Contour drawings

Orbital density pictures are probably the most comprehensive views we can draw, but they require much time and care. An electron contour drawing provides a simplified orbital picture. In this representation, we draw a contour surface that encloses almost all the electron density. Commonly, almost all means 90%. Thus, the electron density is high inside the contour surface but very low outside the surface. Figure 7-19c shows a contour drawing of the 2s orbital. 

The drawback of contour drawings is that all details of electron density inside the surface are lost. Thus, if we want to convey the maximum information about orbitals, we must use combinations of the various types of depictions. 

The quantum number / — 1 corresponds to a p orbital. A p electron can have any of three values for Jitt/, so for each value of tt there are three different p orbitals. The p orbitals, which are not spherical, can be shown in various ways. The most convenient representation shows the three orbitals with identical shapes but pointing in three different directions. Figure 7-22 shows electron contour drawings of the 2p orbitals. Each p orbital has high electron density in one particular direction, perpendicular to the other two orbitals, with the nucleus at the center of the system. The three different orbitals can be represented so that each has its electron density concentrated on both sides of the nucleus along a preferred axis. We can write subscripts on the orbitals to distinguish the three distinct orientations Px, Py, and Pz Each p orbital also has a nodal plane that passes through the nucleus. The nodal plane for the p orbital is the J z plane, for the Py orbital the nodal plane is the X Z plane, and for the Pz orbital it is the Jt plane. 

Contour drawings of the three 2p orbitals. The three orbitals have the same shape, but each is oriented perpendicular to the other two. 

The quantum number 1 — 2 corresponds to a d orbital. A d electron can have any of five values for M/(- 2, -1, 0, +1, and + 2), so there are five different orbitals in each set. Each d orbital has two nodal planes. Consequently, the shapes of the d orbitals are more complicated than their s and p counterparts. The contour drawings in Figure 7-23 show these orbitals in the most convenient way. In these drawings, three orbitals look like three-dimensional cloverleaves, each lying in a plane with the lobes pointed between the axes. A subscript identifies the plane in which each lies dxy, dxz, and dyz. A fourth orbital is also a cloverleaf in the... 

Contour drawings of the d orbitals. Four of the five have two preferred axes and two nodal planes at right angles to each other. The... 

C07-0025. Construct contour drawings of s, p, and d orbitals. Label the coordinate axes. 

C07-0027. Construct an accurately scaled composite contour drawing on which you superimpose a 2s orbital, a 2 Px orbital, and the outermost portion of a 3 Px orbital of the same atom. (Use different colors to distinguish the different orbitals.) What does your contour drawing tell you about the importance of the = 2 orbitals for chemical interactions when the 3 Px orbital contains electrons ... 

C07-0085. Constract contour drawings for the orbitals graphed in Figure 7-20. appropriately scaled to illustrate the size differences among these orbitals. 

The contour drawing capabilities of most graphics software finds the regions of constant peak amplitude through interpolation. Typically, 10-20 contour amplitudes are picked from maximum to minimum. This interpolation can be sophisticated and may include a noise minimizing basis function. However, the use of this filtering may sometimes distort the data presentation. Because 2DLC data are typically performed... 

Contour drawing of the potential energy surface for reaction of 113. (Adapted from reference 220.)... 

Orbital correlation diagram for the reaction od Cp2Zr(ethylene)2 to a zirconacyclopentane complex. The contour drawings and the molecular structure for the reactant, transition state, and product are derived from calculations at the B3LYP level. 

You can specify the number and values of visible contour lines. You specify the total number of contour lines to be shown by simple stating the number, n>0. You normally specify the values of the contour lines as default values. For this case, HyperChem computes the maximum and minimum values on the grid and then draws contours at these values plus n-2 contour lines evenly spaced in between these maximum and minimum values. If you need non-default values, you can specify the starting value and then an increment to define the other n-1 evenly spaced contour lines. If default values were computed previously, HyperChem suggests the starting value and increment of the previous default computation for the new non-default option. 

Local Site Condition Evaluation. In addition to visiting the site, drawing up a contour map and geology reports, acquiring sod-bearing information, and a knowledge of boundaries, setbacks, local requirements, utdity tie-in locations, sewer connections, access to roadways, pipelines, radroads, etc, may be needed to make a fliU assessment. 

Comparison of Alignment Charts and Cartesian Graphs. There are typically fewer lines on an alignment chart as compared to Cartesian plots. This reduces error introduced by interpolation and inconsistency between scales. For example, to find a point (x,j) on a Cartesian graph one draws two lines, one perpendicular to each axis, and these reference lines intersect at the point x,j). This point (x,j) may correspond to some finite value found by rea ding a contour map represented by a family of curves corresponding to different values of the function. 

An alignment chart is used by drawing one reference line through the two axes. This reference line, which need not be perpendicular to either axis, crosses a result axis at a unique finite value. This result axis represents the contour map on a Cartesian graph. Thus each line on an alignment chart represents a point on a Cartesian graph. 

The main reason for extruding polystyrene is to prepare high-impact polystyrene sheet. Such sheet can be formed without difficulty by vacuum forming techniques. In principle the process consists of clamping the sheet above the mould, heating it so that it softens and becomes rubbery and then applying a vacuum to draw out the air between the mould and the sheet so that the sheet takes up the contours of the mould. 

Isotherm A line in a flow system or on a graph connecting points of equal temperature, or a mathematical or graphical relationship between two variables at constant temperature. Or a display using lines on a drawing to show constant-temperature contour lines, as from thermal imaging with infrared techniques. 

Now imagine that we rotate the molecule about the internuclear axis. The curved contour will trace out a surface. If we draw a unit outward normal vector to this surface, it will be everywhere perpendicular to the gradient vector (because the gradient vector points along the trajectory). 

ELF can be visualized with different kinds of images. Colored sections through a molecule are popular, using white for high values of ELF, followed by yellow-red-violet-blue-dark blue for decreasing values simultaneously, the electron density can be depicted by the density of colored points. Contour lines can be used instead of the colors for black and white printing. Another possibility is to draw perspective images with iso surfaces, i.e. surfaces with a constant value of ELF. Fig. 10.2 shows iso surfaces with ELF = 0.8 for some molecules from experience a value of ELF = 0.8 is well suited to reveal the distribution of electron pairs in space. 

To obtain pictures of the orbital ip = R0< >, we would need to combine a plot of R with that of 0, which requires a fourth dimension. There are two common ways to overcome this problem. One is to plot contour values of ip for a plane through the three-dimensional distribution as shown in Figures 3.8a,c another is to plot the surface of one particular contour in three dimensions, as shown in Figures 3.8b,d. The shapes of these surfaces are referred to as the shape of the orbital. However, plots of the angular function 0 (Figure 3.7) are often used to describe the shape of the orbital ip = RQ because they are simple to draw. This is satisfactory for s orbitals, which have a spherical shape, but it is only a rough approximation to the true shape of p orbitals, which do not consist of two spheres but rather two squashed spheres or doughnut shapes. 

Figure 2. Contour maps of the electron density of (a) SCI2 and (b) H2O. The density increases from the outermost 0.001 au isodensily contour in steps of 2 x 10", 4 x 10", and 8 x 10" au with n starting at 3 and increasing in steps of unity. The lines connecting the nuclei are the bond paths, and the lines delimiting each atom are the intersection of the respective interatomic surface with the plane of the drawing. The same values for the contours apply to subsequent contour plots in this paper.

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