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JChemEd.chem.wisc.edu • Vol. 79 No. 9 September 2002 • Journal of Chemical Education 1141

Understanding and Interpreting Molecular Electron DensityDistributionsC. F. Matta and R. J. Gillespie*Department of Chemistry, McMaster University, Hamilton, ON L8S 4M1, Canada; *[email protected]

Advances in the power and speed of computers have madeab initio and density functional theory (DFT) calculationsan almost routine procedure on a PC (1). From these calcu-lations the equilibrium geometry, the energy, and the wavefunction of a molecule can be determined. From the wavefunction one can obtain all the properties of the molecule,including the distribution of electronic charge, or electrondensity. However, the electron density is often not calculatedor discussed, perhaps because it is not widely realized thatvery useful information on bonding and geometry can beobtained from it. It seems particularly important to discusselectron densities in introductory chemistry courses becausestudents can grasp the concept of electron density much morereadily than the abstract mathematical concept of an orbital. Itis also not as widely understood as it should be that orbitalsare not physical observables but only mathematical constructsthat cannot be determined by experiment (2). In contrast, theelectron density distribution in a molecule or crystal can beobserved by electron diffraction and X-ray crystallography(3); and it can also, and often more readily, be obtained fromab initio and density functional theory calculations.

This article gives a simple introduction to the electrondensities of molecules and how they can be analyzed to ob-tain information on bonding and geometry. More detaileddiscussions can be found in the books by Bader (4 ), Popelier(5), and Gillespie and Popelier (6 ). Computational detailsto reproduce the results presented in this paper are presentedin Appendix 1.

The Electron Density

Quantum mechanics allows the determination of theprobability of finding an electron in an infinitesimal volumesurrounding any particular point in space (x,y,z); that is, theprobability density at this point. Since we can assign a prob-ability density to any point in space, the probability densitydefines a scalar field, which is known as the probability densitydistribution. When the probability density distribution ismultiplied by the total number of electrons in the molecule,N, it becomes what is known as the electron density distributionor simply the electron density and is given the symbol ρ(x,y,z).It represents the probability of finding any one of the Nelectrons in an infinitesimal volume of space surroundingthe point (x,y,z), and therefore it yields the total number ofelectrons when integrated over all space. The electron densitycan be conveniently thought of as a cloud or gas of negativecharge that varies in density throughout the molecule. Such acharge cloud, or an approximate representation of it, is oftenused in introductory texts to represent the electron densityψ2 of an atomic orbital. It is also often used incorrectly todepict the orbital ψ itself. In a multielectron atom or moleculeonly the total electron density can be experimentally observedor calculated, and it is this total density with which we are

concerned in this paper. A more formal discussion of electrondensity is presented in Appendix 2.

The electron density is key to the bonding and geometryof a molecule because the forces holding the nuclei togetherin a molecule are the attractive forces between the electrons andthe nuclei. These attractive forces are opposed by the repulsionsbetween the electrons and the repulsions between the nuclei.In the equilibrium geometry of a molecule these electrostaticforces just balance. The fundamentally important Hellman–Feynman theorem (4–7) states that the force on a nucleus in amolecule is the sum of the Coulombic forces exerted by the othernuclei and by the electron density distribution ρ. This meansthat the energy of interaction of the electrons with the nucleican be found by a consideration of the classical electrostaticforces between the nuclei and the electronic charge cloud. Thereare no mysterious quantum mechanical forces, and no otherforce, such as the gravitational force, is of any importance inholding the atoms in a molecule together. The atoms are heldtogether by the electrostatic force exerted by the electronic chargeon the nuclei. But it is quantum mechanics, and particularlythe Pauli principle, that determines the distribution of elec-tronic charge, as we shall see.

The Representation of the Electron Density

The electron density (ρ) varies in three dimensions (i.e.,it is a function of the three spatial coordinates [x,y,z]), so afull description of how ρ varies with position requires a fourthdimension. A common solution to this problem is to show howρ varies in one or more particular planes of the molecule.Figure 1a shows a relief map of the electron density, ρ, of theSCl2 molecule in the σv(xz) plane. The most striking featuresof this figure are that ρ is very large in an almost sphericalregion around each nucleus while assuming relatively verysmall values, and at first sight featureless topology, betweenthese nuclear regions. The high electron density in the nearlyspherical region around each nucleus arises from the tightlyheld core electrons; the relatively very small and more diffusedensity between these regions arises from the more weaklyheld bonding electrons. In fact, it was necessary to truncatethe very high maxima in Figure 1a (at ρ = 2.00 au)1 to makeit possible to show the features of the electron density distri-bution between the nuclei. In particular, there is a ridge ofincreased electron density between the sulfur atom and eachof the chlorine atoms. The electron density has values of3.123 × 103 and 2.589 × 103 au at the S and Cl nuclei, re-spectively, but a value of only 1.662 × 10�1 au at the minimumof the ridge between the peak around the sulfur nucleus andeach of the chlorine nuclei. This ridge of increased electrondensity between the S atom and each of the Cl atoms, small asit is, is the density in the bonding region that is responsible forpulling the nuclei together. Along a line at the top of this ridgethe electron density is locally greater than in any direction

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1142 Journal of Chemical Education • Vol. 79 No. 9 September 2002 • JChemEd.chem.wisc.edu

away from the line. This line coincides with the bond betweenthe S and Cl atoms as it is normally drawn and is called abond path (4–6, 10). The point of minimum electron densityalong the bond path is called the bond critical point.

Figure 1b shows a relief map of the electron density of thewater molecule. Its features are similar to those of the densitymap for SCl2, but the electron density around the hydrogenatoms is much smaller than around the oxygen atom, as wewould expect. The electron density at the maximum at theoxygen nucleus has a value of 2.947 × 102 au, whereas thatat the hydrogen nucleus is only 4.341 × 10�1 au, which is onlyslightly greater than the value of 3.963 × 10�1 au at theminimum at the bond critical point. The very small electrondensity surrounding the hydrogen nucleus is due to less thanone electron because the more electronegative oxygen atomattracts electron density away from the hydrogen atom so thatit has a positive charge.

Another common way to represent the electron densitydistribution is as a contour map, analogous to a topographiccontour map representing the relief of a part of the earth’ssurface. Figure 2a shows a contour map of the electron densityof the SCl2 molecule in the σv(xz) plane. The outer contour hasa value of 0.001 au, and successive contours have values of2 × 10n, 4 × 10n, 8 × 10n au; n starts at �3 and increases insteps of unity. Figure 2b shows a corresponding map for theH2O molecule. Again we clearly see the large concentration ofdensity around each nucleus. The outer contour is arbitrarybecause the density of a hypothetical isolated molecule extendsto infinity. However, the 0.001 au contour corresponds ratherwell to the size of the molecule in the gas phase, as measuredby its van der Waals radius, and the corresponding isodensitysurface in three dimensions usually encloses more than 99%

of the electron population. Thus this outer contour showsthe shape of the molecule in the chosen plane. In a condensedphase the effective size of a molecule is a little smaller. Wesee more clearly here that the bond paths (the lines alongthe top of the density ridges between the nuclei) coincidewith the bonds as they are normally drawn.

Figure 3 shows the electron density contour maps forthe period 2 fluorides LiF, BF3, CF4, OF2, and for the iso-lated B atom. In LiF each atom is almost spherical, consis-tent with the usual model of this molecule as consisting ofthe ions Li+ and F �. The volume of the lithium atom is muchsmaller than that of the F atom, again consistent with the ionicmodel. We will see later that we can also obtain the atomiccharges from the electron density and that the charges on thetwo atoms are almost ±1, again consistent with the represen-tation of these atoms as ions. Moreover, there is a very smalldistortion of the almost spherical density of each atom towardits neighbor, giving a very low ridge of density between thetwo nuclei indicating that the amount of electronic chargein the bonding region is very small. Thus the bonding in thismolecule is close to the hypothetical purely ionic model,which would describe the molecule as consisting of twospherical ions held together by the electrostatic force betweentheir opposite charges.

As we proceed across period 2 the electron density ofthe core of each atom remains very nearly spherical but the

Figure 1. Relief maps of the electron density of (a) SCl2 and (b) H2Oin the plane of the nuclei (density and distances from the origin ofthe coordinate system in au). Isodensity contour lines are shown inthe order 0.001, 0.002, 0.004, 0.008 (four outermost contours);0.02, 0.04, 0.08 (next three); 0.2, 0.4, 0.8 (next three). The densityis truncated at 2.00 au (innermost contour). These contours areshown in blue, violet, magenta, and green, respectively, on thefigure in the table of contents (p 1028).

Figure 2. Contour maps of the electron density of (a) SCl2 and (b)H2O. The density increases from the outermost 0.001 au isodensitycontour in steps of 2 × 10n, 4 × 10n, and 8 × 10n au with n start-ing at �3 and increasing in steps of unity. The lines connecting thenuclei are the bond paths, and the lines delimiting each atom arethe intersection of the respective interatomic surface with the planeof the drawing. The same values for the contours apply to subse-quent contour plots in this paper.

a

b




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outer regions of the atom become increasingly distorted froma spherical shape, stretching out toward the neighboring atomto give an increased electron density at the bond critical point(ρb) (see Table 1). Figure 4 shows the electron density plots

for some chlorides of period 2. We see similar changes in theelectron density distribution for these molecules as we sawfor the fluorides.

For a three-dimensional picture of the electron densitydistribution we can easily show a particular isodensity envelope(i.e., a three dimensional surface corresponding to a givenvalue of the electron density). The 0.001-au envelope gives apicture of the overall shape of the molecule as shown by theexamples in Figure 5. Making the outer 0.001-au envelopetransparent as in Figure 5 reveals an inner envelope, butshowing additional envelopes becomes increasingly difficult.The particular inner surface shown in Figure 5 corresponds tothe bond critical-point isodensity envelope (ρb envelope), thesingle envelope just encompassing all the nuclei. All isodensityenvelopes with ρ < ρb will form a continuous sheath ofdensity surrounding all the nuclei in the molecule, and allisodensity envelopes with ρ > ρb will form a discontinuoussurface surrounding each nucleus separately. Thus the ρbenvelope is just about to break into separate surfaces, onesurrounding each atom, at higher values of ρ.

The ρb envelopes are also shown for some period 2fluorides and chlorides in Figure 6. These surfaces show thedistortion of the electron density from a spherical shape evenmore clearly than the contour maps in Figures 3 and 4. Forexample, in Figure 6 one can see the distinctly tetrahedralshape assumed by the part of the ρb envelope surrounding thecarbon atom in CCl4 owing to the distortion of the electron

Figure 3. Contour maps of the electron density of LiF, CF4, a freeground state boron atom, BF3, OF2 in the σv(xz) plane (the planecontaining the three nuclei) and in the σv′ (yz) plane (the planebisecting ∠ FOF perpendicularly to the σv[xz] plane). (See legendto Fig. 2 for contour values.)

NOTE: Data, taken from ref 6, were obtained using DFT/B3LYPfunctional and a 6-311+G(2d,p) basis set.

2doireProfataDdetaleRdnashtgneLdnoB.1elbaTsedirolhCdnasediroulF

eluceloMmp/htgneLdnoB

ρb ua/q ua/

dclaC ltpxE X A

FiL 3.751 4.651 570.0 � 29.0 � 29.0

FeB 2 8.731 0.041 541.0 � 88.0 � 57.1

FB 3 4.131 7.031 712.0 � 18.0 � 34.2

FC 4 6.231 9.131 903.0 � 16.0 � 54.2

FN 3 2.831 5.831 413.0 � 82.0 � 38.0

FO 2 4.041 5.041 592.0 � 31.0 � 72.0

F2 9.931 8.141 882.0 � 0 � 0

lCiL 2.202 1.022 740.0 � 19.0 � 19.0

lCeB 2 8.971 — 790.0 � 48.0 � 86.1

lCB 3 0.571 2.471 751.0 � 46.0 � 39.1

lCC 4 7.971 1.771 281.0 � 90.0 � 53.0

lCN 3 1.971 9.571 671.0 � 80.0 � 42.0

lCO 2 8.271 0.071 481.0 � 32.0 � 64.0

lCF 5.661 8.261 781.0 � 83.0 � 83.0

Figure 4. Contour maps of the electron density of LiCl, BCl3, SCl2in the σv(xz) plane (the plane containing the three nuclei) and inthe σv′(yz) plane [the plane bisecting ∠ Cl-S-Cl perpendicularly to theσv(xz) plane], NCl3 in the σv plane (plane containing one N–Clbond and bisecting ∠ Cl-N-Cl formed by the remaining two bonds),and Cl2. (See legend to Fig. 2 for contour values.)








density in the four tetrahedral directions. In addition to thedistortion of the electron density toward each neighboringatom we can see other changes. Proceeding across period 2the ligand atoms have an increasingly squashed “onion” shape,flattened on the opposite side from the central atom. Thesechanges can be understood in the light of the Pauli principle,which is an important factor in determining the shape of theelectronic charge cloud. The Pauli principle is discussedbelow and more formally in Appendix 3.

The Pauli Principle

The many-electron wave function (Ψ) of any system isa function of the spatial coordinates of all the electrons andof their spins. The two possible values of the spin angularmomentum of an electron—spin up and spin down—aredescribed respectively by two spin functions denoted as α(ω)and β(ω), where ω is a spin degree of freedom or “spincoordinate”. All electrons are identical and therefore indis-tinguishable from one another. It follows that the interchangeof the positions and the spins (spin coordinates) of any twoelectrons in a system must leave the observable properties ofthe system unchanged. In particular, the electron density mustremain unchanged. In other words, Ψ2 must not be altered

when the space and spin coordinates of any two electronsare interchanged.

This requirement places a restriction on the many-electronwave function itself. Either Ψ remains unchanged or it mustonly change sign. We say that Ψ must be either symmetric orantisymmetric with respect to electron interchange. In fact, onlyantisymmetric wave functions represent the behavior of anensemble of electrons. That the many-electron wave functionmust be antisymmetric to electron interchange (Ψ → � Ψon electron interchange) is a fundamental nonclassical prop-erty of electrons. They share this property with other elemen-tary particles with half-integral spin such as protons, neutrons,and positrons, which are collectively called fermions. Ensemblesof other particles, such as the α particle and the photon, havesymmetric many-particle wave functions (Ψ → Ψ on particleinterchange) and are called bosons.

The requirement that electrons (and fermions in general)have antisymmetric many-particle wave functions is calledthe Pauli principle, which can be stated as follows:

A many-electron wave function must be antisymmetricto the interchange of any pair of electrons.

No theoretical proof of the Pauli principle was given originally.It was injected into electronic structure theory as an em-pirical working tool. The theoretical foundation of spin wassubsequently discovered by Dirac. Spin arises naturally in thesolution of Dirac’s equation, the relativistic version ofSchrödinger’s equation.

A corollary of the Pauli principle is that no two electronswith the same spin can ever simultaneously be at the samepoint in space. If two electrons with the same spin were at thesame point in space simultaneously, then on interchangingthese two electrons, the wave function should change sign asrequired by the Pauli principle (Ψ → � Ψ). Since in this case thetwo electrons have the same space and spin coordinates (i.e.,

Figure 5. Three-dimensional isodensity envelopes of (a) SCl2, (b)H2O, and (c) Cl2. The outer envelope has the value of 0.001 au, thevan der Waals envelope; the inner one is the bond critical pointdensity envelope (ρb-envelope).

c

a

b

Figure 6. Three-dimensional isodensity envelopes of the bond criticalpoint density (ρb-envelope) of LiF, BF3, NF3, BeCl2, CCl4, and OCl2.








are indistinguishable), the many-electron wave function willbe unaffected by their interchange (Ψ → Ψ). This amountsto saying that if two electrons have the same space and spincoordinates, then Ψ = � Ψ, and the only way this could betrue is for Ψ to vanish. In other words, it is impossible to findelectrons of the same spin at the same location at the sametime. However, this restriction does not apply for electronsof opposite spins. Electrons of opposite spin can be at thesame point in space simultaneously.

Since two electrons of the same spin have a zero prob-ability of occupying the same position in space simulta-neously, and since ψ is continuous, there is only a small prob-ability of finding two electrons of the same spin close to eachother in space, and an increasing probability of finding theman increasingly far apart. In other words the Pauli principlerequires electrons with the same spin to keep apart. So themotions of two electrons of the same spin are not independent,but rather are correlated, a phenomenon known as Fermicorrelation. Fermi correlation is not to be confused with theCoulombic correlation sometimes referred to without its quali-fier simply as “correlation”. Coulombic correlation resultsfrom the Coulombic repulsion between any two electrons,regardless of spin, with the consequent loss of independenceof their motion. The Fermi correlation is in most cases muchmore important than the Coulomb correlation in determiningthe electron density.

Electron Distribution in an OctetAs a simple but very important example, consider an

atom with a valence shell octet of electrons, four of onespin (α electrons) and four of the opposite spin (β electrons).

The most probable distribution of the four α electrons—thedistribution that keeps them as far apart as possible—is atthe vertices of a tetrahedron (Fig. 7a). The most probablearrangement of the four β electrons is also at the vertices of atetrahedron (Fig. 7b). In a free atom these two tetrahedraare independent, so they can have any relative orientationgiving, an overall spherical density.

In a molecule, electrons are attracted toward a neigh-boring nucleus but only electrons of opposite spin can beattracted close to each other. In the valence shell of any singlybonded ligand, not all the electrons are formed into pairs, aswas first pointed out by Linnett (11) and discussed in refs 6and 12. Thus in a diatomic molecule such as HF or ClF thetwo tetrahedra of α- and β-spin electrons on each atom arebrought into approximate coincidence at one vertex, forming abonding pair of electrons and increasing the electron densityin this region (Fig. 8).

Figure 8 shows only the most probable position of theelectrons, not their actual position. The increase in the prob-ability of finding an electron in the bonding region dependson the attractive force exerted by the neighboring atom—that is, on its electronegativity. Thus the extent to which twoelectrons of opposite spin are localized in the bonding regionvaries from molecule to molecule, with the electronegativitiesof the atoms. In diatomic molecules the two tetrahedra oneach atom are still free to rotate around this shared vertex,so that the six nonbonding electrons are most probablydistributed in a circle around the direction of the bond andon the opposite side from the bond, forming a torus ofincreased density. This leaves a region of relatively depletedelectron density at the back of the atom opposite the bond,which accounts for the flattening of the surface of constantelectron density in this nonbonding region (Fig. 6). In the Cl2molecule, for example, the distance to the outer 0.001-aucontour from a Cl nucleus in a direction perpendicular to thebond is 210 pm, but in the direction opposite to the bond ithas the appreciably smaller value of 185 pm. This characteristicshape is found for any singly bonded atom whose electrondensity is deformed toward a neighboring atom. It can alsobe clearly seen in the contour maps in Figures 3 and 4.

Electron Density and the VSEPR ModelWhen there are two ligands, as in H2O, SCl2, OF2, or any

other AX2E2 molecule, the two tetrahedra (Figs. 7a and 7b) ofsame-spin electrons lose their independence of one another andare brought into approximate coincidence at all four vertices(Fig. 7c). This results in the formation of two partially localizedbonding pairs (one to each of the two ligands X) and twopartially localized nonbonding pairs (E); the four pairs adoptan approximately tetrahedral geometry (Fig. 7c). There istherefore a concentration of electron density in the bondingregions, which we observe as the distortion of the sphericalelectron density distribution in the bonding directions as seenin the σv(xz) plane of SCl2 and H2O (Fig. 2) and OF2 (Fig. 3).The increased electron density due to the two localized lonepairs is seen as bulges in the electron density in the σv′(yz)plane of OF2 (Fig. 3) and SCl2 (Fig. 4). In OF2 the distanceto the outer 0.001 au contours from an F nucleus in a direc-tion perpendicular to the bond pointing toward the open sideof the molecule in the σv(xz) plane is 166 pm, but in thedirection opposite to the bond it is 155 pm. In SCl2 those

Figure 7. Pauli principle for octet. (a) The most probable relativearrangement of four α-spin electrons. (b) The most probable relativearrangement of four β-spin electrons. Both arrangements may adoptany relative orientation in space. (c) In the presence of the nuclei oftwo combining ligands (X1, and X2), as in H2O or SCl2, the twotetrahedra are brought into approximate coincidence at two apexes(sufficient to bring all of the four apexes into coincidence) formingtwo bonding pairs and two nonbonding pairs (E).

AA

α β αβ

a b c

E

E

X2

X1

A

bondingpairs

non-bonding

pairs

Figure 8. Pauli principle for a diatomicmolecule (e.g., HF). In any diatomic mol-ecule, the two tetrahedra (Figs. 7a and 7b)of opposite spin electrons in the valenceshell of an atom are brought into coinci-dence at only one apex, leaving the mostprobable locations of the remaining sixelectrons equally distributed in a ring.

FH






values are 212 and 186 pm, respectively. In the σv(xz) planeof the NCl3 molecule (Fig. 4) we see the distortion of theelectron density toward the chlorine ligand and a bulge inthe electron density in the lone pair region. The influence ofthe Pauli principle on the distribution of electrons in a valenceshell provides the physical basis for the VSEPR model (6,12), according to which the electrons in the valence shell ofan atom are in pairs, either bonding or nonbonding, that stayas far apart as possible.

The Atoms in Molecules Theory

So far we have considered the shape of the electrondensity of a limited inner region of each atom but not of thecomplete atom. How do we find the shape of the completeatom? In other words, how do we find the interatomic sur-faces that separate one atom from another and define the sizeand shape of each atom? The atoms in molecules (AIM)theory developed by Bader and coworkers (4 ) provides amethod for doing this.

The AIM theory, which is solidly based on quantummechanics, differs from orbital-based theories in that it isbased directly on the electron density and interprets this den-sity to provide information on bonding. The density may beobtained experimentally or from theoretical electronic structurecalculation. Experimental densities of sufficient quality tobe analyzed by the AIM theory can be obtained from low-

temperature X-ray diffraction experiments using thenonspherical multipolar refinement procedure pioneered byCoppens (3). Theoretical densities must be calculated by high-level ab initio or DFT methods (see Appendix 1). It has beenconsistently shown that the numerical results based on AIMconverge to limiting values as the size of the basis set used inthe calculation increases.

For a homonuclear diatomic molecule such as Cl2 theinteratomic surface is clearly a plane passing through themidpoint between the two nuclei—in other words, the pointof minimum density. The plane cuts the surface of the electrondensity relief map in a line that follows the two valleys leadingup to the saddle at the midpoint of the ridge between thetwo peaks of density at the nuclei. This is a line of steepestascent in the density on the two-dimensional contour mapfor the Cl2 molecule (Fig. 9).

In all molecules other than homodiatomics the inter-atomic surfaces are not planar, as seen in Figure 9 for CO andFCl. These surfaces can be found by computing the gradientpaths of the electron density. In a relief map of the density ina particular plane these are the paths of steepest ascent start-ing at infinity and leading up to the maximum or peak ateach nucleus. Gradient paths are always orthogonal to thecontours of constant electron density and they never cross eachother. Figure 10 shows two gradient paths up an idealizedmountain. These are two of the infinite number of steepestpaths up the mountain. Throughout their length they areperpendicular to the circular contours of equal height. Figure10 also shows a longer, less steep path that is not perpendicularto the contours. It not is not a gradient path.

The collection of all the gradient paths of the electrondensity constitutes the gradient vector field of the molecularelectron density. Figure 11 shows the gradient vector field ofthe electron density of the BCl3 molecule in the σh plane.The collection of gradient paths that terminate at a givennucleus defines a discrete region of space surrounding eachnucleus that is called the atomic basin. These basins definethe atoms as they exist in the molecule. Among the gradient

Figure 10. Gradient paths for an idealized mountain. Two gradientpaths (lines of steepest ascent) are shown (a), together with anarbitrary path (b) that is not a line of steepest ascent but representsan easier route up the mountain. The gradient paths cross the con-tours at right angles.

1000 m

a

a

b

800 m

600 m

400 m

200 m

Figure 9. Contour maps of the electron density of (a) Cl2, (b) CO,and (c) FCl. (See legend to Fig. 2 for contour values.)

a

c

b







paths that constitute the gradient vector field are two sets ofunique paths: those that start at infinity but terminate at thepoint of minimum density on the ridge of electron density(the bond critical point) between two nuclei, and those thatstart at this point and terminate at a nucleus. The completeset of the gradient paths in three dimensions that terminateat a bond critical point constitute the surface between theatoms, the interatomic surface. This is also called a zero-fluxsurface because no gradient paths cross it.

Figure 11 shows a few of the infinite number of gradientpaths that lie in the σh plane. The set of two gradient pathsthat terminate at a bond critical point indicate the intersectionof the interatomic surfaces with this plane. In Figure 11 thereare six gradient paths that start at the bond critical point andterminate at a nucleus. The two that start at the same bondcritical point trace a bond path between the boron atom anda fluorine atom. Bond paths are found between every pair ofatoms in a molecule that we usually consider to be bondedto each other, and not between atoms that are not bondedtogether. The existence of a bond path between the nuclei oftwo atoms that share an interatomic surface constitutes a clearand rigorous definition of a bond between the two atomsaccording to the AIM theory.

The interatomic (zero-flux) surfaces partition the moleculeinto separate nonoverlapping atoms (atomic basins), which

extend to infinity on the open side of any exterior atom. Forthe purpose of representing the electron density as a contourmap or an isosurface, or for determining the atomic volume,it is convenient to take the 0.001-au envelope of constantdensity as the practical representation of the surface of anatom on its open (nonbonded) side. Normally, the outer0.001-au isodensity envelope encloses more than 99% of theelectron population of the molecule (4 ). Integrating anyproperty density, such as the electron population, energy, orvolume, over the atomic basin yields the contribution of thisatom to the corresponding molecular property. The sum ofthese atomic properties gives the corresponding molecularproperty with high accuracy. For example, the sum of thecharges is accurately zero for a neutral molecule and the sum ofthe atomic populations is accurately equal to the total numberof electrons. Moreover the properties of atoms or functionalgroups, as defined by AIM, are often almost constant (i.e.,transferable) when their immediate surrounding is similar asis the case in homologous series for example. Therefore, theseatoms and groups contribute constant additive amounts toevery molecular property in a molecule, and they have beenshown to recover the empirical additivity schemes for severalexperimental quantities, the heat of formation being anexample (13). An impressive recent example of using thetransferability and additivity of the properties of atoms andfunctional groups is the calculation of the properties of a verycomplex biological molecule, an opioid, from the propertiesof a number of smaller fragments (14 ).

AIM Atomic Charges and Dipoles

The charge on each atom is the difference between thetotal electron population, obtained by integrating the electrondensity over the volume occupied by the atom, the atomicvolume, and the nuclear charge. The atomic charges for thefluorides and chlorides of the period 2 elements are given inTable 1. The charge on the fluorine ligand decreases acrossthe period from a value of nearly �1 in LiF to zero in F2. Thecharge on the chlorine ligand similarly decreases from a valueof nearly �1 in LiCl to almost zero in CCl4 and then becomesincreasingly positive from NCl3 to FCl. Figure 12 shows thatthe fluorine and chlorine charges correlate well with the elec-tronegativity of the elements (15), bearing in mind the ap-proximate nature of the electronegativity values. For example,the charge on each of the atoms in CCl4 is nearly zero,consistent with the very similar values of their electro-negativity (Cl, 2.8; C, 2.5). The charge on the central atomincreases across period 2 with the increasing number ofligands to a maximum at carbon for the fluorides and atboron for the chlorides.

It is important to understand that the atomic chargesrefer to atoms that are not spherical. Consequently the centroidof electronic charge of an atom does not in general coincide withthe nucleus, and each atom therefore has an electric dipolemoment—or, more generally, an electric dipolar polarization(since only the dipole moment of electrically neutral atomsis origin independent).

The total dipole moment of a molecule is the resultantof the vector sum of the atomic dipolar polarization (�ap) ofall the atoms in the molecule and of all the charge transferdipoles arising from the transfer of charge between bonded

Figure 11. (a) The density and (b) its corresponding gradient vectorfield of BCl3. (See legend to Fig. 2 for contour values.)

b

a








atoms (�ct). A charge transfer dipole is the product of theposition vector of the nucleus by its charge. Removal of theorigin dependence from �ct is discussed elsewhere (16 ). Fora diatomic molecule, say CO, this is given by the vector sumof the two atomic polarization dipoles and the two chargetransfer dipoles; that is, the molecular dipole moment =�ct(O) + �ct(C) + �ap(O) + �ap(C).

In the past the measured dipole moment of a diatomicmolecule was often assumed to be equal to the charge transferdipole moment, so that the atomic charges could be calcu-lated if the bond length was known. However, this method doesnot give correct atomic charges because it ignores the atomicpolarization dipoles, which may be very significant. In theCO molecule the two atomic dipoles are large (�ap[O] +�ap[C] = 2.523 au, negative end pointing to the C atom) andthey oppose the charge transfer dipoles (�ct[O] + �ct[C] =2.485 au, negative end pointing to the O atom), almostcanceling them out, giving a very small overall moleculardipole moment (0.038 au = 0.096 D)2 (see Fig. 13). Thenegative end of the molecular dipole (experimental and cal-culated) points to the C atom. In other words, the moleculehas the observed polarity C � O+, an apparent anomaly that isonly resolved if one takes into account both the charge transfer

and the atomic polarization terms. Even though the atoms bearsignificant charges consistent with their respective electro-negativities (C+1.170 O�1.170), when these charges are used tocalculate the �ct terms and are added to the �ap terms theyrecover the magnitude and direction of the observed moleculardipole. The dipole moment of any molecule can be expressedand recovered from group contributions calculated in thismanner (16 ) using available Windows-based software (17 ).

Comparison between AIM Theory and ConventionalModels for Describing Bonding

We now compare how the atoms in a molecule and thebonds between them are defined in the AIM theory and inconventional bonding models.

AtomsThere is no clear rigorous definition of an atom in a

molecule in conventional bonding models. In the Lewismodel an atom in a molecule is defined as consisting of itscore (nucleus and inner-shell electrons) and the valence shellelectrons. But some of the valence shell electrons of each atomare considered to be shared with another atom, and how theseelectrons should be partitioned between the two atoms so as todescribe the atoms as they exist in the molecule is not defined.

Conventionally, the bonding electrons are arbitrarilydivided in two ways. One is to assume that the bonds are fullyionic, which gives atomic charges that are called oxidationnumbers; the other is to assume that the bonds are fully co-valent, which gives charges that are called formal charges.Although both of these concepts have proved useful they donot give real atomic charges. For molecules with polar bondsthere is no clearly defined method for partitioning the bondingelectrons between the bonded atoms that reflects the unequalsharing of electrons or partial electron transfer. In localized

Figure 13. Contour plot of the electron density of CO, showing themagnitudes and directions of atomic and charge transfer dipoles(arrow length is proportional to magnitude). Arrow heads point tothe negative end. The molecular dipole moment is given by thevector sum of charge transfer terms (�c.t.) and the atomic polariza-tions (�a.p.). Values were obtained at the DFT level using the B3LYPfunctional and the 6-311+G(3df) basis set. The SCF moleculardipole = 0.096 D; the computed molecular dipole (� c.t.[O] +�a.p.[O] + �c.t.[C] + �a.p.[C]) = 0.038 au = 0.096 D, close to theexperimental value of 0.110 D (15).

Figure 12. Electronegativity difference between the halogen atom(X) and the atom to which it is bonded (A) in the halides of period2 (AnX) as a function of the atomic charge of the halogen atomq(X). (a) X = Cl; (b) X = F. Atom A of the halide is shown in thefigures. Charges were obtained at the DFT level using the B3LYPfunctional and a 6-311+G(2d,p) basis set.

Ele

ctro

nega

tivity

Diff

eren

ce

Charge of the Chlorine Atom

Li

Be

B

C

N

O

F1.2

0.6

0.0

-0.6

-1.2

-1.8

0.0-0.2 0.4 0.60.2-0.4-0.6-0.8-1.0

iEle

ctro

nega

tivity

Diff

eren

ce

Li

Be

B

CN

O

F-0.2

-0.8

-1.4

-2.0

-2.6

-3.2

Charge of the Fluorine Atom0.0-0.2-0.4-0.6-0.8-1.0

a

b







orbital models such as the valence bond model, a free atomis defined in terms of the atomic orbitals used to describe it;but in molecule formation some of the orbitals are consideredto overlap with those of a neighboring atom to give bondingorbitals, so again the atoms in the molecule are not clearlydefined. In the molecular orbital model the whole moleculeis described in terms of molecular orbitals and no attempt ismade to define the individual atoms.

In contrast, the AIM theory provides a clear definitionof an atom in a molecule as a space-filling object, from whichall its properties can be obtained. The properties of these atomsare additive to give the corresponding molecular property.

BondsAccording to the Lewis model, a covalent bond consists

of a pair of shared electrons—that is, a pair of electrons thatbelongs to the valence shell of each of the bonded atoms. Inother words the two valence shells are considered to overlap.The electrostatic attraction of this pair of electrons for the twonuclei is considered to provide the attractive force holdingthe two nuclei together. According to the ionic model, twoions (charged atoms) are held together by the Coulombicattraction between their opposite charges. In other words, anionic bond is the electrostatic attraction between two ions(charged atoms) with opposite charges.

These definitions are clear, but they do not apply tothe vast majority of real molecules in which the bonds areneither purely ionic nor purely covalent. Lewis recognizedthat a pair of electrons is generally not shared equally betweentwo electrons because the atoms generally have different powersof attracting electrons, that is, they have different electro-negativities, giving charges to both atoms. Such bonds areconsidered to have some covalent character and some ioniccharacter and are known as polar bonds.

Polar bonds range from bonds between atoms that havelarge but slightly less than integral charges and are thereforeclose to the ionic limit to pure covalent bonds between

atoms of equal electronegativity such as the C–C bond inethane. Almost all bonds are polar. A pure ionic bond is anideal concept that is never observed; and pure covalent bondsare very rare, inasmuch as the atoms in the vast majority ofmolecules have different electronegativities and therefore havenonzero charges. The attractive force in a polar bond can bethought of as being due to both the shared electrons and theatomic charges. In other words, a polar bond has both covalentand ionic character. However, these terms have not beenclearly defined, so it is not possible to quantitatively evaluatethe covalent and ionic character of any given polar bond. Aproposed method for doing this based on determining atomiccharges from the dipole moment of a diatomic molecule is notvalid because it assumes that atoms are spherical and ignoresatomic dipoles. In short, the widely used terms ionic characterand covalent character cannot be clearly defined and thereforecannot be measured, so they have only a rather vague andapproximate meaning.

In contrast, the AIM theory provides clear, unambiguousvalues for the charges on atoms, which at first sight appearto give us a clear definition of the ionic character of a bond.However, considering the atomic charges of the period 2 fluo-rides we see that, although the charge on fluorine decreasesacross the period, the charge on the central atom first increasesup to CF4 before decreasing. The decreasing charge on theligand might be interpreted as showing that the bonds arebecoming less ionic, but the increasing charge on the centralatom could be interpreted as showing that the bonds arebecoming more ionic. Clearly, a knowledge of the atomiccharges does not enable us to define ionic character. All thatwe know with certainty is the atomic charges, which, it seemsreasonable to assume, make a contribution to the strength ofthe bonding in proportion to their product.

In addition to the attractive force provided by oppositeatomic charges, the electronic charge accumulated betweenthe nuclei of the two bonded atoms must also contribute tothe attraction between the two atoms. But because the regionin which charge is accumulated is not sharply defined, theamount of the accumulated charge is not known. The AIMtheory does, however, provide a value of the electron densityat the bond critical point, ρb. We see in Table 1 that ρb forthe period 2 fluorides increases up to CF4 and then becomesessentially constant. The value of ρb is an indication, but nota quantitative measure, of the amount of electronic chargein the bonding region and so can only be regarded as a roughmeasure of the covalent character of a bond—which, as wehave said, has not been precisely defined.

Clearly the concepts of ionic and covalent character haveonly an approximate qualitative significance. They cannot bedefined and therefore measured in any quantitative way.Although they are widely used terms and have some qualitativeusefulness if used carefully they have caused considerablemisunderstanding and controversy. The AIM theory does,however, provide properties that we can use to characterize abond quantitatively, such as the bond critical point density andthe atomic charges. It seems reasonable to assume that thestrength of a bond depends on both these quantities, increasingas ρb and the product of the atomic charges increase.

These assumptions are consistent with the very largebond strength of the BF bond in BF3, which is larger than thatof any other single bond. It has a bond dissociation enthalpy

Figure 14. Contour plot of the electron density of B2H6 in the planeof the bridging hydrogen. Each hydrogen is connected to the twoboron atoms by a bond path to each. In contrast, the boron atomsdo not share a bond path linking them to one another. (See legendto Fig. 2 for contour values.)







of 613 kJ mol�1, compared, for example, to the C–C bonddissociation enthalpy of only 348 kJ mol�1. Even though ρbfor the BF bond (0.217 au) is little smaller than that for theC–C bond in ethane (0.249 au), the atomic charges on Band F are +2.43 and �0.81, respectively, whereas the chargeson carbon in ethane are almost zero and of the same sign.Although it is not consistent with current usage, the BF bondcan be described as having both a large covalent characterand a large ionic character, as has been pointed out in tworecent papers in this Journal (18, 19).

Bond paths are observed between bonded atoms in amolecule and only between these atoms. They are usuallyconsistent with the bonds as defined by the Lewis structureand by experiment. There are, however, differences. There isonly a single bond path between atoms that are multiplybonded in a Lewis structure because the electron density isalways a maximum along the internuclear axis even in a Lewismultiple bond. The value of ρb does, however, increase withincreasing Lewis bond order, as is shown by the values forethane (0.249 au), ethene (0.356 au), and ethyne (0.427 au),which indicate, as expected, an increasing amount of electrondensity in the bonding region.

In molecules that cannot be described by a single Lewisstructure, such as B2H6, and in which the electrons are there-fore not as localized in individual bonds as a Lewis structureassumes, there are nevertheless bond paths between the boronatoms and each of the bridging hydrogen atoms (Fig. 14).There is no direct bond between the boron atoms or betweenthe bridging hydrogen atoms. Moreover, electron density isnot accumulated in the center of each BHB triangle as mightbe assumed from the 3-center, 2-electron model for thesebonds (6 ). A recent use of the concept of the bond path toestablish which atoms are bonded in a molecule when dis-tance alone provides ambiguous answers was for a titaniumcyclopentadienyl complex, as discussed in ref 20.

Summary and Conclusions

1. Unlike an orbital, the electron density of a moleculeis a physical observable that can be obtained by experimentand also by calculation using ab initio or density functionaltheory methods.

2. The electron density is high in an almost sphericalregion surrounding each nucleus and much lower and morediffuse in the bonding region between.

3. The electron density can be most easily shown bycontour maps of suitably chosen planes or as envelopes ofconstant density.

4. The electron density distribution is determined by theelectrostatic attraction between the nuclei and the electrons,the electrostatic repulsion between the electrons, the Fermicorrelation between same spin electrons (due to the operationof the Pauli principle), and the Coulombic correlation (dueto electrostatic repulsion).

5. The result of the operation of the Pauli principle is thatelectrons with the same spin tend to keep apart and electronsof opposite spin may come close to each other to form anopposite-spin pair under the attraction of a nucleus.

6. The AIM theory provides a clear and rigorous defini-tion of an atom as it exists in a molecule. It is the atomicbasin bounded by the interatomic surfaces. The interatomic

surfaces arise naturally from the topology of the electrondensity.

7. The properties of an atom in a molecule, such as itsenergy and charge, can be rigorously defined and evaluatedand are additive to give the property for the molecule.

8. The concept of a bond has precise meaning only interms of a given model or theory. In the Lewis model a bondis defined as a shared electron pair. In the valence bond modelit is defined as a bonding orbital formed by the overlap oftwo atomic orbitals. In the AIM theory a bonding interactionis one in which the atoms are connected by a bond path andshare an interatomic surface.

9. Bond paths are normally found in cases in which thereis a bond as defined by Lewis. There is only one bond pathfor a multiple bond irrespective of the bond order. The bondorder is, however, reflected in the value of ρbcp. Bond paths arealso found in molecules for which a single Lewis structurecannot be written.

10. The concepts of ionic and covalent character of a bondare vague and ill defined. The well-defined AIM-derivedquantities such as the integrated atomic charges and the bondcritical point density provide a quantitative characterizationof bonding (4 ).

Acknowledgment

We are grateful for the constructive criticisms raised by oneof the referees, whose comments helped improve this manuscript.

Notes

1. The abbreviation au stands for “atomic units”, which is asystem of units meant to simplify the equations of molecular andatomic quantum mechanics. The units of the au system are combi-nations of the fundamental units of mass (mass of the electron),charge (charge of the electron), and Planck’s constant. By settingthese three quantities equal to unity one gets simpler equations. Theau system has a simple relation to the SI and Gaussian (cgs) systems ofunits. For example, 1 au of length = a0 (Bohr radius) = 5.29 × 10�9

cm =0.529 Å; 1 au of charge = e = 1.602 × 10�19C = 4.803 × 10�10 esu;1 au of charge density = e/a0

3 = 6.748 eÅ�3 = 1.081 × 1012 C m�3.For a formal discussion of how the au system of units naturallyarises in quantum chemistry, see refs 8 and 9.

2. The experimental value for the magnitude of the C�O+

dipole is 0.110 D.

Literature Cited

1. Foresman, J. B.; Frisch, A. Exploring Chemistry with ElectronicStructure Methods, 2nd ed.; Gaussian: Pittsburgh, 1996.

2. Scerri, E. R. J. Chem. Educ. 2000, 77, 1492.3. See Coppens, P. X-ray Charge Densities and Chemical Bond-

ing; Oxford University Press: New York, 1997. Koritsanszky,T. S.; Coppens, P. Chem. Rev. 2001, 101, 1583.

4. Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Ox-ford University Press: Oxford, 1990.

5. Popelier, P. Atoms in Molecules: An Introduction; Prentice Hall:London, 2000.

6. Gillespie, R. J.; Popelier, P. L. A. Molecular Geometry andChemical Bonding: from Lewis to Electron Densities; OxfordUniversity Press: New York, 2001.




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7. Musher, J. I. Am. J. Phys. 1966, 34, 267.8. Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: Intro-

duction to Advanced Electronic Structure Theory; Dover: NewYork, 1989.

9. Levine, I. N. Quantum Chemistry; Prentice Hall: Upper SaddleRiver, NJ, 1991.

10. Bader, R. F. W. J. Phys. Chem. A 1998, 102, 7314.11. Linnett, J. W. The Electronic Structure of Molecules: A New Ap-

proach; Methuen: London, 1964.12. Gillespie, R. J.; Hargittai, I. The VSEPR Model of Molecular

Geometry; Allyn and Bacon: Boston, 1991.13. Wieberg K. B.; Bader, R. F. W.; Lau C. D. H. J. Am. Chem.

Soc. 1987, 109, 1001. Bader, R. F. W.; Keith T. A.; Gough,K. M.; Laidig, K. E. Mol. Phys. 1992, 75, 1167.

14. Matta, C. F. J. Phys. Chem. A 2001, 105, 11088.15. Allred, A. L.; Rochow, E. G. J. Inorg. Nucl. Chem. 1958, 5,

264.16. Bader, R. F. W.; Matta, C. F. Int. J. Quantum Chem. 2001,

85, 592.17. Matta, C. F. FRAGDIP01; Department of Chemistry Indi-

ana University: Bloomington, 2001; Program No. QCMP201,Quantum Chemistry Program Exchange; http://qcpe.chem.indiana.edu/ (accessed Mar 2002).

18. Gillespie, R. J. J. Chem. Educ. 2001, 78, 1688–1691.19. Haaland, A.; Helgaker, T. U.; Ruud, K.; Shorokhov, D. J. J.

Chem. Educ. 2000, 77, 1076.20. Bader, R. F. W.; Matta, C. F. Inorg. Chem. 2001, 40, 5603.21. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.;

Johnson, B. G; Robb, M. A.; Cheeseman, J. R.; Keith, T.;Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J.B.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres,J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D.J.; Binkley, J. S.; Defrees, D. J.; Baker, J., Stewart, J. P., Head-Gordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94; GaussianInc.: Pittsburgh PA, 1995.

22. Biegler-König, F. W.; Nguyen-Dang, T. T.; Tal, Y.; Bader, R.F. W.; Duke, A. J. J. Phys. B 1981, 14, 2739–2751.

23. Biegler-König, F. W.; Bader, R. F. W.; Tang, T.-H. J. Comp.Chem. 1982, 13, 317–328.

24. Atoms in Molecules Download Page; http://www.chemistry.mcmaster.ca/aimpac/ (accessed Mar 2002); this is Bader’s Website, from which the AIMPAC program suite can be down-loaded free of charge.

25. Born, M. Z. Physik 1926, 37, 863.26. Born, M. Z. Physik 1926, 38, 803.

Appendix 1. Computational Details

Unless otherwise stated, the results presented in this paperwere obtained as follows. Full geometry optimizations of themolecules were performed at the density functional theory B3LYP/6-311++G(3df,2pd) level, and the wave functions were obtainedat the same level as implemented in the program Gaussian94 (21).The resulting densities were analyzed using the program Extremeto locate the critical points. PROAIM was then used to integratethe atomic properties up to the zero-flux surfaces, and GRIDV,GRIDVEC, and CONTOR were used to obtain the plots. Theseare all parts of the suite of programs known as AIMPAC (21–24).

Appendix 2. The Electron Density

A time-independent wave function is a function of the positionin space (r ≡ x,y,z) and the spin degree of freedom, which can beeither up or down. The physical interpretation of the wave functionis due to Max Born (25, 26 ), who was the first to interpret thesquare of its magnitude, |ψ(r)|2, as a probability density function,or probability distribution function. This probability distributionspecifies the probability of finding the particle (here, the electron)at any chosen location in space (r) in an infinitesimal volume dV =dx dy dz around r. The probability of finding the electron at r isgiven by |ψ(r)|2dV, which is required to integrate to unity overall space (normalization condition). A many-electron system, suchas a molecule, is described by a many-electron wave functionΨ(r1,r2,r3,…,rN), which when squared gives the probability den-sity of finding electron1 at r1, electron2 at r2, …, electronN at rN

simultaneously (i.e., the probability of a particular instantaneousconfiguration of all electrons in the system). The probability offinding, say, electron1 at r1 without specifying the location of theN – 1 remaining electrons is found by integrating the many-electronwave function over the coordinates of all electrons except electron1.In other words, the probability of finding electron1 at r1, irrespectiveof the positions of the remaining electrons in the molecules, is givenby ∫ |Ψ(r1,r2,…,rN)|2dr2 …drN, an integration that also implies asummation over all spins. However, since electrons are indistinguish-able and thus cannot be labeled, what is true for electron1 is true forany electron in the system, and if we multiply the latter integral bythe number of electrons in the system we obtain a one-electrondensity function, commonly known as the electron density:

ρ(r) = N∫Ψ2(r1,r2,…,rN)|2dr2 …drN (2.1)

This is the probability of finding a single electron, no matter which,at r (i.e., at the specific spatial position x,y,z and having the specificspin s) weighted by the total number of electrons in the system.Integrating eq 2.1, the density, over all space with respect to thecoordinates of electron1, is

∫ρ(r)dr1 = N∫Ψ2(r1,r2,…,rN)dr1dr2 …drN (2.2)

That is, the integral of the density over all space yields the totalnumber of electrons in the molecule.

Appendix 3. The Pauli Principle

The Pauli exclusion principle requires that no two electronscan occupy the same spin-orbital is a consequence of the moregeneral Pauli antisymmetry principle:

Any many-electron wave function must be antisymmetricto the interchange of the spacial coordinates and spin(collectively referred to as the vector q) of any pair ofelectrons i and j.

This is written: ↓–––↓

Ψ(q1,q2,…,qi,…q j,…qn) = �Ψ(q1,q2,…,q j,…qi,…qn) (3.1)

where qk is the space and spin coordinates of the kth electron. Iftwo electrons have the same space and spin coordinates (i.e., q j =qi, then:

↓–––↓Ψ(q1,q2,…,qi,…q i,…qn) = �Ψ(q1,q2,…,q i,…q i,…qn) (3.2)

Which implies that the wave function is equal to its negative. Re-arranging gives 2Ψ(q1,q2,…,q i,…q i,…qn) = 0, or simply Ψ = 0.




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The physical consequence of this is that two electrons ofthe same spin have zero probability of occupying the sameposition in space; that is, two same-spin electrons exclude eachother in space. Since Ψ is continuous, there is only a smallprobability of finding two electrons of the same spin close toeach other in space; that is, the Pauli antisymmetry requirement

forces them to avoid each other as much as possible and as aresult two electrons of the same spin tend to maximize theirseparation in space. In other words, the motion of two electronsof the same spin is not independent but correlated, a correlationknown as Fermi correlation, which is a direct consequence of thePauli principle.




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