Polar coordinate frame

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 ... 

For the spherical case the gradient of magnitude q is applied along the polar axis of the spherical polar coordinate frame. The boundary is at a radial distance r = a from the sphere center ... 

Then, a new, polar coordinate frame was introduced based on the director n as a polar axis. To obtain the single molecular potential as a function of the first molecule orientation with respect to n, one has to take three successive averages of VF12 (a) over all orientations of intermolecular vector r, (b) over all orientations of molecule 2, and (c) over all intermolecular separations Irl. 

We naturally chose to work in a polar coordinate frame of reference where the trajectory is represented by the polar coordinates (r, 6, 4>). Moreover, it is convenient to define a frame of reference with O located at the origin of the coordinate system with z axis parallel to g2i- The 2 molecule approaches 0 with velocity g2i, whose perpendicular distance to O is called the impact parameter b. The polar coordinates of the point molecule 2 thus give its position relative to a fixed reference point O and a given polar axis z. The radial coordinate r is the distance from O to the molecule... 

In the polar coordinate frame, the in-plane moment of the external force M = a x F) is given by ... 

Using equation (2) the static balance equations in the XjZj and Xjyj planes can be written. Depending on the attitude of the peg, its weight will effect the static balance and must be accounted for in the calculations. The weight vector W is always in -z direction in the world coordinate frame. Its effect expressed in the polar coordinate frame (peg) is ... 

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... 

In the principal axis frame of the alignment tensor, A, the dipolar coupling between two nuclei, P and Q, as a function of the polar coordinates, 6 and tp, is given by... 

Fig. 2. The angular coordinates used to describe the orientation of the magnetic field, B0, and an internuclear vector, r, relative to an arbitrarily chosen coordinate frame which is fixed to the molecule. Description in terms of (A) direction cosines and (B) polar angles.

Figure 30. Schematic charge-cloud distribution of atomic />-state excited by linearly polarized light with electric vector EA . Different relative alignments of Eh lead to Ict), ) states with respect to fixed coordinate frame.

In order to illustrate the mixed state, an example with five sample wavelets will be discussed in detail. Each wavelet is represented by its components ax and ay in the Cartesian basis (optical definition, see Section 9.2.2). If the polarization vector is described by a polarization ellipse with major and minor axes a = cos y and b = sin y, by a tilt angle X of this ellipse against a fixed coordinate frame (see Fig. 1.15), and by the direction of rotation of the electric field vector indicated by the sign of y, the components ax and cty follow from... 

This expression disentangles the properties of the light polarization (coefficients /9, ( 1)), the geometry of two-electron emission (coefficients B t2(Ka, Kb)) and the dynamical parameters of the double photoionization process (coefficients A(ku k2, k)). The pkq(El) are the statistical tensors of the incident light which describe its polarization properties in the electric dipole approximation represented by 1. For linearly polarized light in which the electric field vector defines the z-axis of the coordinate frame, one has only two non-vanishing components given by (see equ. (8.99b))t... 

It is possible to transform this expression into a form which contains the polar and azimuthal angles 0 and O as defined in the tilted coordinate frame of Fig. 1.15. Applying spherical trigonometry, one then has the following relation between these angles... 

Figure 5.32 Angular correlation between 4d5/2 photo- and N5-02,302>3 1S0 Auger electrons in xenon. A polar plot of the coincident intensity is shown for the following experimental conditions photon energy 94.5 eV, Stokes parameter = 0.957, S2 = 0, S3 unknown both electrons are detected in a plane perpendicular to the incident photon beam, the photoelectron at fixed position (ephe) with ° = 90°, O = 150°, the Auger electron at 2 = 90° and 2 = variable (these angles refer to the tilted coordinate frame of Fig. 1.15). Points with error bars are experimental data the solid line is a least-squares fit (see main...

Here we have partitioned the sums over all atoms a and /3 in the molecules P and P in the following manner. First, we sum over equivalent atoms within the same class a E a and (3 E b, which have the same chemical nature X = Xa and Xp = X and the same distance da = da and dp = db to the respective molecular center of mass. Next, we sum over classes a E P and b E P. The orientations da and dp of the position vectors of the atoms d and dp, relative to the molecular centers of mass, are still given with respect to the global coordinate frame. If we denote the polar angles of da and dp in the molecule fixed frames by d°a and dp and remember that the molecular frames are related to the global frame by rotations through the Euler angles o)P and to/., respectively, we find that... 

If the magnetic-field gradient is applied in direction r in polar coordinates, which forms an angle

Cartesian reference frame, the projection P (r, spin density Mo(r, 5) along direction s, which is perpendicular to r (cf. Fig. 5.4.1),... 

The Euler angle f does not appear in the kinetic energy, and consequently it is a cyclic or ignorable variable. In the body frame xyz, the other Euler angles 9, (p are the polar coordinates of the Z axis. The determinant of fhe matrix connecting the two alternative representations in Eq. (7) is D = sin 9 jabc. 

This Lagrangian should be thought of as dependent on 3Ne + 6 generalized coordinates, qt, and velocities, g, respectively. These are the 3Ng coordinates be, Ce which describe the relative positions of the Ne electrons with respect to the nuclear frame three coordinates Xo, Vo and Zo which describe the position of the molecular center of mass as referred to the laboratory coordinate system, and three Eulerian angles 6, and x which describe the instantaneous orientation of the molecular coordinate system with respect to the space fixed X-, Y- and Z-axes. There are numerous ways of specifying Eulerian angles. Because of later reference we will follow the choice used by Wilson et where and 6 are the ordinary polar coordinates of the molecular c-axis O d n 0 < < 2n) and x is the angle between the nodal line N and the positive b axis as is illustrated in Fig. IV.2. x is positive for clockwise rotation about the c axis. 

Figure 5 Equipotential energy surface for a system of three atoms in symmetrized internal coordinates. The OXK = OYk, OY = OZ, and OZx = OXK define two related nonphysical internal nuclear coordinate frames and spaces. A point P in those spaces has spherical polar coordinates p, cox, yx in the OXkYZx frame and p, 0,

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