Distance and orientational dependence

Consider two rigid molecules A and B, both of arbitrary shape. Let R = (R, Q) = (R, 0,0) be the vector pointing from the center of mass of A to the center of mass of B. The coordinates of ft are measured with respect to a space-fixed frame. Let the orientation of molecule A be described by the Euler an es = (a, Pa yx) whidh are 

TaMe 1. Angles and quantum mimbm qxdfying the orientational dqpendcnce of the interaction energy AE , formula (1) 

Cji(Q) is a Racah spherical harmonic, in the phase of Condon and Shortley, which can also be written as a special (K = 0) rotation matrix  

This property (3) and the presence of the 3-j symbol in (lb) makes the angular functions scalar, i.e. invariant under rotations of the total system (see Appendix). They span the complete space of scalar functions depending on and Q due to the 

For linear molecules A and B, where the interaction energy does not depend on the Euler angjes and yg, only terms with = Kg = 0 contribute and one can use (3) to obtain a simplified expression for (1) (see so table 1)  

A B angular coordinates ( a, b- C) volume of angular coordinate space V quantum numbers A 

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A number of different techniques have been applied to test the distance and orientation dependence of ET reactions (Closs and Miller, 1988 Closs et al, 1989 Liang et al., 1990 Reimers and Hush, 1990 Fox and Chanon, 1988 Wasielewski, 1989 Paddon Row and Jordan, 1988 Joachim et al, 1990 McConnell, 1961). Our method of analysing the mode of charge distribution in charged species is esr spectroscopy, which defines the timescale of the detectable dynamic species (Gerson, 1967 Kurreck et al, 1988 Wertz and Bolton, 1972). If an electron transfer is slow relative to the esr timescale (<10 7s) the spectrum corresponds to that of monomeric model compounds with a single electrophore. If the hopping process is rapid on the esr timescale, one will detect an effective delocalization. 

The characteristics of the superexchange or though-bond mechanism for ET have, until recently, remained somewhat obscure. Characteristics primarily describe the distance and orientation dependence of the dynamics of long-range ET, and how this dependency is affected by the nature and composition of the intervening medium. It is this aspect of long-range ET processes that has captured the attention of a broad cross section of the chemical community, partly because a deeper under-... 

Saini S, Srinivas G, Bagchi B (2009) Distance and orientation dependence of excitation energy transfer from molecular systems to metal nanoparticles. J Phys Chem B 113 1817-32... 

Wong KF, Bagchi B, Rossky PJ (2004) Distance and orientation dependence of excitation transfer rates in conjugated systems beyond the Forster theory. J Phys Chem A 108 5752-5763... 

There are implicit distance and orientation dependence to SEF that can be expressed as... 

Because of these considerations, most investigations of distance and orientation dependence of ET dynamics have been carried out using rigid bridges. It is also helpful if the D-B-A dyads possess symmetry. This facilitates the analysis of the electronic coupling between the chromophores and the bridge, as well as enabling one to explore the effect of orbital symmetry on ET dynamics. 

The irreducible tensor product between two (spherical) vectors is defined in Eq. (37). An important feature of this Hamiltonian is that it explicitly describes the dependence of the coupling constants J, Am, and T, on the distance vectors rPP between the molecules and on the orientations phenomenological Hamiltonian (139). Another important difference with the latter is that the ad hoc single-particle spin anisotropy term BS2y, which probably stands implicitly for the magnetic dipole-dipole interactions, has been replaced by a two-body operator that correctly represents these interactions. The distance and orientational dependence of the coupling parameters J, A, , and Tm has been obtained as follows. 

Dick, L.A., Haes, A.J., and Van Duyne, R.P. (2000) Distance and orientation dependence of heterogeneous electron transfer a surface-enhanced resonance Raman scattering study of cytochrome c bond to carboxylic acid terminated alkanethiols adsorbed on silver electrodes. Journal of Physical Chemistry B, 104, 11752-11762. 

Uie rate of electron transfer (ET) from a donor (D) to an acceptor (A) held at fixed distance and orientation depends on of temperature (7), reaction driving force (-AG°) a nuclear reorganization parameter (X), and an electronic coupling matrix element The reorganization parameter... 

The interaction between small molecules and oxide surfaces is characterized by a delicate balance of attractive and repulsive contributions which have different physical and chemical origin. In most cases, several of these contributions have the same order of magnitude, but different distance and orientation dependence. Reliable results for adsorption geometries and energies can be only obtained if all of these contributions are properly taken into account and are quantitatively correctly described. Otherwise, one will get results of doubtful quality, maybe accidentally even the correct answer. 

In the presence of a paramagnetic species, pseudocontact shifts can provide additional restraints since these shifts are both distance and orientation dependent. Gaponenko et a/. have attached a Co + paramagnetic probe to the N-terminal domain of the protein STAT4. The attachment is achieved by modifying a cysteine residue into a thiol-reactive chelator. With the additional restraints derived from pseudocontact shifts, the NMR structures are improved based on the rmsd value of the family of structures obtained. This approach is likewise illustrated in the work of Barbieri et on the protein Calbindin D9k. Here, the ions employed are Ce, Yb " , and Dy ". ... 

Recently a more detailed, orientation-dependent PMF - the potential that depends on the relative orientation of the two amino acid chains (see Figure 15.6) - has been proposed. This potential between any two amino acid residues was derived from the statistical analysis of the experimental native structures deposited in the Protein Data Bank (PDB). In this model potential amino acid side-chains are represented by a single ellipsoidal of revolution. The PMF between two ellipsoidals of revolution was obtained by considering all the interacting sites in them. The site-site potentials were then calculated from the statistics of their distance of separation obtained from the crystal structures available in the PDB [11]. These site-site potentials were then summed up to obtain the distance and orientation-dependent potential between all amino acid residues. The PMFs so obtained show many interesting features. Two of the PMFs are shown in Figure 15.7. 

The internal pressure primarily responds to rupture of London dispersion and dipole-dipole interactions, whereas the ced also includes breaking of the less distance- and orientation-dependent H-bond interactions. For water the ced is extremely high but the... 

This expression is particularly convenient, because (a) it separates the distance and orientation dependencies of the potential and (b) the coefficients of the expansion, the are found to decrease... 

Wigner rotation matrix. This conversion allows us to perform the electrostatic parametrization of a molecule within a conveniently chosen local frame, and to include the transformation from the local to the global interaction frame in a tensor that takes care of both the distance and orientation dependence ... 

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