Research: Science and Education
Lewis Structures Are Models for Predicting Molecular Structure, Not Electronic Structure Gordon H. Purser Department of Chemistry, The University of Tulsa, Tulsa, OK 74104
Even with the advent of easy-to-use, powerful programs that can calculate electronic structures from quantum mechanical principles and then graphically display the results of these calculations (1), the simply derived Lewis structure1 is still a powerful tool that can be used to predict molecular structure. The large number of publications in this Journal over the past quarter of a century testifies to the importance placed on drawing appropriate Lewis structures (2–17 ). In spite of their importance, a review of twelve general chemistry textbooks leads to the conclusion that the presentation of Lewis structures, formal charges, and the issue of expanded octets is treated more inconsistently than almost any other material (18). An article in this Journal reported recently that Lewis structures that emphasize the reduction of formal charge at the expense of the octet rule are often not the “most accurate” representations of molecules and are not worth the confusion they cause first-year students (15). The authors add that Lewis structures with expanded octets actually are “misrepresentations” of the molecules they describe. The article is beginning to impact the presentation of Lewis structures in general chemistry textbooks. For example, one textbook author states, “Although some unresolved questions about expanded octets may seem unsettling to you, the point to keep in mind is that the unmodified octet rule works perfectly well for most of our uses of Lewis structures (18l).” Another author states, “Recent theoretical calculations, based on quantum mechanics, suggest that the structure on the left [referring to a structure that does not minimize formal charge] is the best single Lewis structure for the phosphate ion. In general, when choosing between alternative Lewis structures, you should choose the one that satisfies the octet rule if it is possible to do so” (18a). One final example is an annotation to instructors that “using the powerful ab initio quantum-chemistry programs now available for solving the Schrödinger equation, the best structures for sulfur dioxide are those that conform to the octet rule. The shortened bond lengths (referring to the sulfur dioxide molecule) result from the highly ionic character of the SO bonds” (18e). Three months later, another article drew a contradictory conclusion about the “best” way to draw Lewis structures (16 ). The author suggested that nothing is gained by considering mesomeric forms that contain monovalent, covalently bonded oxygen atoms, and that by minimizing formal charge, structures are obtained for which molecular properties are consistent with observed properties. The conflicting conclusions presented in the two papers were drawn from the same data, but different criteria were used to define the “best” Lewis structure. In the case for adherence to the octet rule, the criterion for the “best” structure was that which would lead to a bonding description requiring the participation only of atomic orbitals involved in the bonding, based on quantum calculations (i.e., s- and p-orbitals but
not d-orbitals). The criterion leading to the contradictory conclusion, that elements beyond the second period should minimize formal charge at the expense of the octet rule, is that the “best” Lewis structure should yield bond lengths and angles consistent with experimental observations. The paradox— between Lewis structures that accurately describe molecular structure but appear to require d-orbital participation where little is found and Lewis structures that appear to avoid dorbital participation but fail to represent correctly the structure of the species in question—can be resolved if it is accepted that Lewis structures, like VSEPR, are classical models; that is, they contain no quantitative information about atomic orbital involvement. Accurate orbital information can be gleaned only from quantum mechanical analysis of a molecule. Lewis structures do not render any orbital information, and they cannot be used to predict the actual electronic structure of a molecule. (However, when used with VSEPR to predict geometry, atomic orbitals that have the appropriate symmetry to combine to form molecular orbitals can be determined.) The term “electronic structure” is used here to refer to the electron distribution, or molecular orbitals, within a molecule or ion. An analogy is that Lewis structures and the VSEPR model are to quantum mechanics what Newtonian mechanics is to relativity. While Newtonian mechanics works at the macroscopic level, it fails when used to describe objects of small mass and large velocity. Lewis structures and the VSEPR model are static pictures, whereas molecular orbital theory takes into account that electrons have wavelike properties. The examples in this paper demonstrate that prediction of orbital participation from Lewis structures is impossible at the current level of understanding. Discussion
Bond Order and Charge Separation in Molecules A covalent bond is produced when nuclei are held together through mutual attraction of electron density. The term bond order frequently is used to describe the amount of electronic interaction between the nuclei. Bond order can be used to predict such quantities as bond length and bond strength, but unlike these latter properties, the bond order itself cannot be measured by experiment. There is no single, fundamental definition for bond order. When using Lewis structures, the bond order is one-half of the number of electrons drawn between two nuclei. Where resonance structures are involved, the bond order can be determined by “averaging” the bond orders obtained using the appropriate resonance forms of the species. When using molecular orbital theory to describe diatomic molecules, the bond order is defined uniquely as one half of the difference between the number of electrons in bonding orbitals and the number of electrons in antibonding orbitals. In quantum
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calculations for polyatomic molecules, multiple definitions of bond order exist from the variety of models that describe the partitioning of electrons within the molecule. Three common methods of calculating bond orders include using Mulliken population analysis (19), natural population analysis (20), and Löwdin population analysis (21). It is beyond the scope of this paper to discuss the details of the various methods of calculating bond orders for electron populations, but it should be realized that each method is inherently deficient. For example, the Mulliken and Lowdin population analysis methods of calculating bond orders tend to overestimate the electron population in high-energy molecular orbitals relative to low-energy molecular orbitals (22). As a result, the calculated values of bond orders can be derived from the wrong set of orbital interactions, and the values of the bond orders will be too high. Calculated natural bond orders suffer a different limitation. Since natural population analysis (from which values of the natural bond order are calculated) forces electron density into orbitals generally localized between two nuclei or on a single nucleus, the calculated values of the bond orders in ions and molecules where electron density actually is delocalized over more than two nuclei will be too low. These limitations do not mean that calculated bond orders are not useful. As long as homologous species are being considered and the same method of calculating bond order is used, the calculated bond order can provide the same qualitative information as bond orders derived from Lewis structures. Like bond order, the charge on an atom in a polyatomic ion or a molecule is not a property that can be defined uniquely or measured directly; but just as bond order is related to measurable quantities (bond length and bond dissociation energy), atomic charges are related to an experimentally measurable property of neutral molecules, the dipole moment. As the term is used here, atomic charge is defined as the apparent point charge on the atoms such that the observed dipole moment results from the known geometry. The advantage of this definition is that atomic charges can be determined unambiguously for atoms in any molecule of C2v symmetry for which the experimental gas phase dipole moment, bond lengths, and bond angles are known. These atomic charges were calculated for H 2O, H2S, NO2, O3, OF2, SF2, and SO2. The data used and the resulting atomic charges (in electrostatic units) are shown in Table 1. From Lewis structures and a knowledge of atomic electronegativities, accurate, qualitative predictions of dipole moments are possible. For example, using only Lewis structures and electronegativities, one would conclude correctly that the oxygen atom in H2O possesses a greater negative charge than does the sulfur atom in H2S. Since quantum mechanical calculations focus on quantitatively identifying the location of electron density, one might expect to calculate atomic charges and definite bond orders accurately, but this is not the case. Calculated values of atomic charge are of value only for making relative comparisons and only when using the same method of calculation. A simple charge distribution analysis of the seven triatomic molecules listed above exemplifies this point. Ab initio calculations of the electronic structures of the molecules were performed using restricted Hartree–Fock theory (unrestricted for NO2) and the 6-31G* or 6-31G** basis set, as appropriate.2 Atomic charges were calculated using (i) fits to the molecular 1014
Table 1. Some Properties of Model Molecules with C2v Symmetry Angle (°)
Bond Length/Å
Dipole Moment/D
Obsd. Charge on Central Atom/esu
SO2
119.5
1.432
1.63
+0.47
H2O
104.5
0.958
1.85
{0.66
O3
116.8
1.280
0.53
+0.16
H2S
93.3
1.346
0.97
{0.22 +0.14
Molecule
NO2
134.1
1.188
0.32
OF2
103.2
1.418
0.30
+0.07
SF2
98.3
1.589
1.05
+0.21
Table 2. Molecular Electrostatic Potential, Population Analysis, and Atomic Charge on Central Atom of Model Molecules with C2v Symmetr y Molecule
Charge on Central Atom/esua
Calculated Charge from Structure/esu Electrostatic
Mulliken
Natural
SO2
+0.47
+0.69
+1.08
+1.86
H2O
{0.66
{0.80
{0.67
{0.97 +0.40
O3
+0.16
+0.32
+0.33
H2S
{0.22
{0.34
{0.13
{0.27
NO2
+0.14
+0.44
+0.67
+0.73
OF2
+0.07
+0.07
+0.30
+0.26
SF2
+0.21
+0.31
+0.84
+1.05
aFrom
Table 1, column 5.
Table 3. Calculated Values of Bond Order Using Population Analysis Population Analysis Modela Molecule SO2
Lowden
Mulliken
Natural
C–T
T–T
C–T
T–T
C–T
T–T
2.05
0.22
1.74
0.12
1.50
0.00
H2O
1.04
0.03
0.88
0.00
1.00
0.00
O3
1.94
0.45
1.35
0.36
1.50
0.00
H2S
1.04
0.01
0.97
0.00
1.00
0.00
NO2
2.12
0.29
1.58
0.17
1.50
0.00
OF2
1.26
0.07
0.93
0.03
1.00
0.00
SF2
1.06
0.05
0.81
0.02
1.00
0.00
aC–T
is the central atom to terminal atom bond order, and T–T is the terminal atom to terminal atom bond order.
electrostatic potential, (ii) Mulliken population analysis, and (iii) natural population analysis. The results are shown in Table 2. Calculated values of bond order using Mulliken population analysis, natural population analysis, and Löwdin population analysis are given in Table 3. It is observed that a unique, calculated bond order does not result from the quantum mechanical treatment, but from a calculation based on an arbitrary definition of a “bond”. Different bond definitions provide different methods of calculation, which in turn yield different bond orders. Examining the absolute, calculated
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Research: Science and Education
separation is much smaller than that calculated by the natural population method, the suitability of this method to explain observed molecular properties is suspect. Further evidence that Lewis structures do not describe a unique electronic structure of a molecule is provided by comparing the localized molecular orbitals obtained from quantum mechanical analysis to the Lewis structures. This comparison for a few simple molecular species is made below.
2.0
Observed Electrostatic
Atomic charge
1.5
Mulliken Natural
1.0
Water The best Lewis structure for water is represented by structure 1.
0.5
O H H
0.0 SO2
H2O
O3
H2S
NO2
OF2
SF2
Molecule Figure 1. Atomic charge on the central atom calculated from (i) the known dipole moment and molecular geometry, (ii) the molecular electrostatic potential, (iii) Mulliken population analysis, and (iv) natural population analysis for the seven molecules: SO2, H2O, O3, H 2S, NO2, OF2, and SF2.
bond order between the central oxygen atom and one of the terminal oxygen atoms in ozone, values of 1.94 (Lowdin), 1.50 (natural) and 1.35 (Mulliken) are observed. However, the general trend in calculated bond order among the different methods is similar: H2O, H 2S, SF2, OF2
