 MHz in excitation frequency (horizontal axis, center = 592.544 nm) and 40 pm in spatial position (axis into the page). (D) Rotation of the data in (c) to show that in the spatial domain, the single molecule maps out the shape of the laser focal spot. Bar, 5 pm. For details, see [33]...

Chiroptical spectroscopies are based on the concept of chirality, the signals are exactly zero for non-chiral samples. In terms of molecular symmetry, this means that the studied system must not contain a rotation-reflection axis of symmetry. This lapidary definition implies that the more known symmetry elements (symmetry plane - equivalent to the one-fold rotation-reflection axis and the center of symmetry - equivalent to the two fold rotation-reflection axis) must also be absent and that the system must be able to exist at least formally in two mirror image-like forms. At first glance this limitation seems to be a disadvantage, however, this direct relation to molecular geometry gives chiroptical properties their enormous sensitivity to even minor and detailed changes in the three-dimensional structure. 

Let us summarize the three important prerequisites for a 3D structure descriptor It should be (1) independent of the number of atoms, that is, the size of a molecule (2) unambiguous regarding the three-dimensional arrangement of the atoms and (3) invariant against translation and rotation of the entire molecule. Further prerequisites depend on the chemical problem to be solved. Some chemical effects may have an undesired influence on the structure descriptor if the experimental data to be processed do not account for them. A typical example is the conformational flexibility of a molecule, which has a profound influence on a 3D descriptor based on Cartesian coordinates. The application in the field of structure-spectrum correlation problems in vibrational spectroscopy requires that a descriptor contains physicochemical information related to vibration states. In addition, it would be helpful to gain the complete 3D structure from the descriptor or at least structural information (descriptor decoding). 

Polyesters derived from maleic anhydride and 2,2-di(4-hydroxyphenyl)pro-pane were copolymerised with styrene and then studied by CP/MAS NMR [39] spectroscopy. The three dimensional-crosslinked network formed by the polymerisation was examined using spin-lattice relaxation times in the rotating frame. A correlation between reaction conditions and the structure of the resulting material was found. The degree of residual unsaturation was determined by subtraction of two relaxation times from a linear additivity model used for erosslinked polymer systems. 

It would be excessive to include in this article a comprehensive treatise on rotational molecular spectroscopy. It is, however, worthwhile to address this important subject for the purpose of demonstrating what kind of theoretical constructions can be handled within the algebraic framework. So far, most applications of algebraic models have been dealt with vibrational rather than rotational spectroscopy. However, it is only a matter of time before the algebraic treatment is applied to rotational spectroscopy since it has a unique association with the three-dimensional model. 

Spin—lattice relaxation is the time constant for the recovery of magnetiTation along the z-axis in a NMR experiment. Various methods are available for the measurement of spin lattice relaxation times. The interested reader is referred to the series of monographs echted by Levy on Carbon-13 NMR spectroscopy [44, 45] for more details. The energy transfer between nuclear moments and the lattice , the three-dimensional system containing the nuclei, provides the mechanism to study molecular motion, e.g. rotations and translations, with correlation times of the order of the nuclear Larmour frequencies, tens to hundreds of MHz. We will limit our chscussion here to the simple inversion-recovery Tj relaxation time measurement experiment, which, in addition to providing a convenient means for the quick estimation of Tj to establish the necessary interpulse delay in two-dimensional NMR experiments, also provides a useful entry point into the discussion of multi-dimensional NMR experiments. 

Rotational spectroscopy arises from the quantized rotations of molecules in three-dimensional space. Atoms do not have rotational spectra. TTowever, diatomic molecules have a relatively simple rotational spectrum, because they can rotate in only two dimensions (a rotation about the internuclear axis will not be observed) and their behavior of rotation is the same for both directions. Nonlinear polyatomic molecules have one (for highly symmetric molecules) to three (for most, less-symmetric species) different rotations in space, complicating a rotational spectrum. 

The energies of rotation of gas-phase molecules can be approximated very well by the three-dimensional rigid rotor ideal system. In fact, we used this approximation in Chapter 14 to derive a basic understanding of rotational spectroscopy. We can apply that understanding to the rotational energy levels and... 

---