Construction of Ligand Group Orbitals for Polyatomics and Transition

Article pubs.acs.org/jchemeduc

Construction of Ligand Group Orbitals for Polyatomics and Transition-Metal Complexes Using an Intuitive Symmetry-Based Approach Adam R. Johnson* Department of Chemistry, Harvey Mudd College, Claremont, California 91711, United States S Supporting Information *

ABSTRACT: A molecular orbital (MO) diagram, especially its frontier orbitals, explains the bonding and reactivity for a chemical compound. It is therefore important for students to learn how to construct one. The traditional methods used to derive these diagrams rely on linear algebra techniques to combine ligand orbitals into symmetry-adapted linear combinations or ligand group orbitals (LGOs). Over the past 10 years, I have developed and refined a simple graphical method for generating ligand group orbitals. The method is an extension of concepts that the students already know from general chemistry: VSEPR and valence bond theory, which are then used to create a generator function. LGOs are prepared by matching symmetry (nodes and phase behavior) of the generator function. The LGOs generated by this technique are qualitatively correct and sufficient for the “back-of-the-envelope” MO diagrams for which they are intended. Through a series of in-class group work and out-of-class problem sets, students learn to derive LGOs for main-group, organometallic, and coordination complexes quickly and correctly. KEYWORDS: Upper-Division Undergraduate, Inorganic Chemistry, Group Theory/Symmetry, MO Theory

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form the MO diagram. For polyatomics, one must consider each atom interacting with all of the other atoms, resulting in a multicomponent problem.4 As a result, inorganic molecules are often not treated as rigorously as they should be. In my inorganic chemistry course, I have developed a qualitative, symmetry-based derivation of LGOs that is then used to construct the corresponding MO diagrams. A major goal of the technique is that it is rapid and easy to apply to a new molecule, especially for transition-metal complexes. The course usually consists of about 15−20 upper-level undergraduate students. Physical chemistry (including symmetry and group theory) is a co-requisite for the course, but most of the students take it the previous year. The course meets in three, 50 min sections per week; two of the meetings are traditional lectures, while the third involves small-group problem work with in-class presentations of solutions. The major studentlearning goal is for them to be able to quickly draw a reasonably accurate (in terms of relative energies of the frontier orbitals) MO diagram for a transition-metal complex of known or predicted geometry. The students are told that this level of theory is not sufficient for “geometry optimizations” such as those performed by computational packages, but it does allow for rationalization and prediction of chemical reactivity given an input geometry.

olecular orbital (MO) theory is an excellent conceptual model for the bonding in molecules. With everimproving computer processors, modern density functional theory,1 and software packages such as Gaussian2 or Spartan,3 it has become straightforward to do high-level calculations on complex molecules even in the undergraduate curriculum. However, due to this emphasis on computers, students are often unable to make “back-of-the-envelope” type predictions when presented with a new molecule in a seminar or during their research. This simple and rough MO theory is used by many practicing inorganic, organic, and organometallic chemists to quickly get an estimate of the frontier orbitals so they can predict, interpret, or rationalize chemical reactivity and bonding. In addition, students are often unable to interpret their computational results, especially for coordination complexes, because they do not know what they are looking for at the frontier energy level. Students typically study MO theory in the introductory and physical chemistry curricula, though a detailed description of the theory is usually limited to diatomic molecules. Making the jump to more complicated molecules, such as polyatomic maingroup compounds or transition-metal complexes, requires the explicit consideration of symmetry and group theory for the formation of symmetry adapted linear combinations (SALCs), also called ligand group orbitals (LGOs). This is often a difficult task for beginning students. For diatomics, one has only to consider the two component atoms that interact to © 2012 American Chemical Society and Division of Chemical Education, Inc.

Published: October 24, 2012 56

dx.doi.org/10.1021/ed300115t | J. Chem. Educ. 2013, 90, 56−62

Journal of Chemical Education

Article

Table 1. Electronic and Molecular Geometries,14 and Valence Bond Hybrid Orbitals for Main-Group Compounds with Increasing Number of Electron Groups around the Central Atom (n)

The technique described herein is based on that published by Verkade,5 but it has been simplified and made more approachable for students beginning their study of MO theory. The textbook by Purcell describes an approach to forming MO diagrams using a similar symmetry-based technique.6 The technique builds steadily through a variety of bonding models that students have seen in prior courses. Beginning with the Lewis structure and an appropriate VSEPR geometry, the LGOs are derived based on concepts from valence-bond theory. Students are quite confident in their abilities by the end of this four to five week process, which provides an MO foundation for the rest of the term. Students can describe the bonding of complicated molecules with a good understanding of not only the molecular nature of the orbitals, but also of the reactivity of a main-group compound or transition-metal complex.

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PART 1. BACKGROUND AND THE FOUNDATION OF THE METHOD Students typically learn Lewis structures and VSEPR in the first-year chemistry course and subsequently learn to deal with more complex bonding models in physical chemistry courses and more complex structures in organic chemistry courses. Lewis theory and VSEPR are reviewed, focusing on the breakdown of the theory for even simple molecules such as CO and O2,7 giving a rationale for continued work on MO theory. Lewis theory is described as useful to explain bonding, but that the simple theory does not always accurately predict chemical behavior, requiring the use of more complex theories. The MO diagram for H2+ is derived in detail using conventional quantum mechanical arguments. The wave function describing the bonding molecular orbital is shown to be identical to that obtained from valence bond theory.8 However, an important distinction in MO theory is the introduction of antibonding orbitals. By including the antibonding orbitals, the bonding in O2 and CO can be more accurately described. However, to describe the bonding in polyatomics, the simple interactions of atoms cannot be readily applied, and the use of symmetry and group theory is necessary. Terms (found in most general or inorganic chemistry textbooks)9−13 that students will use and apply for the construction of LGOs include the basic principles of VSEPR: molecular and electronic geometries,14 spn valence bond hybrid orbitals, the parent geometries for MLn molecules (n = 2−6, Table 1), and deviations from the ideal based on lone pair or bonding pair repulsions. The less common dmspn valence bond hybrids are not available in some textbooks and are required for some geometries (for example dz2 sp3 for trigonal bipyramidal geometry). Thus, a version of Table 1 is provided for student use.

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n

Electronic Geometry14

2 3

linear (D∞h) trigonal planar (D3h)

4

tetrahedral (Td)

4 5

Square planara (D4h) trigonal bipyramidal (D3h)

5

square pyramidala (C4v) octahedral (Oh)

6

6 7a 8a

trigonal prismatica (C3v) pentagonal bipyramidal (D5h) square antiprism (D4d)

Molecular Geometry linear (D∞h) trigonal planar (D3h) bent (C2v) Tetrahedral (Td) trigonal pyramidal (C3v) bent (C2v) square planar (D4h) trigonal bipyramidal (D3h) seesaw (C2v) t-shape (C2v) linear (D∞h) square pyramidal (C4v) Octahedral (Oh) square pyramidal (C4v) square planar (D4h) trigonal prismatic (C3v) pentagonal bipyramidal (D5h) square antiprism (D4d)

Hybrid Orbitals spz sp2 (s, px, py) sp3

dsp2 (dx2−y2) dsp3 (dz2)

dsp3 (dz2) d2sp3 (dz2, dx2−y2)

d2sp3 (dxz, dyz) d3sp3 (dx2−y2, dxy) d4sp3 (dz2, dxy, dxz, dyz)

a

Less common for main-group compounds, but useful to consider for transition-metal complexes.

two aspects of the character table are needed: the Mulliken symmetry labels, used simply as labels for orbitals, and the functions, which are needed for the formation of LGOs. Most inorganic textbooks introduce symmetry and character tables in sufficient detail for the use of this method;15−17 more advanced texts are available for students with interest.18−23 Generation of LGOs for Polyatomics

The most important and conceptually most difficult aspect of forming an MO diagram for a polyatomic is deciding what groups of orbitals will interact. For a diatomic, a single atom interacts with only another single atom, and the MO diagram can be constructed directly from the two components. For polyatomics, a single central atom interacts with all of the ligand atoms at once, resulting in a multicomponent problem. The problem is typically simplified using symmetry; the ligands are arranged into symmetrical groups (symmetry adapted linear combinations). The concept is relatively easy to apply to the σbonding interactions in main-group binary compounds such as BH3, SiCl4, PF6 or SF4; π-bonding is either treated separately or ignored. Hydrogen ligand atoms use their 1s orbitals for σbonding, whereas nonhydrogen ligand atoms are modeled to have a single lobe pointed at the central atom (easily modeled as an spn hybrid orbital). Graphical combinations of the bonding lobes are then created that match the symmetry of a derived generator function24 using nodal features to guide their formation using the following process outlined in Table 2. The process is described as layering on techniques that the students already know. The student draws a Lewis structure and assigns its geometry from VSEPR. Then, they determine the point group of the molecule and the appropriate orbitals that would be used from a valence bond approach. Next, the

PART 2. APPLICATION OF TECHNIQUE TO POLYATOMICS

Symmetry and Group Theory

To describe the bonding in polyatomic molecules, symmetry and group theoretical considerations are usually used to reduce the multicomponent problem to a two-component problem. Symmetry operations are introduced (as review for most of the students) at a level such that students are able to assign a point group to a molecule. Once assigned, the appropriate character table can be used to aid in the construction of the MO diagram. For the purposes of the technique developed in this article, only 57



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Table 2. Process for Forming Ligand Group Orbitals (LGOs) Step

Procedure

I II III

Draw a Lewis structure and assign VSEPR geometrya Assign a point group to the molecular geometry Determine the VB hybrid orbitals for the electronic geometry of the central atom Use the VB hybrid orbitals as generator functions Generate the LGOs by combining the ligand σ-orbitals (lobes) to obtain an orbital with the same symmetry as the generator function Assign symmetry labels to the LGOs from the character table by matching the label of the generator function

IV V VI a

For transition-metal complexes, use the Kepert model (see below).

functions of these orbitals (found in the character table) are used as generator functions to generate the LGOs, and finally, symmetry labels are assigned. The process is best seen with examples. The simplest case is a molecule without lone pairs, such as borane (BH3), where the molecular and electronic geometries are the same. Because there are three ligand hydrogen atoms, we need to generate three LGOs. The predicted VSEPR geometry (Table 1) is trigonal planar, in the D3h point group. The boron orbitals that would be used for bonding to the hydrogen atoms (using valence bond theory) are the sp2 hybrids derived from s, px, and py. Each orbital (or the corresponding function, [x2 + y2 + z2], and [x,y]) on the central atom is used as a generator function to generate an LGO with appropriate symmetry for bonding to the central atom. The LGO is generated by taking a linear combination of the three hydrogen 1s orbitals on the ligand atoms. Contributions of positive, negative, or zero are chosen such that the resulting LGO has the same symmetry properties as the generator function; this is done by inspection of the phase and nodal properties of the generator function. For BH3, an LGO of a1′ symmetry can be generated from the s orbital on boron. Just as the generator function, this LGO will have no nodes and will be all in-phase. It is obtained by summing the three s orbitals on the three hydrogen atoms. Next, two LGOs with e′ symmetry can be generated from the px and py orbitals on boron. Using the py generator function (with a node along the x axis), an LGO is obtained by summing two of the s orbitals in-phase and one of them out-of-phase; the node in the generator orbital is the same as the node in the LGO. The third LGO is generated by the px generator function and both contain a node along the y axis. It is obtained by summing one s orbital out-of-phase with respect to a second; the third s orbital does not contribute to the LGO as it lies in the nodal plane. The three resulting LGOs, shown in Figure 1, have the same symmetry as the generator functions s, px, and py. From the D3h character table, s is a1′, whereas (x,y) is e′. A pencast demonstrating this procedure is available.25 Dealing with lone pairs using this technique is slightly more complicated, and water is a good introductory example. Following the technique as above (Table 2), water is predicted to be bent, in the point group C2v. Water has four electron groups around it and its electronic geometry is tetrahedral; however, with only two ligands, students will generate only two LGOs. The valence bond hybrid orbitals appropriate for bonding in water are the sp3 hybrids, s, px, py, and pz, and two of these orbitals will be used as generator functions to guide the construction of LGOs. The complication for any molecule containing lone pairs is deciding which subset of the valence bond orbitals will be used

Figure 1. The a1′ and e′ generator functions and the resulting LGOs for bonding in BH3 in the D3h point group. This diagram emphasizes phase and nodal properties of the generator functions and the resulting LGOs with dotted lines and shading, respectively.

as generator functions. For the current technique, it suffices to use those that overlap with the bonded atoms (the two hydrogens, in this case) to generate the LGOs. The remaining functions are used to predict the symmetry of the resulting lone pairs. For water, either the s or the pz can be used as one of the generator functions as either can overlap with the ligand hydrogens. Depending on the axis system chosen, either the px or the py could be the second generator function (convention places the molecule in the yz plane, though many texts use the xz plane),18 while the other is used to predict the symmetry of the second lone pair. The net result is that there are two lone pairs and two LGOs. For students learning the technique, it is helpful to construct a table or decision tree showing the interaction of the generator functions with the molecule in question, as shown in Table 3. Table 3. Interaction of Generator Functions with Ligands or Lone Pairs as a Tool for Assigning the Symmetries of the LGOs in H2O (in yz plane) Can Interact with: Generator Function

Symmetry Label

Ligands

Lone Pairs

s pz px py

a1 a1 b1 b2

yes yes no yes

yes yes yes no

Based on inspection of the table, for water lying in the yz plane, the px orbital must correspond to a lone pair, while the py orbital must correspond to a ligand group orbital. One each of the s and pz orbitals (for the current purpose, the decision is arbitrary) corresponds to an LGO and a lone pair; the symmetry of each is a1. The two resulting LGOs, shown in Figure 2, have the same symmetry as s and py. From the C2v character table, s is a1 and py is b2. A final example, SF4, illustrates the case where the electronic geometry and molecular geometry of the molecule have different axis systems. For this molecule, with nonhydrogen ligands, we model the ligand atoms as having a lobe pointing toward the central atom (a p or sp3 hybrid orbital in the case of fluorine). The molecular geometry is see-saw with a C2v point group, but the molecule has a trigonal bipyramidal electronic 58



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lone pair using this technique, but instead predict that a lone pair of that symmetry will result when we construct the MO diagram. Many students with sufficient group theory backgrounds are comfortable using the reducible representation technique for the generation of LGOs, and students are encouraged to use the technique if they wish. However, the mathematical derivation of the LGOs often detracts from the simplicity and rapidity of the graphical method. Several other examples of LGO formation are given in the Supporting Information.

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PART 3. APPLICATION OF TECHNIQUE TO TRANSITION-METAL COMPLEXES Applying the technique to transition-metal complexes is in some ways easier than doing so for main-group compounds, as there are generally no longer stereochemically active lone pairs to consider. The geometries of the complexes are predicted by the Kepert model that considers only ligands,26 rather than VSEPR. The basic procedure is identical to that for main-group complexes (Table 2): given the geometry, use the VB hybrid orbitals as generators for the LGOs. Three examples, [Fe(CN)6]4‑ in Oh, and [CoCl4]2‑ in both Td and D4h, illustrate the process. In an octahedral complex, such as [Fe(CN)6]4‑, the molecular geometry is Oh, and valence bond theory predicts d2sp3 hybridization (Table 1). With six ligands, six LGOs are required, and the s, px, py, pz, dz2, and dx2−y2 orbitals are used as generator functions. It is simplest to assume that sp hybrid orbitals on each cyano ligand point toward the central iron atom. Combination of those six sp hybrid orbitals in an appropriate way by matching the phase and nodal properties of the generator functions results in the expected LGOs with a1g, t1u, and eg symmetry (Figure 4). The six LGOs thus constructed have the same symmetry as those of the six generator functions (not shown). In a similar way, the LGOs for a tetrahedral complex, [CoCl4]2‑ can be derived. The four generator functions are s, px, py, and pz. In the Td point group; the x, y, and z axes point between the ligand dihedrals. Considering sp3 hybrid orbitals on the four chlorines as the lobes pointing toward the central atom, the four generator functions generate four LGOs with a1 and t2 symmetry (Figure 5). As a final example, the same cobalt complex could be considered in the square planar geometry, with D4h symmetry. In this geometry, the generator functions are s, px, py, and dx2−y2. The resulting LGOs, with a1g, eu, and b1g symmetry are shown in Figure 6.

Figure 2. The a1, b1, and b2 generator functions, the two resulting LGOs of a1 and b2 symmetry, and the predicted symmetries of the lone pairs (a1 and b1) for H2O in the C2v point group.

geometry. Because valence bond theory predicts dsp3 hybridization using the dz2 orbital (Table 1), the orbitals that are used as generator functions are s, px, py, p,z, and dz2. To assign the symmetry of the resulting LGOs, use the VB hybrid orbitals from the electronic geometry in the orientation of the molecular geometry (C2v in this example). The decision tree for lone pair assignment for SF4 is given in Table 4; the px and Table 4. Interaction of Generator Functions with Ligands or Lone Pair as a Tool for Assigning the Symmetries of the LGOs in SF4 Can Interact with: Generator Function

Symmetry Label

Ligands

Lone Pairs

s pz px py dz2

a1 a1 b1 b2 a1

yes yes yes yes yes

yes yes no no yes

py orbitals must be used to generate LGOs, while two of the three remaining a1 symmetry orbitals are used as generator functions. Two a1 LGOs are required for SF4, and it is arbitrary which two of the three a1 generator functions are used; for simplicity, the s and pz generator functions are used to generate the LGOs (Figure 3). Again, as the case for water, although the symmetry of the lone pair is described, we do not generate a

Figure 3. The 3 a1, b1, and b2 generator functions, the three resulting LGOs of a1 and b1 and b2 symmetry, and the predicted symmetries of the lone pair (a1) for SF4 in the C2v point group. 59



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LGOs for similar molecules. The tradeoff of reduced accuracy in the bonding picture is made up for by the rapidity of the derivation. The student can quickly predict the frontier orbital set of a compound under study, and follow up with more detailed calculations using a computer program if necessary. Although the technique is easy to apply and useful for the quick preparation of qualitatively correct MO diagrams, it is not without flaws. First, although the procedure for selecting generator functions for main-group compounds with lone pairs works, students do not understand what it means to predict the symmetry of a lone pair until later when we construct MO diagrams.29 Students are shown the results from computations on molecules such as water and SF4 to show them that their predictions of lone pair symmetry are correct, even if the shape of the lone pair is not familiar (see Supporting Information). Second, the technique does not accurately predict the correct “lobe size” for the e′ orbitals in BH3 and similar molecules. Those students with interest can derive the correct coefficients by recalling that the squares of the coefficients of the lobes sum to unity. This is not necessary for a qualitative bonding picture, and the lack of coefficients does not interfere with the ability to generate LGOs with the correct symmetry and symmetry label. Third, although π bonding can be successfully layered on top of this technique, generation of π LGOs by this technique is not straightforward. However, during construction of an MO diagram, once σ bonding is “out of the way,” the orbitals left on the ligands that can overlap with the central atom in a π fashion are relatively easy to see. Fourth, the technique does not address issues of orbital mixing (s−p, s−d, p−d) at the central atom, although it could be readily modified to include this idea (see Supporting Information for an example of including sp mixing in water). Finally, the focus of this technique is on its simplicity, and as such, it cannot be used predictively. It is possible to generate LGOs for any structure, even if it is not the lowest-energy geometry. Early in the development of the technique, the terminology of Verkade was used, and the generator functions were called “generator orbitals”. However, this sometimes led to confusion. Some students assumed that the central atom needed to have the necessary orbital present in its valence shell to be used as a generator. For example, in SF4 (Figure 3), students would not always derive the correct LGOs. As the course emphasizes that sulfur does not have valence d orbitals,30−33 students would only consider generator orbitals based on s and p orbitals and would either obtain three LGOs and a lone pair or four LGOs and no lone pair. Similarly, in square planar methane (see Supporting Information), students would only derive the three LGOs based on the s and p orbitals, neglecting to derive the fourth derived from the dz2 orbital. This student confusion has largely gone away with the change in terminology. In the interest of learning more about student performance, student learning was assessed over a multiyear period using the ACS standardized exam in inorganic chemistry. This exam is a 60 question multiple choice exam given in a 100 min period.34 Questions related to MO theory (7 questions) were selected, and the percentage correct in six student cohorts (Figure 7) was plotted. Because of the difficulty of the exam (39 correct is in the 91st percentile) and the inexact match between the course and the standardized exam, a performance above 60% on a subsection of the ACS exam is considered to be acceptable and a performance above 80% is superior. The molecular orbital theory topic, which was taught every year, underwent significant revision and updating between 2001

Figure 4. LGOs of a1g, t1u, and eg symmetry, derived from s, (px, py, pz) and (dx2−y2, dz2) generator functions for [Fe(CN)6]3‑ in the Oh point group.

Figure 5. LGOs of a1 and t2 symmetry, derived from s, and (px, py, pz) generator functions for [CoCl4]2‑ in the Td point group.

Figure 6. LGOs of a1g, eu, and b1g symmetry, derived from s, px, py, and dx2−y2 generator functions for [CoCl4]2‑ in the D4h point group.

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PART 4. ASSESSMENT OF THE TECHNIQUE After three weeks of lectures, in-class exercises,27 and several problem set questions, students are well-equipped to generate LGOs (and their corresponding MO diagrams) using this graphical approach. They generate LGOs for essentially any main-group, organometallic or coordination complex and almost all students are able to do so on examinations with little difficulty. A report from this Journal demonstrated the construction of molecular orbital diagrams for molecules lacking a center of symmetry by using the fragment MO approach and a traditional approach to forming LGOs.28 Students can apply the generator function technique to derive 60



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friend. I am indebted to my Chem 104 students, especially those that asked probing questions as I developed the method.

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(1) Koch, W.; Holthausen, M. C. A Chemist’s Guide to Density Functional Theory, 2nd ed.; Wiley-VCH: New York, 2001. (2) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A.; Jr..; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö .; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Revision A.1; Gaussian, Inc.: Wallingford, CT, 2009. (3) Spartan ’10; Wavefunction, Inc.: Irvine, CA, 2010. (4) Housecroft, C. E.; Sharpe, A. G. Inorganic Chemistry, 3rd ed.; Pearson Education Limited: Harlow, U.K., 2008; p 122. (5) Verkade, J. G. A Pictorial Approach to Molecular Bonding and Vibrations; Springer-Verlag: New York, 2002. (6) Purcell, K. F.; Kotz, J. C. Inorganic Chemistry; Saunders College Publishing: Philadelphia, PA, 1977; pp 163−182. (7) Following the rules for drawing Lewis structures, the best structure for CO has only a double bond between the atoms (forming the third bond breaks two common rules, requiring the formation of formal charge from a neutral species and putting a positive formal charge on oxygen, the more electronegative element). For O2, the Lewis structure predicts a diamagnetic ground state, whereas the true ground state is a triplet diradical. (8) Karplus, M.; Porter, R. N. Atoms and Molecules; Benjamin/ Cummings: Reading, MA, 1970; pp 304−305. (9) Housecroft, C. E.; Sharpe, A. G. Inorganic Chemistry, 3rd ed.; Pearson Education Limited: Harlow, U.K., 2008; pp 88−99. (10) Miessler, G. L.; Tarr, D. A. Inorganic Chemistry, 4th ed.; Prentice Hall: New York, 2011. (11) Atkins, P.; Overton, T.; Rourke, J.; Weller, M.; Armstrong, F. Shriver and Atkins’ Inorganic Chemistry, 5th ed.; Oxford University Press: Oxford, 2009. (12) Oxtoby, D. W.; Gillis, H. P.; Campion, A. Principles of Modern Chemistry, 7th ed.; Thompson Brooks/Cole: Belmont, CA, 2012. (13) Laird, B. University Chemistry, McGraw-Hill: New York, 2009. (14) The concept I call electronic geometry is defined differently by different authors and is sometimes called the shape of the electron groups around the central atom or the electron group scaffolding predicted by VSEPR. (15) Shriver, D.; Atkins, P. Inorganic Chemistry, 3rd ed.; W. H. Freeman and Company: New York, 1999; pp 117−129. (16) Housecroft, C. E.; Sharpe, A. G. Inorganic Chemistry, 3rd ed.; Pearson Education Limited: Harlow, U.K., 2008. (17) Miessler, G. L.; Tarr, D. A. Inorganic Chemistry; Pearson Prentice Hall: Upper Saddle River, NJ, 2004; pp 76−102. (18) Cotton, F. A. Chemical Applications of Group Theory, 3rd ed.; Wiley-Interscience: New York, 1990. (19) Kettle, S. F. A. Symmetry and Structure, 2nd ed.; John Wiley and Sons: New York, 1995. (20) Vincent, A. Molecular Symmetry and Group Theory, 2nd ed.; John Wiley and Sons: New York, 2001. (21) Carter, R. L. Molecular Symmetry and Group Theory, John Wiley and Sons: New York, 1998. (22) Bishop, D. M. Group Theory and Chemistry; Dover: New York, 1973.

Figure 7. Plot of student performance on the molecular orbital theory questions on the ACS standardized exam (number of students by year: 2001 (24), 2002 (17), 2003 (20), 2004 (10), 2005 (15), and 2011 (16)).

and 2003, but has remained essentially unchanged since that time. The performance by students on the molecular orbital theory portion of the exam showed steady improvement during the years that the generator function method was developed, rising to above 60% in 2003 and staying there through 2011. The results show that the students have an acceptable understanding of MO theory using the methods described herein.

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CONCLUSIONS The generator function technique is a simple and straightforward method for deriving LGOs and thus the MO diagrams, for main-group polyatomics and transition-metal complexes. The intuition gained by the techniques allows students to more deeply understand the results of high-level quantum chemical calculations, but is easy enough to be drawn on the back of an envelope. Students regularly derive quick MO diagrams during seminars, and several students have commented that this approach made it easy for them to do so.

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ASSOCIATED CONTENT

* Supporting Information S

Several additional examples of deriving LGOs using this technique; an example of incorporating sp mixing at the central atom; several sample MO diagrams that illustrate lone pairs and their symmetry prediction; graphical output from Gaussian calculations on water and SH4; a rough schedule showing how the technique is staged; and a student handout. This material is available via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS My interest in MO theory and inorganic chemistry was sparked by Martin Ackermann, Oberlin College; he is a true mentor and 61



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(23) Harris, D. C.; Bertolucci, M. D. Symmetry and Spectroscopy; Dover: New York, 1978. (24) Verkade calls these generator orbitals, a terminology which students often find confusing, as described in the Assessment section. (25) Johnson, A. R. Generating LGOs and constructing MO diagramspencast. https://www.ionicviper.org/class-activity/ generating-lgos-and-constructing-mo-diagrams-pencast (accessed Oct 2012). (26) Housecroft, C. E.; Sharpe, A. G. Inorganic Chemistry, 3rd ed.; Pearson Education Limited: Harlow, U.K., 2008; pp 620−621. (27) Johnson, A. R. Generating LGOs (SALCs). https://www. ionicviper.org/class-activity/generating-lgos-salcs (accessed Oct 2012). (28) Cass, M. E.; Hollingsworth, W. E. J. Chem. Educ. 2004, 81, 997− 1005. (29) Calculated molecular orbitals for several molecules with lone pairs are included in the Supporting Information. (30) Burdett, J. K. Chemical Bonds: A Dialog; John Wiley & Sons: New York, 1997. (31) Mitchell, T. A.; Finocchio, D.; Kua, J. J. Chem. Educ. 2007, 84, 629. (32) Coleman, W. F. J. Chem. Educ. 2010, 87, 999−910. (33) Suidan, L.; Badenhoop, J. K.; Glendening, E. D.; Weinhold, F. J. Chem. Educ. 1995, 72, 583−586. (34) ACS − Division of Chemical Education − Examinations Instutute − About Us. http://chemexams.chem.iastate.edu/about/ index.cfm (accessed Oct 2012).

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Construction of Ligand Group Orbitals for Polyatomics and Transition

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