Centrosymmetric molecule

The example of COj discussed previously, which has no vibrations which are active in both the Raman and infrared spectra, is an illustration of the Principle of Mutual Exclusion For a centrosymmetric molecule every Raman active vibration is inactive in the infrared and any infrared active vibration is inactive in the Raman spectrum. A centrosymmetric molecule is one which possesses a center of symmetry. A center of symmetry is a point in a molecule about which the atoms are arranged in conjugate pairs. That is, taking the center of inversion as the origin (0, 0, 0), for every atom positioned at (au, yi, z ) there will be an identical atom at (-a ,-, —y%, —z,). A square planar molecule XY4 has a center of symmetry at atom X, whereas a trigonal planar molecule XYS does not possess a center of symmetry. 

When one of the cartesian coordinates (i.e. x, y, or z) of a centrosymmetric molecule is inverted through the center of symmetry it is transformed into the negative of itself. On the other hand, a binary product of coordinates (i.e. xx, yy, zz, xz, yz, zx) does not change sign on inversion since each coordinate changes sign separately. Hence for a centrosymmetric molecule every vibration which is infrared active has different symmetry properties with respect to the center of symmetry than does any Raman active mode. Therefore, for a centrosymmetric molecule no single vibration can be active in both the Raman and infrared spectrum. 

Centrosymmetric molecules represent a limiting case as far as molecular symmetry is concerned. They are highly symmetric molecules. At the other extreme, molecules with very low symmetry should produce a set of Raman frequencies very similar to the observed set of infrared frequencies. Between these two extremes there are cases where some vibrations are both Raman and infrared active and others are active in Raman or infrared but not in both. Nitrate ion is an example of a molecule in this intermediate class. 

Now look at octahedral complexes, or those with any other environment possessing a centre of symmetry e.g. square-planar). These present a further problem. The process of violating the parity rule is no longer available, for orbitals of different parity do not mix under a Hamiltonian for a centrosymmetric molecule. Here the nuclear arrangement requires the labelling of d functions as g and of p functions as m in centrosymmetric complexes, d orbitals do not mix with p orbitals. And yet d-d transitions are observed in octahedral chromophores. We must turn to another mechanism. Actually this mechanism is operative for all chromophores, whether centrosymmetric or not. As we shall see, however, it is less effective than that described above and so wasn t mentioned there. For centrosymmetric systems it s the only game in town. 

In addition to these possible blue shifts, there is a general rule that the intensities of the d-d spectra of centrosymmetric molecules decrease with cooling while... 

Few examples of chemical or enzymatic desymmetrizations of centrosymmetric molecules (point group S2 = Q) have been described [150]. The PLE-catalyzed hydrolysis of a centrosymmetric cyclohexanediacetate gave rise to an enantiomeri-cally pure (>99.5% ee) cyclohexanediol monoacetate in high yield [151] (Figure 6.57). 

In 1985, Astheimer et al, [134] described the crystal structure of the mesogenic 4,4 -di(7S0-hexoxybenzalazine, Here, the centrosymmetric molecule is fully elongated with the hexoxy group in an aU-trans conformation and the phenyl rings are exactly coplanar. 

These structure-function relationships provide extremely useful guidance for the future rational design of molecules and polymers with even higher optical nonlinearities. For non-centrosymmetric molecules such as 95, very high first hyperpolarizabilities /3 that determine the second-order nonfinear optical properties were also measured [140]. 

Similarly, the first-order expansion of the p° and a of Eq. (5.1) is, respectively, responsible for IR absorption and Raman scattering. According to the parity, one can easily understand that selection mles for hyper-Raman scattering are rather similar to those for IR [17,18]. Moreover, some of the silent modes, which are IR- and Raman-inactive vibrational modes, can be allowed in hyper-Raman scattering because of the nonlinearity. Incidentally, hyper-Raman-active modes and Raman-active modes are mutually exclusive in centrosymmetric molecules. Similar to Raman spectroscopy, hyper-Raman spectroscopy is feasible by visible excitation. Therefore, hyper-Raman spectroscopy can, in principle, be used as an alternative for IR spectroscopy, especially in IR-opaque media such as an aqueous solution [103]. Moreover, its spatial resolution, caused by the diffraction limit, is expected to be much better than IR microscopy. 

Hyper-Raman spectroscopy is not a surface-specific technique while SFG vibrational spectroscopy can selectively probe surfaces and interfaces, although both methods are based on the second-order nonlinear process. The vibrational SFG is a combination process of IR absorption and Raman scattering and, hence, only accessible to IR/Raman-active modes, which appear only in non-centrosymmetric molecules. Conversely, the hyper-Raman process does not require such broken centrosymmetry. Energy diagrams for IR, Raman, hyper-Raman, and vibrational SFG processes are summarized in Figure 5.17. 

Eq. (58) extends over all values of the quantum number J. It should be noted, however, that in the case of centrosymmetric molecules the rote of nuclear spins must be considered. 

If the molecules possess different excited state permanent and ground state permanent dipole moments, the D-terms can contribute to 2PA. Centrosymmetric molecules do not have permanent dipole moments in both ground and excited states, so their D-terms are zero. 

The structure of the centrosymmetric molecule Mo6010-(0-i-Pr)12 (3 1) has an S-chain of six molybdenum atoms and is... 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

The X-ray crystal structure of Pb2(o-tolyl)6 shows one centrosymmetric molecule per unit cell. Figure 53 clearly shows that there is no expansion of coordination of the lead atom in Pb2(o-tolyl)6- The bond distances of the tetrahedral coordinated lead atom to the carbon atoms are in the range of 2.242-2.249 A the Pb—Pb distance was found to be... 

The molecule adopts a planar and completely conjugated structure which exhibits exceptional stability. The unit cell contains two centrosymmetric molecules. 

It is important to note that the exponents of x for centrosymmetric molecules cannot be even numbers (i.e. materials composed by centrosymmetric molecules cannot generate effects of second order, fourth order, etc.). In fact, if one applies a field +E, the first non-linear term would induce a polarization of + fl-E2 (or + y 2)-E2). If one were to apply instead a field of-.E, the mathematical expression would lead to an induced polarization again of + fi-E2, whereas, as shown in Figure 38a, the induced polarization is -fi-E2. This contradiction can only be resolved if / (or /2)) = 0. 

The connection between the non-linear polarization response and the generation of optical harmonics by NLO materials can be understood by considering Figure 39, which shows a typical non-linear optical response produced by a non-centrosymmetric molecule. 

In the case of a centrosymmetric molecule the Fourier decomposition would be described by polarizations of frequency co, 3co, 5co, etc. 

One of the more interesting applications of non-linear optical effects is the generation of the second harmonic. This phenomenon results when a laser beam passes through a material having second-order NLO properties (hence, composed by non-centrosymmetric molecules) the light emitted has a frequency double that of the incident radiation (or the wavelength has been halved). 

Second harmonic generation (SHG) is one of the most intensively studied nonlinear optical effects that have ever been combined with near-held scanning optical microscopy (Shen et al. 2000 Zayats and Sandoghdar 2000 Zayats and Sandoghdar 2001 Takahashi and Zayats 2002). SHG, which is an even-order nonlinear process, is forbidden in centrosymmetric media under the dipole approximation (Shen 1984). Non-centrosymmetric molecules and lattices are allowed to exhibit SHG light. The second-order nonlinear polarization for SHG (T shg) is given in a scalar form by... 

Crystal structures of two hexitols, galactitol and D-mannitol, have been published. Galactitol is meso, but the permissible intramolecular center of symmetry is not utilized in the crystal.28 The molecules crystallize as enantiomorphic pairs that, in conformation, are almost centrosymmetric molecules the difference therefrom is of the same order of magnitude as the thermal motion of the atoms. The carbon atoms and terminal oxygen atoms form an approximately planar chain. All of the oxygen atoms are both donors and acceptors for an intricate network of hydrogen bonding. 

Comprehensive experimental investigations of 2PA processes in fluorene derivatives were performed by Hales et al. [53,56-59] with open aperture Z-scan [26], two-photon induced fluorescence [60] and femtosecond white-light continuum pump-probe methods [61]. For degenerate two-photon excitation, the experimental 2PA spectra of symmetrical and asymmetrical fiuorenes are presented in Figs. 15 and 16. These spectra were obtained with the combination of open aperture Z-scan and two-photon fluorescence methods [57]. For centrosymmetric molecules, two-photon transitions from... 

Certain crystals have been found to give intensity distribution curves which do not fit either the theoretical centrosymmetric or non-centro-symmetric types. In some crystals composed of centrosymmetric molecules, the centres of symmetry of the molecules are not utilized in the crystal arrangement, but the molecules are associated in pairs related centrosymmetrically to each other. The intensity distribution curves of these crystals lie above the centrosymmetric curve they have been called hypercentric. Other deviations from normal distributions rnav occur, and may be informative on molecular symmetry or arrangement (Lipson and Woolfson, 1952). 

A question which may sometimes be asked is this If an enantio-morphous crystal- -that is, one possessing neither planes, nor inversion axes, nor a centre of symmetry—is dissolved in a solvent, does the solution necessarily rotate the plane of polarization of light The answer to this question is, Not necessarily . If the molecules or ions of which the crystal is composed are themselves enantiomorphous, then the solution will be optically active. But it must be remembered that enantiomorphous crystals may be built from non-centrosymmetric molecules which in isolation possess planes of symmetry—these planes of symmetry being ignored in the crystal structure such molecules in solution would not rotate the plane of polarization of light. (A molecule of this type, in isolation, may rotate the plane of polarization of light (see p. 91), but the mass of randomly oriented molecules in a solution would show no net rotation.) An example is sodium chlorate NaC103 the crystals are enantiomorphous and optically active, but the solution of the salt is inactive because the pyramidal chlorate ions (see Fig. 131) possess planes of symmetry. 

Recall the mutual exclusion rule stated in Section 6.7. The rule follows from the fact that the integral over all space of an odd ( ) function is zero. The functions x, y, and z belong to u representations of molecules with a center of symmetry, since inversion converts each to its negative. Hence one of the functions vjb and ib must belong to a g representation and one to a u representation if the integrand of (9.189) is not to be odd. Thus only g<->u IR transitions are allowed in molecules with a center of symmetry. In contrast, the functions (9.196) are all even (g), so that for centrosymmetric molecules only g<->g and u u Raman transitions are allowed. This proves the mutual exclusion rule. 

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