This journal is c The Royal Society of Chemistry 2013 Chem. Commun., 2013, 49, 9521--9525 9521

Cite this: Chem. Commun.,2013,49, 9521

Chemical bonding involving d-orbitals

Thomas R. Cundari

The landmark 1954 J. Chem. Soc. paper by Craig et al. exploring the role of d-orbitals in chemical

bonding is discussed. This paper set the agenda for discussions that continue until the present about

topics in inorganic chemistry such as hypervalency, arguably the most significant bonding phenomenon

to differentiate the chemistry of the 2p main group elements from their heavier congeners.

Being invited to provide a Viewpoint on the classic 1954 paper‘‘Chemical Bonds involving d-Orbitals. Part I.’’ by Craig, Maccoll,Nyholm, Orgel and Sutton1 was in a sense like running into anold friend that one had not seen in a long time. Happily, thefriend had aged gracefully. I first read this paper as a graduatestudent in the late 1980’s; it was in a stack of ‘‘airplane reading’’for a Christmas trip home. The manuscript discusses maingroup and transition metal chemistry. As I was being trained intransition metal chemistry, the primacy of d orbitals in bonding,structure and reactivity seemed central dogma amongst thoseengaged in this branch of inorganic chemistry. But, what of ourbrethren researching the main group arena? Hence, my maininterest in the paper at that time was in the authors’ discourseon the role of d orbitals in main group chemistry.

In 1954, the topic of d-orbital participation in main groupchemistry and its relevancy to the bonding of hypervalentcompounds was central to inorganic chemistry. Hypervalencyremains one of the most important discriminants between thechemistry of heavier p-block elements and their 2p congenersgiven the overwhelming reluctance of the latter to form hyper-valent compounds. Their chemistry can be contrasted withanother class of Octet rule violators, electron-deficient com-pounds, which reveals them to act with earnest to assuage theirelectronic shortcomings. For example, Lewis acidic boranes,BX3, often react avidly with Lewis bases to form adducts ordimerise to help better satisfy the octet requirement aboutboron. However, hypervalent compounds are common, oftenreasonably stable, and some even seek to further expand theircoordination numbers about the central atom. To wit, hyper-valent SbF5 is a Lewis superacid, indicating its great desire toaccept a sixth electron pair at antimony, as in the F5Sb:FHconjugate superacid. What bonding principles explain suchgreedy behaviour for a compound that is already in possessionof an electron surplus?

One of the earliest answers to the above conundrum wasthat higher energy nd orbitals, beyond the ‘‘normal’’ ns andnp valence for the 3p and heavier p-block elements, permitexpanded valence beyond the limits set forth by Lewis’ OctetRule. The economics of chemical bonding require that theenergetic investment expended to promote electrons to the higherenergy nd manifold is recouped from additional covalent bondsthat can be made, for example, in PCl5 versus PCl3. Craig et al.couch their arguments largely in the form of orbital overlap,specifically, what we might today refer to as the maximum overlapprinciple.

Craig et al. extend the work of another classic paper, the1949 Mulliken, Riecke, Orloff and Orloff work on overlapintegrals.2 {Note that Jaffe3 had independently published over-lap integrals involving d orbitals in 1953.} The authors firstderive the necessary equations for overlap involving d orbitals.The paper discussed here is Part I of II; the following papergives the mathematical specifics for overlap integrals involvingd orbitals.4 Therefore, Craig et al. invest considerable space inthe initial part of the paper to evaluate the overlap integral asthe best measure of bonding strength. They admit that othermeasures might be more desirable, e.g., employing what onemight today refer to as multiple-zeta basis sets instead of asingle exponent representation of the atomic orbitals. In the1950’s, perhaps only the most visionary futurist could haveforeseen the great facility with which modern computers solvebillions, if not trillions, of overlap integrals with very largeatomic orbital basis sets. However, 1954 was not the Dark Agessome might think as Craig et al. do thank ICI, Ltd. for the use ofan electric calculating machine! One presumes it was used tohelp evaluate necessary d-orbital overlap integrals of s and pvariants; even d bonding is discussed, some 10 years beforeCotton’s landmark work on metal–metal multiply-bonded com-plexes.5 Sub-types of d-orbital s bonding such symbatic overlapwith the ‘‘donut’’ of the dz2, Fig. 1, are also presented.

Once the intellectual edifice is securely constructed, orbitalexponents are needed for calculation of overlap integrals involvingd orbitals, and these are derived from Slater’s rules. The latter had

Department of Chemistry, Center for Advanced Scientific Computing and Simulation

(CASCaM), University of North Texas, Denton, Texas 76203, USA.

E-mail: t@unt.edu

Received 10th July 2013,Accepted 1st September 2013

DOI: 10.1039/c3cc45204b

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9522 Chem. Commun., 2013, 49, 9521--9525 This journal is c The Royal Society of Chemistry 2013

been in the literature for more than a generation,6 and whilstCraig et al. readily admit other techniques might be preferable,the limitations of the time require them to put forth theirhypotheses using single-zeta, Slater rule-derived orbital expo-nents. It would, for example, be more than a decade beforelandmark publication of the works of Clementi on basis sets.7

The first molecule to be subjected to the collective analysisof the five authors – each of whom went on to further signi-ficant accolades and achievements in their careers – is hyper-valent PCl5. Two key deductions are put forth here. First, Craiget al. conclude that there is little difference in s overlapintegrals among 3s–3p, 3p–3p and 3p–3d orbitals at relevantP–Cl bond distances (measured by p, p = 1

2(aA + aB)�r, where a arethe Slater exponents and r is the A–B bond distance in a.u.)assuming the P–Cl bond to be effectively homonuclear (t = 0;t measures the asymmetry of the A–B bond, t = (aA – aB)/(aA + aB)).Second, given the magnitude of the overlap integrals, Craiget al. argue against significant d-orbital hybridization to thedegree implied by the valence bond sp3d description of phos-phorus in PCl5.

The Craig et al. paper, like any science, must be viewed in anappropriate historical context. At this time, d-orbital hybridiza-tion to rationalize hypervalency was well known thanks to thebook The Nature of the Chemical Bond.8 Pauling’s argumentswere often couched in the valence bond language of hybridiza-tion. Others did seek to explain hypervalency without the needto excite electrons from valence ns and/or np to higher energynd orbitals.9,10 A few years earlier in 1951 Pimentel published apaper discussing the bonding in hypervalent trihalide ions,X3�, in terms of multi-centre ps–ps bonding.9

The arguments of Hach and Rundle as well as Pimentel canbe extended to other hypervalent species. Taking the trigonalbipyramidal PCl5 example, longer P–Clax versus P–Cleq bondswould arise in a sans-nd model from multi-centre (three-centre,four-electron) bonding in the Clax–P–Clax moiety versus a triadof two-centre, two-electron P–Cleq bonds for the P(Cleq)3 sub-system. Three-centre, four-electron bonding within MO theoryusing a basis of ns and np orbitals necessitates occupation ofone molecular orbital that is bonding; the third and fourthelectrons occupy a formally non-bonding MO, Fig. 2. The non-bonding nature of the latter molecular orbital rationalizes thegreater bond length of P–Clax versus P–Cleq as arising from aformal reduction in the bond order for the P(Clax)2 sub-system.

As a numerical example, consider that geometry optimiza-tion of D3h-PCl5 with a Hartree–Fock/STO-3G level of theory(GAMESS;11 d orbitals on neither P nor Cl) yields the correctbond asymmetry: P–Clax = 2.20 Å, P–Cleq = 2.15 Å. The differ-ence of B2% in bond lengths is close to experiment as quotedby Craig et al. Hoffmann et al. indicated the same asymmetry

via bond overlap populations derived from extended Huckelcalculations on PH5.12 Thus, the simplest MO methods supportthe analysis made by Craig et al. more than a half-century ago:3d orbitals need not be invoked to explain hypervalence.

P–Cl bond asymmetry is also viewed through the lens oforbital overlap arguments and the conclusion rendered byCraig et al. is as before: overlap integrals suggest minimald-orbital participation in this pentavalent phosphorus com-pound. SF6, another archetype among hypervalent compounds,is the next compound submitted to analysis. Again, the conclu-sion is similar to that advanced for PCl5; d-orbital participationin bonding is not evidenced, due in part to the large nuclearasymmetry, i.e., large t value (vide supra), and the small magni-tude of ds–ps overlap integrals at reasonable S–F bondingdistances. As with PCl5, Craig et al. conclude that d-orbitalparticipation is not essential to describe the structure of SF6.

Craig et al. generously offer a myriad of extenuating circum-stances to justify their proposals, which many readers at thetime may have thought heretical, particularly devotees of thePauling school. For example, as alluded to above they admitthe possible inadequacy of Slater’s rules for calculating orbitalexponents. As noted above, their paper was published a decadebefore the work of Clementi and his coworkers at IBM.7 Thefield of theoretical chemistry would also have to wait a furtherdecade for the popularization of Gaussian style basis sets.13

Craig et al. raise the issue of whether or not the exponentsused to provide a numerical representation of the ns, np andnd atomic orbitals became more equal when P moves from anatomic to a bonding state. In Slater’s recipe for deriving effectivenuclear charge, the ns and np orbitals are treated identically.6

Slater’s rules thus yield identical overlaps involving ns and npAOs which differ from the corresponding overlaps involving nd.Craig et al. used a(3s) = a(3p) = 1.71 and a(d) = 0.33 for P in PCl5.As the authors recognized, phosphorus is different as an atom orin P4 as compared to when it participates in PCl3, which in turnis different from the P atom in PCl5. Craig et al. raise two salientquestions. First, to what extent do the AO exponents of phos-phorus differ among the three sub-shells that comprise thethird shell? And, how do these exponents (and hence the radialextent of the orbitals) change upon going from the atomic tomolecular environment?

Consider a numerical ‘‘experiment’’ using a correlated wave-function (FORS, Full Optimized Reaction Space14) for P (4S groundstate) with a large, augmented-sextuple-zeta basis set (aCC615).

Fig. 1 Depiction of symbatic s bonding from Craig et al.1 Reproduced withpermission.

Fig. 2 Bonding (left) and non-bonding (right) Clax–P–Clax HF/STO-3G orbitals ofPCl5. Isovalue = 0.05.

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The calculated |R(r)| values, Fig. 3, indicate that the 3s, 3p and3d are commensurate in size and decay similarly. One mayinfer from Fig. 3 an answer to the first question posited above:similar Slater exponents can be used for the 3s, 3p and 3dorbitals given the assumption that the P in PCl5 resembles aphosphorus atom (akin to the t = 0 assumption of Craig et al.).

For the second question, we accept the validity of the Pauling’selectroneutrality principle:16 atomic charges in stable compoundslie in the range �1,17 and do a calculation on P+ (3P). Calculatingthe radial expectation values for the n = 3 shell of P+ (3P) with thesame FORS wavefunction and aCC6 basis set used above yields:(hRi (a.u.) = 1.85 (3s), 2.25 (3p) and 2.03 (3d)) for P+ (3P) ascompared to hRi = 1.94 (3s), 2.45 (3p) and 2.16 (3d) Bohr for P (4S).The expected contraction of the orbitals upon ionization iscomputed. The computed hRi for P and P+, as well as Fig. 3,suggest that Craig et al. were quite prescient to worry about thedifference in valence orbital exponents among the three sub-shells that comprise the third shell of phosphorus.

Finally, Craig et al. consider that ionic bonding contri-butions will mitigate d-orbital participation. Ionic bonding andionic resonance structures may be invoked to circumvent theneed to invoke hypervalency in such compounds. As a simpleexample, hypervalent PF5 could be formulated as PF4

+F�, whichsatisfies the Octet Rule. Within the context of valence bondtheory, ionic resonance structures mitigate the need to exciteelectrons from valence ns and np to higher energy nd orbitals.

In the end, the authors realize that extant data are notsatisfactory to definitively confirm or refute any of theseextenuating circumstances and one is left with the impressionthat one could simply dispense with d-orbital participation todescribe hypervalent compounds. Perhaps more importantly,beyond important science, the reader is treated to a distin-guishing feature of this classic, one that present day theoristswould do well to emulate – Craig et al. treat their own assump-tions and conclusions as critically as they do those of others.

Given their thesis that orbital overlap integrals do notsupport significant d-orbital participation in PCl5 or SF6, Craiget al. take the next logical step and rebut Mulliken’s proposalthat the greater strength of bonds among elements of the

second short period18 (what most would today refer to as the3p elements) compared to 2p congeners is due to significantd-orbital hybridization. One is reminded of the aforementionedpaper by Hoffmann et al.:12 ‘‘Far too often have 3d orbitals beeninvoked as a kind of theoretical deus ex machina to account forfacts apparently otherwise inexplicable.’’

I admit sheepishly that as a younger chemist I startedreading the paper somewhat less assiduously once the authorsturned their collective gaze to transition metal complexes. Perhapshaving downplayed the importance of d-orbital participation inhypervalent main group compounds, I was afraid they would dothe same with the transition metal chemistry I was working hardto master. However, the magnitudes of the d-orbital overlapintegrals that they churned out on their loaner electric calculatingmachine support the importance of metal d orbitals in coordi-nation complexes for both the s and p manifolds. One wouldassume the former conclusion may have been de rigueur amongpracticing coordination chemists of the time. However, theirdiscussion of metal–ligand p-bonding was more avant-garde. Anoddity of the Craig et al. paper is that authors make no obviousconnection between their theses on metal–ligand p bondingto the work of Dewar and Chatt–Duncanson. The former waspublished in 1951,19 whilst the latter published their workin 1953.20 To be fair, Craig et al. cite papers by Chatt beforehis landmark 1953 paper with Duncanson, which includes apersonal communication from the five authors of the paperunder consideration here. Moreover, Chatt worked for ICI,generous lenders of the aforementioned calculating contrap-tion, so one must assume they were familiar with each other’sresearch.

Conclusions

Craig et al. collate their major theses from this paper, whichwith the reader’s indulgence, I quote verbatim below.

(1) If all the orbital exponents are equal, then (a) a s-bondformed by a 3s3p33d2 octahedral hybrid orbital with a 3ps-orbitalis about 10–20% stronger than one between a 3s3p3 tetrahedralhybrid orbital and a 3ps one; (b) a s-bond formed by a 3d24s4p3

octahedral hybrid orbital with a 3ps one is about 10% strongerthan a bond between a 4s4p3 tetrahedral hybrid orbital and a 3ps

one, and has almost exactly the same strength as a bond betweena 3d4s4p2 square hybrid orbital and a 3ps one.

(2) d-hybridisation with s- and p-orbitals to form s-hybridorbitals will be ineffective if the ratio of the exponent of thed-orbital to that of the s- and the p-orbitals is less than 0.5.

(3) This condition is not satisfied by d-orbitals in the samemain group as the s- and the p-orbitals in a free atom; so,if d-hybridisation is to be effective, the perturbation by theother atoms in the molecule must make the exponents match.Polar perturbation, by highly electro-negative ligands, couldprobably confer such conditional stability. Increasing thenumber of bonds may also do it, although the more d-orbitalsthat have to be matched the more is the promotion energyrequired.

(4) In phosphorus pentachloride, the bonds are all of nearlythe same strength; and the axial may be somewhat (ca. 10%)

Fig. 3 Plot of |R(r)| (y-axis) versus r (a.u.) for P atom for 3s (blue), 3p (red) and3d (green). Plot derived from FORS14 wavefunction permitting correlation amongthe 5 electrons in the 3s, 3p and 3d AOs of P (4S). The aug-cc-pV6Z basis set15

was used.

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longer than the radial bonds (the reported difference is in thissense but is only 3%).

(5) Hybridisation between penultimate d-orbitals and ultimates- and p-orbitals is permitted by the probable exponent values.Highly electronegative ligands are not necessary.

(6) Penultimate d-orbitals on a central atom may formmoderately strong p-bonds with free p-orbitals on ligands.

(7) In square bond arrangements, two dp bonds can beformed at right angles and, with little increase of this angle,they could be notably strengthened if the central atom has avacant pp-orbital with which they can hybridise.

(8) p-Bonding, by one d-orbital, can also arise in the plane ofthe square bonds if suitable ligands are present.

(9) An ultimate d-orbital can form a strong p-bond with app-orbital on another centre even when it is considerably morediffuse than the latter. Polar character in the s-bond couldimprove the overlap between very disparate orbitals if it tendedto equalise the exponents. Such conditionally stable bonds arelikely to be common and important.

(10) p-Bonding between two dp-orbitals is possible, but isunlikely to be important unless at least one of them is a penulti-mate orbital.

(11) dz2–dz2-bonding is possible but not probable.(12) d-Bonding, between two atomic dxy-orbitals, is unlikely

to be of importance.Some might quibble with a few of these points, particularly

(11) and (12), in light of chemistry published in the ensuing60 years since this treatise was written. However, many of theconclusions still define the basic tenets of those engaged in21st century inorganic chemistry teaching and research. Thisalone must surely qualify the Craig et al. paper as a classic.

A classic paper should generate agreement among the scientificcommunity, and it should generate discord. The subject ofd-orbital participation in hypervalent compounds remainedcontentious even with the advent of modern hardware andsoftware. For example, in 1982 Pietro et al.21 state ‘‘The conceptof d-orbital participation in the bonding of hypervalent compoundshas been repeatedly confirmed by quantitative molecular orbitalcalculations.’’ Less than a decade later, Reed and Schleyer22

state that ‘‘. . .the theoretical evidence against the traditional dsp3,d2sp3 bonding models has become substantial. . .’’ A survey of thestatistics of the Craig et al. paper reveals almost B450 citations,respectable for a non-methodology paper in theoretical chemistry.So, clearly, the scientific community has viewed this paperas worthy of reading, discussion and, one suspects from theaforementioned papers, disagreement.

A distinguishing feature of the Craig et al. treatise vis-a-vismany current papers in computational chemistry is the impressivebreadth of topics covered by these five authors. Having labouredwith an electric calculating machine as a youngster long afterits obsolescence, I envisage the authors’ machinations toextract overlap integrals from this confounded gadget. Today,I suspect many theorists take things like overlap integralsfor granted as our computers churn out mega (106), giga(109), tera (1012), peta (1015) or even exa (1018) flops. The easewith which numbers can be generated by modern computersseems, paradoxically, to have narrowed our focus in theory papers.

Overarching principles in bonding, structure and reactivity areoften overlooked amidst the minute dissection of a catalyst,detailed analyses of a specific molecule’s orbitals, or whether ornot a method can reproduce a specific database of enthalpies.Back in the day, to use the vernacular, numbers were moreprecious, and so chemists like Craig et al. naturally tried toextract maximum scientific value from hard-won numericaldata. For example, after considering d-orbital participation inboth main group and transition metal chemistry, they ruminateon the implications of their research for the structure offerrocene, now the quintessential organometallic, then theprogenitor of a new class of compounds.

One suspects that the ‘‘founding fathers’’ had more time tothink, theorize and hypothesize whilst waiting for new-fangledcomputing machines of the day to slowly spit out numbers.Despite the growth in power of equipment and algorithms,computational chemistry is still more commonly used to explainto refine existing systems rather than identify and designcompletely novel chemical entities. Editors and referees shouldencourage or tolerate hypothesizing in computational papersonce the blow-by-blow description of a specific system has beengiven. It would be desirable to see a return by authors – full-time aficionados and the growing legion of experimentalistswho use DFT as an adjunct to their studies – to follow the leadset forth by Craig, Maccoll, Nyholm, Orgel and Sutton in thisclassic 1954 J. Chem. Soc. paper: pay more attention to thechemistry behind the numbers rather than how well the com-puted numbers do or do not reproduce a particular spectrum ora particular bond length.

Acknowledgements

TRC acknowledges the hard work of undergraduate, graduate,and postdoctoral scientists, and support of their research bythe U.S. National Science Foundation (CHE-1057758) and U.S.Department of Energy, Catalysis Sciences Program, (DE-FG02-03ER15387). Discussions with Paul Bagus, Wes Borden andGeorge Schoendorff (UNT Chemistry, CASCaM), and Mike Schmidt(ISU, Ames Lab) are gratefully acknowledged.

Notes and references1 D. P. Craig, A. Maccoll, R. S. Nyholm, L. E. Orgel and L. E. Sutton,

J. Chem. Soc., 1954, 332–353.2 R. S. Mulliken, C. A. Rieke, D. Orloff and H. Orloff, J. Chem. Phys.,

1949, 17, 510 and 1248.3 H. H. Jaffe, J. Chem. Phys., 1953, 21, 258.4 D. P. Craig, A. Maccoll, R. S. Nyholm, L. E. Orgel and L. E. Sutton,

J. Chem. Soc., 1954, 354.5 F. A. Cotton, N. F. Curtis, C. B. Harris, B. F. G. Johnson, S. J. Lippard,

J. T. Mague, W. R. Robinson and J. S. Wood, Science, 1964, 145,1305–1307.

6 J. C. Slater, Phys. Rev., 1930, 36, 57.7 E. T. Clementi and D. L. Raimondi, J. Chem. Phys., 1963, 38,

2686.8 L. Pauling, The Nature of the Chemical Bond and the Structure of

Molecules and Crystals: An Introduction to Modern StructuralChemistry, Cornell University Press, Ithaca, NY, 1960.

9 G. C. Pimentel, J. Chem. Phys., 1951, 19, 446.10 R. J. Hach and R. E. Rundle, J. Am. Chem. Soc., 1951, 73, 4321.11 M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert,

M. S. Gordon, J. J. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen,

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S. Su, T. L. Windus, M. Dupuis and J. A. Montgomery, J. Comput.Chem., 1993, 14, 1347.

12 R. Hoffmann, J. M. Howell and E. L. Muetterties, J. Am. Chem. Soc.,1972, 94, 3047.

13 W. J. Hehre, R. F. Stewart and J. A. Pople, J. Chem. Phys., 1969,51, 2657.

14 The FORS wavefunction allows mixing of a reference state with allpossible excited states (which are allowed to relax) of a givenmultiplicity for a space of ‘‘active’’ electrons and orbitals. The FORSwavefunction is spherically symmetric or can be made so via state-averaging. For the example of the P (4S) atom, defined parameters

for the FORS wavefunction derived from the 3s23p33d0 referencestate are: multiplicity = 4, active electrons = 5, active orbitals = 9.

15 T. Van Mourik and T. H. Dunning, Int. J. Quantum Chem., 2000, 76, 205.16 L. Pauling, J. Chem. Soc., 1948, 1461.17 Pauling suggests �1

2 in ref. 16, but �1 in ref. 8 ( p. 172).18 R. S. Mulliken, J. Am. Chem. Soc., 1950, 72, 4493.19 M. J. S. Dewar, Bull. Soc. Chim. Fr., 1951, 18, C71.20 J. Chatt and L. A. Duncanson, J. Chem. Soc., 1953, 2939.21 W. J. Pietro, M. M. Francl, W. J. Hehre, D. J. DeFrees, J. A. Pople and

J. S. Binkley, J. Am. Chem. Soc., 1982, 104, 5039.22 A. E. Reed and P. v. R. Schleyer, J. Am. Chem. Soc., 1990, 112, 1434.

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