Technology Report pubs.acs.org/jchemeduc
Interactive Web-Based Pointillist Visualization of Hydrogenic Orbitals Using Jmol Shane P. Tully,† Thomas M. Stitt,† Robert D. Caldwell,† Brian J. Hardock,† Robert M. Hanson,‡ and Przemyslaw Maslak*,† †
Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802, United States Department of Chemistry, St. Olaf College, Northfield, Minnesota 55057, United States
‡
S Supporting Information *
ABSTRACT: A Monte Carlo method is used to generate interactive pointillist displays of electron density in hydrogenic orbitals. The Web applet incorporating Jmol viewer allows for clear and accurate presentation of three-dimensional shapes and sizes of orbitals up to n = 5, where n is the principle quantum number. The obtained radial probability distributions can be compared with theoretical curves on two-dimensional plots. The applet also produces orbital cross sections and contour plots and their three-dimensional isometric projections, illustrating the standing-wave nature of the orbitals. The classical boundary surfaces with various predetermine cutoffs may also be displayed for size comparisons. Nodal surfaces are available for all displays. KEYWORDS: First-Year Undergraduate/General, Chemoinformatics, Physical Chemistry, Internet/Web-Based Learning, Atomic Properties/Structure
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common values.21,22 Such displays usually give a false impression of an abrupt ending of electron cloud, resulting in “smooth-skin” orbitals where the details of internal probability distribution cannot be seen easily. As was recognized almost 50 years ago: “The ideal model of an atomic or molecular orbital would be a cloud-like structure showing the probability of finding an electron at all points in space relative to the nucleus”.7 Along these lines, recently, 3D rotatable dot plots were reported.29 We have now developed 3D point-density plots that we refer to as pointillist displays.30−32 With the powerful capabilities of Jmol viewer33 and its built-in hydrogenic functions, we can accurately represent electron density in terms of orbital shapes and relative sizes and underscore its probabilistic nature and the “fuzziness” of orbital boundaries. We have incorporated the pointillist representations into a Web based applet (Figure 1) that allows one to compare various orbital display modes mentioned above and interactively probe their relative sizes and nodal properties in 2D and 3D representations.
xploration of the shapes and relative sizes of hydrogenic orbitals is a standard topic in the undergraduate chemistry curriculum.1−6 The challenge in devising visualization methods for such orbitals is finding an adequate method of representing four-dimensional constructs (electron density as a function of three independent positional variables) in two-dimensional (2D) drawings or in three-dimensional (3D) models or computer displays. Many methods of orbital representation have been developed, and their pedagogical advantages have been described on the pages of this Journal over the years.7−21 Two-dimensional plots of wave functions (ψ) or electron density (ψ2) are often used.3,4,7,17 Other options include plots of radial wave function (R), radial density (R2), or radial probability distribution (r2R2), all as functions of r (separation from the nucleus).3,4,16,17,22,23 Another dimension-reducing technique is to present angular or radial cross sections along especially informative planes. These images can be shown as contour-plots,7,10,13,17,18,22 dot-density diagrams,9,12,24−26 or gradient cross sections.19 One weakness of such two-dimensional displays is that several such figures may be needed to sufficiently represent spatial characteristics of the orbital. To mitigate that shortcoming, the third dimension may be added to contour or cross-section plots, giving isometric projections that illustrate the standing-wave nature of the orbitals.11,16,18,20,22 For other types of 3D models, it is common to draw a boundary surface with predetermined, typically low values of ψ.16,21,22,27 On the solid or semitransparent isosurface, or mesh, color, or dashed or solid line, markings are applied that correspond to the algebraic signs of the underlying wave functions. For orbital size comparisons, the boundaries are either selected to include some significant percent of total electron density (90% or 95%)28 or are set at predetermined © 2012 American Chemical Society and Division of Chemical Education, Inc.
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POINTILLIST REPRESENTATIONS The display is based on a Monte Carlo algorithm one of us developed and used previously in a program called ORBITAL.34 In the Jmol version presented here, points are randomly generated within a cube (with rounded corners) whose size matches the extent of the orbital. A point is rejected if the wave function value at that point is smaller than a preset value. The selection process is repeated until the desired number of points (dots) is collected. The dots are colored according to the sign (phase) of the wave function or colorPublished: October 24, 2012 129
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Technology Report
Figure 1. Screen capture of the Monte Carlo applet pointillist display of 4d(z2) orbital with its conical nodes and two-dimensional radial plots with 90% radial density radii marks.
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gradient mapped with electron density values. The resulting displays clearly illustrate the overall shape of the orbitals, the overall decline in electron density as distance from the nucleus increases, the undefined nature of the boundary of the orbitals (i.e., their fuzziness), and the nodal properties.8 These dots are assigned to spherical shells of preset thickness and are counted within each layer. The resulting (“experimental”) bar graph is normalized and plotted in flot.35 The obtained radial probability distribution can be compared to direct plots of the theoretical radial distribution functions (r2R2). The distance (r) at which the maximum of the distribution occurs and the average nucleus−electron separation are compared for both the Monte Carlo and the theoretical models. As expected, the agreement between the two sets of data improves with the number of dots generated. The applet also includes the more standard modes of orbital display. All orbitals can be shown on a common scale or sized to fit the display. The “standard” isosurface (|ψ| = 0.0026 cutoff) was selected to maximize the visible details of inner lobes for all orbitals. For comparison, 50%, 90%, and 95% of total electron density (ψ 2 ) boundary surfaces and the corresponding spheres for the radial density36 can also be displayed over the “standard” isosurfaces. Within given display modes, orbitals can be selected and updated by changing the quantum numbers. All nodal surfaces (organized into spherical, conical, and planar)8 are available for inspection with adjustable translucency. A tour of a 3d orbital, highlighting the capabilities of the applet is also included.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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ASSOCIATED CONTENT
S Supporting Information *
Description of the methods used in constructing the applet and the listing of applet functionality. Compete directory of all files needed to run the applet on a server. This material is available via the Internet at http://pubs.acs.org. 130
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