Research: Science and Education
Electron Densities, Atomic Charges, and Ionic, Covalent, and Polar Bonds R. J. Gillespie Department of Chemistry, McMaster University, Hamilton, ON L8S 4M1, Canada;
[email protected] In a recent paper in this Journal, Haaland et al. (1) disputed the conclusion of an earlier paper of mine (2) in which I stated that the bonds in BF3 and SiF4 are much more “ionic” than has hitherto been generally believed. In their recent paper the authors state that “Questions concerning the mean bond energies, the bond distances … are scientific questions to which there are definite answers. … The question of whether gaseous BF3 or SiF4 should be described as ionic or as polar covalent depends on our choice of words.” I agree with this statement, and the purpose of this paper is to attempt to clarify the meaning of the words that we use to describe bonds by basing the discussion on an analysis of the electron density distribution in a molecule. The importance of such an analysis is that the electron density is a real observable property of a molecule that can be determined with considerable accuracy both experimentally and by modern ab initio or density functional theory calculations. Conventional descriptions of bonds are based on two limiting models—the ionic model and the covalent model. The ionic model assumes that molecules and crystals are composed of spherical ions with integral charges held together by the Coulombic attraction between these ions. However, there are no purely ionic bonds even in crystals, since the ions are not truly spherical and there is always a small amount of density shared between the atoms. The covalent model assumes that atoms are held together by the “sharing” of electron density. According to Lewis, an electron pair is shared if it is used to complete the valence shell of both bonded atoms in a Lewis diagram. In terms of the electron density of a molecule the meaning of “sharing” is not so clear. It is generally taken to mean that in a covalent bond electron density is accumulated between the two atoms and it is the electrostatic attraction between this density and the two nuclei that holds the two atoms together. The only purely covalent bonds are those between identical atoms in symmetrical molecules such as the C–C bond in ethane and the Si–Si bond in disilane. Thus the very large majority of bonds have a character intermediate between that of an ionic bond and that of a covalent bond. Such bonds are usually described as polar covalent, so that almost all bonds may be described as polar covalent. But this description leaves unanswered the question how ionic and how covalent? Bonds are often said to have more or less “ionic character” and more or less “covalent character”. Unfortunately these terms are vague and cannot be rigorously defined. When a bond is said to have a large ionic character it is usually assumed that this means it has a small “covalent character”. However, as we shall see this is not necessarily the case. Ultimately all bonding, no matter how we describe it, is the result of the attractive electrostatic force between the electron density and the nuclei and the repulsive force between the nuclei. So the understanding and description of bonding ultimately depends on analyzing and understanding 1688
Table 1. Atomic Charges (q) and Bond Critical Point Densities (b) of Some AF Molecules q /au Molecule
AIM
LiF
F ᎑0.94
BeF2
᎑0.90
BF3
ρb /au
APT A 0.94
F ᎑0.86
A 0.86
0.075
1.81
᎑0.66
1.32
0.145
᎑0.86
2.58
᎑0.57
1.70
0.217
CF4
᎑0.74
2.96
᎑0.52
2.06
0.309
NF3
᎑0.36
1.09
᎑0.39
1.17
0.314
OF2
᎑0.12
0.23
᎑0.22
0.44
0.295
F2
0.00
0.288
0.94
0.00 ᎑0.88
0.00
NaF
0.00 ᎑0.94
0.88
0.051
MgF2
᎑0.91
1.83
᎑0.75
1.50
0.080
AlF3
᎑0.88
2.65
᎑0.64
1.90
0.115
SiF4
᎑0.86
3.42
᎑0.60
2.39
0.154
PF3
᎑0.84
2.51
᎑0.59
1.78
0.168
SF2
᎑0.71
1.43
᎑0.47
0.95
0.182
ClF
᎑0.50
0.50
᎑0.28
0.28
0.187
the electron density distribution. This analysis can be carried out, as we shall see, using the atoms in molecules (AIM) theory. I based my statement that the bonds in BF3 and SiF4 are much more ionic than is commonly supposed on the large calculated (AIM) atomic charges in these molecules (Table 1), which are much closer to the full ionic charges than the zero charges expected for a covalent molecule (3, 4 ). If we take the AIM charges on fluorine as an indication of “the ionic character” of the bonds we could reasonably say that the BF and SiF bonds are over 80% ionic so that we can justifiably describe these bonds as having a large ionic character. Haaland et al. discuss ATP charges calculated by the atomic polar tensor method (Table 1) that are smaller than the AIM charges. Using these smaller charges we would still have to conclude that the bonds are over 50% ionic. The values of atomic charges vary with the method used to determine them. We use the AIM charges because, as we shall see in the next section, the AIM theory provides a clear unambiguous definition of an atom in a molecule and it is only on the basis of a clear and unambiguous definition of an atom that its charge can be calculated. Other methods such as the ATP method have been proposed for obtaining atomic charges but none are based on a clear unambiguous definition of an atom in a molecule. The AIM Theory and Atomic Charges The AIM theory (4–6 ) partitions the electron density of a molecule into individual non-overlapping atomic fragments by rigorously defined interatomic surfaces as illustrated
Journal of Chemical Education • Vol. 78 No. 12 December 2001 • JChemEd.chem.wisc.edu
Research: Science and Education
for the BF3 molecule in Figure 1. The interatomic surfaces are the surfaces of zero flux in the gradient vector field of the molecular density (see Appendix). Various properties of each of these atomic fragments can then be derived from its electron density distribution. We consider here just the simple case of the atomic charge, which is easily obtained by integrating the electron density over the space occupied by the atom as defined by its interatomic surfaces. The total charge of the molecule, which is zero for a neutral molecule, and the appropriate integral charge for a polyatomic ion is simply the sum of the atomic charges. Other properties of these atoms such as atomic volumes, dipoles, polarizabilities, and energies are similarly additive to give the corresponding molecular property. The importance of the AIM theory is that it leads to precisely defined non-overlapping atoms. Without going into the theoretical justification for the interatomic surfaces defined by the AIM theory we can see in Figure 1 that they follow the valleys of minimum density between the atoms. This is clearly a natural way to divide up the density of the molecule into atomic fragments. Atoms defined in this way are not spherical, as we can see from Figure 1. Any other arbitrary method of partitioning the density of a molecule into atomic densities would lead to the density of one or more of the atoms decreasing from the nucleus and then increasing again before its surface was reached. This would be the case for the B atom in BF3, for example, if it were larger than indicated by the AIM interatomic surfaces as suggested by Haaland et al. (1). This clearly would be an unnatural and unreasonable way to partition the density. Haaland et al. criticize the AIM definition of an atom in a molecule, and the corresponding charge, stating that “Most chemists would probably describe a molecule as consisting of approximately spherical overlapping atoms or ions.” They state that “the space allotted to the B atom [by the AIM theory] is very small” and that “the allotted space includes only the K shell of the atom; electrons in the 2s or 2p atomic orbitals would presumably be found in those regions of space assigned to the F atoms. Such a division of space may have led to the overestimation of the negative charge assigned to the F atom and the positive charge assigned to the B atom.” In order to obtain an atomic charge, or any other property of an atom in a molecule, the atom must be clearly and rigorously defined. Any method that defines atoms in such a way that they overlap is not a rigorous definition because there is no clear way to partition the density in the overlap region between the two atoms. Such a method cannot give true additive atomic properties. Indeed, to say that atoms overlap is to say that individual atoms do not exist in a molecule. Since atomic orbitals occupy all of space, the F orbitals clearly overlap those of the B atom, so that atomic orbitals do not provide a method of defining atomic charges. It is therefore incorrect to claim on this basis that the B atom as defined by the AIM theory is too small so that its atomic charge is underestimated. AIM clearly defines an atom in a molecule, and defined in this way its properties such as charge are also clearly defined and additive. The atoms in a molecule as defined by the AIM theory are not spherical as Haaland et al. would prefer to think of them. If they were, the atoms in a molecule would not occupy all the space in a molecule or they would overlap extensively. If
Figure 1. Contour map of the electron density of the BF3 molecule in the molecular plane. The outer contour line corresponds to a density of 0.001 au and the next contour lines correspond to values increasing according to the pattern 2 × 10n, 4 × 10n, 8 × 10n, where n varies from ᎑3 to 2. Also shown are the interatomic surfaces, the bond paths, and the bond critical points (see text).
there are regions where these atoms do not overlap, what is in this space—a vacuum? The electron density of a molecule is spread continuously through the molecule so there is no empty space. If the atoms overlap extensively how do we divide the density in these overlap regions between the atoms? There is no clear unambiguous way to do this—any such division must be arbitrary and based on some criterion other than the density. The ATP method is not based on any clear definition of an atom in a molecule so it cannot give meaningful charges. Moreover, the charge of the boron or silicon atoms in BF3 and SiF4 cannot be determined by this method. These charges must be calculated from the charge of the fluorine atoms, so it cannot be demonstrated that these charges are additive as true atomic charges should be. AIM Theory and Bond Critical Point Densities If we look at the electron density distribution of the BF3 molecule (Fig. 1) we can see that there is a line between any pair of bonded atoms along which the electron density is greater than along any other line between these two atoms. We can also see that in any direction away from this line the density decreases. Along this line the density increases towards one of the nuclei and it is a minimum at a point between. This point lies on the interatomic surface and is called the bond critical point. The line is called a bond path. It is the electron density accumulated along this line and in its vicinity that we can regard as shared between the two atoms. A useful measure, but not a rigorous definition, of the amount of shared density is provided by the value of the density at the bond critical point, ρb. Values of ρb are given in Table 1 for the fluorides of periods 2 and 3. They increase from 0.075
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Research: Science and Education
au in LiF to 0.217 au in BF3 to 0.288 au in F2, and from 0.051 au in NaF to 0.154 au in SiF 4 to 0.187 au in ClF (3, 4 ). On the basis of these values the BF bonds and SiF bonds have considerable covalent character. We have to conclude that these bonds have both considerable ionic character, as measured by the atomic charges, and covalent character as measured by the bond critical point density. It is perhaps time to abandon these ill-defined terms “ionic and covalent character” and to refer only to the clearly defined and measurable atomic charges and bond critical point density. The term “polar covalent” is particularly vague, as it embraces almost all bonds and gives no idea of the relative ionic and covalent character of the bond. The large atomic charges in BF3 and SiF4 are important to understanding the great strength of these bonds—which are the two strongest known single bonds, with bond dissociation enthalpies of 613 and 565 kJ mol᎑1, respectively (7). In comparison, the C–C bond in ethane has a bond dissociation enthalpy of only 345 kJ mol᎑1. The great strength of the BF and SiF bonds arises from both the relatively large amount of shared density and the large atomic charges. So it is not strictly correct to say that these bonds are “predominately ionic”; if we are forced to use these terms we have to say that they are both strongly covalent and strongly ionic. It was this strong ionic character as measured by the atomic charges that I wished to draw attention to in my earlier paper (2) because this aspect of these bonds has been largely unrecognized. Both the term polar covalent and the conventional representation of these molecules with bond lines unjustifiably emphasize the covalent character at the expense of the ionic character. Summary and Conclusions 1. The widely used terms “ionic character” and “covalent character” are vague and can be misleading because they cannot be rigorously defined. 2. Before we can assign a charge or another property to an atom in a molecule the atom must be clearly defined. 3. The AIM theory provides a rigorous and clear definition of an atom in a molecule. The atoms defined by AIM have strictly additive properties such as charge, volume, and dipole moment. 4. The atomic charges in BF3 in SiF4 are large, justifying the statement that the bonds in these molecules have a large ionic character. We must recognize, however, that ionic character is a vague, ill-defined quantity. 5. The atoms in a molecule as defined by the AIM theory together occupy all the space in the molecule and therefore are not spherical. 6. Any model that considers a molecule to contain spherical overlapping atoms cannot be used as the basis for the calculation of atomic charges or any other property of an atom in a molecule. 7. Atoms in molecules cannot be rigorously defined in terms of atomic or localized orbitals because no orbitals are fully localized. So they must overlap each other and the contribution of each orbital to these overlap regions can only be decided by some arbitrary method. 8. The analysis of the electron density by the AIM theory provides a value of the bond critical point density, ρb, which 1690
Figure 2. The gradient vector field in the plane of the BF3 molecule. The interatomic surfaces defined by the gradient vector field and the bond paths and bond critical points are also shown.
can be taken as measure of the covalent character of a bond. The values of ρb show that the bonds in BF3 and SiF4 have considerable covalent character. 9. The BF and SiF bonds have both large atomic charges and large critical point densities so, using the conventional terms, we can say that they have considerable ionic character and considerable covalent character. This is the reason for their exceptional strength. 10. The possibility that a bond may be both strongly ionic and strongly covalent has not been previously widely recognized. The term polar covalent is therefore particularly vague, as it gives no idea of the individual contributions of the atomic charges and the shared density to the bond, which may both be large. Literature Cited 1. Haaland, A.; Helgaker T. U.; Ruud, K.; Shorokhov, D. J. J. Chem. Educ. 2000, 77, 1076. 2. Gillespie, R. J. J. Chem. Educ. 1998, 75, 923. 3. Gillespie, R. J.; Johnson, S. A.; Tang, T.-H.; Robinson, E. A. Inorg. Chem. 1997, 36, 3022. 4. Gillespie, R. J.; Popelier, P. L. A. Chemical Bonding and Molecular Geometry: from Lewis to Electron Densities; Oxford University Press: New York, 2001. 5. Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Clarendon: Oxford, 1990. 6. Popelier, P. L. A. Atoms in Molecules: An Introduction; Pearson Education: Harlow, England, 2000. 7. Huheey, J. E.; Keiter, E. A.; Keiter, R. L. Inorganic Chemistry, 4th ed.; HarperCollins: New York, 1993.
Appendix: The AIM Analysis of the Electron Density The AIM analysis of the electron density is based on the gradient vector field of the density. This is illustrated for
Journal of Chemical Education • Vol. 78 No. 12 December 2001 • JChemEd.chem.wisc.edu
Research: Science and Education
the BF3 molecule in Figure 2. The gradient vector field can be constructed by drawing gradient paths (i.e., lines of steepest ascent) starting from any point at infinity. These lines are perpendicular to the contour lines of constant density in a two-dimensional plot and are perpendicular to surfaces of constant density in three dimensions. These gradient paths do not intersect and, with an important exception that is discussed next, they all terminate at a nucleus. The gradient paths that terminate at a particular nucleus partition the molecule into regions each of which contains one nucleus, which is called the atomic basin of that nucleus and constitutes the atom in the molecule. The exceptions are
the two gradient paths that start at a point on each interatomic line (the bond critical point) and terminate at a nucleus, and the two that start at infinity and terminate at the bond critical point. The two that start at each bond critical point and terminate at two different nuclei form what is called the bond path, a path along which density is greater than in any direction away from this line. The two gradient paths that terminate at each bond critical point form an interatomic boundary in this two-dimensional contour map, and the complete set of such paths in three dimensions forms an interatomic surface. It is called a zero-flux surface because no gradient paths cross this surface.
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