Coordinate, polar

Since xq = -y/l + Jq, we obtain A = arcsinh jo- Thus, we can express the hyperbolic functions in terms of the shaded area A  

Cartesian coordinates locate a point (x, y) in a plane by specifying how far east (x coordinate) and how far north (y coordinate) it lies from the origin (0, 0). A second popular way to locate a point in two dimensions makes use of plane polar coordinates, (r, 6), which specifies distance and direction from the origin. As shown in Fig. 10.4, the direction is defined by an angle 0, obtained by counterclockwise rotation from an eastward heading. Expressed in terms of Cartesian variables x and y, the polar coordinates are given by 

Integration of a function over two-dimensional space is expressed by 

In Cartesian coordinates, the plane can be tiled by infinitesimal rectangles of width dx and height dy. Both x and y range over [— oo, oo]. In polar coordinates, tiling of the plane can be accomplished by fan-shaped differential elements of area with sides dr mArdO, as shown in Fig. 10.5. Since r and 9 have ranges [0, oo] and [0, 27t], respectively, an integral over two-dimensional space in polar coordinates is given by 

FIGURE 10.5 Alternative tilings of a plane in Cartesian and polar coordinates. 

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The parameters of this matrix are the image / and the vector d written by [dx, dy] in cartesian coordinates or [ r, 0] in polar coordinates. The number of co-occurrence on the image / of pairs of pixels separated by vector d. The latter pairs have i and j intensities respectively, i.e. 

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. 

One possibility is to use hyperspherical coordinates, as these enable the use of basis fiinctions which describe reagent and product internal states in the same expansion. Hyperspherical coordinates have been extensively discussed in the literature [M, 35 and 36] and in the present application they reduce to polar coordinates (p, p) defined as follows ... 

Figure Bl.9.5. Geometrical relations between the Cartesian coordmates in real space, the spherical polar coordinates and the cylindrical polar coordinates.

If the scattering system is isotropic, equation (Bl.9.54) can be expressed in spherical polar coordinates (the derivation is similar to equation (B 1.9.32)) ... 

Similarly, the expression for the effective kinetic energy operator in polar coordinates will be. 

Hence, the expression of Eq. (5) indicates that, in a polar coordinate system, Eq. (4) will remain unchanged even if the position of the conical intersection is shifted from the origin of the coordinate system. 

We have used the above analysis scheme for all single- and two-surface calculations. Thus, when the wave function is represented in polar coordinates, we have mapped the wave function, it a i(, 0 r, t) in each... 

The numerical calculations have been done on a two-coordinate system with q being a radial coordinate and <() the polar coordinate. We consider a 3 x 3 non-adiabatic (vector) mabix t in which and T4, aie two components. If we assume = 0, takes the following form,... 

An example that is closely related to organic photochemishy is the x e case [70]. A doubly degenerate E term is the ground or excited state of any polyatomic system that has at least one axis of symmetry of not less than third order. It may be shown [70] that if the quadratic tenn in Eq, (17) is neglected, the potential surface becomes a moat around the degeneracy, sometimes called Mexican hat, The polar coordinates p and <(>, shown in Figure 20, can be used to write an expression for the energy ... 

Here, we discuss the motion of a system of three identical nuclei in the vicinity of the D3/, configuration. The conventional coordinates for the in-plane motion are employed, as shown in Figure 5. The noraial coordinates Qx, Qy, Qz), the plane polar coordinates (p,(p,z), and the Cartesian displacement coordinates (xi,yhZi of the three nuclei (t = 1,2,3) are related by [20,94]... 

By using the plane polar coordinates defined in Eq. (D.l), one obtains... 

Transformation of the Hamiltonian into polar coordinates then leads to... 

It has to be emphasized that in this framework J is the angular momentum operatoi in ordinary coordinate space (i.e., configuration space) and 0 is a (differential) ordinary angular polar coordinate. 

Here, 8(5) is the Dirac 6 function and /(0) is an arbitrary function to be determined [it can be shown that any function of the type/(g,0) leads to the same result because of the h q) function]. By considering Eq. 046) for the z component, we obtain (employing polar coordinates) ... 

In what follows, the 2D space is assumed to be a plane, and therefore we apply either the polar coordinates q, 0) or the Cartesian coordinates (x,y). 

We start heating the curl equation expressed in terms of polar coordinates ... 

To shift it to some arbirtrary point ( yo,0jo) we first express Eq. (161b) in terms of Cartesian coordinates, and then shift the solution to the point of interest, namely, to (xjo, o)[= ( T/Oi 0yo)]- Once completed, the solution is transformed back to polar coordinates (for details see Appendix F). Following... 

Reference [73] presents the first line-integral study between two excited states, namely, between the second and the third states in this series of states. Here, like before, the calculations are done for a fixed value of ri (results are reported for ri = 1.251 A) but in contrast to the previous study the origin of the system of coordinates is located at the point of this particulai conical intersection, that is, the (2,3) conical intersection. Accordingly, the two polar coordinates (adiabatic coupling term i.e. X(p (— C,2 c>(,2/ )) again employing chain rules for the transformation... 

Next, we are interested in expressing this equation in terms of polar coordinates (q, 0). For this purpose, we recall the following relations ... 

The PCM algorithm is as follows. First, the cavity siuface is determined from the van der Waals radii of the atoms. That fraction of each atom s van der Waals sphere which contributes to the cavity is then divided into a nmnber of small surface elements of calculable surface area. The simplest way to to this is to define a local polar coordinate frame at tlie centre of each atom s van der Waals sphere and to use fixed increments of AO and A(p to give rectangular surface elements (Figure 11.22). The surface can also be divided using tessellation methods [Paschual-Ahuir d al. 1987]. An initial value of the point charge for each surface element is then calculated from the electric field gradient due to the solute alone ... 

Governing equations in two-dimensional polar coordinate systems... 

After the substitution of pressure via the penalty relationship the flow equations in a polar coordinate system are written as... 

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