Transition-moment vector

The form of the Plmn will be discussed later, because it is instructive to develop the argument by considering next the information which is obtained from any spectroscopic technique. Figure 2a shows a direction within a unit of structure which is defined by the polar and azimuthal angles (, r ). For example, this could be the direction defining the change in dipole moment (the transition moment vector) in an infra-red spectroscopic measurement. The spectroscopic measurements provide... 

For the case where the transition moment vector makes an angle with the chain axis, the chain axis average [Pg.89]

Figure 4.17 A. Fluorescence polarization spectrum. A general case of a molecule with two absorption bands a and b with transition moment vectors at right angles to each other. (1) Absorption spectrum (2) Polarization of flourescence excitation spectrum. 

We noted earlier (Section I. 1.) that the intensity of an absorption band is proportional to the square of the changing dipole moment in the molecule (i. e., transition moment) during the corresponding normal vibration. The intensity also depends upon the direction that the electric vector in the incident radiation makes with the transition moment. In particular, the intensity is proportional to the square of the scalar product of the transition moment and electric field vectors. This implies, for example, that if the electric field vector is perpendicular to the transition moment vector no absorption will occur. This fundamental relationship is the basis for the utilization of polarized infrared radiation as a powerful tool in the study of the spectra and structure of oriented polymers. We consider below some aspects of this technique. 

Figure 2 Conventional representation of micelles formed by an ionic surfactant, such as sodium dodecyl sulfate. The inner core region consists of the methylene tails of the surfactants. The Stem layer consists of surfactant headgroups and bound counterion species. The diffuse double layer consists of unbound counterions and coions which preserve the electrical neutrality of the overall solution. Also pictured are the transition moment vectors for the S-O stretching modes of sodium dodecyl sulfate.

The driving force for formation of rod shaped SDS micelles is the elimination of water from die micellar core/water interface (31). The reduction in average headgroup area reflects the removal of water molecules between the SDS headgroups, and should affect the bands due to the asymmetric S-O stretching vibrations, as indicated in the discussion of the transition moment vectors above. 

The transition moment vector of va, S-O points between surfactant molecules, and is therefore extremely sensitive to surfactant-surfactant interactions. The enhanced perturbation of vas S-O indicates that the electrostatic interactions between the dissimilar headgroups reduces a0 sufficiently to "drive" rod micelle formation in mixtures of DTAC and SDS. 

Figure 27. UV spectra of anthracene and the ring fused bis-anthracene (ref. 2). On the anthracene structure are shown the in-plane orientations of the electric dipole transition moment vectors associated with the intense short wavelength UV bands (1Bb) near 250 nm and the weak long wavelength bands ( LJ near 360 nm. The Xmax for the exciton bands of bis-anthracene are red shifted from the corresponding bands in the parent, as the vectors intersect at >90°.

Figure 34. (a) Dihydroxylation of benzo[a]pyrene. (b) Coupling constant and exciton chirality CD CEs of the bis-p-dimethylaminobenzoate. (c) (-) Exciton chirality and relative orientation of the transition moment vectors, hence the absolute configuration of the diol. 

IR absorptions involve elastic or Rayleigh45 or constant-energy scattering of light in more detail, the electric field vector E of the input light must couple with the transition electric dipole moment fi,f as E fi,f. If E Lp,if, then no IR transition is seen. Allowed IR transitions require that the transition moment vector fii be nonzero—i.e., that is, that the static electric dipole moment fi of the molecule change during the IR absorption. 

The ellipticity, or the intensity of the circular dichroism (CD) spectrum, is fundamentally characterized by the rotatory strength R, which is given by the imaginary part of the inner product between the electronic and magnetic dipole transition moment vectors ... 

Here i//0 is the ground vibrational wave function and ij/ is the wavefunction corresponding to the first excited vibrational state of the th normal mode /< is the electric dipole moment operator Qj is the normal coordinate for the /th vibrational mode the subscript 0 at derivative indicates that the term is evaluated at the equilibrium geometry. The related rotational strength or VCD intensity is determined by the dot product between the electric dipole and magnetic dipole transition moment vectors, as given in (2) ... 

Most interactions of electromagnetic radiation with matter contain a geometric, as well as an energetic component. For visible and ultraviolet absorption this is because the fundamental relationship governing the absorption of light is the transition moment integral, (6.1), in which r is the transition moment vector defining... 

In 1958, Beer et al. re-examined the infrared dichroism situation and remeasured the dichroic ratio for several of the peptide bands in collagen. On the basis of studies on model compounds, they calculated the directions of the transition moments for the principal bands in the peptide link and, using the atomic coordinates for various proposed collagen models, the inclination of these transition moment vectors to the fiber axis. Since Beer (1956) had shown that the dichroism of a partially oriented polymer may be considered equivalent to that of a sample containing fully aligned and completely random portions, a disorientation parameter (/) characteristic of the degree of order of the sample could be calculated for each band. The spread of the values obtained for this parameter, which is characteristic of the sample only and not of the bands used to determine it,... 

FIGiJflE 3.22 Drawing of the top view and transition-moment vectors at 365 and 532 nm of tiie closed form of DE obtained by CNDO/S AMI MORAC molecular orbital calculations.The 532 nm is parallel to the long axis of the molecule. After reference 42, redrawn by permission of ACS. 

Since the electric dipole moment operator is a vector operator, the electric dipole transition moment will also be a vector quantity. The probability of an electric dipole transition is given by the square of the scalar product between the transition moment vector in the molecule and the electric field vector of the light, and is therefore proportional to the squared cosine of the angle between these two vectors. Thus, an orientational dependence results for the absorption and emission of linearly polarized light. The orientation of the transition moment with respect to the molecular system of axes is... 

From Equation (1.33) it is seen that the transition moment is a vector quantity. The square of its absolute value determines the transition probability, whereas its direction is called the polarization direction. If one of the principal axes of a molecule is the long axis, the transition as well as the corresponding absorption band is often called parallel or long axis polarized if the direction of the transition moment vector is perpendicular to the long axis, the transition and the absorption band are called perpendicular or short-axis polarized. The longest wavelength transition of s-cis-buta-... 

Integration over the coordinates of those electrons that occupy the same orbital in o and Xi —/ gives unity, <(/// (///> = 1, so the transition moment simplifies to Mei y = , which involves only the orbitals xj/i and ij/j that participate in the transition. The operator Lri=X(xi, V/, z() and the electronic transition moment MeU j are vector quantities the coordinates xh yt and zt define the positions of the electrons e. To obtain the individual components of the electronic transition moment vector (Equation 4.20), we must integrate the products where u stands for x, y or z. The... 

Fluorescence depolarization measurements of aromatic residues and other probes in proteins can provide information on the amplitudes and time scales of motions in the picosecond-to-nanosecond range. As for NMR relaxation, the parameters of interest are related to time correlation functions whose decay is determined by reorientation of certain vectors associated with the probe (i.e., vectors between nuclei for NMR relaxation and transition moment vectors for fluorescence depolarization). Because the contributions of the various types of motions to the NMR relaxation rates depend on the Fourier transform of the appropriate correlation functions, it is difficult to obtain a unique result from the measurements. As described above, most experimental estimates of the time scales and magnitudes of the motions generally depend on the particular choice of model used for their interpretation. Fluorescence depolarization, although more limited in the sense that only a few protein residues (i.e., tryptophans and tyrosines) can be studied with present techniques, has the distinct advantage that the measured quantity is directly related to the decay of the correlation function. 

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