Article

Cite This: J. Chem. Educ. 2019, 96, 2467−2475 pubs.acs.org/jchemeduc

Representation of Three-Center−Two-Electron Bonds in Covalent Molecules with Bridging Hydrogen Atoms Gerard Parkin*

Department of Chemistry, Columbia University, New York, New York 10027, United States

*S Supporting Information

ABSTRACT: Many compounds can be represented well in terms of the two-center−two-electron (2c−2e) bond model. However, it is well-known that this approach has limitations; for example, certain compounds require the use of three- center−two-electron (3c−2e) bonds to provide an adequate description of the bonding. Although a classic example of a compound that features a 3c−2e bond is provided by − fi diborane, B2H6,3c 2e interactions also feature prominently in transition metal chemistry, as exempli ed by bridging hydride compounds, agostic compounds, dihydrogen complexes, and hydrocarbon and silane σ-complexes. In addition to being able to identify the different types of bonds (2c−2e and 3c−2e) present in a molecule, it is essential to be able to utilize these models to evaluate the chemical reasonableness of a molecule by applying the octet and 18-electron rules; however, to do so requires determination of the electron counts of atoms in molecules. Although this is easily achieved for molecules that possess only 2c− 2e bonds, the situation is more complex for those that possess 3c−2e bonds. Therefore, this article describes a convenient approach for representing 3c−2e interactions in a manner that facilitates the electron counting procedure for such compounds. In particular, specific attention is devoted to the use of the half-arrow formalism to represent 3c−2e interactions in compounds with bridging hydrogen atoms. KEYWORDS: Second-Year Undergraduate, Graduate Education/Research, Inorganic Chemistry, Analogies/Transfer, Problem Solving/Decision Making, Covalent Bonding, Lewis Structures, Main-Group Elements, Organometallics, Transition Elements

■ INTRODUCTION deficient”5,7 to indicate that there is an insufficient number of 1,2 2a,3 electrons for all bonds to be 2c−2e. The octet and 18-electron rules, together with the “ fi ” construction of so-called “Lewis structures”,4 are of pivotal The structures of so-called electron de cient compounds importance because they provide a rudimentary means to may, however, be rationalized by the use of multicenter − evaluate the chemical reasonableness of a molecule. Central to bonding descriptions, of which the three-center two-electron − 8−10 the application of these rules is the necessity of determining (3c 2e) model is the simplest. In view of this explanation the electron count (or electron number, EN) of the atoms in Downloaded via SAINT EDWARD'S UNIV on July 28, 2020 at 17:00:55 (UTC). the molecule of interest. For many molecules (e.g., H2 and CH4), the bonding can be described well in terms of a two- − −

See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. center two-electron (2c 2e) model, and the electron counts of atoms in such molecules are typically easily evaluated from the Lewis structure. In contrast, the bonding in some molecules cannot be described by using only 2c−2e bonds, as exemplified by those with bridging hydrogen atoms, of 5,6 which diborane, B2H6 (Figure 1), is an excellent example. For example, whereas ethane, C2H6, has a total of 14 valence electrons derived from two carbon atoms and six hydrogen Figure 1. Comparison of the structures of C2H6 and B2H6.C2H6 has 14 valence electrons, which is sufficient to form seven 2c−2e bonds, atoms (i.e., (2 × 4) + (6 × 1)) and can hence possess seven whereas B2H6 has only 12 valence electrons and can only form a − − 2c 2e bonds to give rise to the familiar structure shown in maximum of six 2c 2e bonds; therefore, B2H6 cannot adopt the same Figure 1,B2H6 has a total of only 12 valence electrons (i.e., (2 structure as C2H6, and multicenter bonding is required to rationalize × 3) + (6 × 1)) and can therefore only support a maximum of its structure. The dotted lines connecting the bridging hydrogen and six 2c−2e bonds; as such, the number of electrons is boron atoms are intended to indicate that these are not 2c−2e bonds. ffi insu cient for B2H6 to adopt a structure akin to that of ff C2H6. Therefore, B2H6 adopts a di erent type of structure in Received: August 10, 2019 which two of the hydrogen atoms bridge the two boron centers. Revised: September 7, 2019 Compounds of this type are often referred to as “electron Published: October 9, 2019

© 2019 American Chemical Society and Division of Chemical Education, Inc. 2467 DOI: 10.1021/acs.jchemed.9b00750 J. Chem. Educ. 2019, 96, 2467−2475 Journal of Chemical Education Article

Figure 4. Two alternative representations for a dative bond, as ff illustrated for H3NBH3. Despite the di erent appearances, the representations are not resonance structures because the electrons are in the same location (recall that resonance structures differ in the locations of the electrons).

■ REVIEW OF THE COVALENT BOND CLASSIFICATION AND THE REPRESENTATION OF 2C−2E BONDS Prior to discussing 3c−2e interactions, it is first pertinent to review the representation of 2c−2e bonds. The molecular σ Figure 2. Molecular orbital diagram for H2. The bonding orbital ( ) orbital description of a 2c−2e bond is straightforward: the is occupied by two electrons, so each hydrogen is assigned a duet overlap of two atomic orbitals on adjacent atoms results in the configuration. The + and − symbols on the orbitals refer to their relative phases, as do their colors. formation of bonding and antibonding orbitals, of which the former is occupied by two electrons, as illustrated by H2 (Figure 2). However, although this is a common feature of all of the bonding, Rundle once quipped, “There are no such 2c−2e bonds, there are two fundamentally different types of things as electron deficient compounds, only theory deficient covalent bond that are differentiated by the number of chemists.”11,12 electrons that each partner contributes to the interaction However, although the bonding in “electron deficient” (Figure 3). Thus, a normal covalent bond is one in which each compounds such as B2H6 may be understood by the neutral partner contributes one electron to the bonding application of theoretical methods, this approach lacks the orbital,14 whereas a dative covalent (or coordinate) bond is convenience of using simple electron counting procedures to one in which a single atom provides both electrons.15 evaluate the chemical reasonableness of a covalent molecule. Common examples of species that provide one electron to a Unfortunately, the electron counts for atoms in molecules with bond are H, Cl and CH3 (i.e., radicals), whereas species that − 3c 2e interactions are not as easily determined as those with provide two electrons are illustrated by H2O, NH3,PR3, and 2c−2e interactions because (i) the structural representations CO (i.e., neutral closed shell molecules with lone pairs). employed often lack clarity, and (ii) there is also the need to Normal covalent and dative covalent bonds are represented evaluate the electron count of more than one atom. The by different symbols. Thus, a normal covalent bond is depicted purpose of this article is, therefore, to describe how to as a line between two atoms, whereas a dative covalent bond is represent the bonding in molecules with 3c−2e bonds in such represented as an arrow with the head pointing to the acceptor a manner that it is simple to evaluate the electron count of atom (Figure 4). The latter can also be denoted by an each atom that is involved in the interaction, and thereby equivalent form that uses a line but also depicts formal charges ff 16,17 utilize e ectively the octet and 18-electron rules for classes of on the atoms, as illustrated for H3NBH3 (Figure 4). It molecules such as bridging hydride, dihydrogen, and agostic must be emphasized that the two depictions of the dative complexes, which are often encountered in undergraduate covalent bond are not resonance structures because the inorganic chemistry courses.13 electrons are in the same location (i.e., between N and B;

Figure 3. Covalent bond classification (CBC) of L, X, and Z ligands. Each partner provides one electron for a normal covalent bond (b), whereas a single partner provides both electrons for a dative covalent bond (a, c).

2468 DOI: 10.1021/acs.jchemed.9b00750 J. Chem. Educ. 2019, 96, 2467−2475 Journal of Chemical Education Article

Figure 4); as such, the two depictions are just different dative covalent bond is provided by the ammonia−borane → fi representations of the same bonding description, and the form adduct, H3N BH3, in which the boron is classi ed as MLX3 fi that is used is often just a matter of convenience. and the nitrogen is classi ed as MX3Z. Thus, both the boron The covalent bond classification18 provides a means to (i.e., EN = 3 + 2 + (3 × 1) = 8) and nitrogen (i.e., EN = 5 + (3 categorize covalent molecules and, for this approach, ligands × 1) = 8) possess octet configurations. Alternatively, a are classified according to the number of electrons that they procedure that is often more convenient for compounds with contribute to the bonding orbital. Thus, a neutral two-electron multidentate ligands is to determine the electron count by the donor is classified as L, a one-electron donor is X, and a zero- expression EN = m + Σ(ligand electrons) − Q±, where electron donor is Z (Figure 3). Σ(ligand electrons) is the sum of the contributions of Many ligands possess more than one L, X or Z function, or individual ligands. For example, the electron count for fi × × combination thereof, and in such cases, the ligand is classi ed Cp2TaH3 is EN = 5 + (2 5) + (3 1) = 18; likewise, fi as [LlXxZz], where l, x, and z are the respective numbers of L, Cp2W(CO) also has an 18-electron con guration (Figure 7). X, and Z functionalities. For example, an η6-benzene ligand is fi “ ” classi ed as L3 on the basis that each of the double bonds may be conceptually viewed as an L donor akin to that of C2H4 (Figure 5). Correspondingly, an η5-cyclopentadienyl ligand is

Figure 7. Alternative procedure for determining the electron counts (EN) for atoms in some compounds with multidentate ligands (EN = m + Σ(ligand electrons) − Q±). fi Figure 5. [LlXxZz] classi cations of two common multidentate fi ligands. A cyclopentadienyl ligand is classi ed as L2X because the Lewis structure contains two double bonds (L) and an alkyl moiety ■ MOLECULAR ORBITAL DESCRIPTION OF 3C−2E fi (X), whereas a benzene ligand is classi ed as L3. INTERACTIONS AND STRUCTURE−BONDING REPRESENTATIONS fi classi ed as L2X on the basis that the cyclopentadienyl radical The simplest molecule that possesses a 3c−2e interaction is + may be conceptually considered to be composed of two [H3] , which has been spectroscopically observed but not “double bonds” (each of which is an L donor) and an alkyl isolated. Both linear and triangular structures may be radical (which is an X donor). In terms of overall electron considered, and the molecular orbitals for the two forms, count, cyclopentadienyl and benzene are respectively consid- which are related by an orbital correlation diagram, are ered to be five-electron and six-electron donors (Figure 5). illustrated in Figure 8.20 Since the system contains a single pair The electron count about an atom (M) in a molecule of the of electrons, the preferred structure is determined only by the Q± type [MLlXxZz] , where l, x, and z are the respective numbers occupancy of the fully bonding orbital (i.e., all orbitals in of L, X, and Z functions, and Q± is the charge on the molecule, phase), which is lowest in energy for the triangular form is, therefore, given by the expression EN = m +2l + x − Q±, because it maximizes the overlap of the three hydrogen 1s where m is the number of valence electrons on a neutral M atom. For the transition metals in groups 3−11,19 m corresponds to the Periodic Table group number, (e.g., Sc, 3; Ti, 4; V, 5; etc.), whereas for the main group elements in groups 12−18, the value of m is the last digit. As an illustration of the determination of the electron count, fi the carbon atom in CH4 is classi ed as MX4 and therefore possesses an octet configuration (i.e., EN = 4 + (4 × 1) = 8, Figure 6). A simple example of a molecule that possesses a

+ Figure 8. Molecular orbitals for linear and triangular [H3] . Each Figure 6. Procedure for determining the electron counts (EN) for hydrogen possesses a duet configuration. The + and − symbols on the atoms in some simple compounds (EN = m +2l + x − Q±). orbitals refer to their relative phases, as do their colors.

2469 DOI: 10.1021/acs.jchemed.9b00750 J. Chem. Educ. 2019, 96, 2467−2475 Journal of Chemical Education Article

represented by the structures shown in Figure 9. Evaluation of each hydrogen atom in turn illustrates how the duet fi con guration may be derived. Thus, HA and HB have a duet configuration because they share a pair of electrons, whereas fi + HC has a duet con guration because HC (with no electrons) − shares the pair of electrons in the HA HB bond. This approach for representing a 3c−2e interaction can also be used if the three atoms are not part of an equilateral triangle such that the interactions between all pairs of atoms are not equal. However, an alternative way to represent this bonding situation is to employ the so-called “half-arrow” notation that Figure 9. (a) Connectivity and (b) three equivalent structure− was introduced in 1980 by Green for bridging hydride + 21,22 bonding representations of [H3] . compounds (Figure 10). With this notation, a half-arrow is drawn from a central bridging hydrogen atom to an outer atom. This representation is very similar to the well-known use of a full-arrow for bridging halide compounds (e.g., Al2Cl6)to indicate the donation of a lone pair of electrons (Figure 10); the purpose of using a half-arrow (rather than a full-arrow) for the bridging hydrogen atom is to emphasize that the arrow does not correspond to donation of a lone pair but rather donation of a bond pair.22 Otherwise, there is no difference in the impact of a half-arrow or full-arrow on the electron count − of the outer atom. Figure 10. Half-arrow representation for a single 3c 2e interaction It is important to emphasize that the two representations for for a bridging hydride compound (top) and comparison with the − representation for a bridging chloride ligand that has two 2c−2e the 3c 2e interaction (i.e., a full-arrow from the center of the bonds because of the availability of lone pairs on chlorine (bottom). bond and a half-arrow from the central atom) are intended to convey the same information with respect to electron counting purposes, and the form that is used is simply a matter of orbitals. Since the pair of electrons in the bonding orbital of + convenience, similar to the manner in which a dative covalent [H3] is associated with all of the hydrogen atoms, each → +− − hydrogen atom possesses a duet configuration; this situation is bond can represented as either A BorA B . For example, the full-arrow from the center of the bond is most commonly analogous to both hydrogen atoms in H2 being assigned a duet configuration because they share a pair of electrons (Figure 2). employed if the angle at the central atom is distinctly nonlinear and the two “outer” atoms are considered to be within a Thus, with respect to electron counting, an important feature “ ” 23 of a 3c−2e interaction is that each atom is assigned a shared bonding distance (i.e., a closed system ), whereas the half- pair of electrons. It should be noted that the same concept arrow representation is used if the angle at the central atom is + significantly obtuse and there is little interaction between the would apply if the linear form of [H3] (or a bent form with an 24 H−H−H angle not equal to 60°) were to be more stable (i.e., outer atoms. each atom would still be assigned a duet configuration). Rather than having to construct an MO diagram to ■ EXAMPLES OF 3C−2E INTERACTIONS − determine the electron counts for atoms involved in a 3c 2e Molecules that feature 3c−2e interactions are rather common, − interaction, however, it is more convenient to use structure and the application of the above structure−bonding fi bonding gures that allow one to evaluate the electron counts. representations to these molecules is illustrated below by + Consider, for example, [H3] being derived from two hydrogen consideration of three classes of compounds, namely, (i) + atoms (HA and HB) and a proton (HC ), as illustrated in the bridging hydride compounds, (ii) agostic compounds, and (iii) + 13 center structure of Figure 9. In valence bond terms, the [H3] dihydrogen compounds. + molecule can be viewed as an adduct of H2 and H , which differs from a conventional adduct in the sense that the donor Bridging Hydride Compounds pair of electrons is derived from an H−H bond rather than a Bridging hydride compounds constitute the largest class of − fi lone pair on a single atom. The bonding can, therefore, be molecules with 3c 2e interactions, as exempli ed by B2H6,as

− − − − Figure 11. Representations of the 3c 2e B H B interaction in diborane. Coordination of BH3 by a B H bond of another molecule allows boron to achieve an octet configuration. The 3c−2e interaction can be represented by either an arrow from the center of the B−H bond or an equivalent form in which a half-arrow is drawn from the hydrogen atom.

2470 DOI: 10.1021/acs.jchemed.9b00750 J. Chem. Educ. 2019, 96, 2467−2475 Journal of Chemical Education Article

means to determine whether any additional direct bonding exists between the outer A atoms. For example, if A was a transition metal, an apparent electron count of EN would indicate that each A atom would require an additional (18 − − Figure 12. Often encountered incorrect representation of B2H6 that EN) A A bonds in order to achieve an 18-electron appears to depict two 2c−2e bonds to hydrogen. configuration, because each A−A bond would increase the electron count by one. As an illustration, the two rhenium μ noted above. The bonding in B2H6 may be considered in terms atoms of [CpReH2]2( -H)2 would possess 17-electron of its formation by dimerization of BH3. Thus, whereas the configurations, and so a direct Re−Re bond is required to fi boronatominBH3 has only a sextet con guration, achieve an 18-electron configuration (Figure 14). In contrast, − μ fi coordination by the pair of electrons in a B H bond of a [(CO)4Re]2( -H)2 possesses an 18-electron con guration for second molecule results in an octet configuration (Figure 11). each rhenium atom, and so no additional Re−Re bond is The half-arrow representation provides a very convenient required (Figure 14). means to evaluate the electron count of each boron center. It A good illustration of the use of the half-arrow to represent − also provides a clear depiction of the 3c 2e nature of the the structure of bridging hydride compounds is provided by interaction, in contrast to representations that simply draw two μ 25 Os3(CO)10( -H)2, which has been depicted in the literature by lines from each hydrogen atom, which can thereby give the a variety of different structures that have ambiguous bonding misleading impression that hydrogen is forming two normal descriptions, with Os−Os bond orders that vary from zero to covalent bonds (Figure 12). two (Figure 15). The ambiguity is also highlighted by the fact Transition metal compounds also feature bridging hydride that some representations simply utilize ill-defined dotted lines ligands, and some examples are presented in Figures 13 and 26 μ − to represent the bonding. The half-arrow representation, 14. For example, [(CO)5W( -H)W(CO)5] can be depicted by the two structure−bonding representations shown in Figure however, clearly indicates that each osmium center associated with the bridging hydrogens adopts an 18-electron config- 13, in which both tungsten centers possess 18-electron − configurations. uration without requiring an additional Os Os bond. Significantly, this description is in accord with theoretical calculations, which indicate that there is no OsOs double bond and that the interaction is best represented as two 3c−2e interactions.27 With respect to the origin of the different bonding μ representations of Os3(CO)10( -H)2 in Figure 15,itis pertinent to note that the derivation of an OsOs double bond is a consequence of essentially apportioning only “half of an electron” of a bridging hydrogen atom to each metal center.28 The problem with this approach can be readily μ Figure 13. Half-arrow bonding representations for [(CO)5W( - ascertained by recognizing that this method would also predict − −  H)W(CO)5] . The 3c 2e interaction allows each tungsten center to aB B double bond in B2H6 in order for boron to achieve an possess an 18-electron configuration without requiring an M−M octet configuration, which is clearly not feasible. More bond. generally, this approach predicts the number of metal−metal bonds (m#) for transition metals by using the formula m# = (18n − N)/2, where n is the number of M atoms, and N is the total valence electron count for the molecule. However, this simple formula does not take into account the 3c−2e nature of bridging hydride interactions, and so the predicted M−M bond order is greater than that determined by computational analysis; in fact, the bond order predicted by the formula does not actually depend on whether the hydrogen atoms are terminal or bridged, which is clearly not a satisfactory situation. In contrast, the half-arrow representation provides a bonding description that is much more in accord with that predicted theoretically. In addition to the above compounds that feature hydrogen bridging between two boron centers and between two μ transition metals centers, hybrid compounds are known in Figure 14. Half-arrow bonding representations for [(CO)4Re]2( - μ − H)2 and [CpReH2]2( -H)2. Whereas the two 3c 2e interactions in which hydrogen bridges boron and a metal. For example, there μ [(CO)4Re]2( -H)2 allow the rhenium centers to achieve an 18- is a large variety of transition metal borohydride compounds in fi μ 1 2 electron con guration, [CpReH2]2( -H)2 achieves only a 17-electron which the metal and boron are bridged by one (κ ), two (κ ), configuration; therefore, a Re−Re bond is required to obtain an 18- or three (κ3) hydrogen atoms (Figure 16).29 The bonding in fi electron con guration for the latter. these compounds may likewise be described in terms of the half-arrow representations, which provides a convenient means One aspect of evaluating the electron count of atoms to evaluate the electron count of a metal center, as illustrated − − − κ2 involved in an A H A3c 2e interaction is that it provides a for Cp2Nb( -H2BH2)inFigure 17.

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μ Figure 15. Structural representations for Os3(CO)10( 2-H)2. Only E and F provide good representations of the bonding.

interaction is provided by the observation that the titanium ethyl compound, (dmpe)TiEtCl3, exhibits a very unusual acute Ti−C−C bond angle of 85.9° because of the interaction of the titanium with the β-C−H bond. This interaction is a consequence of the titanium center of (dmpe)TiEtCl3 being Figure 16. Coordination modes of metal borohydride compounds. highly electron deficient (with a formal electron count of 12 in the absence of an interaction), and the electronic unsaturation is partially relieved by interaction of the titanium center with the electron density associated with a C−H bond. Dihydrogen Compounds and Other σ-Complexes Dihydrogen compounds compose an extremely interesting class of molecules, first discovered by Kubas,31 in which the − H H bond of H2 remains intact upon coordination to a metal η2 ’ center, [M]( -H2), as illustrated in Figure 19. Prior to Kubas s Figure 17. Example of a κ2-borohydride complex. discovery, the only known interaction between H2 and a metal center was oxidative addition resulting in a dihydride Agostic Compounds compound in which the H−H bond has been cleaved, − An interesting class of compounds is one in which there is an [M]H2. The 3c 2e interaction can be viewed as donation of interaction between a C−H bond and a metal center. Such the pair of electrons associated with the H−H bond into an compounds, and their interactions, are referred to by the term empty orbital on the metal center (cf. [H ]+); for this reason, “ ” 13,30 3 agostic . Agostic interactions are recognized to be an dihydrogen compounds are typically represented by drawing important feature of organometallic chemistry, and they can an arrow from the center of the H−H bond to the metal center play important roles in modifying the structure and stability of (Figure 20). a molecule in which the metal center would otherwise be fi The ability to isolate a dihydrogen compound, rather than a electron de cient. Classic examples of compounds that possess dihydride derivative, requires that there be little backbonding agostic interactions are provided by the bis- from the metal center into the H−H σ* orbital, because such (dimethylphosphino)ethane titanium alkyl compounds transfer of electron density promotes cleavage of the H−H (dmpe)TiMeCl3 and (dmpe)TiEtCl3, in which the titanium centers interact, respectively, with α- and β-C−H bonds bond. For this reason, dihydrogen compounds are typically (Figure 18).30 The significance of the agostic C−H−M associated with metal centers that have ligands that remove electron density via backbonding (e.g., CO).32

Figure 18. Examples of compounds that feature α- and β-agostic Figure 19. Dihydrogen and dihydride tautomers. The former interactions. Coordination of the C−H bond to Ti increases the possesses one 3c−2e interaction, whereas the latter possesses two electron count by two. 2c−2e interactions.

2472 DOI: 10.1021/acs.jchemed.9b00750 J. Chem. Educ. 2019, 96, 2467−2475 Journal of Chemical Education Article

Figure 20. Examples of transition metal dihydrogen complexes. The dihydrogen molecule donates its bonding electrons to the metal to − fi Figure 22. Example of a bridging methyl compound with 3c 2e result in 18-electron con gurations. interactions. Dihydrogen complexes are a subset of a class of molecules referred to as σ-complexes, so named because the compounds compounds, which include bridging hydride and agostic are considered to be derived by donation of the electron complexes, and furthermore provides a means to evaluate density of an X−H σ-bond to a metal. Specific examples are metal−metal bond orders in bridging hydride compounds. provided by hydrocarbon33 and silane34 adducts, in which − − C H and Si H bonds interact with a metal center. However, ■ ASSOCIATED CONTENT such compounds are much less commonly encountered than *S Supporting Information dihydrogen complexes. Alkane σ-complexes are particularly unstable but are often invoked as intermediates in reactions The Supporting Information is available on the ACS involving the oxidative-addition and reductive-elimination of Publications website at DOI: 10.1021/acs.jchemed.9b00750. C−H bonds. By comparison, silane σ-complexes are more Worksheets to help students practice the concepts stable than alkane σ-complexes, as illustrated by (PDF) η2 Mo(dppe)2( -HSiH2Ph), which has been isolated in the Worksheet answers and comments for instructors (PDF) solid state. Since the 3c−2e interactions are asymmetric, with longer M−C and M−Si bonds than M−H bonds, these ■ AUTHOR INFORMATION compounds are frequently represented by using either the half- Corresponding Author arrow or full-arrow representations, although both are intended * to convey the same information with respect to electron E-mail: [email protected]. counting purposes (Figure 21). ORCID Gerard Parkin: 0000-0003-1925-0547 Notes The author declares no competing financial interest. ■ ACKNOWLEDGMENTS The National Science Foundation (CHE-1465095) is thanked for support.

Figure 21. Two alternative representations of silane σ-complexes, ■ REFERENCES both of which convey the same information with respect to the (1) (a) Lewis, G. N. Valence and The Structure of Atoms and purpose of electron counting. Molecules; The Chemical Catalog Company, 1923. (b) Lewis, G. N. The atom and the molecule. J. Chem. Phys. 1933, 1,17−28. (2) (a) Langmuir, I. Types of valence. Science 1921, 54,59−67. − Other Examples of 3c 2e Interactions (b) Langmuir, I. The structure of atoms and the octet theory of Many examples of compounds with 3c−2e interactions feature valence. Proc. Natl. Acad. Sci. U. S. A. 1919, 5, 252−259. a bridging hydrogen because the nondirectional spherical (3) (a) Jensen, W. B. The origin of the 18-electron rule. J. Chem. Educ. 2005, 82,28−28. (b) Pyykkö, P. Understanding the eighteen- nature of the 1s orbital facilitates an interaction with more than − one atom. Despite this observation, a bridging hydrogen atom electron rule. J. Organomet. Chem. 2006, 691,43364340. − (c) Rasmussen, S. C. The 18-electron rule and electron counting in is not a requirement for the existence of a 3c 2e interaction, as transition metal compounds: Theory and application. ChemTexts illustrated by Al2Me6, which possesses bridging methyl groups 2015, 1, 10. (d) Mingos, D. M. P. Complementary spherical electron (Figure 22). In addition, although less common, a variety of density model and its implications for the 18 electron rule. J. other ligands, such as PR3 and MeCN, are also known to Organomet. Chem. 2004, 689, 4420−4436. (e) Mingos, D. M. P. participate in 3c−2e interactions.8 Development of bonding models based on the Periodic Table. Chimia 2019, 73, 152−164. (f) Landis, C. R.; Weinhold, F. 18-Electron rule ■ SUMMARY and the 3c/4e hyperbonding saturation limit. J. Comput. Chem. 2016, − In summary, the half-arrow notation provides a means to 37, 237 241. represent the structures of molecules with 3c−2e interactions (4) Gillespie, R. J.; Robinson, E. A. Gilbert N. Lewis and the chemical bond: The electron pair and the octet rule from 1916 to the in such a manner that the electron count of each atom can be present day. J. Comput. Chem. 2007, 28,87−97. conveniently evaluated. Therefore, in conjunction with the (5) (a) Wade, K. Electron Deficient Compounds; Springer, 1971. octet and 18-electron rules, the approach affords a simple (b) Wade, K. Boranes: Rule-breakers become pattern-makers. New process to evaluate the chemical reasonableness of a molecule. Scientist, June 6, 1974, pp 615−617 (c) Stone, F. G. Electron-deficient This notation has broad applicability to a variety of classes of compounds. Endeavour 1961, 20,61−67.

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2475 DOI: 10.1021/acs.jchemed.9b00750 J. Chem. Educ. 2019, 96, 2467−2475