Symmetric coordinate system

Another approach to the A + BC - AB + C reaction has been developed by Diestler (1971) making use of two symmetric coordinate systems ( ab>xbc) which is appropriate for reactants, and (xAC,xAB) for products. The common coordinate xAB is initially large, decreases during collision and increases again whether direct or atom-exchange events take place. The total wavefunction is written in the form... 

For symmetric top species, Pave hes along the symmetry axis of the molecule, so the orientation of Pave can again be described in terms of 0 and (j), the angles used to locate the orientation of the molecule s symmetry axis relative to the lab-fixed coordinate system. As a result, the El integral again can be decomposed into three pieces ... 

Equation (13.20) corresponds to a symmetrical orthogonalization of the basis. The initial coordinate system, (the basis functions %) is non-orthogonal, but by multiplying with a matrix such as S the new coordinate system has orthogonal axes. 

In the above treatment of the problem of the particle in a box, no consideration was given to its natural symmetry. As the potential function is symmetric with respect to the center of the box, it is intuitively obvious that this position should be chosen as the origin of the abscissa. In Fig. 4b, x =s 0 at the center of the box and the walls are symmetrically placed at x = 1/2. Clearly, the analysis must in this case lead to the same result as above, because the particle does not know what coordinate system has been chosen It is sufficient to replace x by x +1/2 in the solution given by Eq. (68). This operation is a simple translation of the abscissa, as explained in Section 1.2. The result is shown in Fig. 4b, where the wave function is now given by... 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

There is an important relationship between vectors and skew-symmetric tensors. Suppose A and B are two vectors in a three-dimensional rectangular coordinate system whose components are connected by... 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

Another problem comes in examining the polarizability. In the physical picture, the spherically symmetric molecule, just like an atom, has isotropic polarizability. In the chemical picture, for a diatomic molecule we have two unique polarizabilities (1) and in the internal coordinate system or (2) dzz = 5 (o xc + (isotropic polarizability) and Aa = — [polar-... 

Although we know that the total force field / is conservative, we lack such knowledge concerning the separate pieces fx or A- Therefore some specific path of integration must be chosen to complete the definition (210). Seemingly the most natural path is along the radius on which the point ro lies. As the center of the coordinate system a position of the nucleus is chosen in the case of single-ion system, the center of symmetry for a symmetrical molecule, and some inner point for molecule or cluster with lower symmetry. Then... 

The Fourier transform of the spherical atomic density is particularly simple. One can select S to lie along the z axis of the spherical polar coordinate system (Fig. 1.4), in which case S-r = Sr cos. If pj(r) is the radial density function of the spherically symmetric atom,... 

The skew-symmetric part S 4 is equivalent to a vector (x t)/2 with components (/. t),/2 = (/.jtk — /.ktj)/2, involving correlations between a libration and a perpendicular translation. The components of S 4 can be reduced to zero, and S made symmetric, by a change of origin. It can be shown that the origin shift that symmetrizes S also minimizes the trace of T. In terms of the coordinate system based on the principal axes of L, the required origin shifts p, are... 

As in a Cartesian coordinate system the tensor product u v, of the vectors u and v, has the elements uxvp, the EFG tensor is a symmetric tensor with elements... 

It can easily be shown that ao.rand is independent of anisotropy for any type of anisotropy, not only axially symmetric [21]. Changing the coordinate system of Eq. (3.21) to the local one determined by the anisotropy direction of each spin. 

Wood Kirkwood (Ref 36a) assumed a curved shock front leading a zone which is cylindrically symmetric. Their coordinates were x, coincident with the axis of the cylindrical chge, and r, the radial distance from the axis. The vector mass velocitytf has an axial component u and a radial component >. Fig 3Oof Ref 66, p 157 is a sketch of the flow in a coordinate system which moves with.the deton wave. Here = space coordinate within reaction wave ... 

Fig 30 Schematic diagram of cylindrically symmetric flow In a detonation wave, with coordinate system at rest in the detonation front... 

In the preceding sections the optical response of matter has been described by a scalar dielectric function e, which relates the electric field E to the displacement D. More generally, D and E are connected by the tensor constitutive relation (5.46), which we write compactly as D = e0e E. The dielectric tensor is often symmetric, so that a coordinate system can be found in which it is diagonal ... 

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