Contours, definition

The results of electrostatic potential calculations can be used to predict initial attack positions of protons (or other ions) during a reaction. You can use the Contour Plot dialog box to request a plot of the contour map of the electrostatic potential of a molecular system after you done a semi-empirical or ab initio calculation. By definition, the electrostatic potential is calculated using the following expression ... 

Figure 23 shows sections of with J fixed, C2 = 0 and various values of L/N, where V = 27 is the total polyad action. There is an obvious conical point, X, at K, L) = (7,0). In addition, the concentric contours degenerate to points Y and Z at K, L) = (—7, V). It is important for what follows to explore the character of the point X. As a preliminary, the Hamiltonian may be reduced from the above seven-parameter form, to one involving two essential parameters, both of which vary with the total action. The appropriate definitions [14, 28, 29], modified to conform with the present (7, K) notation, are... 

The definition of the radius of an ion in a crystal as the distance along the bond to the point of minimum electron density is identical with the definition of the radius of an atom in a crystal or molecule that we discuss in the analysis of electron density distributions in Chapter 6. The radius defined in this way does not depend on any assumption about whether the bond is ionic or covalent and is therefore applicable to any atom in a molecule or crystal independently of the covalent or ionic nature of the bond, but it is not constant from one molecule or crystal to another. The almost perfectly circular form of the contours in Figure... 

The three quantum numbers may be said to control the size (n), shape (/), and orientation (m) of the orbital tfw Most important for orbital visualization are the angular shapes labeled by the azimuthal quantum number / s-type (spherical, / = 0), p-type ( dumbbell, / = 1), d-type ( cloverleaf, / = 2), and so forth. The shapes and orientations of basic s-type, p-type, and d-type hydrogenic orbitals are conventionally visualized as shown in Figs. 1.1 and 1.2. Figure 1.1 depicts a surface of each orbital, corresponding to a chosen electron density near the outer fringes of the orbital. However, a wave-like object intrinsically lacks any definite boundary, and surface plots obviously cannot depict the interesting variations of orbital amplitude under the surface. Such variations are better represented by radial or contour... 

As indicated in Table 4.2, the eigenvalues of the Hessian matrix of fix) indicate the shape of a function. For a positive-definite symmetric matrix, the eigenvectors (refer to Appendix A) form an orthonormal set. For example, in two dimensions, if the eigenvectors are Vj and v2, v[v2 =0 (the eigenvectors are perpendicular to each other). The eigenvectors also correspond to the directions of the principal axes of the contours of fix). 

Representation of the trust region to select the step length. Solid lines are contours of fix). Dashed lines are contours of the convex quadratic approximation of fix) at x. The dotted circle is the trust region boundary in which 8 is the step length. x0 is the minimum of the quadratic model for which H(x) is positive-definite. 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

The reason for this definition can be seen in Fig. 1.2, where we see that the effect of Eq. 1.18 is to rotate the contours of / into those of Og f- the value of the function is preserved by the symmetry operation. 

As seen, the formation of a closed contour for unpaired electron delocalization is a widespread phenomenon that detains the electron in the framework of a definite molecular fragment and really defines the physical and chemical properties of ion radicals. 

Point M in the figure is the optimum that by one definition of the research problem objective should be determined. Each intersection line in the plane is a line of constant response values and is called contour lines-contour diagram. 

Graph edges oriented in a definite direction which indicates the order of interconnection between the nodes are called directed arcs. Their orientation is indicated by the arrows placed either on the arcs or near them. A graph, a cycle, and a tree containing directed arcs are called directed. A directed cycle is also called a contour. 

Now we are able to define contour or contour-ordered Green function - the useful tool of Keldysh diagrammatic technique. The definition is similar to the previous one... 

In the simulation definition, the downward axial velocity was calculated as a negative value, and the upward axial flow (the backflow) was calculated as positive axial velocity. To clearly describe the backflow, the positive axial velocity contour is shown in Fig, 6. Figure 6 shows that the backflow occurred between screws with >50% of the bottom zone showing backflow. 

Theoretically, the radius of an ion extends from the nucleus to the outermost orbital occupied by electrons. The very nature of the angular wave function of an electron, which approaches zero asymptotically with increasing distance from the nucleus, indicates that an atom or ion has no definite size. Electron density maps compiled in X-ray determinations of crystal structures rarely show zero contours along a metal-anion bond. 

The figure really represents a three dimensional map with distance between carbon and bromine, r<>Br, plotted north and south, distance between bromine and bromine, rBr-Br, plotted east and west and energy of the system plotted at right angles to the plane. Contour lines of definite energies are projected down onto the plane of the map. The two atoms of bromine are just alike, as are also the two atoms of carbon. Accordingly it is sufficient to trace the course of only one bromine and one carbon atom. An... 

This definition of the CSM implicitly implies invariance to rotation and translation. Normalization of the original shape prior to the transformation additionally allows invariance to scale (Figure 1). We normalize by scaling the shape so that the maximum distance between points on the contour and the centroid is a given constant (in this chapter all examples are given following normalization to 1 however, CSM values are multiplied by 100 for convenience of handling). The normalization presents an upper bound on the mean-squared distance moved by... 

Figure 5 illustrates more generally various cases that can occur for simple quadratic functions of form q x) — JxTHx, for n = 2, where H is a constant matrix. The contour plots display different characteristics when H is (a) positive-definite (elliptical contours with lowest function value at the center) and q is said to be a convex quadratic, (b) positive-semidefinite, (c) indefinite, or (d) negative-definite (elliptical contours with highest function value at the center), and q is a concave quadratic. For this figure, the following matrices are used for those different functions ... 

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