Separation vector

If the interaction Hamiltonian in the Coulomb term is expanded in a series about the separation vector, the first term of the expansion is a dipole-dipole interaction, the second a dipole-quadrupole interaction, etc.<4> Again reverting to a classical analog (dipole oscillators), the energy of interaction between the two dipoles is inversely proportional to the third power of the... 

The constitution of molecules is given by lists of atoms and bonds (connectivity lists topological representation )6 . In addition, the number of free electrons for each atom is also carried in a separate vector. This is necessary as some reaction generators may transform free electrons into bonds, or vice versa. Thus, by working with free electrons and the electrons involved in bonds, all valence electrons of a molecule are explicitly specified. 

FIGURE 3.2 Track formation according to Mott (1930). Simultaneous excitation of atoms at 1 and 2 has negligible probability in second-order perturbation theory unless the interatomic separation vector R is well aligned with the incoming and outgoing momentum vectors of the incident particle. Reproduced from Mozumder (1969), by permission of John Wiley Sons, Inc. ... 

The reptation model (225) also appears to produce a Rouse spectrum at long times. In order to renew its configurations a chain must diffuse out of the tunnel defined by the fixed obstacles along its length. De Gennes calculates the autocorrelation function for the end-separation vector, obtaining... 

For molecules with inversion symmetry, like H2, the expansion parameter X must be even, Eqs. 4.15 through 4.17. (It also must be non-negative.) In order to relate the expansion coefficient Axl to the Cartesian dipole components calculated in a body-fixed frame, we choose the unit separation vector, R, to be parallel to the z-axis, hence M = 0, Yw = [(2L + 1)/4k] /2, and... 

Nitroaromatic explosives and other nitrated organic explosives are under the normal conditions neutral compounds and therefore cannot be separated directly by capillary zone electrophoresis (CZE) technique. Another separation vector must be introduced in order to achieve the resolution between the solutes. Micellar electrokinetic chromatography (MEKC) is typically employed on microchip scene for separation of nitroaromatic explosives. 

The right-hand side of Eqs. (3) and (4) consist each of two terms. The first term represents the effect of distillation (separation vector), and the second the effect of the chemical reaction (reaction vector). For nonreactive systems only the separation vector plays a role. For kinetically controlled systems, both vectors can dominate the system behavior, depending on the value of Da. For Da —> °°, the liquid mixture approaches chemical equilibrium. 

Many entries in a BB-matrix are zero. Therefore, for the internal representation of molecules in the program a more compact form was chosen. The constitution of molecules and ensemble of molecules is given by doubly indexed lists of bonds. The free electrons on the atoms are kept in a separate vector, as are the atomic numbers (see Fig. 4). 

Fig. 3.11. Production of chimeric antibodies. Chimeric genes or cDNA for the heavy and light chains may be assembled into two separate vectors as shown, or may be assembled into a single vector.

For atomic liquids with only spherically symmetric interactions, the pair distribution function will contain no angular dependence and hence the structure in the system (at the pairwise level) is completely given by the radial distribution function, g(r), where r=lrl is simply the magnitude of the separation vector. For a molecular system, the radial distribution function is obtained from the full angle average of the pair distribution function. 

The local frame for rigid molecules, once chosen, will always be clearly defined. The necessary transformation of the separation vector from the laboratory to the local frame is usually accomplished by multiplication by the rotation matrix of the central molecule. The construction of this rotation matrix is usually a straightforward task. In fact, it will already be available in any program that describes molecular orientations in terms of quaternions (coordinate-transformed eulerian angles) [3,24]. 

Figure 1 (a) Radial distribution functions for soft sphere (solid line) and dipolar soft-sphere fluids at the same temperature and density, (b) The spatial distribution function g(r,0), where 0 is the angle between the dipole and separation vectors, for the dipolar soft-sphere in (a). 

I For conciseness, the sccilar acoustic and vector wavefield equations have been combined into single equations in this summary, with alternative subscripts a and v to distinguish them it is understood, of course, that d , A , and Rv n (but not the adjoint operator F. ), would be printed in boldface in separate vector wavefield equations since they refer to 3-D vector elements in... 

Let again R be the center-of-mass separation vector at a point of the trajectory, v the relative velocity and n the unit vector perpendicular to the equipotential contour at the point in question (Figure 5). In the case of a diatomic reagent the formulas given in Section 2 assume the following simpler form ... 

Figure 5. Schematic representation of O + H2 and O + HO IDCI) collisions. The heavy line repesents a part of the critical dividing surface, lighter lines represent equipotential contours. Ellipsoidal approximations are used for both kind of surfaces as described in the text, v is the relative velocity of the collision partners, R is their center-of-mass separation vector, y is te Jacobi angle, n is the normal to the equipotential energy surfeces. The coordinate origin is at the center of mass of the molecule axis x coincides with its longitudinal axis. In the present model calculations, straight line trajectories up to the critical dividing surface are assumed...

Let s make these ideas more precise. Suppose that we let transients decay, so that a trajectory is on the attractor. Suppose x(z) is a point on the attractor at time t, and consider a nearby point, say x(z) + 5(Z), where S is a tiny separation vector of initial length 5(, II = IO" , say (Figure 9.3.4). 

Before describing the numerical methods used to locate quaslperiodic or periodic resonant orhlts, we will define several sets of coordinates that are useful in specifying the size and shape of the three atom triangle. Let 5 be the scaled) vector from the center-of-mass of H2 to the F atom, and let r be the (scaled) H2 separation vector... 

The length of the vectors M(V grows or decreases exponentially in time as exp(AH). After long time the sum (2.72) will be dominated by the term corresponding to the largest Lyapunov exponent. Thus, for almost all initial directions of the separation vector the distance... 

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