Hands-On Hybridization: 3D-Printed Models of Hybrid Orbitals

Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

pubs.acs.org/jchemeduc

Hands-On Hybridization: 3D-Printed Models of Hybrid Orbitals Riccardo de Cataldo, Kaitlyn M. Griffith, and Keir H. Fogarty* Department of Chemistry, High Point University, One University Parkway, High Point, North Carolina 27268, United States

J. Chem. Educ. Downloaded from pubs.acs.org by KAROLINSKA INST on 08/16/18. For personal use only.

S Supporting Information *

ABSTRACT: Introductory chemistry students encounter the concept of hybrid orbitals as a transition from atomic orbitals to molecular bonding. The principal purpose of learning hybridization in the undergraduate curriculum is to impart an understanding of the origins of molecular bonding and geometry. Physical models of both individual hybrid orbitals and combinations of hybrid orbital types have the potential to aid in student visualization of molecular geometry. 3D printing can serve to generate physically accurate models in a costeffective manner. The use of a freely available JavaScript applet (CalcPlot3D) enables the generation of 3D-printing files (.stl/.3mf files) that can be subsequently printed on a 3D printer. The procedure is low-cost and relatively flexible and produces mathematically accurate hybrid orbital models that can serve as hands-on pedagogical tools. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Physical Chemistry, Inorganic Chemistry, Hands-On Learning/Manipulatives, Multimedia-Based Learning, Atomic Properties/Structure, Quantum Chemistry

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constituent atomic orbitals.16,17 Hybrid orbitals are thus models of a “geometrical rearrangement” of atomic orbitals in preparation for molecular bond formation.16−19 Teaching hybrid orbitals is not without controversy, as some contend that general and organic chemistry courses should not teach hybrid orbitals for two reasons: (1) curricula introduce hybrid orbitals on the heels of atomic orbitals, adding a layer of complexity to orbitals, and (2) hybrid orbitals do not satisfactorily model electron behavior in upper-level chemistry courses; both cases lead to potential confusion.21,22 Nevertheless, it remains common practice for chemistry curricula to introduce hybrid orbitals in general chemistry to build the foundation for using orbitals as a reasoning mechanism for more sophisticated concepts.23−25 For example, organic chemistry has three such topics: (1) determining relative acidity and basicity, (2) determining whether electrons contribute to resonance stabilization, and (3) prediction and rationalization of relative bond lengths (e.g., lengths of C−H bonds with sp-, sp2-, and sp3-hybridized carbon).26,27 Any tangible 3D representation of these hybrid orbitals would thus help students understand the geometric and steric factors that must be considered for these types of reasoning problems. Despite the pervasiveness of hybrid orbitals in chemical education, there are not many available physical models of hybrid orbitals. When available, hybrid orbitals are typically purchased in the same kit as atomic orbitals.28 Artists render these commercial orbital models qualitatively, focusing on molecular geometries with snap-in components.29 Our procedure utilizes the mathematics of quantum mechanics to

INTRODUCTION 3D printing is finding increasing use as a learning tool in chemistry classrooms. Learning applications of 3D printing in chemical education include molecular orbitals,1,2 crystallographic models,3−5 molecular models (including molecules, DNA, proteins, and metal complexes),6−9 potential energy surfaces,10−12 analytical methods (i.e., IR and NMR spectroscopy and HPLC),13 understanding wavelength dispersion,14 and other manipulatable conceptual models.15 The undergraduate chemistry curriculum typically introduces students to three types of electronic orbitals: atomic, hybrid, and molecular. The conceptually abstract nature of these orbitals sometimes causes students difficulty when they must envision the three-dimensional (3D) structures of orbitals given only two-dimensional (2D) representations commonly used in textbooks.16−19 We previously demonstrated the generation of mathematically accurate 3D-printed atomic orbitals, which bridge this gap by providing physical models to help students envision 3D orbital structure.20 The work described in this paper represents an effort to extend our same procedure for generating atomic orbitals to hybrid orbitals. In addition, inclass worksheets were developed to use the 3D-printed atomic and hybrid orbital models in a general chemistry class. The atomic orbital worksheet and answer key along with the hybrid orbital worksheet and answer key, can be found in the Supporting Information. As detailed below, student reception of the 3D-printed orbitals in the general chemistry course was positive, with students commenting on the particular helpfulness of the 3D models in differentiating orbital types. Hybrid orbitals represent linear combinations of atomic orbitals and serve as models that exhibit appropriate geometry for atomic bonding.16,19 One may envision these orbitals as arising from constructive and destructive interference of © XXXX American Chemical Society and Division of Chemical Education, Inc.

Received: February 5, 2018 Revised: July 11, 2018

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DOI: 10.1021/acs.jchemed.8b00078 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

Article

Figure 1. Three phases of the 3D printing process. (A) Graph of the sp21 hybrid orbital in CalcPlot3D along with three axes and a conical base. (B) Repaired .stl file for the same hybrid orbital in Simplify 3D, a program that converts .stl files into .gcode files, which are read by 3D printers. (C) Fully printed physical sp21 orbital. Visible on the physical model in (C) is a seam that arises from printing the orbital as two halves. These two halves are then glued together to create the complete model. This method is used to save 3D-printing materials by eliminating support structures, as discussed in more detail below.

This paper outlines the printing procedure using as an example a carbon sp2 orbital. We use a numbered subscript convention to differentiate among hybrid orbitals, as not all hybrid orbitals (e.g., sp3) neatly align with axes. SI Table 2 provides a key for specific identification of the numbered hybrid orbitals. The Schrödinger probability density equation for the carbon sp21 hybrid orbital (aligned with an arbitrary x axis) is40

produce morphologically accurate models focusing on individual stand-alone orbitals. The latter approach allows greater appreciation of the three-dimensional entirety of orbital structure rather than just the exterior-facing lobes of the snapin model. Furthermore, our procedure can generate multinode models, which are impossible with the common snap-in/out models provided by some companies.29,30 With the increasing availability of 3D printing technologies, our models currently cost ∼$0.85 per model, while some of the kits may range from $65 to $250+.28−30 3D-printed hybrid orbital models allow students to better visualize their shapes, which hopefully helps them construct a more solid foundation in their chemical reasoning abilities. These models are meant to complement online virtual orbital modeling tools, which are also very useful in improving student understanding of orbitals.31,32 Research indicates that students who interact with both physical and virtual models involving complex spatial concepts are better able to mentally visualize them at later dates.33−36 This work builds on the procedure we developed for 3D printing of atomic orbitals,20 outlining necessary variations below. In addition, the Supporting Information provides print-ready 3Dprinting files of hybrid and atomic orbitals.

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(Ψsp2 )2 = 1

1 (Ψ2s + 3

2 Ψ2p )2

(1)

x

The 1/3 and 2 are included as normalization constants for the hybrid orbital equation. The sp21 hybrid orbital is a linear combination of the Schrödinger wave equations for the 2s and 2px orbitals (denoted as Ψ2s and Ψ2px, respectively): Ψ2s =

1 b3/2(2 − br )e−br /2 32π

(2)

Ψ2p =

1 b5/2r e−br /2 sin θcos φ 32π

(3)

x

where 1 32π is a normalization constant, b = Z/a0, where Z is the effective nuclear charge and a0 is the Bohr radius, and r is the distance from the nucleus. Figure 1 shows the three stages for producing a 3D-printed model using eq 1: (A) plotting the hybrid orbital equation in CalcPlot3D, (B) exporting the CalcPlot3D results as a .stl file and prepping that file for 3D printing, and (C) 3D printing the model. These three steps will be discussed in more detail below.

GENERATION OF VIRTUAL MODELS

Hybrid Orbital Theory

Schrödinger wave equations, Ψ, model electron orbitals, and the squares of these wave equations, Ψ2, produce shapes representing the corresponding probability distributions of electron density.17 It is common to visualize orbitals by plotting the surface representing a specific value of Ψ2 low enough to enclose 90−95% of the total electron density associated with that orbital.16,17,37 The defined volume of an isosurface representation allows for the printing of the physical models. As in our previous work, we utilized the freeware CalcPlot3D to generate stereolithography (.stl) files necessary to produce 3D-printed models.20,38,39 This paper extends our procedure to hybrid orbitals by demonstrating the linear combinations of constituent atomic orbitals necessary to generate the corresponding hybrid orbitals. The Supporting Information has two tables that present the equations used to generate the hybrid orbitals and another table listing the orbitals for which print-ready 3D-printing files are provided.

Input into CalcPlot3D

CalcPlot3D is the free-access online graphing tool utilized to generate the stereolithography files of each orbital model.38 CalcPlot3d has been updated since our previous work,20 transitioning to a JavaScript app with a more intuitive user interface. A guide for producing orbitals in the new version of CalcPlot3D is provided in the Supporting Information. Within CalcPlot3D, a drop-down menu allows either an “implicit surface” or a “parametric surface” to be added to the plotting area. The implicit surface is used to plot the orbital, while the parametric surface selection is used to add the three axes and the cone. The parametric surfaces provide not only visual context for the orientation of the orbital in space but also structural support, as discussed in our previous work.20 B



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Equation 4 thus shows how the wave equation for (Ψsp21)2 is typed into CalcPlot3D’s implicit surface tool: a = (c((2 − bρ)e

−bρ /2

+

2 (bρe

−bρ /2

2

)sinϕcosθ ))

The 3D-printed models of both atomic and hybrid orbitals were used in workshops conducted in an introductory general chemistry course. One and a half lecture periods (100 min/ lecture period) were devoted to the workshops: a full lecture for the atomic orbital workshop and half a lecture for the hybrid orbital workshop. The Supporting Information contains four documents pertaining to these workshops: the atomic orbital worksheet and answer key and the hybrid orbital worksheet and answer key. In each workshop, students worked in groups of three to four to complete a worksheet. Students confirmed their answers on the worksheets with the professor and were awarded class participation points on the basis of worksheet completeness and correctness (the point total for both worksheets was 3 points out of 30 total class participation points for the semester, which amounted to 0.5% of their final grade in the course). The workshop class periods were separated by ∼2 weeks, as determined by the appearance of atomic and hybrid orbitals in the curriculum. The atomic orbital workshop was run the lecture period after the students were introduced to the concepts of orbitals, quantum numbers, nodes, etc. The workshop alternated between students running computer simulations of orbitals32 and students inspecting and identifying 3D-printed models without the aid of text or simulation. For example, students explored ndxy orbitals (n = 3, 4) in the simulation portion of the assignment, after which they came to the front of the room and had to correctly identify the suite of five 3d orbital models as well as the 4dxy and 4dz2 models through deductive reasoning. Students were started on the task by being given the identity of the 3dxy model. On average, because of the portions of the assignment that used the simulation, there was rarely a “traffic jam” of student groups waiting to interact with the one set of 3D-printed models at the front of the room. The hybrid orbital workshop was conducted during the second half of a lecture period, the first half of which was an introduction to hybrid orbitals, molecular geometry, and valence shell electron pair repulsion (VSEPR). The previous week’s lectures were devoted to the learning of Lewis structures, including expanded octets. The workshop related the shape and orientation of hybrid orbital models to Lewis structures. Once again, students worked in groups of three to four. The different structure of the worksheet relative to the atomic orbital workshop meant that there were sometimes several groups interacting with the one set of hybrid models concurrently. In such a scenario, individual hybrid models would be passed from group to group to ensure forward progress. There was also time set aside for class-wide discussion to arrive at consensus, during which individual orbital models would be displayed using the classroom’s document camera.

(4)

Table 1. Parameters Appearing in CalcPlot3D (Equation 4)a Identity

a

(Ψsp21)2

b

Z/a0

c

1 3

jij 1 jj k 32π

Resulting Value 0.03b

zyzijjj Z yzzz zzjj zz {k a0 {

EXAMPLE PEDAGOGY

3D-Printed Orbitals Used in General Chemistry

It should be noted that several changes have been used to translate eqs 1, 2, and 3 into CalcPlot3D. First, ϕ and θ have been switched relative to the convention used in typical chemistry texts.16−18 In addition, the distance from the nucleus, r, has been replaced by ρ. Next, the equation has been simplified using the parameters a, b, and c. If a parameter is typed into the CalcPlot3D equation input, a parameter window with an adjustable slider opens that can be used to tailor the orbital’s appearance to the desired configuration. Table 1 displays the identities and resulting values for the three

Parameter

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6.05 (UNC/Å)c 3/2

0.856 (UNC/Å)3/2c

a

The parameter a represents the probability density cutoff used to generate the isosurface. The parameter b contains the nuclear charge, Z, and the Bohr radius, a0. The nuclear charge of 3.2 used for the calculation was for a carbon atom, taking into account 1s shielding.41 The Bohr radius was set to 0.5292 Å. The parameter c combines constants that are contained in eqs 1, 2, and 3. bThe probability cutoff of the isosurface, a, is unitless and is set to a user-defined value. cThe units of the parameters b and c contain the abbreviation “UNC”, which stands for “unit nuclear charge”, and were calculated based on literature values.

parameters in eq 4. The parameters b and c were set to the defined values indicated in the table. These values were found by assuming that the sp2 hybrid orbital is on a carbon atom, with a shielded nuclear charge of 3.2.41 The Bohr radius was set to 0.5292 Å, and thus, the plotted orbitals are on an angstrom scale. The value for a was chosen on the basis of the appearance of the hybrid orbital in CalcPlot3D, with the resulting scale compared to the diameter of a carbon atom. With the value of 0.03 for a, a hybrid orbital with a maximum length of ∼2.5 Å was generated, which can be compared to the diameter of a carbon atom (from ∼1.4 Å to 3.4 Å).42,43 The resultant CalcPlot3D plot can be seen in Figure 1A. Because of their curved structures, the lobes cannot stand upright without the use of support structures. The parametric function tool of CalcPlot3D permits the graphing of these structures: (1) three cylindrical axes (X, Y, and Z) cross into the lobes, and in particular, the Y axis extends into (2) a conical base that allows the model to stand upright. CalcPlot3D has the capability to export this five-function graph (i.e., the orbital, three axes, and the conical base) as a stereolithography (.stl) file. A procedure for generating the .stl file for the orbital with axes and cone in CalcPlot3D is outlined in the Supporting Information, along with instructions concerning preparation of the .stl file for 3D printing and techniques to decrease the amount of 3D-printing material needed to produce each model.

Potential for Use in Upper-Level Chemistry Classes

To date, the 3D-printed atomic and hybrid orbital models have been used primarily as “show and tell” objects in colleagues’ inorganic and quantum physical chemistry courses at our institution. We have discussed the creation of a physical chemistry laboratory or take-home assignment in which students engage with the mathematics of quantum mechanics in CalcPlot3D and then learn the technique of 3D printing and print models. Such an assignment would have threefold utility: (1) students must have a mastery of orbital equations to C



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Figure 2. Results of the in-class survey regarding the pedagogical efficacy of atomic and hybrid orbital models. Undergraduate introductory chemistry students (n = 34) completed a survey ranking eight question topics as either “Very Helpful”, “Somewhat Helpful”, or “Not Helpful”.

of students on questions that were felt to be relevant to student use of 3D-printed orbitals. Students were asked questions concerning atomic orbitals on both homework and exams. Hybrid orbitals are a topic introduced at the end of the semester, after all of the scheduled exams, so students encountered hybrid orbital questions only on homework. For atomic orbitals, students who participated in the 3D-printoriented workshops (cohort 1, n = 34) outperformed their peers who had not participated (cohort 2, n = 33) on relevant exam questions: cohort 1, 73.9 ± 5.3%; cohort 2, 51.5 ± 8.7%. For homework questions related to atomic orbitals, the two cohorts’ performances were more in line: cohort 1, 83.2 ± 7.3%; cohort 2, 83.5 ± 9.7%. For hybrid-orbital-related homework, cohort 1 once again outperformed their peers who had not participated in 3D-printed orbital workshops: cohort 1, 76.3 ± 8.3%; cohort 2, 58 ± 15%. We would like to stress that these data are largely qualitative; we did not use a formal instrument to assess before and after conception of orbitals relevant to the 3D-printed models presented. In addition, homework and exam questions differ from year to year, so comparison of average grade data is strictly qualitative. The goal of our 3D-printing work is the exposition of an easy method to make a variety of hands-on, quantitatively accurate models that ideally will aid student spatial reasoning. Our preliminary data are hopeful, but more work needs to be done to accurately quantify learning outcomes.

effectively replicate orbitals in CalcPlot3D; (2) students learn a skill, 3D printing, that is increasingly relevant in the modern laboratory;44−46 and (3) a supply of models is generated for use in introductory courses.

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STUDENT RECEPTION Thirty-four undergraduate introductory general chemistry students completed an anonymous survey consisting of eight questions that gauged the efficacy of the 3D-printed atomic and hybrid model workshops in clarifying concepts of quantum mechanics, such as quantum numbers, differentiating among orbital shapes, orientations, and identities. Three of these questions assessed student understanding of hybrid orbital concepts: molecular bonding, molecular geometry, and molecular VSEPR theory. In Figure 2, the eight questions and responses are presented in a graphical manner. For example, the pie chart at the top left with the title “Shape” shows data obtained from the question paraphrased as, “Do 3D-printed models aid your understanding of atomic orbital shape?” Responses were scored as “Very Helpful”, “Somewhat Helpful”, and “Not Helpful”. An internal Institutional Review Board approved of this study, and all participating students gave informed consent for the results to be published. The most common positive student feedback indicated that handson models greatly aided student understanding of orbital shape and orientation in three-dimensional space. The most common critique of the orbitals involved students wishing to have spent more time handling the models and a more obvious system of labeling. If labeling of orbitals is desired, we suggest the use of a paint pen. Our worksheets involved discussion-centered student identification of model orbital types, so our models were deliberately unlabeled. A blank template of the in-class survey is provided in the Supporting Information. While a formal assessment of learning outcomes associated with 3D-printed orbital models has not been performed, we compared homework and exam performance from two cohorts

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CONCLUDING REMARKS Some students have difficulty conceptualizing the intricate three-dimensional structure of hybrid orbitals, which several courses in the chemistry curriculum apply to reason through problems. Existing model kits do not accurately portray hybrid orbitals and their resulting geometries. The adaptation of our previous work discussed in this paper demonstrates how the capabilities of 3D printers to print fine details eliminates the complexity of orbital shape as an impediment to model D



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creation.20 The 3D-printed models were subsequently used for general chemistry in-class workshop activities centered on orbitals. A student survey found that the 3D-printed models helped students’ perceived understanding of both atomic and hybrid orbitals. The Supporting Information provides worksheets used for the in-class activities, a detailed procedure for the generation of these structures in CalcPlot3D, and tables with relevant orbital equations. In addition, over 100 3D-printready atomic and hybrid orbital .stl files are provided. We believe that 3D printing is a relatively easy and cost-effective tool for producing hands-on models of atomic and hybrid orbitals for use in the chemistry curriculum.20

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(3) Scalfani, V. F.; Vaid, T. P. 3D Printed Molecules and Extended Solid Models for Teaching Symmetry and Point Groups. J. Chem. Educ. 2014, 91 (8), 1174−1180. (4) Rodenbough, P. P.; Vanti, W. B.; Chan, S.-W. 3D-Printing Crystallographic Unit Cells for Learning Materials Science and Engineering. J. Chem. Educ. 2015, 92 (11), 1960−1962. (5) Casas, L.; Estop, E. Virtual and Printed 3D Models for Teaching Crystal Symmetry and Point Groups. J. Chem. Educ. 2015, 92 (8), 1338−1343. (6) Scalfani, V. F.; Turner, C. H.; Rupar, P. A.; Jenkins, A. H.; Bara, J. E. 3D Printed Block Copolymer Nanostructures. J. Chem. Educ. 2015, 92 (11), 1866−1870. (7) Rossi, S.; Benaglia, M.; Brenna, D.; Porta, R.; Orlandi, M. Three Dimensional (3D) Printing: A Straightforward, User-Friendly Protocol To Convert Virtual Chemical Models to Real-Life Objects. J. Chem. Educ. 2015, 92 (8), 1398−1401. (8) Meyer, S. C. 3D Printing of Protein Models in an Undergraduate Laboratory: Leucine Zippers. J. Chem. Educ. 2015, 92 (12), 2120− 2125. (9) Van Wieren, K.; Tailor, H. N.; Scalfani, V. F.; Merbouh, N. Rapid Access to Multicolor Three-Dimensional Printed Chemistry and Biochemistry Models Using Visualization and Three-Dimensional Printing Software Programs. J. Chem. Educ. 2017, 94 (7), 964−969. (10) Kaliakin, D. S.; Zaari, R. R.; Varganov, S. A. 3D Printed Potential and Free Energy Surfaces for Teaching Fundamental Concepts in Physical Chemistry. J. Chem. Educ. 2015, 92 (12), 2106−2112. (11) Lolur, P.; Dawes, R. 3D Printing of Molecular Potential Energy Surface Models. J. Chem. Educ. 2014, 91 (8), 1181−1184. (12) Teplukhin, A.; Babikov, D. Visualization of Potential Energy Function Using an Isoenergy Approach and 3D Prototyping. J. Chem. Educ. 2015, 92 (2), 305−309. (13) Higman, C. S.; Situ, H.; Blacklin, P.; Hein, J. E. Hands-On Data Analysis: Using 3D Printing To Visualize Reaction Progress Surfaces. J. Chem. Educ. 2017, 94 (9), 1367−1371. (14) Piunno, P. A. E. Teaching the Operating Principles of a Diffraction Grating Using a 3D-Printable Demonstration Kit. J. Chem. Educ. 2017, 94 (5), 615−620. (15) Smiar, K.; Mendez, J. D. Creating and Using Interactive, 3DPrinted Models to Improve Student Comprehension of the Bohr Model of the Atom, Bond Polarity, and Hybridization. J. Chem. Educ. 2016, 93 (9), 1591−1594. (16) Atkins, P. W.; De Paula, J.; Friedman, R. S. Physical Chemistry: Quanta, Matter, and Change; Oxford University Press: Oxford, U.K., 2014. (17) Laidler, K. J.; Meiser, J. H. Physical Chemistry; Benjamin/ Cummings: Menlo Park, CA, 1982. (18) Engel, T.; Reid, P. Physical Chemistry, 3rd ed.; Pearson Education: Glenview, IL, 2013. (19) Tro, N. J. Chemistry: A Molecular Approach, 3rd ed.; Pearson Education: Upper Saddle River, NJ, 2014. (20) Griffith, K. M.; de Cataldo, R.; Fogarty, K. H. Do-It-Yourself: 3D Models of Hydrogenic Orbitals through 3D Printing. J. Chem. Educ. 2016, 93 (9), 1586−1590. (21) Grushow, A. Is It Time To Retire the Hybrid Atomic Orbital? J. Chem. Educ. 2011, 88 (7), 860−862. (22) Bent, H. A. Should orbitals be x-rated in beginning chemistry courses? J. Chem. Educ. 1984, 61 (5), 421. (23) Tro, N. J. Retire the Hybrid Atomic Orbital? Not So Fast. J. Chem. Educ. 2012, 89 (5), 567−568. (24) Landis, C. R.; Weinhold, F. Comments on “Is It Time To Retire the Hybrid Atomic Orbital? J. Chem. Educ. 2012, 89 (5), 570− 572. (25) DeKock, R. L.; Strikwerda, J. R. Retire Hybrid Atomic Orbitals? J. Chem. Educ. 2012, 89 (5), 569−569. (26) Hiberty, P. C.; Volatron, F.; Shaik, S. In Defense of the Hybrid Atomic Orbitals. J. Chem. Educ. 2012, 89 (5), 575−577. (27) McMurry, J. Organic Chemistry, 8th ed.; Brooks/Cole: Belmont, CA, 2012.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00078. Detailed instructions for generating .stl files of orbitals using CalcPlot3D, generation of physical models, equations for atomic and hybrid orbitals, and a list of files included in the SI (PDF, DOCX) Atomic orbital worksheet for use with 3D printed models in a general chemistry course (PDF, DOCX) Key to the atomic orbital worksheet, with instructor notes (PDF, DOCX) Hybrid orbital worksheet for use with 3D printed models in a general chemistry course (PDF, DOCX) Key to the hybrid orbital worksheet, with instructor notes (PDF, DOCX) Blank survey discussed in this paper (PDF, DOCX) Zip folder containing all of the ready-to-print hybrid orbital and atomic orbital.stl files (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Keir H. Fogarty: 0000-0001-8953-4778 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project was supported by the chemistry and physics departments of High Point University. We thank Aaron Titus, Patrick Vigueira, Briana Fiser, Martin Dewitt, Brad Barlow, Alan Vasquez, and Simeon Simeonides for their help in the utilization of the 3D printer at High Point University. In addition, we thank Brian Augustine and Chris Fowler for valuable advice and input concerning the project. Finally, we thank the CalcPlot3D developers for their help with this project, particularly Paul Seeburger, Associate Professor of Mathematics, Monroe Community College, Rochester, NY.

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REFERENCES

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Hands-On Hybridization: 3D-Printed Models of Hybrid Orbitals

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