Letter pubs.acs.org/jchemeduc
Comment on “Visualizing Three-Dimensional Hybrid Atomic Orbitals Using Winplot: An Application for Student Self Instruction” Ian. J. Rhile* Department of Chemistry and Biochemistry, Albright College, Reading, Pennsylvania 19612-5234, United States ABSTRACT: The graphical representations of hybrid orbitals in “Visualizing Three-Dimensional Hybrid Atomic Orbitals Using Winplot: An Application for Student Self Instruction” do not accurately represent their isosurfaces.
KEYWORDS: Computer-Based Learning, Quantum Chemistry, Distance Learning/Self Instruction, Mathematics/Symbolic Mathematics, Physical Chemistry, Upper-Division Undergraduate read with interest the recent technology report “Visualizing Three-Dimensional Hybrid Atomic Orbitals Using Winplot: An Application for Student Self Instruction” by Saputra and coworkers.1 The authors propose a visualization of hybrid orbitals. In the derivation, the authors use the square of the sum of the weighted 2s and 2p angular wave functions, and plot a threedimensional surface based on this function. The authors ignore the radial distribution in this derivation and state, “The radial part only gives the information about the orbital size, but it does not have a direct effect on the overall orbital shape.”1 The usual representation of an orbital is an isosurface,2,3 a surface of the wave function with a constant value of ψ or ψ2 (or ψ*ψ). A contour line is the plane curve of the same function. Plots for the visualization and for the contour lines for the three sp2 hybrid orbitals are presented in Figure 1. The angular wave function and contour line plot shapes are different. In these cases, the wave function ψ has the form of the following equation, with coefficients c, radial wave functions R, and angular wave functions Y.4
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the 2s and 2p orbitals (i.e., R2s(ρ) and R2p(ρ)) are different,17 and therefore, the contributions of the 2s and 2p angular wave functions to ψ depend on distance to the origin (except where they are coincidently the same, at ρ = 1 au).14 Hence, the sum of the angular wave functions (without the radial wave functions) has only a limited mathematical relationship to the overall wave function. However, the proposed visualizations accurately reflect the directions of maximum electron density for the orbitals and do qualitatively resemble the actual orbital isosurfaces. Inasmuch as isosurfaces are the standard shape used for orbitals, the proposed visualizations are inaccurate.
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Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ψ (ρ , θ , ϕ) = c 2sR 2s(ρ)Y2s(θ , ϕ) + c 2p R 2p(ρ)Y2p (θ , ϕ) x
x
y
z
z
The radial wave functions are the product of linear and exponential terms in distance, which affect isosurface shape. The angular wave functions for various atomic orbitals have been plotted, providing the value of the function at each pair of θ and φ.5−8 These “polar plots” have been criticized since the values of |Y(θ,φ)| or [Y(θ,φ)]2 are not distances from the nucleus as the graphs suggest.9−16 The plots for atomic orbitals do have some use in that ψ is proportional to Y(θ,φ) on sphere of a specific radius since R(ρ) is constant on the sphere. In the proposed visualization, however, the radial wave functions for © XXXX American Chemical Society and Division of Chemical Education, Inc.
REFERENCES
(1) Saputra, A.; Canaval, L. R.; Sunyono; Fadiawati, N.; Diawati, C.; Setyorini, M.; Kadaritna, N.; Budi Kadaryanto, K. Visualizing ThreeDimensional Hybrid Atomic Orbitals Using Winplot: An Application for Student Self Instruction. J. Chem. Educ. 2015, 92, 1557−1558. (2) Ogryzlo, E. A.; Porter, G. B. Contour Surfaces for Atomic and Molecular Orbitals. J. Chem. Educ. 1963, 40, 256−261. (3) Ramachandran, B.; Kong, P. C. Three-Dimensional Graphical Visualization of One-Electron Atomic Orbitals. J. Chem. Educ. 1995, 72, 406−408. (4) Mortimer, R. G. Physical Chemistry, 3rd ed.; Elsevier Academic Press: Burlington, MA, 2008; pp 870, 872−873.
+ c 2p R 2p(ρ)Y2p (θ , ϕ) + c 2p R 2p(ρ)Y2p (θ , ϕ) y
AUTHOR INFORMATION
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DOI: 10.1021/acs.jchemed.5b00626 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Letter
Figure 1. (a−c) Contour lines in the xy-plane of the sp2 hybrid orbital visualization in ref 1, the squares of sums of angular wave function, with the distance from the origin representing [Y(θ,φ)]2 at each angle φ. (d−f) Orbital isosurfaces of ψ2 at 0.001 increments from 0 to 0.008 using wave functions in ref 17 and coefficients in refs 1 and 4. All plots were generated in gnuplot as in ref 7. (5) Friedman, H. G., Jr.; Choppin, G. R.; Feuerbacher, D. G. J. Chem. Educ. 1964, 41, 354−358. (6) Liebl, M. Orbital Plots of the Hydrogen Atom. J. Chem. Educ. 1988, 65, 23−24. (7) Moore, B. G. Orbital Plots Using Gnuplot. J. Chem. Educ. 2000, 77, 785−789. (8) Chung, W. C. Three-Dimensional Atomic Orbital Plots in the Classroom Using Winplot. J. Chem. Educ. 2013, 90, 1090−1092. (9) Cohen, I. The Shape of the 2p and Related Orbitals. J. Chem. Educ. 1961, 38, 20−22. (10) Ogryzlo, E. A. On the Shapes of f Orbitals. J. Chem. Educ. 1965, 42, 150. (11) Perlmutter-Hayman, B. The Graphical Representation of Hydrogen-like Wave Functions. J. Chem. Educ. 1969, 46, 428−430. (12) Linnett, J. W.; Bordass, W. T. A New Way of Presenting Atomic Orbitals. J. Chem. Educ. 1970, 47, 672−675. (13) Levine, I. N. Quantum Chemistry, 3rd ed.; Allyn and Bacon: Boston, MA, 1983; pp 125−129. (14) Kikuchi, O.; Suzuki, K. Orbital Shape Representations. J. Chem. Educ. 1985, 62, 206−209. (15) Barth, R. Where the Electrons Are. J. Chem. Educ. 1995, 72, 401−403. (16) Peacock-López, E. On the Problem of the Exact Shape of Orbitals. Chem. Educator 2003, 8, 96−101. (17) Pauling, L.; Bright, W. E. Introduction to Quantum Mechanics, International Student ed.; McGraw Hill: New York, 1935; pp 134− 136.
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DOI: 10.1021/acs.jchemed.5b00626 J. Chem. Educ. XXXX, XXX, XXX−XXX
