Cartesian coordinates intersections

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

In Figure 1, the force balance in Cartesian coordinates for a body not intersected by phase boundaries is... 

Figure 14. Classical trajectories for the H + H2(v = l,j = 0) reaction representing a 1-TS (a-d) and a 2-TS reaction path (e-h). Both trajectories lead to H2(v = 2,/ = 5,k = 0) products and the same scattering angle, 0 = 50°. (a-c) 1-TS trajectory in Cartesian coordinates. The positions of the atoms (Ha, solid circles Hb, open circles He, dotted circles) are plotted at constant time intervals of 4.1 fs on top of snapshots of the potential energy surface in a space-fixed frame centered at the reactant HbHc molecule. The location of the conical intersection is indicated by crosses (x). (d) 1-TS trajectory in hyperspherical coordinates (cf. Fig. 1) showing the different H - - H2 arrangements (open diamonds) at the same time intervals as panels (a-c) the potential energy contours are for a fixed hyperradius of p = 4.0 a.u. (e-h) As above for the 2-TS trajectory. Note that the 1-TS trajectory is deflected to the nearside (deflection angle 0 = +50°), whereas the 2-TS trajectory proceeds via an insertion mechanism and is deflected to the farside (0 = —50°).

Coordinate Systems The basic concept of analytic geometry is the establishment of a one-to-one correspondence between the points of the plane and number pairs (x, y). This correspondence may be done in a number of ways. The rectangular or cartesian coordinate system consists of two straight lines intersecting at right angles (Fig. 3-12). A point is designated by (x, y), where x (the abscissa) is the distance of the point from the y axis measured parallel to the x axis,... 

A semi-infinite solid slab is dipped vertically into a large pool of liquid open to the atmosphere. As a result of surface tension, the liquid rises above the pool level to meet the slab on its face at the contact angle 0 <7t/2. The geometry is two-dimensional, and a rectangular Cartesian coordinate system is chosen with origin at the intersection of the slab face and the undisturbed pool surface. The coordinate y is measured vertically upward from the undisturbed pool level, and the coordinate x is measured into the raised liquid. 

We will consider the general class of orthogonal curvilinear coordinates, designated q, q2, and <73, whose coordinate surfaces always intersect at right angles. The Cartesian coordinates of a point in three-dimensional space can be expressed in terms of a set of curvilinear coordinates by relations of the form... 

Fig. 9.1 The tilted Cartesian coordinate system with 60° angle between x and y axes and the tilted network in the x y-plane. A polyhex corresponding to the carbon skeleton of dibenzo[b,g] phenanthrene is placed onto this network in such a way that x,y intersections appear in the centre of each of its hexagons...

To characterize the neighborhood of X it is convenient to define intersection adapted coordinates, x,y,Wi,i = f-(fV " - 2), where is the number of internal coordinates. In this cartesian coordinate system the x-and j/-axes are chosen as unit vectors along the gradients g and h that is,... 

Intersection-Adapted Cartesian Coordinates 3.3.1. in Intersection Adapted Coordinates... 

The above Cartesian coordinates do not provide the optimal description of the conical intersection. We will show below that a set hyperspherical coordinates provides the best representation. We begin by defining spherical polar coordinates QO, (f>C) for the z, x,y (g , h ) axes and... 

In Eq. (7.1), states A and B are the two electronic states ground and excited states associated with the conical intersection, is the yth mass-weighted Cartesian coordinate of the t th atom, the index i labels the N atoms and y the Cartesians components, x, y, and z. These quantities are in principle obtainable only from a theoretical calculation. Nevertheless, as we shall discuss subsequently, they have a simple interpretation and one can often make a reasonable guess as to the nature of these two vectors using qualitative valence bond theory. 

In a nematic preparation, between a horizontal slide and coverslip, the locus of vertical directors is made up of one or several lines, corresponding to intersections of surfaces n x, y, z)=0 and n ix, y, z)=0 (within a Cartesian coordinate system x, y, z attached to the preparation, z being normal to... 

Consider the origin of the Cartesian coordinate system, Jtyz, at the intersection of the line of symmetry and the die lips (Fig. 9.17). The original width of the polymer film (in the X direction and at the die lips) is Wq and the extension length is L. The width of the film decreases along the z direction, because the film is being drawn in that direction by the chill-roll. Its width at the roll is wl. Similar drawing takes place in... 

In the mid-seventeenth century the French mathematician Rene Descartes proposed a simple method of relating pairs of numbers as points on a rectangular plane surface today called a rectangular Cartesian coordinate system. A typical two-dimensional Cartesian coordinate system consists of two perpendicular axes, called the coordinate axes. The vertical or y-axis is called the ordinate, while the horizontal or jc-axis is called the abscissa. The point of intersection between the two axes is called the origin. In designating a point on this coordinate system, the abscissa of the point always is given first. Thus, the notation (4,5) refers to the point whose abscissa is 4 and whose ordinate is 5, as shown in Fig. 1-1. 

---