Relative orientation

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

Consider the case of two neutral, linear, dipolar molecules, such as HCN and KCl, in a coordinate system with its origin at the CM of molecule A and the z-axis aligned with the intemiolecular vector r pointing from the CM of A to the CM of B. The relative orientation of the two molecules is uniquely specified by their spherical polar angles 0, 03 and the difierence <]) = - <])3 between their azimuthal angles. The leading temi in the... 

A more natural way to account for the anisotropy is to treat tire parairreters in an interatomic potential, such as equation (A 1.5.64). as fiurctioirs of the relative orientation of the interacting molecules. Comer [131] was perhaps the first to use such an approach. Pack [132] pointed out that Legendre expansions of the well depth e and equilibrium location of the interaction potential converge more rapidly tirair Legendre expansions of the potential itself... 

From these relations it follows that is related to the angular momentum modulus, and that the pairs of angle a, P and y, 8 are the azimuthal, and the polar angle of the (J ) and the (L ) vector, respectively. The angle is associated with the relative orientation of the body-fixed and space-fixed coordinate frames. The probability to find the particular rotational state IMK) in the coherent state is... 

Figure 1. The space-fixed (ATZ) and body-fixed (xyz) frames. Any rotation of the coordinate system XYZ) to (xyz) may be performed by three successive rotations, denoted by the Euler angles (a, 3, y), about the coordinate axes as follows a) rotation about the Z axis through an angle a(0 < a < 2n), (b) rotation about the new yi axis through an angle P(0 < P < 7i), (c) rotation about the new zi axis through an angle y(0 Y < 2n). The relative orientations of the initial and final coordinate axes are shown in panel (d).

The explicit definition of water molecules seems to be the best way to represent the bulk properties of the solvent correctly. If only a thin layer of explicitly defined solvent molecules is used (due to hmited computational resources), difficulties may rise to reproduce the bulk behavior of water, especially near the border with the vacuum. Even with the definition of a full solvent environment the results depend on the model used for this purpose. In the relative simple case of TIP3P and SPC, which are widely and successfully used, the atoms of the water molecule have fixed charges and fixed relative orientation. Even without internal motions and the charge polarization ability, TIP3P reproduces the bulk properties of water quite well. For a further discussion of other available solvent models, readers are referred to Chapter VII, Section 1.3.2 of the Handbook. Unfortunately, the more sophisticated the water models are (to reproduce the physical properties and thermodynamics of this outstanding solvent correctly), the more impractical they are for being used within molecular dynamics simulations. 

Let us now consider the effect of placing another molecule with a linear charge distributioi (charges c(i and < 2) with its centre of mass at the point P. The relative orientation of the twc... 

A molecular fitting algorithm requires a numerical measure of the difference between two structures when they are positioned in space. The objective of the fitting procedure is to find the relative orientations of the molecules in which this function is minimised. The most common measure of the fit between two structures is the root mean square distance between pairs of atoms, or RMSD ... 

When fitting two structures, the aim is to find the relative orientations of the two molecules in which the RMSD is a minimum. Many methods have been devised to perform this seemingly irmocuous calculation. Some algorithms, such as that described by Ferro and Hermans [Ferro and Hermans 1977] use an iterative procedure in which the one molecule is moved relative to the other, gradually reducing the RMSD. Other methods locate the best fit directly, such as Kabsch s algorithm [Kabsch 1978]. 

The E field lies parallel to the Aq veetor, and the H field is perpendieular to Aq both are perpendieular to the direetion of propagation of the light k/ k. E and H have the same phase beeause they both vary with time and spatial loeation as sin (cot - k r). The relative orientations of these veetors are shown below. 

In the gas/liquid spray form of nebulizer, a stream of gas interacts with a stream of liquid. Depending on the relative velocity of the two streams and their relative orientation, the liquid flow is broken down into a spray of droplets, as in the common hair sprays. 

Flowever, the values of the total orbital angular momentum quantum number, L, are limited or, in other words, the relative orientations of f j and 2 are limited. The orientations which they can take up are governed by the values that the quantum number L can take. L is associated with the total orbital angular momentum for the two electrons and is restricted to the values... 

Interesting structures can be formed by combinations of ring and side-chain substituents in special relative orientations. As indicated above, structures (28) contain the elements of azomethine or carbonyl ylides, which are 1,3-dipoles. Charge-separated species formed by attachment of an anionic group to an azonia-nitrogen also are 1,3-dipoles pyridine 1-oxide (32) is perhaps the simplest example of these the ylide (33) is another. More complex combinations lead to 1,4-dipoles , for instance the pyrimidine derivative (34), and the cross-conjugated ylide (35). Compounds of this type have been reviewed by Ramsden (80AHCl26)l). 

It is up to the researcher to decide whether to use a Cartesian similarity measure or a dihedral measure and what elements to include in the summation [29]. It should be stressed that while the RMS distances perfonn well and are often used, there are no restrictions against other similarity measures. Eor example, similarity measures that emphasize chemical interactions, hydrophobicity, or the relative orientation of large molecular domains rather than local geometry may serve well if appropriately used. 

Figure 1 The principal sources of structural data are the NOEs, which give information on the spatial proximity d of protons coupling constants, which give information on dihedral angles < i and residual dipolar couplings, which give information on the relative orientation 0 of a bond vector with respect to the molecule (to the magnetic anisotropy tensor or an alignment tensor). Protons are shown as spheres. The dashed line indicates a coordinate system rigidly attached to the molecule.

Figure 3.S Schematic diagram of packing side chains In the hydrophobic core of colled-coll structures according to the "knobs In holes" model. The positions of the side chains along the surface of the cylindrical a helix Is pro-jected onto a plane parallel with the heUcal axis for both a helices of the coiled-coil. (a) Projected positions of side chains in helix 1. (b) Projected positions of side chains in helix 2. (c) Superposition of (a) and (b) using the relative orientation of the helices In the coiled-coil structure. The side-chain positions of the first helix, the "knobs," superimpose between the side-chain positions In the second helix, the "holes." The green shading outlines a d-resldue (leucine) from helix 1 surrounded by four side chains from helix 2, and the brown shading outlines an a-resldue (usually hydrophobic) from helix 1 surrounded by four side chains from helix 2.

Chothia, C., Janin, J. Relative orientation of close-packed p-pleated sheets in proteins. Proc. Natl. Acad. Sci. USA 78 4146-4150, 1981. 

The variable domains associate in a strikingly different manner. It is obvious from Figure 15.11 that if they were associated in the same way as the constant domains, via the four-stranded p sheets, the CDR loops, which are linked mainly to the five-stranded p sheet, would be too far apart on the outside of each domain to contribute jointly to the antigen-binding site. Thus in the variable domains the five-stranded p sheets form the domain-domain interaction area (Figure 15.11). Furthermore, the relative orientation of the p strands in the two domains is closer to parallel than in the constant domains and the curvature of the five-stranded p sheets is such that they do not pack... 

The previous analysis has shown that the properties of unidirectional fibre composites are highly anisotropic. To alleviate this problem, it is common to build up laminates consisting of stacks of unidirectional lamina arranged at different orientations. Clearly many permutations are possible in terms of the numbers of layers (or plies) and the relative orientation of the fibres in each... 

Inspired by experimental observations on bundles of carbon nanotubes, calculations of the electronic structure have also been carried out on arrays of (6,6) armchair nanotubes to determine the crystalline structure of the arrays, the relative orientation of adjacent nanotubes, and the optimal spacing between them. Figure 5 shows one tetragonal and two hexagonal arrays that were considered, with space group symmetries P42/mmc P6/mmni Dh,), and P6/mcc... 

Catalysis occurs because the catalyst in some manner increases the probability of reaction. This may result from the reactants being brought closer together [catalysis by approximation, or the propinquity effect ], or somehow assisted to achieve the necessary relative orientation for reaction. Noncovalent interactions may be responsible for the effect. Covalent bond changes may also take place in catalysis. In a formal way, the manner in which catalysis occurs can be described by schemes such as Schemes I and II. 

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