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Computational Chemistry in Organic Chemistry Lecture using WebMO

Author(s): 

Brian J. Esselman,* Department of Chemistry, University of Wisconsin, Madison, WI

Aubrey J. Ellison,* Department of Chemistry, University of Wisconsin, Madison, WI

Jia Zhou, Department of Chemistry, University of Wisconsin, Madison, WI

Abstract: 

Advances in computational software and hardware have allowed computational chemistry to become a more integrated component of undergraduate organic chemistry curriculum.  With a few exceptions, most of this attention has been given to small lecture activities or individual laboratory exercises.  To allow students to fully utilize the insights of computational chemistry, it must become fully integrated into all aspects of instruction.  Here we present out on-going efforts in integrating computational chemistry into the entire lecture curriculum via carefully chosen examples on lecture slides, problem sets, and assessments.  Rather than having students perform these calculations themselves, as they do in our laboratory course, we have taken advantage of the HTML-export feature of WebMO to provide students with direct links to computational data.  As students work through our curriculum, their learning is supported by the ability to view relevant molecular geometries, charge distributions, orbitals, vibrational modes, etc.  We are confident that use of these tools leads to better learning outcomes.

 

Introduction

Coinciding with the availability of student-friendly software, computational chemistry has been introduced in many graduate and undergraduate courses over the past few decades.1-7  Many excellent examples of lecture and laboratory exercises have been presented in the literature across college-level instruction.  Given the obvious ability of computational chemistry to help students understand and visualize the three-dimensional nature of molecules and highlight the connection between structure and reactivity, it is unsurprising that instructors of general, organic, and inorganic chemistry have seized upon this capability.  Importantly, computational results can provide chemical insights to students without the need to understand the details of how the calculations were performed.  Hence, many published exercises designed for early undergraduate chemistry students are able to avoid delving deeply into the details of quantum mechanics and computational chemistry.  This demonstrates the ability of computational results to provide chemical insights to students without the need to understand the details of how the results were obtained.  While there is definitely substantial value in a proper understanding of quantum mechanics, levels of theory, basis sets, optimization routines, etc., these topics can be reserved for upper-level physical chemistry and computational chemistry courses without sacrificing substantial student understanding of structure and bonding in the introductory courses.

Stand-alone computational chemistry exercises have the inherent drawback of time, cognitive load, and effort required to learn how to perform the calculations.  This burden, in cases of infrequent implementation, can easily overshadow the insights derived from the exercise that would be useful to the early college student.  A few university courses8-10 and department-wide chemistry programs11-13 have embedded computational chemistry throughout the curriculum.  These implementations reduce the cognitive load on students in any particular computational exercise as students become increasingly familiar with the software due to repeated exposure.  Additionally, students and instructors inherently see the benefit of mastering the new software necessary to complete their calculations, knowing that this skill will serve them well in their future coursework.  Undoubtedly, such repeated utilizations can provide a more meaningful and longer-lasting impact on students’ understanding of three-dimensional structures and chemical bonding.

In organic chemistry, there are two very natural avenues for computational chemistry to support the curriculum: supporting analysis of experimental data and elucidating how the electronic or molecular structure drives the chemical outcome.  In the organic laboratory, many authors have demonstrated that computational chemistry can help students assign IR or NMR spectra.4, 14-22  This support can allow students to make assignments and perform analyses that may be beyond their abilities without computational predictions.  Further, a number of authors have demonstrated the utility of calculations to help students predict or rationalize the outcome of an organic reaction in lecture and laboratory settings,23-26 helping students obtain a deeper understanding of many important concepts.  Student exercises have been used to investigate conformational isomerism,9-10, 27-31 structure and relative energy of reactive intermediates,19, 21, 25-26, 32-37 the selectivity of an organic reaction,19, 25-26, 38-42 assessing relative acidity,43-44 evaluation of a reaction mechanism,45 etc.  All of these examples provide other educators a template for how they can enhance their curricula and lead to better student learning.

We wish to take this process one step further and integrate computational outputs throughout the entire organic lecture curriculum.  Focusing on interpreting computational results will provide the benefits of a deeper conversation about structure, bonding, and reactivity, without substantially adding to the cognitive load and time necessary for students to perform the calculations.  Based upon previous work,30, 46-47 we have embedded WebMO7-HTML exports into our course materials: student handouts, problem sets, lecture slides, discussion activities, and exams.  In a manner similar to Springer,46 we present computationally generated images of molecules alongside more common two-dimensional images in lecture.  We have extended this use of three-dimensional images from WebMO to include all course content, allowing students to view structures and orbitals in a virtual model kit.  Alongside these computer-generated structures, we encourage students to use wedge-dash notation for molecules drawn on paper and traditional model kits to help build their representational translation.  While we have just begun this modernization of our curricular materials, we intend to make virtually all images of molecules presented in course content accessible via student engagement with HTML links.  Even during this early adaptation and implementation, we have observed several important anecdotal outcomes.  Our students seem to have a deeper grasp of three-dimensional structures, an improved understanding of how atomic and molecular orbitals influence chemical reactivity, and a greater mechanistic focus on why and how reactions occur.  As an intended consequence, students are able to move away from pattern recognition and memorization and toward utilizing key concepts to rationalize reactivity.  We have changed our own teaching frame of mind to be more focused on how the molecular and electronic structure leads to the function of the molecule.  This article presents some of our initial work and reflections.

Curricular Implementation

The representative examples shown here are from 1st and 2nd semester organic chemistry lectures (CHEM 343 and CHEM 345) at the University of Wisconsin–Madison.  This is the main sequence of organic chemistry and has between 200 and 300 students per lecture in a wide range of majors.  Together, these courses cover the topics presented in the 6th edition of Organic Chemistry by Loudon and Parise.48  The course does not have a laboratory component, and students may take the laboratory course (CHEM 344) concurrently with or subsequent to CHEM 345.  Students may have some exposure to computational chemistry through use of computational outputs in general chemistry or later in CHEM 344 where they run calculations to support their understanding of molecular structure and properties.9-10  Regardless of their level of exposure to output files prior to CHEM 343, virtually none of these students have used any computational chemistry software to perform calculations themselves.  These representative examples demonstrate the usage of WebMO-HTML exports in lecture notes, problem sets, and on course assessments.  Each figure, in the following section, is an authentic image of computational results used in our courses and instruction to support student learning.  These figures include examples of geometry optimizations, vibrational frequencies, NMR chemical shift predictions, and Natural Bond Orbital (NBO) calculations.  The NBO calculation generates and visualizes the molecular orbitals and natural bond orbitals; we use whichever is most impactful for illustrating a concept to students.  In general, MOs are useful for depicting conjugated systems as MOs inherently show the delocalization of electrons throughout a molecule.  NBOs are useful for depicting lone pairs and bonds using the hybridizations and localized notions of bonding common in organic and general chemistry.

1st semester examples

Starting with the very first lecture, students are introduced to computational depictions of chemical structures.  The early introduction is done in such a way that computational models are intentionally presented in parallel with Lewis structure drawings.  This pairing serves to build student confidence and understanding in translating two-dimensional representations into accurate chemical geometries, as well as to demonstrate the utility of computational models.  Figure 1 shows the WebMO output of methane embedded directly into the lecture slide.  Incorporation of the live output allows its manipulation by the instructor, demonstrating geometric shape at the same time as providing initial familiarity with the WebMO interface.  Adjacent to the active WebMO window are traditional, instructor-drawn depictions of methane.  The juxtaposition of the three-dimensional visualization and two-dimensional drawing can then be used to demonstrate how wedge-dash notation better aligns with geometric reality than a simple two-dimensional Lewis structure.  As the semester progresses, students will continue to have static images or live outputs of WebMO molecules presented in lecture in conjunction with two-dimensional hand drawings.

Figure 1. The first usage of WebMO to support a review lecture on Lewis structures.  The methane WebMO output is embedded directly into the lecture slide to allow the instructor to manipulate the WebMO image live without the need to switch between applications.  The slide also contains a clickable link to the methane WebMO output for student use after lecture through the pdf export.

While model kits have been used for conveying three-dimensionality of molecular structures for quite some time, computational molecular depictions add to the depth and breadth of topics that can be presented by including optimized geometries, molecular orbitals, and more accurate charge distributions.  As an example, Figure 2 shows the reactive carbocation intermediate from HBr addition to 1-methylcyclohexene.  Use of WebMO images can quickly highlight the loss of a π bond and subsequent change of the C-atom hybridizations throughout the reaction, supporting the electron-pushing mechanism.  In the first step, WebMO can be used to display the NBO-depiction of an empty π orbital on the carbocation intermediate, focusing student attention on this (formally) empty orbital.  With the depiction of the empty π orbital of the carbocation, students can more easily visualize the structural and electronic factors that contribute to regio- and stereochemical outcomes.  For molecules like 1-methylcyclohexene, the regiochemistry of cation formation is largely controlled by the relative stability of the carbocation.  For aliphatic carbocations, hyperconjugation is dominant stabilization factor, application of which requires students to visualize the overlap of filled orbitals with the carbocation π orbital.  The computational depiction reinforces this concept because of the plainly visible alignment of the C–H and C–C bonds with the empty π orbital in the optimized carbocation structure.  Additionally, visualization of the geometry changes, otherwise difficult to convey with hand-drawn structures, as well as depiction of the NBO of the empty π orbital on C7H13+ showing the orbital shape, helps students to rationalize the possibility of a nucleophilic addition to either face of the intermediate.  Such visualization of computational results promote a better connection between reactivity and stereochemistry than an electron-pushing mechanism can do alone.

Figure 2. Lecture slide NBO depiction of the formally empty (occupancy = 0.43 e-) π orbital on C7H13+ formed in the electrophilic addition of HBr to 1-methylcyclohexene.

As shown in Figure 3, student understanding of the regiochemistry and stereochemistry of halohydrin formation can be enhanced by providing computational geometries for each mechanistic step together with the relevant orbitals.  The acceptor orbitals on each electrophile (σ*Br–Br on bromine and σ*C–Br on the bromonium ion) are displayed, corresponding to each of the σ bonds that break during the course of the reaction.  The initial addition of bromine to isobutene is more complex than depicted here, as the orbitals shown are for only one donor-acceptor interaction.  More correctly, this step involves two simultaneous donor-acceptor interactions in a concerted three-membered ring formation.  Students can rationalize that as the π orbital electrons are transferred to the σ*Br–Br orbital, the bond order of Br2 is lowered to zero and the corresponding σ bond is broken.  Students can see that the π bond on isobutene has a symmetric orbital density above and below the plane of the molecule, either lobe of which can react with bromine.  Two parallel reactions are displayed to show how the two enantiomeric bromonium intermediates are formed as a result of the orbital symmetry of isobutene.  The two enantiomeric intermediates react via an SN2 reaction with a water nucleophile.  When bromonium ions are depicted in most textbooks, they are shown as having two equivalent σC–Br bonds.  In contrast, the WebMO image showcases the lengthening of the σC–Br bond connected to the more substituted carbon atom.  This difference in bond length coincides with an increase in positive charge at the more substituted carbon atom and allows students to rationalize the regioselectivity of the SN2 reaction.  Additionally, it is apparent from the location and shape of the σ*C–Br orbital that the nucleophilic water must approach the electrophilic carbon from the back side of the σC–Br bond.  Finally, there is a subtle feature of this SN2 reaction that becomes more apparent with the computationally predicted structure: the geometry around the C-atoms of the three-membered ring are flattened (the C-atoms are more sp2-like) allowing the nucleophile to effectively approach the σ*C–Br orbital by reducing the impact of the steric repulsion of alkyl substituents on the more substituted C-atom.  Once again, the teaching of halohydrin formation by simultaneously using the electron-pushing mechanism and computational outputs helps students rationalize the Markovnikov regiochemistry and anti stereochemistry far more effectively than a combination of electron-pushing formalism and model kit usage.

Figure 3. Lecture slide depicting electrophilic addition of bromine to isobutene.  Occupied orbital on isobutene depicted in red/blue and unoccupied orbitals on bromine and the bromonium ion depicted in yellow/green.

Later in the semester, the course delves into the SN2-ring opening of epoxides, which is closely tied conceptually to halohydrin formation.  With the familiarity of bromonium ions already presented in lecture and student handouts, we ask students to analyze the structure and derive reactivity from WebMO images of the protonated and non-protonated epoxide of 2,2-dimethyloxirane (Figure 4).  Structurally, there are subtle differences between the protonated and non-protonated epoxide that are not always well-captured in two-dimensional drawings.  To assist in student interpretation, calculated bond lengths and partial charges are provided for each structure.  In 2,2-dimethyloxirane, the C–O bond lengths are nearly identical.  The protonated 2,2-dimethyloxirane, however, bears a strong similarity to a bromonium cation in the depletion of electron density on the tertiary carbon, resulting in a longer, weaker bond on the more substituted side.  This tertiary carbon also shows a flatter geometry, as it is more sp2-like, reducing steric repulsion for nucleophilic substitution.  Using these structures, students are able to predict that the unprotonated epoxide reacts via a normal SN2 reaction at the least substituted carbon atom, while the protonated epoxide displays a reversal of regiochemistry.  This exercise helps students rationalize the difference in regioselectivity for epoxides reacting under acidic versus basic conditions, drawing from their previous conceptual understanding.

Figure 4. Annotated problem set answer key discussing of the geometry changes of 2,2-dimethyloxirane and protonated 2,2-dimethyloxirane

2nd semester examples

Our 2nd semester course begins with spectroscopy and spectrometry (1H-NMR, 13C-NMR, 13C-NMR APT, HSQC, IR, and EI-MS).  As has been previously reported, there are numerous ways of supporting student analysis and interpretation of the experimental spectra with computational chemistry.4, 14-22  Figure 5A shows a typically-assigned IR spectrum of vinyl propionate.49  Using WebMO’s HTML-export feature and editing of line of code, it is possible to directly link the assigned stretches (C(sp2)–H, C(sp2)=O, C(sp2)=C(sp2), and C(sp2)–O), giving students a quick visual connection between vibrational mode and the absorption observed in the IR spectrum.  Figure 5B shows the 13C-NMR spectrum of ethyl acetate with the isotropic NMR chemical shifts provided.  While most of our students can assign C1 and C2 with ease, the computational NMR prediction allows students to confidently assign each of the methyl group 13C-NMR signals (C3 and C4).  For IR and NMR analysis, students are made aware that the predicted relative position of signals is more reliable than the absolute absorption frequency or chemical shift.  For example, students do not seem confused by the vinyl propionate computationally-predicted IR stretches of the C=O (1840 cm-1) and C=C (1736 cm-1) being too high compared to experimental values (1762 and 1649 cm-1, respectively).  Nor has anyone raised concern about the over-prediction of the ethyl acetate carbonyl 13C-atom NMR chemical shift by 10 ppm.  Our hope is that students are learning to use these tools as guides or to confirm their assignments.  Furthermore, we expect that students are increasing their awareness of the difference between a predicted or estimated value and its empirical or experimental counterpart.  It is critical to emphasize to students that the experimental value is the real one and should be trusted.

Figure 5. Annotated problem set answer keys with linked computational predictions of associated spectra.  A)  The optimization and vibrational frequency calculation allows students to view each of the vinyl propionate key functional group stretches identified in the experimental IR spectrum on the answer key.  B)  An NMR calculation provides estimated chemical shift data for each 1H- (not shown) and 13C- atom nucleus referenced to TMS for support in assigning the experimental 13C NMR spectrum.

Electrophilic aromatic substitution (EAS) reactions are a standard part of second semester organic chemistry, where aromatic π systems react as nucleophiles.  While some instructors have explored EAS activation and directing effects via computational investigations of the aryl substrate, we employ computational chemistry to help students rationalize the regioselectivity of these reactions using the arenium cation intermediates.  The charge distribution in the aryl substrate or the electron density distribution of its highest-energy occupied molecular orbital (HOMO) can indicate the more nucleophilic carbon atoms of the ring.  While this is correlated to their relative reactivity, it is a missed opportunity to explore the potential energy surface (PES) of this reaction and connect the relative rates of product formation to the transition states of their rate-determining steps.  Using a more rigorous approach, our students use Hammond’s Postulate and the energies of the high energy cationic intermediates (A and B, Figure 6) to rationalize the outcome of these kinetically controlled EAS reactions.  An example exam question is provided in Figure 6, where students were asked to rationalize the experimental spectra provided from the reaction of bleach with 4'-methoxyacetophenone in acidic conditions.50  As part of that analysis, students were instructed to rationalize the energy difference between the two regioisomeric arenium cations (A and B) using resonance arguments.  Computational data were provided to highlight the combined impact of the π conjugation represented by the associated resonance structures, steric interactions, and dipole-dipole interactions on the potential energy surface rather than simply asking students to depict the π conjugation.  For exams, where students do not have access directly to a computer, the data is provided and organized/pre-analyzed for them to allow the students to use the data to make useful conclusions.  (For a graduate course, where academic misconduct was less of a concern, students were given computational data to use directly via HTML exports during the exam.)

Figure 6. Annotated exam answer key with the two regioisomeric chlorination arenium cations of 4'-methoxyacetophenone (A and B).  Resonance structures are provided for the lower-energy arenium cation A.

Arguably the most important reaction of 2nd semester organic chemistry lecture courses is the attachment of a nucleophile to a carbonyl.  We have adapted a recently published lab exercise10 into a problem set question for the lecture course where students use computational outputs depicting the relevant HOMO and LUMO orbitals to rationalize the attachment of cyanide to acetone.  Figure 7 shows a small portion of this question in which the trajectory of attachment (Bürgi-Dunitz Angle51) corresponds to a favorable overlap of the large lobes of the nucleophile’s HOMO and electrophile’s LUMO.  This also highlights the nucleophilicity and electrophilicity of the bond-forming carbon atoms in this reaction.  Students can view the motion of the atoms at the transition state and visualize how the acetone geometry changes as the cyanide approaches.  Throughout the course of the question, students view the optimized structures of cyanide, acetone, the two molecules coming together in the transition state, and the resulting tetrahedral intermediate.  As the question progresses, students view the relevant orbitals (HOMO of cyanide and LUMO of acetone) of this reaction and compare the LUMO on isobutylene to acetone to rationalize the differences in reactivity of πC–O and πC–C bonds.  It is hard to imagine students making these comparisons in the absence of computational data to support and assist in their analysis.

Figure 7. Annotated problem set answer key displaying the reaction of acetone with a cyanide nucleophile.

With the understanding of the fundamentals of organic reactivity, molecular comparisons become central to rationalizing reactivity as the course progresses.  Computational chemistry can be used to showcase the differing electrophilicity of the carbonyl and b C-atoms of a,b-unsaturated carbonyl-containing compounds.  Figure 8 presents a lecture slide with NBO charge data and depictions of the LUMO for chalcone and protonated ethyl 2-butenoate.  While resonance structures are traditionally drawn to show the electron-withdrawing effect of the O-atom on the carbonyl and b C-atoms in both molecules, the inclusion of the charge data and LUMO provides additional depth.  These data allow students to clearly see that the carbonyl C-atom will be more electrophilic in a neutral or protonated system than the b C-atom, which is difficult to ascertain from resonance structures alone.  This allows students to rationalize the preference for nucleophiles to attach to the carbonyl C-atom for a,b-unsaturated carbonyl-containing compounds, which further allows them to rationalize 1,2- versus 1,4-conjugate additions for these systems without relying almost entirely upon memorization of reaction rates.

Figure 8. LUMO acceptor orbitals and NBO charge data of two a,b-unsaturated carbonyl compounds: chalcone (top) and protonated ethyl 2-butenoate (bottom)

Computational Methods

Calculations

All of the calculations here were obtained using Gaussian 0952 or Gaussian 1653 using WebMO installed on the University of Wisconsin–Madison Chemistry Department educational computer cluster (Sunbird).  Nearly all geometry optimization and vibrational frequency calculations were carried out using B3LYP/6-31G(d).  Subsequent NMR or NBO calculations were carried out to obtain NMR chemical shift data or depictions of relevant orbitals.  For most of the molecules presented in undergraduate organic chemistry textbooks, the optimization will take a few minutes to a few hours, and NMR or NBO calculations add only a small fraction of the optimization time to the total time to generate the desired computational output.  A significant advantage of the WebMO-based approach presented here is that it can be implemented for little to no cost.  At institutions with a computer cluster running any modern computational software, WebMO can be added for a relatively low cost.  For an institution without the necessary infrastructure to maintain a computational cluster, the free version of WebMO can be linked to GAMESS2-3 (also free) to generate computational data from an individual PC.

The B3LYP/6-31G(d) level of theory and basis set is adequate for calculation of most geometries and properties of organic molecules for the undergraduate level.  B3LYP/6-31G(d) calculations do not handle all molecules with experimental accuracy and instructors should use all computational data with caution.  As needed, both the level of theory and the basis set can be improved for better modeling of experimental data in a one-time calculation.  Additionally, instructors must use care when translating two-dimensional structures into three dimensions.  Many of the images drawn in textbooks and on websites of organic molecules are not the low-energy conformation or even a stable structure.

HTML export and embedding

As shown in Scheme 1, once the desired computational output is obtained, selecting the HTML-export feature of WebMO creates a .tar file containing the necessary files to open that output in a modified local HTML WebMO version, similar to what the user sees in WebMO itself.  These files can be given directly to students via a learning management system or via the construction of a simple web site with links to the hosted HTML files.  Alternately, we have found it convenient to place links to the relevant computational outputs directly into the documents we use for lecture presentations, discussion notes, problem sets, exams, etc.  By editing one line of the Javascript code, it is possible to generate clickable items that direct students to specific content (orbitals, dipoles, vibrational modes, charge distribution, etc). in the output file, further minimizing student barriers for usage of WebMO.  Students can click on these links at the moment that they are most relevant and be taken directly to the three-dimensional structure, important orbital, or property.

Scheme 1. Work flow for generating and posting computational outputs for students.

Results and Discussion

Initial Reflections on Implementation

The representative examples shown here provide students with an easily accessible resource that will enhance their understanding of organic chemistry.  Having the WebMO images embedded in all aspects of the course materials provides a framework that allows instructors to focus on how reactivity is driven by physical organic principles.  Instruction carried out in this manner forces students to move further away from memorizing reactions, patterns of reactivity, or oversimplified criteria for predicting reaction outcomes.  We have found that exposure to computational chemistry results, even without students performing the calculations, improves student ability to focus on key concepts: π conjugation, hyperconjugation, electrophilicity, nucleophilicity, bond strength, etc.  Finally, student intuition about molecular and electronic structure is enhanced by repeated exposure to more realistic depictions of molecules and orbital geometries

Conclusion and Future Directions

We believe that the benefits of including computational chemistry in the early undergraduate curriculum have not been fully explored or understood.  Given the relatively recent wide-spread availability of these tools, instructors have not fully grasped their utility in teaching students to think about the reactions of organic chemistry and the properties of molecules using computational results.  Certainly, organic chemistry instructors have been focusing on the relationship between electronic structure and the reactivity of molecules since before use of computational chemistry in the classroom.  Computational chemistry just makes that focus easier for instructors to engender in their students.  Students can see depictions of σ bonds, π bonds, lone pairs, acceptor orbitals, etc.  This ability to visualize the electronic structure directly can reduce the cognitive load of trying to predict the reactivity of a molecule, in a manner more similar to an expert who can fluidly move between two- and three-dimensional representations.  To understand the behavior of an organic molecule, students must view a two-dimensional image, assign the hybridization of the atoms using a valid method, determine the three-dimensional geometry of each atom, and then use the hybridization and three-dimensional structure to understand how the orbitals might lead to a particular reactivity.  The value of the computational outputs is to provide the scaffolding necessary for students to translate organic molecules in such a way that they can predict reactivity.  With exposure to these images and their repeated use, students should become more able to predict the reactivity of analogous molecules without the computational outputs.

It is easy to imagine a future in which computational chemistry is an embedded component of all chemistry textbooks from the high school level onward.  As the migration of textbooks to an online platform continues to accelerate, it would be easy for all depicted molecules to become clickable links highlighting their structure, properties, or relevant orbitals.  This external support for courses would allow educators to routinely enhance instruction with readily-available data and to easily develop curricula where computational chemistry is a natural and authentic component.

While we will continue to grow the organic chemistry curriculum described to include more direct application of computational chemistry outputs, it is clear that students would benefit from these types of implementations at the general chemistry and high school levels.  Currently, two related projects are underway, providing computational outputs to high school teachers and their students via WebMO-HTML exports.

Associated content

Supporting Information

Examples of lecture notes, problem sets, and assessments associated with the chapters of Organic Chemistry48 are available upon request.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: Brian J. Esselman (esselman@wisc.edu), Aubrey J. Ellison (ajellison@wisc.edu)

Acknowledgments

The authors gratefully acknowledge the input of the many teaching assistants and students that have helped refine the curricular approach.  We also thank the participation of Nicholas Hill, Maria Zdanovskaia, Asif Habib, Cara Schwarz, and Amy Van Aartsen on this project and related projects that have improved our approach to implementing calculations within the organic curriculum.  We thank Paul McGuire, Alan Silver, and J. R. Schmidt for their assistance with WebMO and computational resources.  Computer resources are partially supported by National Science Foundation Grant CHE-0840494.

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Date: 
05/06/19 to 05/08/19