DOI: 10.1002/chem.200802134

Topology of Electron Charge Density for Chemical Bonds from ValenceBond Theory: A Probe of Bonding Types

Lixian Zhang,[a] Fuming Ying,[a] Wei Wu,*[a] Philippe C. Hiberty,*[b] and Sason Shaik*[c]

Introduction

Molecular electron density is a physical observable thatplays a key role in chemical bonding and chemical reactivity,and as such has drawn much attention from theoreticalchemists.[15] The theory of atoms in molecules (AIM) of

Bader[6] is a powerful topological theory of chemical struc-ture and reactivity. Based on the AIM theory, the electrondensity, 1, and its associated Laplacian, 521, provide impor-tant information on the chemical bonding of molecules. Ifone defines a bond path, the formation of a bond resultsfrom competition between the contraction of electron densi-ty perpendicular to the bond path and the parallel expansionof electron density along the path and away from the intera-tomic surface. In the framework of this theory, the bondingof two atoms may be classified into two groups: shared andclosed-shell interactions, according to the distribution of 1and its associated 521 at the bond critical point (BCP), atwhich the Hessian of 1 has one positive and two negative ei-genvalues. The shared interaction has a large value for 1and a negative value for 521 at the BCP, whereas theclosed-shell interaction has a small value of 1 and a positivevalue of 521 at the BCP. Thus, the values of 1 and 521 atthe BCP can characterize the nature of the bonding.While AIM theory considers the densities, valence bond

(VB) theory provides an alternative, and perhaps comple-mentary, view of chemical bonding, based on the wavefunc-tion that is represented in terms of classical VB structures.In Paulings classical VB theory, electron-pair bonding isclassified as either covalent or ionic, with a continuous spec-trum of intermediate situations stretching between the two

Abstract: To characterize the nature ofbonding we derive the topologicalproperties of the electron charge densi-ty of a variety of bonds based on abinitio valence bond methods. The elec-tron density and its associated Lapla-cian are partitioned into covalent,ionic, and resonance components in thevalence bond spirit. The analysis pro-vides a density-based signature ofbonding types and reveals, along withthe classical covalent and ionic bonds,the existence of two-electron bonds inwhich most of the bonding arises from

the covalentionic resonance energy,so-called charge-shift bonds. As expect-ed, the covalent component of the Lap-lacian at the bond critical point isfound to be largely negative for classi-cal covalent bonds. In contrast, forcharge-shift bonds, the covalent part ofthe Laplacian is small or positive, inagreement with the weakly attractive

or repulsive character of the covalentinteraction in these bonds. On theother hand, the resonance componentof the Laplacian is always negative ornearly zero, and it increases in absolutevalue with the charge-shift character ofthe bond, in agreement with the de-crease of kinetic energy associated withcovalentionic mixing. A new interpre-tation of the topology of the total den-sity at the bond critical point is pro-posed to characterize covalent, ionic,and charge-shift bonding from the den-sity point of view.

Keywords: ab initio calculations bond theory charge-shift bonding electron pairing valence bonds

[a] L. Zhang, F. Ying, Prof. W. WuState Key Laboratory of Physical Chemistry of Solid Surfaces andDepartment of ChemistryCollege of Chemistry and Chemical Engineering, Xiamen UniversityXiamen 361005 (P.R. China)Fax: (+86)592-2184708E-mail : weiwu@xmu.edu.cn

[b] P. C. HibertyLaboratoire de Chimie PhysiqueGroupe de Chimie Thorique, Universit de Paris-Sud91405 Orsay Cdex (France)Fax: (+33)1-69-41-61-75E-mail : philippe.hiberty@lcp.u-psud.fr

[c] Prof. S. ShaikDepartment of Organic Chemistry andThe Lise Meitner-Minerva Center for Computational QuantumChemistryThe Hebrew University, Jerusalem, 91904 (Israel)Fax: (+972)2-6584680E-mail : sason@yfaat.ch.huji.ac.il

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extremes. Covalent bonding is formed by sharing two elec-trons between the two atoms, while ionic bonding is formedby transferring one electron from one atom to the other,and is stabilized by electrostatic attraction between the neg-ative and positive charge centers. Roughly speaking, cova-lent bonds bind together neutral fragments and ionic bondsbind fragments bearing opposite static charges. However,recent modern VB studies[710] have shown that there existsa third class of electron-pair bonding, called charge-shift(CS) bonding, which transcends considerations of staticcharge distributions and is characterized by exceptionallylarge resonance energies associated with the mixing of thecovalent and ionic components. In CS bonds, which may behomopolar as well as heteropolar, the resonance energy isthe major component of the bonding energy, while covalentcoupling and/or ionic attraction are of minor importanceand in some notable cases are altogether repulsive. To illus-trate the striking differences between the CS and covalentbonds, we show in Figure 1 the covalent-only and full disso-ciation energy curves for two typical examples taken fromour previous study.[9] Thus, in H2, which has a typical cova-lent bond, the covalent pairing of the electrons leads to anenergy well close to the full energy well of the bond. Bycontrast, in the F2 molecule, which possesses a typical CSbond, the covalent interaction is completely repulsive andthe entire bonding results from the resonance interaction.Consideration of CS bonds as a specific class of bonds on

its own is not only of abstract theoretical interest. In fact,charge-shift bonding has been shown to display peculiar ex-perimental manifestations, such as depleted electronic densi-ty in the middle of the bond,[9] large barriers, for example inthe HC+FH!HF+HC exchange reaction,[10] the reluctanceof the SiX bond to undergo heterolysis in solution,[11] andso on. As such, classifying the different facets of chemicalbonding requires not only consideration of covalencyionici-ty (by consideration of weights of the corresponding VBstructures), but also the covalentionic resonance energiesof the bonds.In an effort to establish a link between the classifications

of AIM theory and VB theory for two-electron bonding,several researchers have studied the topological propertiesof electron density in diatomic molecules as calculated byab initio VB methods. Cooper and co-workers[12,13] used thespin-coupled VB method to examine the change from cova-lent character at long range to a substantial ionic characterat equilibrium distance. However, both spin-coupled VB aswell as GVB calculations do not allow a clear-cut distinctionto be made between the covalent and ionic components ofthe bond. Rincn and Almedia,[14] being cognizant of thisfeature, projected the GVB wavefunctions of bonds onto abasis of classical VB structures constructed with purely localhybrid orbitals. In so doing, they could calculate the respec-tive coefficients of the covalent and ionic structures for aseries of diatomic molecules, and could quantify the degreeof covalent/ionic character of the corresponding bonds. Sub-sequently, by analyzing the topology of the electron chargedensities, these authors found that in most of the molecules

the shared interaction could be interpreted as a covalentbond, while the closed-shell interaction type was naturallyfound in ionic bonds. There were, however, some outstand-ing exceptions. For example, two polar molecules, HF andLiH, were both found to have a slightly dominant covalentcharacter on the basis of VB structure coefficients, but thetopology of electron densities designated the HF bond ascovalent (shared interaction) and LiH as ionic (closed-shell interaction). Moreover, the NaH bond had a domi-nant covalent character in a VB sense, but a closed-shellcharacter in the AIM sense. Finally, two homonuclear mole-cules, F2 and Cl2, which certainly do not have ionic bonds,were found to have a density topology typical of the closed-shell interaction type. In the face of this contradiction,Rincn and Almedia[14] concluded that the VB theory clas-sification of the electronic pair bond as covalent and ionicand AIMs classification of the atomic interaction as sharedand closed-shell are complementary rather than equivalent.

Figure 1. Dissociation energy curves for H2 (top) and F2 (bottom). *:purely covalent VB structure. &: one of the two identical ionic structures.* denotes the covalent + ionic exact ground state. For F2, the energiesare up-shifted by 198 a.u.

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Moreover, they found that it is impossible to infer topologi-cal properties of the electron density at the BCP from VBstructural coefficients.Thus, the above results mean that for quite a few simple

bonds (like FF, ClCl, HF, NaH, etc.) considerations ofdensities, on the one hand, and of wavefunctions and ener-gies on the other, lead to different classifications of the two-electron bonds. At the same time, for many other bonds, thetwo approaches lead to the same conclusions. This puzzlingdichotomy requires clarification, and this can be accom-plished by linking densities and their Laplacians to the prop-erties of VB wavefunctions in a more detailed way than hasbeen done before. Accordingly, the purpose of this paper isto unify the density-based and VB-based views of electron-pair bonding by analyzing the wavefunctions of 18 represen-tative molecules by using ab initio VB methods, then parti-tioning the so-obtained densities into contributions from co-valent, ionic, and resonance components. Special attentionwill be paid to the contribution of the resonance componentto the Laplacian of electron density, 1, with the purpose offinding the fingerprints of the VB-derived CS bond type interms of the topological properties of the electron density.

Computational Methods

VB methods : A many-electron wavefunction Y in VB theory is ex-pressed as a linear combination of HeitlerLondonSlaterPauling(HLSP) functions, FK, in Equation (1), in which FK corresponds to aclassical VB structure, built with local hybrid atomic orbitals, and CK isthe respective structural coefficient.

Y X

K

CKFK 1

The coefficients CK in Equation (1) are subsequently determined by solv-ing the usual secular equation HC=EMC, in which H and M are the ma-trices of the Hamiltonian and overlap between the VB structures, respec-tively, and E is the total energy. The procedure of the evaluation of Ham-iltonian and overlap matrices is described in detail elsewhere[1517] andwill not be addressed further in this paper.

For a given VB method based on the above wavefunction, the normal-ized structural weights are calculated by means of the familiar ChirgwinCoulson definition,[18] Equation (2), which is used for estimating the con-tribution of a given VB structure to the system.

WK C2K X

L 6KCKCLMKL 2

Obtaining a VB wavefunction of the type in Equation (1) can be ach-ieved by a number of methods: in the VB self-consistent field(VBSCF)[19] procedure, both the VB orbitals and structural coefficientsare optimized simultaneously to minimize the total energy. As such, theVBSCF method is equivalent to the CASSCF method if all the VB struc-tures are involved, and takes care of the static electron correlation, butlacks dynamic correlation. The breathing orbital valence bond (BOVB)method[20] uses different orbital sets during the optimization for differentVB structures. Thus, the resulting orbitals respond to the instantaneouscharges of the individual VB structures rather than to an average field ofall the structures. As has been amply shown,[20] the BOVB method ac-counts for that part of the dynamic electron correlation which varies

along a bond dissociation curve, while at the same time the BOVB wave-function is as compact as in the VBSCF method.

The BOVB method has several levels of sophistication. At the most basicone, all orbitals are strictly localized on their respective fragments. Somefurther accuracy can still be gained by allowing the active doubly-occu-pied orbitals (those involved in the bond in the ionic VB structures) tobe split into two singly-occupied ones, or by allowing the orbitals not in-volved in the bonds that are broken or formed in a chemical process (e.g.dissociation) to be delocalized. This higher level, referred to SD-BOVB(SD= split-delocalized), has been used for the VB calculations through-out this work.

The energies of the ionic structures in Figure 1 are the corresponding di-agonal elements in the BOVB 33 Hamiltonian matrix of the groundstate. The covalent curves, however, represent optimized covalent struc-tures. The covalentionic resonance energies are defined relative to theseoptimized covalent structures, and are hence quasi-variational quanti-ties.[9]

Density properties of VB wavefunctions : Given the many-electron wave-function Y, the electron density of an electronic system is defined asgiven in Equation (3), in which x stands for the spatial space (r) and spin(s) variables, and N is the number of electrons for the system.

1xl NZYx1,x2, . . . xNY*x1,x2, . . . xNdx2dx3 . . . dxN 3

By integrating over spin, the probability of finding an electron in dr1without reference to spin is given by Equation (4).

1r1 Z1x1ds1 4

Combining Equations (1) and (4) gives the density function in the VBframework, Equation (5), in which 1KL(r) is the transition density func-tion, given by Equation (6).

1r X

K,L

CKCL1KLr 5

1KLr NZFKF*Lds1dx2 . . . :dxN 6

A common method to reduce the computational effort required by theVB calculations is to freeze the core electrons with molecular orbitals(MOs) obtained from a HartreeFock calculation, and to treat only thevalence electrons in the VB scheme. With this core-valence separation,the electron density can be partitioned as the sum of the core and va-lence densities, Equation (7).

1KLr MKL1corer 1valKL 7

As we are investigating the nature of single bonds in diatomic AB mol-ecules, the active space to be treated in a VB way is a simple two-elec-tron two-orbital space, which requires only three VB structures: one co-valent and two ionic. The VB wavefunction can thus be written as inEquation (8), in which the subscripts correspond to the covalent andionic situations ACBC, A+B, and AB+ , respectively.

Y CcovFcov Cion1Fion1 Cion2Fion2 8

The electron density can be divided into three components: covalent,ionic, and resonance, as shown in Equation (9); 1cov, 1ion, and 1res aregiven by Equations (10)(12), respectively.

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1 1cov 1ion 1res 9

1cov Wcov1cov 10

1ion W ion11ion1 W ion21ion2 11

1res X

K 6LCKCL1KLMKL1K 12

In the above equations, 1cov, 1ion1, and 1ion2 are diagonal elements definedin Equation (6). In accord, the Laplacian of electron density is also parti-tioned as in Equation (13), in which the three components are defined ina similar fashion to the density, as shown in Equations (14)(16).

r21 r21cov r21ion r21res 13

r21cov Wcovr21cov 14

r21ion W ion1r21ion1 W ion2r21ion2 15

r21res X

K 6LCKCL1KLMKL1K 16

The definition of the resonance electron density, 1res, and that of its asso-ciated Laplacian, 521res, bear some resemblance to the definition of thereduced resonance integral,[21] defined as HKLEKMKL, which is usuallyused for estimating resonance effect in VB theory.

Especially interesting is the topology of the electron density at the BCP.For the molecules studied in this work, the BCP is determined by findingthe point at which the derivative of 1 is zero. Based on the local expres-sion of the virial theorem, Equation (17), in which G(r) is the positivedefinite form of the kinetic energy density and V(r) is the negative poten-tial energy density, one may determine whether the kinetic or the poten-tial energy density dominates the bonding interaction from the sign of521.

h2

4mr21r 2Gr Vr 17

For the closed-shell interaction the kinetic energy density is dominant for521, and thus 521>0, whereas for the standard shared interaction mostof the contribution to 521 is from the potential energy density, and thus521

0.55 to 0.69 in the series C2H6, N2H4, O2H2, F2. This tenden-cy can be rationalized by considering the energies requiredto transform the two neutral fragments (2XC) into an ionpair (X+ +X), that is, the difference between the ionizationpotential (IP) and the electron affinity (EA) of the X frag-ment. As is experimentally known, this difference (IPEA)increases in the series CH3, NH2, OH, F, thus accounting forthe increase of covalent weight in the XX bond in thisseries. The covalent weight is also dominant in Cl2. The twolast homonuclear molecules, Li2 and Na2, depart from theother ones with an almost exclusive covalent character, esti-mated to be 96% in both cases. This almost complete lackof ionic contributions in these two weak bonds can be relat-ed to two factors that differentiate these two moleculesfrom the others: 1) the interatomic equilibrium distances aremore than one Angstrom longer than diatomic molecules ofthe same row, and since the contribution of ionic structuresvanishes at long distances, the ionic weights of these longbonds is marginal; 2) a large part of the bonding energy inNa2 and Li2 is due to dispersion terms (here represented byformally covalent interactions of p type), which are non-ionic.Entries 914 in Table 1 display diatomic heteronuclear

molecules of moderate polarities, for which the covalentweight remains dominant. In these molecules, the weights ofthe two ionic structures are now different, one being verysmall and the other significant, according to the differentelectronegativities of the fragments. In NaH and LiH, thevery unfavorable ionic structures NaH+ and LiH+ werefound to have a small negative weight (0.03) in each case.As negative weights are unphysical [though formally possi-ble according to the definition of weights by Eq. (2)], thesetwo ionic structures were removed in the final VB calcula-tions. In agreement with the conclusions of Rincon and Al-meida,[14] we found that both HF and LiH have a dominantcovalent character. While this statement is rather consensualfor HF, the covalent versus ionic character of LiH has beensubject to great controversy in the past.[6b,2426] Some au-thors[6b] considered LiH as an ionic molecule, in apparentagreement with its high dipole moment (5.90 D). On theother hand, Jug et al.[24,25] concluded from studies using a va-riety of basis sets that HF and LiH should have comparableionicities. Mo et al. also concluded that there is a major co-valent character in LiH,[26] arguing that the covalent struc-ture greatly contributes to the dipole moment, owing to thepolarization of the hybrid atomic orbital of lithium involvedin the covalent bond. Indeed, some of us[7] have argued thatthe nature of the LiH bond depends critically on environ-mental perturbations that can tip it easily towards the ionicor covalent directions. Be that as it may, what matters forthe present study is that for the isolated LiH molecule, thecovalent structure is lower in energy than the ionic one, andthat the resonance energy will be calculated with referenceto the former. This energy ordering of VB structures in LiHpersists at the level of complete configuration interaction ofVB type in the 6-311++G** basis set.[27] Of course, NaHdisplays similar features to LiH.

As the last category of diatomic molecules, the alkali fluo-rides and chlorides (entries 1518 in Table 1) have dominantionic weights, and are referred to in what follows as ionicbonds.

Covalentionic resonance energies : The resonance energies(RE) that arise from the mixing of the covalent and ioniccomponents in the various bonds are displayed in column 7of Table 1. These quantities are defined as the energy differ-ence between the ground state and the lowest-lying VBstructure, hereafter called the dominant structure. The domi-nant structure is the covalent one in homonuclear and mod-erately polar bonds (entries 114 in Table 1), and the lowestionic structure in ionic bonds (last four entries). To appreci-ate the contribution of the resonance energy to the bondingof a molecule, a relative resonance energy (RRE) quantityis also listed in the table, defined as the resonance energy di-vided by the total bonding energy. Thus, for example, anRRE smaller than 50% means that the resonance energy isa minor contribution to the bonding energy (BDE) com-pared to the bonding energy of the dominant structure, andthe bond finds its place in the traditional classification, thatis, as a covalent, polar covalent, or ionic bond, as the casemay be. On the other hand, an RRE exceeding 50% meansthat the resonance energy is the major bonding factor thatlinks the atoms together. In such a case, the bond cannot beconsidered any more as covalent or ionic, and is termed acharge-shift (CS) bond,[9] in reference to the fact that co-valentionic mixing is associated with fluctuation of theelectron pair from the average reference population.It appears from Table 1 that the HH bond (entry 1) is a

perfect archetype of the classical covalent bond, with a reso-nance energy of only 9.2 kcalmol1 in relation to a totalbonding energy of 105.0 kcalmol1, which means that thebond is almost entirely sustained by the covalent interaction(see also Figure 1 above). The other homonuclear bondshave highly variable contributions of the resonance energy.Moving from left to right in the second row of the periodictable, both RE and RRE increase in the series C2H6, N2H4,O2H2, F2. The CC bond is still a classical covalent bond,with a minor contribution of the resonance energy. On theother hand, the NN bond in N2H4 is already a charge-shiftbond, albeit a somewhat limiting case, with an RRE slightlylarger than 50%. Reaching OO and FF, the RRE be-comes even larger than 100%, which means that the reso-nance energy is larger than the total bonding energy, or inother words that the dominant structure, here the covalentone, is unbound at equilibrium geometry. The largestcharge-shift character is found in the FF bond, with anRRE of 184%, meaning that the covalent component has astrongly negative bonding energy (i.e. repulsive) of28.4 kcalmol1 (see also Figure 1). Note that the charge-shift character of homonuclear bonds is a unique property,which is related neither to the weights of the ionic struc-tures, nor to the covalentionic energy gap.[30] For example,it can be seen in Figure 1 that this gap is larger in F2 than in

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H2 at equilibrium distance, but nevertheless it is F2 that hasthe larger RE.As was first mentioned by Sanderson,[28] the weakening of

the covalent bonding energies from left to right in a row ofthe periodic table is a consequence of the interatomic repul-sions exerted by the electron pairs adjacent to the bond, es-pecially when these are lone pairs, hence the name of lone-pair bond-weakening effect (LPBWE).[28,29] As has beenshown in a previous study,[30] the weakening of covalentbonding coincides with an increase of the resonance energy,because the repulsive lone pair-bond interactions have theeffect of increasing the kinetic energy and tipping the virialratio off balance, while covalentionic mixing lowers the ki-netic energy without changing the potential energy toomuch, thus restoring the correct virial ratio. According tothis reasoning, both Li2 and Na2 molecules, which are devoidof side electron pairs, are expected to have fully bonding co-valent components and very weak resonance energies.Table 1 confirms these qualitative predictions, as most of thebonding in the two molecules is covalent, with negligiblecontributions coming from the resonance energy. On theother hand, Cl2, which has six lone pairs, is logically foundto be a CS bond with an unbound covalent component(RRE larger than 100%).Turning to the polar molecules (entries 914), one can see

large differences in the resonance energies, which do notappear to be related to the polarities of these bonds. Thus,HCl, and especially HF, have large resonance energies (34.9and 90.8 kcalmol1, respectively), while all the others haveresonance energies smaller than 11 kcalmol1. With an RREof 73%, HF is clearly classified as a heteronuclear CS bondthat displays some features different from the homonuclearCS bonds. Indeed, in HF, the covalent curve is attractive, asshown in Figure 2, at variance with the repulsive covalentcurves of O2H2 and F2. However, the contribution of cova-lent bonding is rather small, 33.2 kcalmol1 at equilibriumgeometry, whereas the covalentionic resonance energy pro-vides the major component (75%) of the bonding interac-tion. By contrast, BH, AlH, LiH, and NaH bonds aretypical polar covalent bonds, with RREs ranging from 7%

to 15%. Interestingly, this indicates that the bonds in LiHand HF are of different natures, in agreement with the den-sity analyses of Rincon and Almeida.[14]The HCl bond liesbetween these two clear-cut categories, but still on the polarcovalent side. On the other hand, LiF, NaF, LiCl, and NaCl(last four entries) genuinely display the characteristics ofionic bonding, with very low resonance energies.Table 1 also lists the value of the overlap S between the

hybrid atomic orbitals involved in the covalent componentof the bond for the molecules that are not fully ionic(column 6, entries 114). Note that the structure overlap be-tween the covalent and ionic structures is proportional tothe orbital overlap S.[21] Interestingly, the overlaps appear tobe rather large for the covalent or polar covalent bonds, inthe range 0.460.74. On the other hand, charge-shift bondsare characterized by smaller overlaps, in the range 0.210.34, and the respective trends of overlaps and RREs followan inverse relationship: the larger the RRE, the smaller theoverlap. These findings are readily explained: the charge-shift character is favored by the presence of lone pairs,which have the side effect of inducing interatomic repulsions(vide supra). These repulsions prevent the two fragmentsfrom approaching one another to a sufficiently short dis-tance to establish optimal overlap. Thus, the resonance ener-gies, the weakening of the covalent bonding componentsand the orbital overlaps are seen to display consistent trendsaccording to the CS character (or lack of CS character) ofthe different bonds.

Electron densities : Table 2 lists the total electron density 1at BCP and the associated Laplacian 521, as well as theircovalent, ionic, and resonance components. Also listed is thelocation of the BCP, characterized by its distance RBCP fromthe less electronegative atom, which can be compared to theinteratomic distance Req displayed in Table 1. For homonu-clear molecules, the BCP is located at the midpoint betweenthe two atoms, for evident symmetry reasons. On the otherhand, polar bonds have two competing effects: the covalentinteraction that tends to establish the BCP at the midpoint,and the ionic interaction that pushes the BCP towards thepositively charged atom. As a result, the location of theBCP is tipped towards the less electronegative atom in allcases (RBCP

may or may not have a BCP, depending on the computation-al method employed. In the present calculations, Na2 indeedpresents a BCP at the center of the bond, like the other ho-monuclear bonds. On the other hand, the density of Li2 ex-hibits a maximum at the center of the bond, which is there-fore not a BCP, and two minima on both sides of the center.However, since the topological properties of Li2 and Na2 arevery similar at mid-bond, we included the values for Li2 inTable 2, despite the absence of BCP in this case.Let us now consider the total density 1 and its Laplacian

521, and recall the interpretation of these quantities accord-ing to AIM theory.[6] In brief, large values of 1 and largenegative 521 values at BCP are considered to characterizeshared interactions, and are generally interpreted as cova-lent bonding (however, this last interpretation is questiona-ble, as will be seen). By contrast, small values of 1 and largepositive 521 values characterize closed-shell interactions, in-terpreted as ionic bonds for the cases at hand. For homonu-clear bonds, this analysis leads one to classify the HH, CC, and NN bonds as classically covalent, which is nearly inagreement with the VB analysis above, if we consider theNN bond as intermediate between a classically covalentand a CS bond. On the side of heteronuclear molecules, theAIM analysis clearly designates LiF, NaF, LiCl, and NaCl asionic bonds, displaying small 1 values and large positive521. AlH, LiH, and NaH also have the characteristics ofionic bonds, but less marked than in the preceding case,which can be interpreted as partial ionic character. BH,with a large positive 1 and a large negative 521, is interpret-ed as mostly covalent, in agreement with the small electro-negativity difference between boron and hydrogen, and inaccord with the VB weights in Table 1.In contrast to the above classical picture, some of the mol-

ecules displayed in Table 2 do not fit well into the standardAIM classification. Among the homonuclear bonds, Cl2 andO2H2 have large 1 and quasi-zero 521 values and, even

worse, F2 has a large 1 value and a large positive 521, whichmakes these molecules unclassifiable by the standard crite-ria. Among the heteronuclear bonds, the exceedingly largenegative 521 in HF and, to a lesser extent, in HCl, are puz-zling with regards to the significant polarities of these twomolecules, which no one would classify as purely covalent.Thus, it appears that almost one third of the typical bondsexamined in this study cannot be characterized in a clear-cutway from properties of the total density, which calls for amore detailed investigation in terms of partitioned densitiesand Laplacians, according to Equations (9)(16).The covalent, ionic, and resonance components of the

densities and the corresponding Laplacians at BCP are dis-played in Table 2, columns 611. Let us consider the homo-nuclear bond first (entries 18). It can be seen that the cova-lent component of the density, 1cov, stretches between 0.08and 0.13 (except in the cases of Li2 and Na2, which are bothlinked by weak bonds), whereas the ionic component of thedensity, 1ion, is about twice as small. This ratio is in accordwith the ratio of covalent versus ionic weights, but does notdistinguish between covalent and CS bonds. More informa-tive is the covalent component of the Laplacian, 521cov. Itappears from the values displayed in column 7 of Table 2that 521cov is largely negative for H2 and C2H6, two typicalclassical covalent bonds, then becomes quasi-zero for N2H4,the limiting case between covalent and charge-shift bonds,and finally becomes largely positive for O2H2 and F2, twotypical charge-shift bonds in which the covalent componentis repulsive.Interestingly, the 521cov values correlate very well with

the covalent bonding energy components for all the homo-nuclear bonds, which rank in the order: H2, C2H6 (stronglybonding)>N2H4, Li2, Na2 (weakly bonding)>O2H2, Cl2, F2(repulsive). This is illustrated in Figure 3, in which 521cov isplotted against the covalent bonding energy Dcov, showingthat the correlation is practically linear, especially if one

Table 2. Topological properties of the electron density at the bond critical point.

Molecule Ground State Cov Ion ResRBCP RBCP/Req 1 521 1cov 521cov 1ion 521ion 1res 521res

HH 0.3669 0.5 0.27 1.39 0.18 0.70 0.08 0.37 0.02 0.31H3CCH3 0.7625 0.5 0.25 0.62 0.14 0.26 0.09 0.01 0.03 0.36H2NNH2 0.7395 0.5 0.29 0.54 0.16 0.02 0.08 0.16 0.05 0.68HOOH 0.7410 0.5 0.26 0.02 0.13 0.46 0.08 0.27 0.05 0.75FF 0.7168 0.5 0.25 0.58 0.13 1.00 0.07 0.41 0.05 0.83ClCl 1.0091 0.5 0.14 0.01 0.08 0.14 0.04 0.13 0.03 0.26LiLi[a] 1.3866 0.5 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00NaNa 1.5853 0.5 0.01 0.00 0.01 0.00 0.00 0.00 0.00 0.00HF 0.1686 0.18 0.38 2.52 0.25 1.82 0.10 0.18 0.03 0.52HCl 0.4058 0.32 0.26 0.81 0.16 0.33 0.07 0.06 0.03 0.42BH 0.5301 0.43 0.19 0.61 0.15 0.59 0.04 0.04 0.00 0.01AlH 0.7986 0.49 0.07 0.21 0.06 0.07 0.02 0.13 0.00 0.01LiH[b] 0.7315 0.45 0.03 0.15 0.03 0.09 0.01 0.06 0.00 0.00NaH[b] 0.9080 0.47 0.03 0.12 0.02 0.07 0.01 0.05 0.00 0.00LiF 0.6219 0.39 0.07 0.62 0.02 0.12 0.04 0.51 0.01 0.01NaF 0.9193 0.46 0.05 0.37 0.02 0.08 0.03 0.27 0.00 0.02LiCl 0.7019 0.34 0.04 0.24 0.02 0.07 0.01 0.16 0.00 0.00NaCl 0.9893 0.41 0.03 0.18 0.01 0.06 0.01 0.13 0.00 0.00[a] Li2 displays no BCP at the center of the bond. [b] 6-31++G** basis set.

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considers molecules belonging to the same row of the peri-odic table.It is interesting to interpret these tendencies with refer-

ence to Equation (17) above, which shows that 521cov at theBCP can be decomposed into a positive kinetic componentand a negative potential component. Thus, any increase (de-crease) of the kinetic energy as the covalent interactiontakes place will result in a positive (negative) value of521cov. Now, as demonstrated elsewhere,[30] and recalledabove, a repulsive covalent interaction, as the one found ina charge-shift bond, increases the kinetic energy, while abonding covalent interaction has the reverse effect of de-creasing the kinetic energy. In full agreement with this pre-diction, it can be seen that positive values of 521cov coincidewith repulsive covalent interactions (F2, Cl2, O2H2), whilenegative values coincide with significantly bonding covalentinteractions (H2, C2H6). In accord, if we divide Figure 3 intoquadrants by tracing dashed lines emanating horizontallyand vertically, from 521cov=0 and Dcov=0, respectively, theplot of 521cov versus Dcov will clearly mark two regions, withthe CS bonds lying on the top left-hand side, and the classi-cal covalent bonds on the bottom right-hand side. Clearlytherefore, both the sign and the energy aspects associatedwith the 521cov quantity give the same conclusions about co-valent versus CS-bonding.The values of the resonance components of the Laplacian

521res, are reported in the last column of Table 2, and plot-ted against the resonance energy in Figure 4. Interestingly,these values are negative or nearly zero in all the cases,which, according to Equation (17) and the reasoning above,should correspond to a decrease of the kinetic energy uponbond formation. This is a nice confirmation of our recentfinding,[30] based on a VB decomposition of the bondingenergy in two-electron bonds, that covalentionic resonance

causes a decrease of the kinetic energy without significantlyaffecting the potential energy. The magnitudes of the 521resvalues display a good correlation with the resonance ener-gies of the homonuclear molecules, if they are consideredwithin their specific row in the periodic table, as shown inFigure 4 (top), in which the points corresponding to second-row molecules are interconnected by straight segments.Thus, 521res increases regularly the from C2H6 to F2, like theresonance energies in the same series, with a quasi-straightcorrelation line. Likewise, Cl2 has a large negative 521resand large resonance energy (48.7 kcalmol1). Finally, 521reshas zero values in Li2 and Na2, in agreement with the negli-gible resonance energies of these two molecules. As awhole, the values of 521cov and 521res in homonuclear mole-cules vary in a consistent way along a series and confirm themechanism, already mentioned above, by which charge-shiftbonding compensates the increase of kinetic energy due tothe repulsive covalent component by a concomitant de-crease of kinetic energy due to resonance.[30]

The ionic components of the Laplacians, 521ion, are posi-tive for N2H4, O2H2, F2, and Cl2, as expected according toAIMs views of ionic interactions. On the other hand, thevery small value of 521ion for C2H6, and especially the nega-tive value for H2, are puzzling. These may be due to the

Figure 3. A plot of the covalent component of the Laplacian versus Dcov,the bonding energy of the covalent component of the wavefunction forhomonuclear bonds. Dcov is calculated as the difference between the totalbonding energy BDE and the resonance energy RE, given in Table 1.

Figure 4. Plots of the resonance component of the Laplacian versus theresonance energy. Top: Homonuclear bonds. Bottom: Heteronuclearbonds.

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W. Wu, P. C. Hiberty, S. Shaik et al.

large overlap that exists between the covalent and ionic VBstructures in these two cases (this overlap is proportional tothe orbital overlap S), making the ionic structure stronglyresemble the covalent one. Be that as it may, the combina-tion of positive covalent and ionic Laplacians in the case ofdefinite CS bonds (entries 46, Table 2) compensates for thenegative resonance Laplacian, so that the full Laplacian(521) becomes either close to zero or positive; this appearsto be a signature of charge-shift bonding in homonuclearbonds.In the heteronuclear molecules, there is no good correla-

tion between 521cov and the bonding energy of the covalentcomponent of the bond, perhaps because the contribution ofthe ionic form has a large effect on the equilibrium distance,which is therefore very different from the optimal distancefor the covalent interaction. The same remark holds for521ion, except for the ionic molecules (last four entries inTable 2), for which 521ion is large and positive, and close tothe Laplacian of the total density, 521. Thus, the two quanti-ties 521cov and 521ion have a clear physical meaning only ifthe bond is either homonuclear (covalent and CS bonds), orfully ionic, but not in the intermediate heteronuclear cases(entries 914, Table 2). The large negative value of 521cov inHF, 1.82, is intriguing for a CS bond. Thus, based on a su-perficial analogy with homonuclear CS bonds (e.g. FF andHOOH) one might have expected a small or positive valuethat would signify a weak or repulsive covalent interaction.However, the HF bond lacks repulsive character in the co-valent component due to the fact that all the lone pairsbelong to the same atom, so that the covalent VB structureinvolves no interatomic lone pair repulsions. Thus, a nega-tive value of 521cov in a heteronuclear CS bond is not sur-prising, even if its magnitude is somewhat unexpected. Onthe other hand, the resonance component of the Laplacian,521res, still correlates well with the resonance energies in allheteronuclear molecules, as in the homonuclear case(Figure 4, bottom). Indeed, the only significant values of521res are found for HCl and HF (0.42 and 0.52, respec-tively), which have clearly larger resonance energies thanthe other heteronuclear molecules of the series.Based on the above analysis of partitioned densities, we

can now better understand the total densities 1 and theirLaplacians, 521, as well as their relationship to the natureof bonding in the set of representative molecules. Before in-terpreting the properties of total densities, an essential firststep is to identify possible charge-shift bonds. For homonu-clear molecules, they are characterized by a combination oflarge positive 1 and positive or zero 521 (small 1 and small521 simply means that we are dealing with a weak bond).For heteronuclear molecules, charge-shift bonds possess acombination of large negative 521 and a small RBCP/Reqratio. According to this criterion, HF is seen to be a CSbond, and HCl also has some CS character, although to alesser extent, at variance with all other heteronuclear mole-cules. Finally, for the molecules that do not have the charac-teristics of CS bonding, the standard AIM interpretation ap-plies well: large values of 1 and large negative 521 values

characterize classical covalent bonds, and small values of 1and large positive 521 values characterize ionic bonds. Inaccord, LiF, NaF, LiCl, and NaCl are viewed as fully ionicbonds, BH is a covalent bond, and AlH, LiH, and NaH aremixed covalentionic bonds. All the information that arisesfrom the values of 1 and 521 is in qualitative agreementwith the analysis based on VB calculations, but an importantissue is that total densities can be calculated by any accuratecomputational method, and thus can provide insight of theVB type independently of any VB method.Finally, one may wonder if the 521res quantity can be re-

lated in a global manner to a single fundamental property ofthe atomic constituents of the bonds. As we argued recent-ly,[9,31] and here too, bonded atoms that are either highlyelectronegative and/or possessing a large number of lonepairs will generally possess large RE. Thus, a general rela-tionship was shown to exist between RE and the averageelectronegativity of the bonded atoms for a series of homo-and heteronuclear bonds (excluding purely ionic bonds): thelarger the average electronegativity, the larger the RE.[9,31]

Figure 5 complements this trend by showing a plot of the521res quantity against the average electronegativity of thebonded atoms. The purely ionic bonds, represented by greydots in Figure 5, exhibit the expected trend corresponding totheir low RE, with 521res being close to zero. For all theother bonds (black dots in Figure 5), one can see a generaltrend essentially similar to the behavior of RE versus theaverage electronegativity.[9,31] Thus, we find a general corre-lation that as the average electronegativity increases, the521res quantity becomes more negative and reaches its larg-est negative values for the CS bonds HF, OO, FF, at thebottom right hand corner of the figure; these bonds, whichpossess large REs, are characterized also by large averageelectronegativity and a large negative value of the 521rescomponent of the Laplacian.

Figure 5. A plot of the resonance component of the Laplacian vs. theaverage electonegativity of AB bonds.

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Conclusion

The topological properties of electron charge density in aseries of 18 molecules, ranging from weak to strong bondsand from homonuclear to strongly polar bonds, have beenstudied by using an ab initio valence bond approach. It hasbeen shown that a new insight is provided by partitioningthe electron density and its associated Laplacian into threecomponents: covalent, ionic, and resonance. The covalentcomponent of the Laplacian, 521cov, correlates very wellwith the purely covalent bonding energy in a series of ho-monuclear bonds, being negative when the covalent interac-tion is attractive, and positive when the covalent interactionis repulsive, as in O2H2 or F2. This confirms, now from thedensity point of view, a previous finding that repulsive cova-lent interactions, due to the lone-pair bond-weakeningeffect, correspond to an increase of the kinetic energy. Onthe other hand, the resonance component of the Laplacian,521res, increases in absolute value as the resonance energyincreases in a series, and is always negative, confirming thatcovalentionic resonance energy corresponds to a loweringof the kinetic energy in the bonding region. Thus, the topo-logical properties of the density representation confirm thereality of charge-shift bonding (CS bonding), a specific cate-gory of bonds in which the covalent contribution is weak orrepulsive, with most of the bonding due to the covalentionic resonance energy. These density probes also confirmthe respective influences of covalent interaction and reso-nance energy on the kinetic energy upon bond formation.Taking CS bonding into account facilitates the interpreta-

tion of the total density 1 and its Laplacian 521 at the BCP.In terms of total densities at the BCP, the signature of CSbonding is a combination of large 1 and small or positive521 values for homonuclear bonds, and a combination ofnegative 521 and proximity of the BCP to the less electro-negative atom (small RBCP/Req) for heteronuclear bonds. Inall other cases, large values of 1 and large negative 521values characterize classical covalent bonds, and smallvalues of 1 and large positive 521 values characterize ionicbonds; intermediate features indicate a mixed covalentionic bond without charge-shift character.In summary, CS bonding was first characterized by VB-re-

lated quantities, as resonance energies and dissociationenergy curves of covalent VB structures.[7,8] This conceptwas then re-established by electron localization function(ELF) analysis of density functional calculations.[9] The pres-ent work characterizes CS bonding by yet another point ofview, that of total electron density, which can be obtainedby any computational method, independently of VB theory.CS bonding is a distinct category of bonding with clear theo-retical signatures and experimental manifestations.[9]

Acknowledgement

The research at XMU is supported by the Natural Science Foundation ofChina (20873106 and 20533020) and The National Basic Research Pro-

gram of China (2004CB719902). S.S. acknowledges support by an ISFgrant (16/06).

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Published online: February 3, 2009

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