electron pairing and chemical bonds: bonding in hypervalent molecules from analysis of fermi holes

Electron Pairing and Chemical Bonds:Bonding in Hypervalent Moleculesfrom Analysis of Fermi Holes

ROBERT PONEC,1 ANTHONY J. DUBEN2

1Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Prague 6, Suchdol, 165 02,Czech Republic2Department of Computer Science, Southeast Missouri State University, Cape Girardeau, Missouri63701-4799

Received 1 September 1998; accepted 23 December 1998

ABSTRACT: Bonding in the hypervalent molecules SF , BrF , PF , and SF was4 5 5 6studied using multicenter bond order indices and examination of the eigenvaluesand the eigenvectors of the Fermi holes of the constituent atoms. Diagonalizationof the Fermi holes provided quantitative validation of Musher’s categorizationof hypervalency with SF and BrF representative of type I, and PF and SF4 5 5 6belonging to type II. The eigenvalues and eigenvectors of type I moleculesdistinguished between classic two-center two-electron bonds and three-centerfour-electron bonds, whereas the results of diagonalization for type II moleculesdemonstrated the presence of substantial reorganization of the valence state ofthe central atom leading to equivalent bonds and the highest expected symmetryof the molecule. Q 1999 John Wiley & Sons, Inc. J Comput Chem 20: 760]771,1999

Keywords: pair density; Fermi hole; chemical bonding; hypervalence

Introduction

he continuing progress in the development ofT new computational tools of quantum chem-istry has reached a stage in which reliable theoreti-

Correspondence to: R. Ponec; e-mail: [email protected] sponsor: Grant Agency of the Czech

Academy of Sciences, contractrgrant number: A4072606r1996;Ž .contractrgrant sponsor: National Research Council USA

cal calculations have become possible on real,chemically interesting systems making computa-tional chemistry a routine part of chemical re-search. The increasing accuracy and sophisticationof quantum chemical methods used has, however,one unpleasant side effect. The complexity of thewave functions makes their interpretation moredifficult. It is harder to find direct links betweenthe results of calculations and classical chemicalconcepts of bonds, bond orders, valences, VBstructures, etc., which chemists usually use. The

( )Journal of Computational Chemistry, Vol. 20, No. 8, 760]771 1999Q 1999 John Wiley & Sons, Inc. CCC 0192-8651 / 99 / 080760-12

ELECTRON PAIRING AND CHEMICAL BONDS

development of new computational methods andprocedures thus needs to be accompanied by theparallel design of auxiliary methods for interpret-ing wave functions and for extracting from themthe desired structural information. Earlier exam-ples of these efforts in relating modern theory andclassical concepts of bond order and valence aredescribed in refs. 1]7.

Our recent studies in which the systematic anal-ysis of pair density and related quantities wasintroduced8 ] 11 as a new efficient way of visualiz-ing molecular structure even in systems with com-plicated bonding patterns12,13 fit into the frame-work of these efforts. Our aim in this study is topursue the philosophy of these previous studiesand to show that the Fermi holes14,15 can be usedadvantageously as a new means of characterizingbonding in hypervalent molecules. The study isdivided into two parts. In the first part, the theo-retical background underlying the introduction ofthe Fermi hole will be reviewed briefly. In thesecond part, the new theoretical technique will beapplied to the investigation of the valence states ofatoms in molecules with emphasis on the analysisof bonding in hypervalent systems. Although themethodology is at present practically applicable atsimple SCF level of theory only, the anticipatedeffects of including electron correlation will bediscussed qualitatively.

Theoretical

Although the methodology of the analysis of theFermi hole was described thoroughly in our previ-ous studies,14,15 it is worthwhile to recapitulatebriefly the basic ideas of the approach. The Fermiholes, first introduced by Wigner16 in the field ofsolid-state physics, describe the effect of mutualcoupling of electrons of the same spin on thebehavior of quantum systems. Our aim in thisstudy is to demonstrate the usefulness of the Fermiholes for the analysis of bonding in molecularsystems. The definition of the Fermi hole beginswith the conditional probability of finding one

Želectron at position r , provided the second refer-1.ence electron is localized at r . This probability is2

Ž .given in terms of pair density r r , r by:1 2

Ž .2 r r , r1 2Ž . Ž .P r s 1r 12 Ž .r r2

The Fermi hole associated with the reference elec-tron in r is then given by:2

Ž . Ž . Ž . Ž .h r s r r y P r 2r 1 1 r 12 2

Ž .where r r is the usual first-order density, which1describes the probability of finding the electron atthe point r . Here, it should be noted that another1

w Ž .xclosely related definition eq. 3 , in which theFermi hole is a negative quantity, is usually pre-ferred:

Ž . Ž . Ž . Ž .H r s P r y r r 3r 1 r 1 12 2

but this simple rearrangement has no impact onthe physics behind the hole. For reasons that will

Žbecome clear later possibility to interpret theeigenvalues of the matrix representing the hole as

.occupation numbers , we prefer, in this study,Ž .definition 2 , in which the Fermi hole is a positive

quantity.The physical meaning of the Fermi hole is thus

that it provides corrected ‘‘net’’ information aboutthe distribution of the first electron of the pair,

Ž .provided the second reference electron is at r .2Ž .The Fermi hole defined by eq. 2 has been ana-

lyzed and discussed by various investigators17 ] 22

and their usefulness for the qualitative discussionof bonding was clearly demonstrated. Our aim inthis study is to follow up the results of thesestudies and to demonstrate that the appropriatelygeneralized quantity, the so-called domain-aver-aged Fermi hole, can be used advantageously as anew, efficient means of characterizing bonding inhypervalent molecules. The idea of introducing theconcept of the domain-averaged Fermi hole is verysimple and goes back to the original definition of

Ž .the conditional probability eq. 1 . If we release thecondition of strict localization of the reference elec-tron in a single point and replace it with a physi-cally much more reasonable condition, allowing itto be found anywhere in some region of space, V,

Ž .the original definition eq. 2 of the Fermi hole ischanged to:

Ž .2H r r , r drV 1 2 2Ž . Ž .P r s 4V 1 Ž .H r rV 2

Based on this definition, the domain-averagedFermi hole associated with the reference electronin the region V is then defined as:

Ž . Ž . Ž . Ž .h r s r r y P r 5V 1 1 V 1

JOURNAL OF COMPUTATIONAL CHEMISTRY 761

PONEC AND DUBEN

Here, it is necessary to stress that, in our previousstudies,14,15 the term ‘‘integrated Fermi hole’’ was

Ž .used for the quantity in eq. 5 ; however, becausethis term was a bit confusing, the referee proposedthe more appropriate term ‘‘domain-averagedFermi hole.’’

This averaged Fermi hole satisfies the universalnormalization:

Ž . Ž .h r dr s 1 6H V 1 1

which holds, irrespective of the form of the regionV. Although the validity of this equation is notinfluenced by the actual size and form of theregion V, it is possible to choose an V that isespecially useful for chemistry. The choice is basedon Bader’s idea of partitioning the total electrondensity into regions associated with individualatoms.23 Adopting this type of partitioning, theregion V is identified with the atomic regions ofBader’s partitioning. The simplest possibility is toidentify V with the atomic region of a single atom.This choice will be emphasized in this study. Inprevious studies14,15 we demonstrated that theFermi hole associated with the atomic region of asingle atom provides valuable information aboutthe valence state of that atom in a molecule. Thequantity that provides this information is not,

w Ž .xhowever, the Fermi hole eq. 5 itself, but ratherŽ .the ‘‘charge-weighted’’ Fermi hole g r relatedV 1

to the original hole by the simple relation:

Ž . Ž . Ž .g r s N h r 7V 1 V V 1

where the proportionality factor N is equal to theV

mean number of electrons in the region V:

Ž . Ž .N s r r dr 8HV 1 1V

The philosophy underlying the introduction ofthese ‘‘charge-weighted’’ Fermi holes is very sim-ple. The ‘‘normal’’ Fermi hole was derived fromthe conditional probability describing the distribu-tion of the first electron of the pair, provided theother was known to be somewhere in V. Thelocalization of a single electron in the region V isan artificial act that does not reflect the fact that inthe real molecule the region V is populated by NV

electrons, rather than by only one. The proportion-Ž .ality factor in eq. 7 is therefore nothing but a

correction for the statistical probability of distribu-tion of electrons in V.

In this connection it is perhaps worth mention-Ž .ing that the quantity, eq. 7 , can be obtained

alternatively by the appropriate integration of theso-called exchange part of the pair density 24 :

Ž . Ž . Ž . Ž .g r s r r r r dr y 2 r r , r drH HV 1 1 2 2 1 2 2V V

Ž . Ž . Ž .s N r r y 2 r r , r dr 9HV 1 1 2 2V

Ž . Ž .The quantities eqs. 5 and 7 are also closelyrelated to the so-called inter-loge correlation termsŽ . 21F V, V9 introduced by Bader :

Ž . Ž . Ž .F V , V9 s g r dr s N r r drH HV 1 1 V 1 1V9 V9

Ž . Ž .y 2 dr r r , r dr 10H H1 1 2 2V9 V

Having introduced the basic quantities, it is alsonow possible to specify briefly the methods oftheir analysis. Despite their being related to pair

Ž .densities, the Fermi hole, whether from eq. 7 orŽ .eq. 5 , is a one-electron quantity and as such can

be analyzed by the same means and procedures asŽ .the ordinary electron density r r . One particu-

larly useful method consists in the diagonalizationof the matrix G , which represents the ‘‘charge-V

Ž .weighted’’ Fermi hole in eq. 7 in the AO basis. Inour previous study,15 we have shown that, whenthe region V is identified with the region of a

wsingle isolated atom, the resulting Fermi hole eq.Ž .x7 can straightforwardly be interpreted in termsof valence state of the corresponding atom, andseveral examples of such an interpretation havebeen presented. These examples involved, how-ever, only several trivial systems like CH , NH ,4 3H O, etc., in which the valence state of the central2atom is well known. But, there are other morecomplex systems for which the exact picture ofbonding is not very clear, such as hypervalentsystems like SF , SF , PF , BrF , etc., which are of4 6 5 5interest due to their lack of conformity to theLewis octet rule. It was therefore of special interestfor us to apply the aforementioned formalism tothese hypervalent systems and to elucidate thebonding in them.

Computations

Although the noted methodology was formu-lated quite generally and can be applied at anylevel of theory, the practical applications discussed

VOL. 20, NO. 8762


in this study are based on some simplifying as-sumptions. The first of these concerns the pairdensity, which in our case was derived fromsemiempirical SCF wave functions generated bythe AM1 method.25 We are, of course, aware of thefact that, in view of the present standards, such anapproach is rather crude, but we used it mainlybecause of the simplicity with which it can beimplemented and subsequently tested at this par-ticular level. Moreover, we believe, and the resultsindeed seem to warrant this belief, that even atthis simple level the conclusions can be regardedas reliable enough and can certainly provide agood starting point prior to extensions to higherlevels of theory. In addition to limiting ourselvesto the semiempirical SCF level of theory there wasyet another approximation used in practical test-ing. This approximation concerns the integrationover Bader’s atomic regions, which, because of thenonavailability of the programs for such an inte-gration, was replaced by appropriately restrictingthe summations over the basis functions in thesense of Mulliken population analysis. Within thisapproach, which was also used in our previousstudies,14,15 an electron is expected to be in theregion of an atom A if it resides in a basis functioncentered on this atom. Thus, for example, the inte-grals over the atomic region of atom A of the typeis approximated as:

occ occ A

² < :2 i i s 2 c c SÝ ÝÝÝA m i n i mvm ni i

A

Ž . Ž . Ž .mms PS s Q A 11Ým

The above analysis was applied to wave functionsdescribing the bonding in several hypervalent sys-tems like SF , PF , SF , and BrF . The calculations4 5 6 5were performed by the semiempirical AM1 methodincorporated into the SPARTAN PCq ; program.Based on these primary data, the pair densitiesand corresponding Fermi holes were generatedusing our own FORTRAN program, which is avail-able upon request. The molecules were in com-pletely optimized molecular geometries. Experi-mental and calculated geometrical parameters aresummarized in Table I. The agreement betweenexperimental and theoretical parameters was not

Ž .excellent in all cases especially bond lengths , butfor the qualitative demonstration of the applicabil-ity of the approach, the reproduction of basic

Ž .structural parameters including the bond anglescan be regarded as satisfactory.

Results and Discussion

Before examining the calculated Fermi holes, itis useful to summarize the state of knowledge ofbonding in hypervalent molecules. Perhaps thebest and the most competent discussion of thisproblem can be found in a relatively recent reviewby Gillespie and Robinson,26 who critically re-viewed all existing contradictory opinions andproposals. Thus, according to Musher,27 the con-cept of hypervalence applies to nonmetals of thegroups V, VI, VII, and VIII of the Periodic Table inany of their valence states other than their loweststable valences 3, 2, 1, and 0, respectively. Accord-ing to this definition, the species SF , PF , SF , BrF ,4 5 6 5

TABLE I.Comparison of Experimental and AM1-Calculated Geometrical Parameters for a Series of RepresentativeHypervalent Molecules.

Geometrical Experimental values Calculated values( ) ( )Molecule parameter distance or angle distance or angle

˚ ˚SF S—F 1.646 A 1.573 A4 ax

˚ ˚S—F 1.545 A 1.545 Aeq/F SF 101.58 103.88eq eq/F SF 173.18 170.48ax ax

˚ ˚SF S—F 1.564 A 1.540 A6

˚ ˚PF P—F 1.58 A 1.549 A5 ax

˚ ˚P—F 1.53 A 1.535 Aeq

˚ ˚BrF Br—F 1.774 A 1.799 A5 eq

˚ ˚Br—F 1.689 A 1.823 Aax/F BrF 84.58 85.28ax eq


PONEC AND DUBEN

ClF , etc., in which there are more than four elec-3tron pairs around the central atom in the classicalstructural formula, ought to be considered hyper-valent. In addition to this phenomenologicalscheme, Musher proposed to introduce an addi-tional classification to distinguish between twotypes of hypervalent molecules. The hypervalent

Ž .molecules of the first kind HV , have been char-Iacterized as being formed from existing molecules

Žby the addition of two collinear monovalent or.single divalent ligands in a manner that preserves

the structure of the original molecule. This class ofmolecules includes SF , ClF , XeF , BrF , etc. Hy-4, 3 2 5

Ž .pervalent molecules of the second type HV haveIIbeen characterized as being formed from existing

Žmolecules by the addition of two monovalent or.single divalent ligands accompanied by deep re-

organization of the electron and geometrical struc-ture with the resulting equilibrium geometry pri-marily governed by steric factors. Molecules likePF , SF , IF , or XeO belong to class HV , and5 6 7 4 IIthey usually display the highest possible symme-

Žtry octahedron in SF , trigonal bipyramid in PF ,6 5.pentagonal bipyramid in IF , etc. .7

To explain the violation of the Lewis octet rulein hypervalent systems, the concept of d-orbitalparticipation in hybridization was invoked ini-tially,28 but this explanation is now considered tobe incorrect. The rejection of the idea of d-orbitalparticipation comes mainly from the results ofmodern of ab initio calculations in which it wasshown that d-orbitals play the role of polarizationfunctions without a major contribution to bonding.Another general observation from these calcula-tions is that of the considerable ionic character ofbonding in these systems. Based on this ionicity, areinterpretation of the original Lewis octet rulewas proposed29,30 in which only electrons in thevalence shell of the central atoms are to be takeninto account instead of including unspecified elec-tron pairs that may be fully or partially shared orunshared in the original model. The ionic natureand polarization of some bonds lowers consider-ably the number of electrons in the valence shell ofthe central atom, and the octet rule is no longerviolated, and these systems are to be regarded as‘‘normal.’’ However, this picture of bonding is alsonot completely satisfactory. For example, the exis-tence of six bond paths in Bader’s analysis in SF6and related systems31 clearly suggests that thereare six bonds around the central atom in thesemolecules. Consistent with this picture are theresults of recent sophisticated post-HF calcula-tions32 ] 35 in which the so-called ‘‘democracy prin-

ciple’’ was formulated. According to this principle,Ž .almost all electrons of central atoms can engagein bonding if provided with a sufficient energeticincentive. The electrons of the central atom origi-nally residing in formally doubly occupied freeelectron pairs decouple and each of the resultingsingly occupied orbitals takes part in bonding witha singly occupied orbital of the ligand. Electroncorrelation is responsible for this phenomenon. Thismeans that such decoupling can be specificallydetected in calculations in which electron correla-tion is properly taken into account. This wasdemonstrated in spin-coupled calculations,32 ] 34

and in application of the GVB method, in whichMessmer examined the importance of radial andangular correlation of lone pairs on bonding.35 Inthis connection it is of interest to note that thesame idea of decoupling free electron pairs wasalready considered by Musher,27 but the demon-stration of decoupling was impossible at the levelof the qualitative MO model with which heworked. The phenomenon thus found its manifes-tation in the form of three-center four-electronbonds and it is just this three-center bonding thatis characteristic of all ab initio calculations re-stricted to the SCF level of theory.29

In view of what has been noted, this means thatthe model of three-center four-electron bondingconsistent with the existing ab initio calculationsshould thus be regarded only as a projection of thebonding situation biased by the neglect of theelectron correlation in SCF methods. On the otherhand, when electron correlation is taken into ac-count, the problem of hypervalent bonding seemsinstead to be consistent with the picture of an

Ž .expanded octet duodecet rule.Having outlined the present state of the theory

of hypervalence, we now present the conclusionsof our analysis. It is necessary to note that theFermi holes generated in this report came from HFpair densities; therefore, a systematic bias from theHF approximation is likely to be reflected in ourconclusions. The model is, however, simple enoughto estimate the possible effects of including theelectron correlation.

As already stated, the approach is based on theanalysis of ‘‘charge-weighted’’ Fermi holes associ-ated with the central atom of the hypervalentsystem. The Fermi holes satisfy:

Ž . Ž .g r dr s N 12H V 1 1 V

VOL. 20, NO. 8764


which ensures the natural normalization of thehole to the total number of electrons in V. In thecase in which the region V is associated with asingle atom, the normalization factor is simplyequal to the electron density on the atom in themolecule. This characterization is, however, rathercrude and says nothing about any internal distri-bution of electrons on the atom. In a previousstudy14 we demonstrated that useful additionalinformation can be extracted from the analysis ofeigenvales and eigenvectors of the matrix G rep-V

w Ž .xresenting the Fermi hole eq. 7 in the AO basis.For example, the diagonalization of the Fermi holeassociated with the carbon atom in CH yields4only four nonzero eigenvalues. The degeneracy of

Žthese eigenvalues h s h s h s 0.903, h s1 2 3 4.1.222 suggests that the corresponding eigenvec-

tors transform according to T - and A -irreducible2 1representations of the T group. This, however, isdinconsistent with the classical picture of fourequivalent tetrahedrally oriented free valences. Theequivalence can be obtained if the primary eigen-vectors resulting from the diagonalization are sub-jected to the isopycnic transformation assessed byCioslowski.36 The result of this transformation,which leaves the Fermi hole invariant, is the finalpicture of the valence state, which indeed showsfour equivalent tetrahedrally oriented spx-hy-bridized orbitals corresponding to free valences ofbroken CH bonds, with the occupation numbers

Ž .h s h s h s h s 0.982 Fig. 1 . The occupation1 2 3 4numbers of these free valences are close to 1 be-cause the polarity of CH bond is low. This can be

FIGURE 1. Eigenvector of the Fermi hole associatedwith the C atom in CH . Eigenvector with the eigenvalue40.982 corresponds to the nonpolar, broken C—H bond.

easily demonstrated by diagonalizing the Fermihole associated with one of the hydrogens. Thisdiagonalization yields one nonzero eigenvalueequal to 1.018, which exactly complements thepopulation of free valence of broken CH bond toyield a nearly nonpolar two-center two-electronCH bond. In a similar way, the analysis of theFermi hole of the N atom in NH also yields four3nonzero eigenvalues. One of them is equal to 2.000and the remaining three are 1.076. Inspection ofthe form of associated eigenvectors shows thatthey correspond to a free electron pair on N andthree tetrahedrally oriented free valences of brokenNH bonds, respectively. Diagonalizing the Fermihole associated with one of the hydrogens con-firms the identification of the NH bonds as two-center two-electron bonds. The diagonalizationyields one nonzero eigenvalue equal to 0.924, com-plementing the population of broken free valenceto complete the nearly evenly shared one-bondingelectron pair.

The simple cases in the previous paragraphstrictly follow the Lewis octet rule. The remainderof this study presents the application of the sameapproach to more complex hypervalent systems.The resulting picture of the valence state of thecentral atom is compared with existing theories ofhypervalence.

SF4

SF , whose structure is shown in Figure 2, be-4longs to the hypervalent system of type HV which,Iaccording to Musher,27 exhibits two types of bonds.One of them corresponds to two equatorial SFbonds and resembles normal covalent bonds. Onthe other hand, the axial SF bonds are quite differ-ent, and are classified as real hypervalent bonds.Let us compare this expectation with the actuallycomputed results. In this analysis, the main focuswill be on the nature of the valence state of thecentral hypervalent S atom. The diagonalization ofthe Fermi hole associated with this atom yieldsfour nonzero eigenvalues. One of these eigenval-ues is equal to 2.00 and inspection of the corre-sponding eigenvector shows that it corresponds to

Ž .free electron pair on S Fig. 3 . There is a degener-ate pair of eigenvalues equal to 0.694 and oneeigenvalue equal to 0.677. Detailed inspection ofthe associated eigenvectors assigns the degeneratepair to two broken valences of equatorial SF bondsŽ .Fig. 4 , and the remaining eigenvector is very

Ž .much a pure p orbital on S Fig. 5 . The deviationof the eigenvalues from unity can be interpreted as


PONEC AND DUBEN

FIGURE 2. Schematic structure of the SF molecule.4

indicating highly polar SF bonds. This interpreta-tion is again supported by the analysis of theFermi holes associated with equatorial fluorine

Ž .atoms. There are eight nonzero eigenvalues 2 = 4Žof which six corresponding to free electron pairs

.on fluorines are equal or very close to 2 and the

FIGURE 3. Eigenvector of the Fermi hole associatedwith the S atom in SF . Eigenvector with the eigenvalue42.000 corresponds to free electron pair on sulfur.

FIGURE 4. Another eigenvector of the Fermi holeassociated with the S atom in SF . Eigenvector with the4eigenvalue 0.694 corresponds to one of the broken polarequatorial SF bonds.

FIGURE 5. Another eigenvector of the Fermi holeassociated with the S atom in SF . Eigenvector with the4eigenvalue 0.677 corresponds to the nearly pure p orbitalon S, which participates in two polar axial SF bonds.

VOL. 20, NO. 8766


Žremaining two corresponding to free valences of.broken equatorial SF bonds are equal to 1.512

each. The electron pairs in equatorial SF bonds areshared much less symmetrically than in the case ofCH or NH bonds, consistent with the picture ofhighly polar bonds with considerable reduction ofthe charge on S. Except for this higher polarity,these SF bonds are quite normal s bonds, whichmay be expected on the basis of Musher’sanalysis.27

Having analyzed the equatorial SF bonds, let usturn our attention to the axial SF bonds. The eigen-value equal to 0.677 is associated with an eigen-vector that looks like a pure p orbital on S. Thisresult is noteworthy because there are two axial SFbonds. This implies that the partially populatedsingle p orbital on S has to be engaged in two axialSF bonds. This result is very important for at leasttwo reasons. First, there is a close correspondencewith the classical model by Hasch and Rundle36

and Pimentel,38 implying the existence of three-center four-electron bonding. The presence ofthree-center four-electron bonds predominantly lo-calized in axial SF bonds is also confirmed by the

11,39 Ž .values of three-center bond indices Table II .The second reason is that the axial SF bonds areeven more polar than the equatorial ones. Todemonstrate this, we compare the eigenvalue of0.677 corresponding to the partially populated porbital on S with the eigenvalues resulting fromdiagonalization of the Fermi hole associated withtwo axial fluorines. This diagonalization yieldseight nonzero eigenvalues of which six are close to

Ž .2.00 free electron pairs on fluorines , and the re-maining two are equal to 1.658 each. Taking into

account that the partially populated p orbital on Sin fact participates in two bonds, the effectivepopulation per one such bond should be 0.677r2f 0.338. This again nearly perfectly complementsthe population on individual fluorines to give thepicture of two unevenly shared electron pairs intwo axial SF bonds. By artificially halving thepopulation of the p orbital on S, one can modelqualitatively the effect of decoupling anticipatedby Musher27 and reflected in the form of angularcorrelation35 in a more sophisticated post-HF cal-culation. Despite being based on the HF level oftheory, the picture of bonding in this type of hy-pervalent molecule can thus support surprisinglywell conclusions from modern post-HF calcula-tions that incorporate the expansion of the valenceshell over the classical Lewis octet.

BrF5

Another example of the HV -type of hyperva-Ilent molecule is bromine pentafluoride, BrF . The5structure of this molecule is a slightly distorted

Ž .square pyramid Fig. 6 , and the calculated equi-librium geometry corresponds to this type of struc-ture. The main focus in discussing the bonding ofBrF will be on the analysis of the valence state of5the central Br atom in terms of its associated Fermihole. The diagonalization of this hole again yieldsfour nonzero eigenvalues. One of them is equal to2.00 and, in addition to another one equal to 1.111,there is a degenerated pair of eigenvalues equal to0.986. Inspection of the corresponding eigenvectorsshows that eigenvalue 2.00 is associated with as-free electron pair on Br opposite to the axial BrF

TABLE II.Calculated Values of Three-Center Bond Indices in Several Hypervalent Molecules.

Number of symmetryaMolecule Three-center bond equivalent indices Multicenter bond index

SF F SF 1 y0.0824 ax axF SF 1 y0.054eq eqF SF 4 0.000ax eq

BrF F BrF 2 y0.1415 3 4F BrF 4 y0.0521 3

PF F PF 6 y0.0345 eq axF PF 3 y0.015eq eqF PF 1 y0.010ax ax

SF F SF 12 y0.0396 1 4F SF 3 y0.0241 3

aNegative values of three-center bond index indicate three-center four-electron bonding limiting value for ideal three-centerfour-electron bond is y0.187.11


PONEC AND DUBEN

FIGURE 6. Schematic structure of the BrF molecule.5

bond. Eigenvalue 1.111 corresponds to an eigen-vector representing the free valence of the brokenaxial BrF bond. This bond is similar in many re-spects to equatorial SF bonds in SF with the ex-4ception of lower polarity. Analysis of the Fermihole associated with the axial fluorine confirmsthis finding. Diagonalization again yields fournonzero eigenvalues of which three are close orequal to 2, corresponding to free electron pairs onF, whereas the remaining value is equal to 1.223,representing the orbital of broken axial BrF bond.

The degenerate pair of eigenvalues correspondsto eigenvectors representing nearly pure p and px yorbitals on bromine. The situation in which twopartially populated p orbitals have to participatein four BrF bonds is analogous to the case of axialbonds in SF . This situation implies the existence4of three-center four-electron bonds, and these

Žbonds localized predominantly in the F BrF and3 4.F BrF bonds can indeed be detected using three-5 6

Ž .center bond indices Table II . Another noteworthyaspect of equatorial BrF bonds is their relativelyhigher polarity compared with the axial BrF bond.This polarity may seem surprising, because theeigenvalues corresponding to p and p orbitalsx yparticipating in these bonds are equal to 0.986;however, as in the case of axial SF bonds in SF ,4one should recall the fact that each of the p orbitalsparticipates in two bonds. The effective populationper one formal bond should thus be close to0.986r2 f 0.493. This picture is also consistent withthe analysis of the Fermi holes associated with

Ž .equatorial fluorines. The analysis yields 16 4 = 4Ž .nonzero eigenvalues, of which 12 3 = 4 , corre-

sponding to free pairs on fluorines, are close to

2.00. The remaining four, corresponding to freevalences of broken BrF bonds, are equal to 1.510.The populations in free valences on Br and F ofbroken equatorial BrF bonds complement eachother quite closely, to give the final picture of fourpolar two-center two-electron bonds.

PF5

An example of a hypervalent system of typeHV is PF . Following Musher’s27 reasoning, weII 5expect that formation molecules of this type re-quire deep electron and geometrical reorganizationyielding the highest possible symmetry as the moststable structure. This is the case with PF which is5a trigonal bipyramid with axial bonds slightlylonger than the equatorial ones. This difference hasbeen qualitatively well reproduced by the calcula-

Ž .tions Table I . In analyzing the bonding in thismolecule, the main focus will be on the analysis ofthe valence state of the central phosphorus atom.The diagonalization of the Fermi hole associatedwith this atom again yields four nonzero eigenval-ues, of which three are equal to 0.551 and theremaining one equal to 0.481. Detailed inspectionof the form of individual eigenvectors shows thattriply degenerate eigenvectors correspond to free

Ž .valences of three equatorial PF bonds Fig. 7 , andthe remaining resembles a pure p orbital oriented

FIGURE 7. Eigenvector of the Fermi hole associatedwith the P atom in PF . Eigenvector with the eigenvalue50.551 corresponds to one of the broken polar equatorialPF bonds.

VOL. 20, NO. 8768


Ž .along axial PF bonds Fig. 8 . The situation issimilar to that of SF in which a single partially4populated p orbital was also shown to participatein two axial bonds. Despite this parallel, there isan important difference. In HV -type molecules,IIthe central atom uses all its valence electrons inbonding so that it attains its maximal possiblevalence. This, of course, also presumes decouplingof all available free electron pairs, but becausethese molecules display the highest possible sym-metry, the decoupling is accompanied by deepelectron reorganization on the central atom. As aresult of this reorganization, the coexistence ofpseudocovalent equatorial and three-center four-electron axial bonds characteristic of HV moleculesIis no longer tenable, and all PF bonds becomemore or less equivalent. This conclusion is sup-ported by the values of three-center bond indicesŽ .Table II , which suggest that the three-centerbonding present here is much weaker than forgenuine three-center four-electron bonds. Themole-cule ought to be regarded as composed offive more or less equivalent polar PF bonds. How-ever, one should note that the expansion of theoctet is indicated only indirectly by the absence of

FIGURE 8. Another eigenvector of the Fermi holeassociated with the P atom of PF . Eigenvector with the5eigenvalue 0.481 corresponds to nearly pure p orbitalsof phosphorus, the decoupling of which is required forthe formation of two polar axial PF bonds.

localized three-center four-electron bonding andby the near equivalency of all five PF bonds. Theexpansion of the octet can, in real molecules, againbe attributed to decoupling of the electron pair in apartially populated p orbital. Because electron cor-relation, which is responsible for this decouplingin real molecules, is not taken into account in HFpair densities, the explicit demonstration of thisexpansion lies outside the scope of the presentapproach. We believe, however, that when elec-tron correlation is properly taken into account, theformalism just examined would be able to detectthe octet expansion directly.

SF6

Another example of the HV hypervalentIImolecule is sulfur hexafluoride, SF . As is typical6for this class of systems, all valence electrons par-ticipate in bonding, implying deep electron andgeometrical reorganization to attain the highestpossible symmetry. Consistent with this expecta-tion, the molecule exists in the form of a regularoctahedron with all SF bonds equal in length. Theanalysis begins again with diagonalization of theFermi hole associated with the S atom. Fournonzero eigenvalues are found in two degeneratepairs equal to 0.773 and 0.677, respectively. Theeigenvectors associated with the first pair of eigen-values correspond to sp x hybrids of axially ori-ented free valences of two broken SF bonds. Oneof these eigenvectors is visualized in Figure 9. The

FIGURE 9. Eigenvector of the Fermi hole associatedwith the central S atom in SF . Eigenvector with the6eigenvalue 0.773 corresponds to one of two brokenpolar SF bonds.


PONEC AND DUBEN

remaining two eigenvectors resemble equatoriallyoriented pure p orbitals that are engaged in fourremaining SF bonds, and one of the symmetryequivalent pairs of these eigenvectors is shown inFigure 10. Such a picture of bonding is reminiscentof what we have met previously with the PF5molecule. Comparable to what is characteristic ofHV -type systems, the decoupling of each of theIIindividual p orbitals required for the expansion ofthe valence shell is accompanied by deep electronreorganization, allowing the molecule to adopt thehigh observed symmetry. As a consequence of thiselectron reorganization, there is thus no differencebetween individual SF bonds, and all these bondsare strictly equivalent and display a quite smallthree-center character. We can thus again see thedifference between HV - and HV -type hyperva-I IIlent systems. Whereas, in HV molecules, the de-Icoupling of more or less pure p orbitals in twobonds is projected at the HF level in the presenceof three-center four-electron bonding, the situationin HV systems is different, and in keeping withIIthe presence of deep electron reorganization thefinal picture of bonding is consistent with theobserved high symmetry, with all bonds com-pletely or nearly equivalent. This can be demon-strated easily by inspecting Table II, in which thevalues of multicenter bond indices are not onlyconsistent with the observed high symmetry, but

FIGURE 10. Another eigenvector of the Fermi holeassociated with the S atom in SF . Eigenvector with the6eigenvalue 0.667 corresponds to one of the two nearlypure orbitals on S, the decoupling of which is required toform two other SF bonds. The remaining two such bondsresult from the decoupling of the secondsymmetry-equivalent p orbital on S.

also are considerably smaller than for genuinethree-center four-electron bonds.

The picture of bonding presented here explainsand rationalizes the results of even much moresophisticated calculations surprisingly well. Onequestion that arises is whether the fact that four,and only four, nonzero eigenvalues are alwaysfound in the diagonalization of the Fermi holeassociated with the heavy atom, is related to thelow flexibility of the AM1 basis set which involvesonly one s and three p orbitals. The answer to thisquestion is ‘‘no,’’ because essentially the sameresults are obtained if analogous calculations arerepeated using the semiempirical SINDO method40

in which d orbitals are involved on the atoms ofthe second row. As Table III shows, the only resultof including d orbitals in the basis is the appear-

Žance of additional nonzero eigenvalues two in PF5and five in SF , which are, however, much smaller6than the four original eigenvalues. This is consis-tent with the generally accepted conclusion that dorbitals play only a negligible role in hypervalentbonding.

Acknowledgment

The authors also thank one of the referees forhelpful suggestions.

TABLE III.Comparison of Eigenvalues of Fermi HolesGenerated by AM1 and SINDO Methods toDemonstrate the Minor Role of d-Orbitalsin Basis Set.

Molecule AM1 SINDO

SF 0.773 1.16060.773 0.8370.659 0.7210.659 0.715

0.2950.289

PF 0.551 0.67850.551 0.6260.551 0.6300.481 0.610

0.1930.1090.1090.0730.073

VOL. 20, NO. 8770


References

1. Coulson, C. A. Proc R Soc 1939, A169, 413.2. Wiberg, K. B. Tetrahedron 1968, 24, 1083.3. Mayer, I. Int J Quantum Chem 1984, 26, 151.4. Jug, K. J Am Chem Soc 1977, 99, 7800.5. Cioslowski, J.; Mixon, S. T. J Am Chem Soc 1991, 113, 4142.6. Gopinathan, M. S.; Jug, K. Theor Chim Acta 1983, 63, 497,

511.7. Jug, K. J Comput Chem 1984, 5, 555.8. Ponec, R.; Strnad, M. Int J Quantum Chem 1994, 50, 43.9. Ponec, R.; Bochicchio, R. Int J Quantum Chem 1995, 54, 99.

Ž .10. Ponec, R.; Uhlik, F. J Mol Struct Theochem 1997, 391, 159.11. Ponec, R.; Mayer, I. J Phys Chem 1997, 101, 1738.12. Ponec, R.; Jug, K. Int J Quantum Chem 1996, 60, 75.13. Bochicchio, R.; Ponec, R.; Uhlik, F. Inorg Chem 1997, 36,

5363.14. Ponec, R. J Math Chem 1997, 21, 323.15. Ponec, R. J Math Chem 1998, 23, 85.16. Wigner, E.; Seitz, F. Phys Rev 1933, 43, 804.17. Boyd, R. J.; Coulson, C. A. J Phys B 1974, 7, 1805.18. Luken, W. L.; Beratan, D. N. Theor Chim Acta 1982, 61, 265.19. Luken, W. L. Croat Chem Acta 1984, 57, 1283.20. Cooper, I. L.; Pounder, C. N. M. Theor Chim Acta 1973, 47,

51.21. Bader, R. F. W.; Stephens, M. E. J Am Chem Soc 1975, 97,

7391.

22. Doggett, G. Mol Phys 1977, 34, 1739.23. Bader, R. F. W. Atoms in Molecules, a Quantum Theory;

Clarendon Press: Oxford, UK, 1994.24. Ruedenberg, K. Rev Modern Phys 1962, 14, 326.25. Dewar, M. J. S.; Zoebish, E. G.; Hailey, E. F.; Stewart, J. J. P.

J Am Chem Soc 1985, 107, 3902.26. Gillespie, R. J.; Robinson, E. A. Inorg Chem 1995, 34, 978.27. Musher, J. J. Angew Chem Int Ed 1969, 8, 54.28. Kwart, H., King, K. G. d-Orbitals in Chemistry of Silicon,

Phosphorus and Sulphur: Reactivity and Structure, Vol 3;Springer: Berlin, 1977.

29. Reed, E. A.; Schleyer, P. v. R. J Am Chem Soc 1990, 112,1434.

30. Cioslowski, J.; Mixon, S. T. Inorg Chem 1993, 32, 3204.Ž .31. Robinson, E. A. J Mol Struct Theochem 1989, 9, 186.

32. Cunningham, T. P.; Cooper, D. L.; Gerratt, J.; Karadakov, P.;Raimondi, M. Int J Quantum Chem 1996, 60, 393.

33. Cooper, D. L.; Cunningham, T. P.; Gerratt, J.; Karadakov, P.;Raimondi, M. J Am Chem Soc 1994, 116, 4414.

34. Cooper, D. L.; Gerratt, J.; Raimondi, M. J Chem Soc PerkinTrans 1989, 2, 1187.

35. Patterson, C. H.; Messmer, R. P. J Am Chem Soc 1990, 112,4138.

36. Cioslowski, J. Int J Quantum Chem 1990, S24, 1833.37. Pimentel, G. C. Physics 1951, 19, 446.38. Hach, R. J.; Rundle, R. E. J Am Chem Soc 1951, 73, 4321.39. Ponec, R.; Uhlik, F. Croat Chem Acta 1996, 69, 941.40. Nanda, D. N.; Jug, K. Theor Chim Acta 1980, 57, 95.


electron pairing and chemical bonds: bonding in hypervalent molecules from analysis of fermi holes

Documents