Three-dimensional orbital contours

Once the job is completed, the UniChem GUI can be used to visualize results. It can be used to visualize common three-dimensional properties, such as electron density, orbital densities, electrostatic potentials, and spin density. It supports both the visualization of three-dimensional surfaces and colorized or contoured two-dimensional planes. There is a lot of control over colors, rendering quality, and the like. The final image can be printed or saved in several file formats. 

Wave functions can be visualized as the total electron density, orbital densities, electrostatic potential, atomic densities, or the Laplacian of the electron density. The program computes the data from the basis functions and molecular orbital coefficients. Thus, it does not need a large amount of disk space to store data, but the computation can be time-consuming. Molden can also compute electrostatic charges from the wave function. Several visualization modes are available, including contour plots, three-dimensional isosurfaces, and data slices. 

We need ways to visualize electrons as particle-waves delocalized in three-dimensional space. Orbital pictures provide maps of how an electron wave Is distributed In space. There are several ways to represent these three-dimensional maps. Each one shows some important orbital features, but none shows all of them. We use three different representations plots of electron density, pictures of electron density, and pictures of electron contour surfaces. 

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... 

C07-0084. The conventional method of showing the three-dimensional shape of an orbital is an electron contour surface. What are the limitations of this representation ... 

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. 

To summarize at this point, it is reiterated that wavefunction tjr r,6,three-dimensional shape of each orbital can be represented by a contour surface, on which every point has the same value off. The three-dimensional shapes of nine hydrogenic orbitals (2s, 2p, and 3d) are displayed in Fig. 2.1.5. In these orbitals, the nodal surfaces are located at the intersections where f changes its sign. For instance, for the 2p orbital, the yz plane is a nodal plane. For the 3d y orbital, the xz and yz plane are the nodal planes. 

How do we depict a probability function One way would be to draw contours connecting regions where there is an equal probability of finding the electron. If F2 for a Is orbital is plotted, a three-dimensional plot emerges. Of course, this is a two-dimensional representation of a three-dimensional plot—the contours are really spherical like the different layers of an onion. These circles are rather like the contour lines on a map except that they represent areas of equal probability of finding the electron instead of areas of equal altitude. 

Fig. 6.13 Representations of the valence m.o.s of HF (only one of the two tt orbitals is shown) by contour plots and a three-dimensional grid in the case of the vacant (anti-bonding) m.o. Plot of DD (density difference) illustrates the detailed structure of the difference map between the total electron density and the atomic contributions if no bond was formed (full lines for increase of electron density and dotted lines for decrease).

Finally, we can represent the orbital as a three-dimensional object by rotating Figure 5.7 about the z-axis. Each of the closed contours in Figure 5.7 will then trace out a three-dimensional isosurface on which all the points (r, 9, 4>) have the... 

The chemist s sketches, which are typically drawn to emphasize directionality of the sp hybrid orbitals, and a contour plot of the actual shape, are shown in Figure 6.44. Each of these contours can be rotated about the x-y plane to produce a three-dimensional isosurface whose amplitude is chosen to be a specific fraction of the maximum amplitude of the wave function. These isosurfaces demonstrate that sp hybridization causes the amplitude of the boron atom to be pooched out at three equally spaced locations around the equator of the atom (see Fig. 6.42). The 2p orbital is not involved and remains perpendicular to the plane of the sp hybrids. The standard chemist s sketches of the sp hybrid orbitals and a contour plot that displays the exact shape and directionality of each orbital are shown in Figure 6.44. The isosurfaces shown in Figure 6.43 were generated from these contour plots. 

Hence, on a circle centered on the y axis and parallel to the xz plane, is constant. Thus a three-dimensional contour surface may be developed by rotating the cross section in Fig. 6.12 about the y axis, giving a pair of distorted ellipsoids. The shape of a real 2p orbital is two separated, distorted ellipsoids, and not two tangent spheres. 

FIGURE 13.13 Cross section of the i7- 2p+, (or i7 2p i) molecular orbital.To obtain the three-dimensional contour surface, rotate the figure about the z axis. The z axis is a nodal line for this MO (as it is for the 2p+i AO.)... 

We saw in Chapter 6 that the probability of finding an election in three-dimensional space depends on what orbital it is in. Look back at Figures 6.19 and 6.22, which show the radial probability distribution functions for the s orbitals and contour plots of the 2p orbitals, respectively, (a) Which orbitals, 2s or 2p, have more electron density at the nucleus (b) How would you modify Slater s rules to adjust for the difference in electronic penetration of the nucleus for the 2s and 2p orbitals ... 

Fig. 43 Three-dimensional contour surfaces of the valence MOs responsible for the cyclic delocalization of the election density in the Uj and U5 rings of the cyc/o-U X ( = 3, 4 X = OH, NH) clusters (figures in italics are the orbital energies, in eV). Reprinted with permission from [185]. Copyright ACS Journal Archives...

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