Technology Report pubs.acs.org/jchemeduc
Visualization of a Large Set of Hydrogen Atomic Orbital Contours Using New and Expanded Sets of Parametric Equations Ian J. Rhile* Department of Chemistry and Biochemistry, Albright College, Reading, Pennsylvania 19612-5234, United States S Supporting Information *
ABSTRACT: Images of isosurfaces and contour lines for a large set of hydrogen atomic orbitals were generated with new and expanded sets of parametric equations. Parametric equations for 4f and 5g orbitals are derived for the first time.
KEYWORDS: Physical Chemistry, Atomic Properties/Structure, Mathematics/Symbolic Mathematics, Quantum Chemistry
A
1. The 2s, 2p, 3d, 4f, and 5g orbitals were generated with a new set of parametric equations. The distance from the origin r was related to the angles θ (the polar angle from the z-axis) and φ (the azimuthal angle around the z-axis) in spherical coordinates. These parametric equations use the Lambert W function. (The Lambert W function is the inverse of the function f(x) = x ex.26,27 The wave functions for these orbitals can be manipulated such that the Lambert W function can be used. More details about the Lambert W function and its implementation are found in the Supporting Information.) These are the first parametric equations for f and g orbitals in the literature, although they cannot be extended to higher principal quantum numbers. 2. The 3pz, 4pz, 5pz, 6pz, 4dz2, 5dz2. and 6dz2 orbitals were generated by using and extending the parametric equations for contour lines derived by Scaife.10 He described contour line parametrizations for the 2pz, 3pz, and 3dz2 orbitals in the xz-plane. In this work, the surface is generated using the cylindrical symmetry of these orbitals around the z-axis; in addition, expressions for the additional orbitals are derived. (The px and py orbitals can be generated by exchanging the variables.) 3. The dx2−y2, dxy, dyz, and dxz orbitals for 4 ≤ n ≤ 6 were generated with new parametric equations. The dx2−y2 orbitals have a similar form as an ellipse in plane geometry for x and z around the y-axis and for y and z around the x-axis at a specific value of r. The other orbitals are generated through rotations of the dx2−y2
tomic orbitals are a theme throughout the undergraduate chemistry curriculum, and visualizing them has been a theme in this journal. Contour plots as isosurfaces or contour lines in a plane are the most familiar representations of the hydrogen wave functions.1−24 In these representations, a surface of a fixed value of the wave function ψ is plotted in analogy to a contour map. The values of ψ and ψ*ψ, the probability density, are closely related. The easiest method to generate an isosurface is to use an equation for the surface. Ramachandran and Kong derived parametric equations to generate atomic orbital isosurfaces for 2s, 2p, 3p, 3dx2−y2, and 3dz2 orbitals in terms of r, the distance from the origin.18 Before that, Scaife introduced parametric equations for contour lines in the xy-, yz-, and yz-planes for orbitals with n ≤ 3,10 which were an expansion from earlier approaches.1,2 To date, no parametric equations have been derived for the 4f or 5g orbitals or any orbitals with n > 3. The alternative to these equations is to generate the isosurfaces or contour lines implicitly through an algorithm that may require programming or software with the ability to implement them. (The marching cubes algorithm is a common algorithm for isosurfaces,25 and other algorithms have been used for computerized plotting of orbitals.8,12,13,21) The use of software has allowed for orbital plotting for pedagogical purposes, with varying levels of complexity and success.7,9,15−20,22,24 Here we generate vector graphic images for a large set of orbitals using parametric equations, many of which are newly described. These representations are generated with gnuplot, a readily available, free, open-source program. The orbitals were generated using three sets of parametric equations: © 2014 American Chemical Society and Division of Chemical Education, Inc.
Published: September 15, 2014 1739
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Technology Report
use of color, the phases and symmetry for the orbitals. In addition, the complex nature of the 4f and 5g orbitals can be presented. In conclusion, orbital image files were generated using gnuplot using parametric equations. All of the parametric equations described here are new or expanded versions of known ones. The images can be used in the classroom. The equations that are described can be used in any software that can generate surfaces parametrically, not just those that have an algorithm for determining them implicitly.
orbitals around axes. (For example, the dxy orbital constitutes 45° rotation around the z-axis.) The derivations and complete details for the parametric equations are given in the Supporting Information. In all, the parametric equations represent 52 orbitals. The parametric equations were used to generate image files for the isosurfaces using gnuplot. (Previous implementation of isosurfaces in gnuplot19 and Winplot,24 a similar program, was limited to the angular wave function. These “polar plots” distort the relationship between position and the value of ψ.3−5,11,12,17,23) Images in scalable vector graphics (SVG) and enhanced metafile (EMF) formats are provided in the Supporting Information. Both formats are vector graphics, consisting of curved and straight lines and text; such file types can be enlarged without pixelation, and SVG images can be edited (for example, with Inkscape or Gimp). For these reasons, SVG images are the preferred image format for line drawings at Wikipedia.28 The gnuplot input files are included; they can be used to generate image files in other formats and can be manipulated in the gnuplot interface. A sample isosurface of the 4fxyz orbital is provided in Figure 1.
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ASSOCIATED CONTENT
S Supporting Information *
Complete derivations of the parametric equations for the orbitals; description of the implementation in gnuplot; gnuplot input files and image files (SVG and EMF) for isosurfaces and contour lines; macro-enabled Excel spreadsheet for contour lines. This material is available via the Internet at http://pubs. acs.org.
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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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Figure 1. 4fxyz orbital isosurface at ψ = ±0.003. The distance units are atomic units.
The parametric equations also provide a means of calculating contour lines in the xy-, yz-, or xz-planes by setting z = 0 or θ = π/2; x = 0 or φ = 0; y = 0 or φ = π/2, respectively.1,2 This complements the approach of Scaife,10 and provides an alternative to contours generated implicitly through algorithms in gnuplot such as described by Moore.19 In parallel to the isosurfaces, these are the first parametric equations for the contour lines for the 4f and 5g orbitals. Gnuplot input files and image files for several contour lines are provided in the Supporting Information, as well as in a macro-enabled Microsoft Excel spreadsheet. The image files and gnuplot input files can be used for pedagogical purposes in several ways. Most textbooks still use artist renditions of orbitals, often with incorrect features such as lobes that touch each other and distorted shapes, which can create misconceptions. Hence, an instructor can use the image files directly to describe atomic orbitals, or generate his or her own image files in different orientations. Students can also load the input files into gnuplot and manipulate the orbitals interactively. In addition, the equations can be used in any mathematical package that plots surfaces parametrically, thus expanding the available options for orbital contour plotting. All the representations show features of atomic orbitals, such as the location and number of nodes, orbital shape, and, through the 1740
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