Communication Cite This: J. Chem. Educ. 2019, 96, 187−190
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Graphical Representation of Hydrogenic Orbitals: Incorporating Both Radial and Angular Parts of the Wave Function Meghna A. Manae and Anirban Hazra* Department of Chemistry, Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pune 411008, Maharashtra, India
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S Supporting Information *
ABSTRACT: There exist several two- and three-dimensional graphical representations of hydrogen-like orbitals. Despite this, connecting the mathematical form of the atomic orbital, a function of both radial and angular variables, to its actual shape is often challenging for students. Here, we present a new graphical representation using bubble plots to show the combined contribution of the radial and angular parts of the wave function to its shape. This representation can be demonstrated on the blackboard, as well as easily plotted by students. Additionally, it is one step away from contour plots of orbitals, which are commonly used in their depiction. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Hands-On Learning/Manipulatives, Misconceptions/Discrepant Events, Quantum Chemistry, Atomic Properties/Structure
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appears in several commonly used textbooks.19−22 Coincidentally, this plot closely resembles the shape of the 2pz orbital itself and causes confusion among students. Our experience teaching two undergraduate courses entitled “Chemical Principles I” and “Quantum Chemistry” at the Indian Institute of Science Education and Research Pune, India (A.H. taught these courses multiple times from 2012 to 2018; M.A.M. was initially a student in the latter course and subsequently a teaching assistant for both courses), indicates that several students incorrectly perceive the shape of the orbitals to be solely due to the angular part of the wave function. While textbooks mention that orbital shapes result from a combination of the angular and radial parts,19−22 this is sometimes overlooked by students. Surprisingly, this misconception exists even in the chemical education literature. For example, Rhile23 has recently pointed out the inaccuracy in representation of sp3 hybrid orbitals if this incorrect assumption is made.24 The method proposed in the present work gives students a hands-on way to understand the origin of orbital shapes, which involves both its radial and angular parts.
INTRODUCTION In general chemistry and introductory quantum chemistry courses, the hydrogen atom is the simplest real chemical system for which quantum mechanical laws are applied and the wave function calculated. Physical observables such as position and angular momentum can then be obtained from the wave function. The hydrogen atom is conceptually rich and forms the basis for understanding more complex systems. Several graphical representations of the one-electron hydrogen-like wave functions have been proposed to aid students’ understanding.1−8 Recently, these include three-dimensional models,9−11 interactive computer programs,12−17 and online applets.18 In this work, we present a simple method to obtain orbital shapes from one-electron wave functions with the following aim: Combine the radial and angular parts of the wave function in an easily plottable graphical representation, which provides students with a clear understanding of the origin of orbital shapes. The connection between the shape of the s orbitals and their wave functions is easy to grasp owing to their spherical symmetry. However, for orbitals with l ≥ 1, the angular part adds complexity. For instance, consider the 2pz orbital; most textbooks handle this complexity by plotting the angular and the radial parts of the wave function separately.19−22 The angular part f(θ) is usually plotted in the polar coordinate system, a two-dimensional coordinate system where the variable θ is the angle made with a reference direction and the absolute value of the function is the distance from the origin along the radial line at that θ, while the sign of the function is indicated with + or −. The resulting “polar plot” showing two tangent circles (Figure 1a) or tangent spheres © 2018 American Chemical Society and Division of Chemical Education, Inc.
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WAVE FUNCTIONS OF THE PARTICLE-ON-A-SPHERE That the cos θ function on its own does not represent the shape of the 2pz orbital can be easily understood by considering the particle-on-a-sphere problem, which is defined as a particle constrained to move on the surface of a sphere Received: May 18, 2018 Revised: November 6, 2018 Published: November 28, 2018 187
DOI: 10.1021/acs.jchemed.8b00372 J. Chem. Educ. 2019, 96, 187−190
Journal of Chemical Education
Communication
Figure 1. (a) Polar plot of f(θ) = cos θ, the angular part of the 2pz wave function in the range 0°≤ θ ≤ 360°, with the sign of the function indicated with + or − as shown. The origin of the plot is the point where the two circles touch. (b) The eigenfunction ψ(θ, φ) = cos θ of the particle-on-asphere represented on a cross section of the sphere (gray circle) passing through the z axis at certain θ values in the range 0°≤ θ ≤ 180°. The values of the eigenfunction are indicated with small red and blue circles or bubbles whose areas are proportional to the function values. Red (blue) represents positive (negative) values. (c) The eigenfunction ψ(θ, φ) = cos θ represented on the cross section of the sphere.
to a one-dimensional function in θ, which is in the comfort zone of students. Table S1 in the Supporting Information (SI) gives the values of the wave function for different r and θ and clearly indicates that the set of values for one particular r (various θ) simply gets uniformly scaled for any other r. The bubble plot of the 2pz orbital on a plane passing through the z axis is shown in Figure 2a, with the area of each of the red and blue bubbles proportional to the positive and negative values of the wave function, respectively. Any arbitrary plane passing through the z axis can be chosen for the plot since the wave function is independent of φ. Mentally visualizing a number of planes with different values of φ (just like in the case of the particle-on-a-sphere) allows one to connect to the threedimensional wave function. The plot along a plane can be viewed in two ways: one, in which a gray circle corresponds to a particular value of r on which the wave function at certain θ values is plotted and, two, in which a radial line starting from the center (dotted line in Figure 2a) corresponds to a particular value of θ, on which the wave function at certain r values is plotted. A student can easily intuit that, for a given r, the value of the function, i.e., the size of the bubble, will be the highest when θ = 0°, zero when θ = 90°, and lowest when θ = 180°, and the function is antisymmetric about the xy plane (θ = 90°). Similarly, for a given θ, the value of the function is zero for r = 0, goes through a maximum for intermediate r, and tends to zero for large r (Figure 2b). To get further insight into the shape of the function, approximate contour plots of the 2pz orbital can be drawn on a plane by joining all bubbles at which the function value is roughly the same (Figure 2c). The bubbles are approximately categorized according to their size, and all bubbles in a group are labeled with the same numeral as shown in Figure 2c. The numerals 1, 2, 3, and 4 correspond successively to bubbles of decreasing size, and prime superscripts on the numerals indicate negative values. This categorization can also be found in Table S1 along with the function values. The bubbles in a group are then joined in Figure 2c to obtain approximate contours. These contours can be directly translated to the three-dimensional shape of the 2pz orbital by spinning them about the z axis. One is now able to appreciate that the shape of the 2pz orbital can be pictured as two slightly flattened spheres which are separated by the xy plane and is in contrast
with a constant potential energy.25 One of the eigenfunctions (l = 1, ml = 0) for this problem in spherical coordinates is ψ(θ, φ) = cos θ. The polar plot of this function (Figure 1a) may give the wrong impression that the particle has a dumbbell shaped probability distribution with different possible distances from the origin, but this is obviously not true because the particle is constrained to have a fixed distance from the origin equal to the radius of the sphere. To represent the wave function in a physically correct manner, we consider a cross section of the sphere passing through the z axis which is the large gray circle in Figure 1b. On this gray circle, the values of the wave function at different θ values (0°≤ θ ≤ 180°) spanning half its circumference are indicated by small circles or bubbles, whose areas are proportional to the value of the function; i.e., the radii of the bubbles, a, are proportional to |cos θ| . Since the wave function is symmetric about the z axis, i.e., it has no φ dependence, the value of the wave function on the rest of the circumference can be obtained by reflecting the semicircle (Figure 1b) about the z axis to arrive at Figure 1c. This bubble plot represents the value of the wave function along any plane passing through the z axis. Mentally visualizing a number of such planes allows one to picture the actual threedimensional wave function. The colors red and blue indicate positive and negative values, respectively. This clearly shows that the particle has a higher probability density for being present near the poles of the sphere (θ = 0°, 180°) as compared to other regions, and zero probability density for being present at regions close to its equator (θ = 90°).
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WAVE FUNCTIONS OF THE HYDROGEN ATOM This idea is now extended to represent one-electron hydrogen atom eigenfunctions. We wish to connect the eigenfunction with its shape and explain that the orbital shape is dependent on both the radial and angular parts. The 2pz orbital wave function is ψ2p (r , θ , ϕ) = z
1 32π
ij Z yz jj zz ka{
5/2
r e−Zr /2acos θ
(1)
where Z is the atomic number of the hydrogen-like atom and a is a constant, approximately equal to the Bohr radius (0.52 Å). With different fixed values of r (r = 2a, 4a, 6a, and 8a represented by gray circles in Figure 2a,c), eq 1 can be reduced 188
DOI: 10.1021/acs.jchemed.8b00372 J. Chem. Educ. 2019, 96, 187−190
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clearly seen. Figure S2 of the Supporting Information shows the representation of the 3dz2 orbital.
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CONCLUDING REMARKS Our experience suggests that, in the classroom, one can use the present approach in the same sequence as described in the paper. The particle-on-a-sphere can first be discussed by the instructor to make the misconception about the angular part and shape explicit. The representation of the hydrogenic 2pz orbital can then be drawn on the blackboard in steps, first on one circle and then on other concentric circles. The bubbles with similar size can be identified by the instructor, and approximate contour plots can be drawn. Extensions to other p orbitals can be discussed. This can be complemented with visualization of the orbitals using any online applet. The students are thus provided a concrete feeling for the shape of the orbitals and the connection of shape to their functional form. In conclusion, this paper presents a simple and intuitive way of representing hydrogen-like atomic orbitals which can be plotted by hand. This demystifies the origin of the threedimensional shapes of atomic orbitals. The angular and radial parts are not separated in this representation, thus emphasizing that their combined contribution gives rise to the shape of the atomic orbitals.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00372. Values of the 2pz wave function, extension of the representation to 2px and 2py orbitals, graphs of the 3pz and 3dz2 orbitals, and additional description to represent the 3px and 3py orbitals (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Anirban Hazra: 0000-0003-2012-381X
Figure 2. (a) Bubble plot of the 2pz wave function on a plane passing through the z axis. The gray circles correspond to r values 2a, 4a, 6a, and 8a. On these circles, the values of the wave function at different θ values are indicated with red and blue bubbles whose areas are proportional to the function values. Red (blue) represents positive (negative) values. The dotted line represents one particular value of θ. (b) The wave function plotted along the dotted line shown in part a. (c) The bubbles are grouped on the basis of their size and sign and labeled with numerals 1, 2, 3, and 4. Primes on the numerals are used for labeling negative values. Bubbles of the same size and sign are connected by the gray curves to generate approximate contours.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS M.A.M. thanks IISER Pune for a research fellowship. REFERENCES
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to the polar plot of only the angular part of the function, which looks like two perfectly tangent spheres (Figure 1a). The bubble plot may, at first glance, seem unfeasible for the 2px and 2py orbitals which depend on both θ and φ. However, it can be easily extended to the 2px and 2py orbitals due to the symmetry in the p orbitals (Section S1 in the SI). p orbitals with higher n values can also be plotted. Figure S1 shows the 3pz orbital followed by a discussion of its extension to the 3px and 3py orbitals. Importantly, the two nodes in the 3pz orbital (one radial node at r = 6a and one angular node at θ = 90°) are 189
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DOI: 10.1021/acs.jchemed.8b00372 J. Chem. Educ. 2019, 96, 187−190
