Orbital isosurfaces

Fig. 4.3 Representation of the Cyg LUMO and degenerate LUMO -t 1 and LUMO -t 3 molecular orbitals (isosurface value 0.02 a.u) where all non-equivalent [6,6] and [5,6] bonds have been marked. Those bonds with favorable orbitals to interact with the HOMO of the diene are marked with ellipses...

While these comments pertain to orbital isosurface plots in general, the situation is particularly dire for diffuse electrons. In such cases, one should demand to know what fraction of the orbital density is encapsulated within a given isosurface plot. In other words, we need to know the fractional electron value. 

In view of these examples, side-by-side comparison of orbital isosurfaces for different molecules, clusters, or orbitals is appropriate only when comparing common values of f, not Unfortunately, this wisdom has... 

Molecular orbitals were one of the first molecular features that could be visualized with simple graphical hardware. The reason for this early representation is found in the complex theory of quantum chemistry. Basically, a structure is more attractive and easier to understand when orbitals are displayed, rather than numerical orbital coefficients. The molecular orbitals, calculated by semi-empirical or ab initio quantum mechanical methods, are represented by isosurfaces, corresponding to the electron density surfeces Figure 2-125a). 

Figure 2-125. Different isovalue-based surfaces of phenylalanine a) isoelectronic density b) molecular orbitals (HOMO-LUMO) c) isopotential surface and d) isosurface of the electron cryo-microscopic volume of the ribosome of Escherichia coii.

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. 

Among the quantities which have proven of value as graphical models are the molecular orbitals, the electron density, the spin density (for radicals and other molecules with unpaired electrons), the electrostatic potential and the local ionization potential. These may all be expressed as three-dimensional functions of the coordinates. One way to display them on a two-dimensional video screen (or on a printed page) is to define a surface of constant value, a so-called isovalue surface or, more simply, isosurface. ... 

This chapter introduces and illustrates isosurface displays of molecular orbitals, electron and spin densities, electrostatic potentials and local ionization potentials, as well as maps of the lowest-unoccupied molecular orbital, the electrostatic and local ionization potentials and the spin density (on top of electron density surfaces). Applications of these models to the description of molecular properties and chemical reactivity and selectivity are provided in Chapter 19 of this guide. 

Figure 24 Isosurfaces at +0.02 of molecular orbitals of the model compounds of polysilanes obtained by semiempirical PM3 calculation.

The manifold energy splitting is accompanied by the formation of delocalized orbitals as shown by the contour and isosurface plots in Fig. 18 a... 

Figure 12 Isosurfaces at fixed value (10% of max. amplitude) of the square modulus of the highest occupied (HOMO) and lowest unoccupied (LUMO) Kohn-Sham orbitals for the Si m cluster in the Si02 matrix.

FIGURE 10.2 Molecular orbitals in hydrogen fluoride, plotted using two isosurfaces, at 0.02 e 1/2 Bohr 3/2. (See the color version of this figure in Color Plates section.)... 

Figure 4. Various representations of spin-coupled orbital <)>, for benzene. Left contours in the horizontal plane 1 bohr above the molecular plane. Centre contours in a vertical mirror plane. Right a representative isosurface (3-D contour).

Fig. 18. Ground state electronic structures for cresyl and o-(methylthio)cresyl phenoxyl radicals. Isosurface representations of molecular orbitals solved by ab initio density functional theory methods for cresyl (ere) and o-(methylthio)cresyl (mtc) phenoxyl radicals. Eigenvalues are listed and for each the SOMO is identihed with an asterisk ( ).

Marsella has proposed a scheme for predicting the preferred direction of attack at carbonyls. He suggests visualizing the computed electrostatic potential on an isosurface of the carbonyl jt orbital (the LUMO). The lobe that is more positive accurately predicts the correct face that is attacked for 13 different reactions. 

Be certain you understand these choices in each image you examine (or create ). These same issues appear in Chapter 5 when we discuss the wave functions for electrons in atoms, called atomic orbitals. Throughout this book, we have taken great care to generate accurate contour plots and isosurfaces for them from computer calculations to guide your thinking about the distribution of electrons in atoms and molecules. 

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... 

FIGURE 5.8 The shapes of the three 2p orbitals, with phases and nodal planes Indicated. The Isosurfaces In (a), (b), and (c) Identify points where the amplitude of each wave function Is 0.2 of Its maximum amplitude. (a) 2p orbital, (b) 2p orbital, (c) 2py orbital. 

Each 2p orbital appears, loosely speaking, as a pair of flattened and distorted hemispheres, with opposite phase, facing each other across their nodal plane. You can see how this shape would change dramatically if we selected for display isosurfaces with other values of amplitude. Sometimes you see the 2p orbitals... 

FIGURE 6.39 Isosurface representation of the electron density in the HF (T bond formed from a pair of electrons initially localized in a 2s orbital on H and in a 2p orbital on F. 

FIGURE 6.44 Exact and approximate representations of the hybrid orbital shapes. For each type of hybrid orbital shown the left column shows typical chemists sketches, the center column shows isosurfaces, and the right column shows contour plots. The top row are the sp hybrid orbitals, the middle row are the sp hybrid orbitals, and the bottom row are the sp hybrid orbitals. 

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. 

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