Lewis Structures of SO₂, SO₃, and SO₄^2- Using Ab Initio Calculations

by

Zeb C. Kramer

Faculty Advisor Dr. Stephen A. Angel, Chemistry

Abstract

Lewis structures of SO₂, SO₃, and SO₄^2-, are inconsistently presented in introductory chemistry textbooks.  These texts and several articles published in the Journal of Chemical Education either favor structures generated using the octet rule or those that reduce the formal charges on the constituent atoms.  It is the purpose of this study to elucidate the most relevant Lewis structure of each molecule using ab initio quantum chemical calculations.  Justification of the model chemistries are assessed by comparing experimental values to calculated vibrational frequencies, force constants, heats of formation, and dipole moments.  Using the empirically valid basis set and method, the relevant Lewis structure may be presented by analyzing bond lengths, population analysis methods, and additional output available in the software packages Gaussian ’03 and Spartan ’04.

Introduction

                Lewis structures of molecules provide easily constructed and essential models of molecules.  Since their introduction by G.N. Lewis in 1916, Lewis structures have performed a vital role in the field of chemistry, relaying information on the stability, physical properties, and the reactivity of molecules.  For some molecules, it is possible to draw more than one Lewis structure.  Each structure corresponds to a configuration with a distinct energy.  A high energy structure corresponds to an unstable arrangement while those of lower energy describe more stable configurations.  The actual structure of a molecule is a hybrid of all possible combinations; however, low energy (stable) structures contribute more heavily to the actual arrangement.

                Despite their importance, the presentation of Lewis structures in introductory chemistry courses is marked with ambiguity.  There are two rules of thumb used to generate Lewis structures.  The first is the octet rule which states that all atoms in a molecule are to be surrounded by eight electrons (with the exception of hydrogen which may be surrounded by only two).  The second is the minimalization of the formal charges on the constituent atoms (assigning the number of electrons to each atom that is sufficient to balance the nuclear charge).  For many molecules, it is possible to generate the most stable Lewis structure using these trends.  However, there exist some molecules and ions for which the two rules support contrasting structures.  These molecules require further experimental or theoretical information in order to compose the most stable structure.  Such is the case for the molecules sulfur dioxide (SO₂), sulfur trioxide (SO₃), and the sulfate ion (SO₄^2-).  The conflicting structures for each are give below:

Freshman chemistry texts often use these molecules as examples for Lewis structure construction.  Some of the texts favor the formal charge structures and others the octet structures.  Several studies published in the Journal of Chemical Education have also reached contradictory conclusions.  In one paper by Gordon H. Purser, the author states “Only Lewis structures that minimize formal charges and expand octets coincide exactly with the results of quantum mechanical modeling” (1).  Another study by Suidan et al. asserts “… the Lewis structures that most accurately represent these molecules are the original Lewis structures, which generally abide by the octet rule” (2).  Moreover, sulfur trioxide is used in the synthesis of sulfuric acid (battery acid), is one of the most potent oxidizing agents known, and is involved in the formation of acid rain.  The sulfate ion is a common constituent of many salts used universally in chemistry laboratories.

                Determining the structures of some compounds (such as SO₃) experimentally can be both expensive and difficult.  An alternative means of studying molecular structure is ab initio molecular orbital theory.  Using only the results of quantum chemical calculations performed by a computer, it is possible to characterize the structure of many molecules.  Common software packages such as Spartan ‘04 and Gaussian ‘03 provide the necessary programs to perform these calculations.  In this study, these packages were employed to investigate the structures of the sulfur-based molecules mentioned above.

                Ab initio calculations depend upon both the method of calculation and the initial conditions employed.  There are several quantum chemical methods used to approximate a molecular system.  These methods vary in both the approximations they make and in their complexity.  The simplest calculation in ab initio theory is the Hartree-Fock Self Consistent Field (SCF) calculation.  More advanced ab initio methods are simply extensions upon the SCF approximation.

                The input for a quantum chemical calculation usually consists of the types of atoms, an initial guess of their relative positions, the overall charge of the molecule or ion, the multiplicity (which is determined by the spins of the electrons in the molecule), and an initial guess for the molecular wave function (which specifies the state of the molecule).  The molecular wave function is often represented as a determinant whose terms are one electron spin-orbitals:

Y = |c₁(i) c₂(i) … c_n(i)|,

where each column corresponds to a spin orbital and each row corresponds to an electron.  Each of these spin-orbitals may be factored into a one electron spatial wave function, called a molecular orbital, and a spin function:

c_j(x,y,z,x) = f_j(x,y,z)a(x),

where x, y, and z signify spatial coordinates and x specifies the spin of the electron.  Each molecular orbital is usually composed of a linear combination of atomic orbitals (called basis functions):

f_j(x,y,z) = c₁j_1s + c₂j_2s + c₃j_12px + c₄j_2py + c₅j_2pz + …,

where each j_k is a atomic orbital and each c_2k is the corresponding coefficient of the linear combination.  The set of functions j_k is the basis set.  The functions of the basis set are themselves approximations of actual atomic orbitals.  These functions are composed of linear combinations of functions called Gaussian Type Orbitals (GTOs) of the form:

G(x,y,z) = Nx^ay^bz^ce^{-zr^2},

where a, b, and c specify the angular momentum, N is a coefficient, and z determines the width of the orbital.  The parameters N and z are determined by fitting to Slater type orbitals or in atomic orbital approximation methods.  GTOs are often used since they are readily integrated, making them less computationally expensive than other relevant functions (3).

                Generally, the more atomic orbitals represented in the basis set, the greater the accuracy of the generated results.  However, larger basis sets slow calculation time; when picking a basis set it is necessary both to provide a sufficient number of functions to describe the system and to use a basis set of reasonable size to reduce computation time.

                The SCF calculation is often used in conjunction with a geometry optimization.  In an optimization, the nuclear positions of the atoms are calculated along with the molecular wave function so that the potential energy of the system is minimized.  Such an arrangement is said to be a global minimum on the system’s potential energy surface (PES).  In a SCF optimization, the nuclear coordinates are initially fixed.  The SCF calculation is begun by taking the derivative of the energy with respect to each coefficient composing the molecular wave function and setting each to zero.  This produces a system of equations called the Hartree-Fock equations, from which can be obtained each molecular orbital.  However, in order to solve for the first molecular orbital (which entails finding the coefficients of the linear combination of the basis set functions for that orbital) it is necessary to know all the other molecular orbitals.  Thus, it is necessary to have an initial guess for these orbitals in order to generate coefficients for the first orbital.  After each molecular orbital has been solved for using the initial guess of the molecular wave function, the molecular orbitals used to generate the first molecular orbital have changed.  Thus, the process of solving for the molecular orbitals is performed again using the new approximations.  The procedure is iterated until the coefficients differ by less than a set tolerance.  When the coefficients are determined, the SCF calculation is completed.  In an optimization, the gradient of the system’s energy with respect to the nuclear coordinates is found.  The opposite direction of the gradient indicates the direction on the PES of a local minimum (or, in some cases, a saddle point).  The nuclear positions are then varied to a point in the direction opposite the gradient and closer to the minimum on the PES.  Varying the nuclear coordinates changes the system and its energy.  With the new positions, a new SCF calculation is performed and the process is begun again.  The optimization is iterated until the root mean square of the gradient is less than some set tolerance (3).

                One critical approximation made when solving the Hartree-Fock equations pertains to electrons with opposite spins.  Rather than accounting for the explicit electron-electron repulsions, the Hartree-Fock equations assume each electron experiences the average repulsion due to each other electron (that is it treats the other electrons as clouds of negative charge rather than as particles).  This approximation causes overestimation of the energy of the system since electrons tend to correlate their motions to avoid repulsive forces.  Advanced calculations attempt to approximate the state of the system by including the energy contribution of electron correlation.  In this way, these computations are simply extensions upon Hartree-Fock calculations.  The general method employed to account for electron correlation is to extend the molecular wave function.  Each determinant corresponds to an electron configuration of the molecule.  However, the molecular wave function cannot be represented by a single configuration.  In order to generate additional configurations, occupied molecular orbitals in the original determinant are replaced with unoccupied (or virtual) orbitals.  This corresponds to promoting electrons from low to high energy orbitals.  Thus, the new molecular wave function is expressed as a linear combination of determinants, each with a corresponding coefficient that must be determined in the computation.  A wave function composed of determinants of all possible promotions is said to be the Full Configuration Interaction wave function; this wave function is the best possible approximation for a given basis set, but is extremely computationally expensive.  In practical calculations, usually only singly and doubly promoted electron configurations are included with the Hartree-Fock wave function.  In this way, electron correlation effects are often accounted for.  Even elementary correlation calculations can be computationally costly if the molecular system is large or if the basis set employed is of significant size (3, 4).

                Once the molecular wave function is determined, the state of the system is specified.  A multitude of information can be extracted from the molecular wave function.  Several notable quantities are the vibrational frequencies, atomic charges, thermochemical data, bond lengths, bond angles, dipole moments, and additional information relevant to the electronic distribution of the molecule.  When generated from an ab initio approximation, each quantity inherits error from the method and basis set from which it was obtained.  Vibrational frequencies, and quantities dependent upon them, also suffer from an additional error if advanced correlation computations are used.  These frequencies are generated from the first derivative of the energy with respect to time.  If this derivative must be evaluated numerically (as often occurs for advanced correlation computations), error propagated from the numerical technique will be introduced into the calculated frequencies.  Atomic charges are also subject to scrutiny.  When calculating atomic charges (in a procedure called population analysis), it is necessary to assign electron density in a molecule to one atom or another.  This assignment is a necessarily arbitrary process for which there exist several techniques.  The most common of these is Mulliken population analysis which assigns half the overlapping electron density between two atoms to each (5).  This simple scheme gives a qualitative picture of the electron environment of atoms in the molecule.  However, Mulliken population analysis tends to overestimate the electron population in high energy molecular orbitals relative to low energy orbitals.  Natural population analysis (another method used in this study) tends to allocate electron density to orbitals located between two nuclei or to a single nucleus.  Mulliken population analysis favors higher bond orders while natural population analysis favors lower bond orders (6).

                The purpose of this study is to identify the most important Lewis structures of SO₂, SO₃, and SO₄^2- using the results of quantum chemical calculations.  Justification of the basis set and method used (termed the model chemistry) will be assessed with comparison of calculated vibrational frequencies, force constants, dipole moments, and thermochemical data with experimental values.  Selection of the appropriate Lewis structure is based upon the bond length and electron distribution obtained from the empirically valid model chemistry.  The intent of this report is to convey the uses of quantum chemical calculations in common chemical applications and to clarify the presentation of Lewis structures in introductory chemistry courses.

Calculations & Procedures

                In these studies, several ab initio calculations were performed upon SO₂, SO₃, and SO₄^2-, varying the basis set and computational algorithm.  The Spartan ‘04 and Gaussian ‘03 software packages provided the necessary programs for this research.  Geometry optimizations were performed using the Hartree-Fock and Møller-Plesset second order (MP2) methods.  Special emphasis was placed upon the larger Pople basis sets, 6-311G* and 6-311+G*.  The second of these sets is reputed to be necessary for calculations performed on anions and molecules with diffuse electrons since they include diffuse orbitals in the basis set.  Moreover, note that these basis sets also include polarization functions (d-orbitals) hypothesized to be important in sulfur bonds.  In the calculations performed, the energy, vibrational frequencies, bond lengths and angles, force constants, selected thermochemical data, and dipole moments were computed.  Mulliken and natural population analysis results were also obtained.  Empirical values for the vibrational frequencies, force constants, dipole moments, and thermochemical data were compared to calculated values to determine the optimum method and basis set.  The relevant Lewis structures were compared with the computed data from the chosen model chemistry.

Results & Discussion

                Comparison of calculated vibrational frequencies of SO₃ with values produced by Chysostom et al. strongly validate the choice MP2/6-311G* as the model chemistry.  The MP2 method optimized structures also show superior agreement to those calculated using only the Hartree-Fock method.  The table below lists relevant results for SO₃:

Vibrational Frequencies of SO₃ (7)

The first two columns specify the method and the basis set used in the calculations.

All vavenumvers have the units cm^-1.

At the time of this report, the values above are the only experimental values to be obtained.  Note the MP2/6-311G* shows much better agreement than the MP2/6-311+G*, although the two basis sets differ only by diffuse functions added to the latter.  It is expected that the MP2 algorithm will use only the necessary functions to describe the system, and so the basis set 6-311+G* should either yield superior, or at least very similar, results to those generated with 6-311G*.  It is expected that SO₄^2- will favor the 6-311+G* basis set, since it is an anion, while SO₂ will mimic the results shown be SO₃.  The next phase of the study is to obtain and compare other experimental values to calculated data.

                The compound SO can be unambiguously represented as a sulfur and oxygen atom connected by a double bond.  It is interesting to compare the bond length and electron density of SO with that of the other molecules.  Below are listed some of the calculated bond lengths:

S-O Bond Lengths of Structures Optimized at the MP2 Level

All values are given in Angstroms.

Note that both SO₂ and SO₄^2- have comparable bond lengths with SO; SO₃ has bond lengths calculated to be much shorter than SO.  This evidence seems to support the formal charge structures, which favor higher bond orders.  However, the length of the bonds may be due to electrostatic effects rather than shared electron density between the sulfur and oxygen atoms (2).  Surfaces of constant electron density of 0.20, 0.25, and 0.30 (all electron densities have units of electron per bohr³) for SO₂, SO₃, and SO₄^2-, respectively, calculated at the MP2/6-311G* level are shown below:

Note that all molecules have interatomic electron density at the density value 0.20.  It is significant that at the density 0.25, SO, SO₂, and SO₄^2- have little or no density between the sulfur and oxygen atoms.  SO₂ has no interatomic density at 0.25, which suggests that the sulfur-oxygen bonds are slightly weaker than the SO bond, as would be expected for the octet structure.  SO₄^2- shows similar results.  Note that SO₃ does have a significant amount of density at 0.25.  This would suggest that SO₃ has a stronger covalent bond than the double bond of SO.  Thus, electron density studies seem to favor the formal charge structure for SO₃.  At 0.30, all electron density is isolated about the atomic nuclei.

                Atomic charges (calculated from almost any model chemistry employing a basis set larger than STO-3G) seem to favor the octet conforming structures.  Since the results of each model chemistry are similar, an example of atomic charges calculated using MP2/6-311+G* is shown below:

Mulliken Atomic Charges of Constituent Atoms

Natural Atomic Charges of Constituent Atoms

                In each method, a large positive charge (an electron deficient environment) is associated with the sulfur, while a negative charge is associated with the electronegative oxygen atoms.  These data favor the octet structures and support the hypothesis that the short bond length is due to electrostatic rather than covalent interactions.  Recall that Mulliken analysis tends to favor high bond orders (high bond orders are associated with the formal charge structure).  It is thus significant that the Mulliken analysis favors the octet structure.

Conclusion

                Evidence for both the formal charge and octet structure has been exhibited.  While bond lengths purport the formal charge structures, the population analysis results favor the octet structures.  Electron density structures favor the octet structures for SO₂ and SO₄^2- and the formal charge structure for SO₃.  In order to judge which results are valid, it will be necessary to obtain further experimental data for SO₂, SO₃, and SO₄^2-.  At this time, only the experimental vibrational frequencies of SO₃ have been obtained. 

                Another phenomenon observed was the success of the 6-311G* basis set over the 6-311+G* to describe the SO3 vibrational frequencies.  Inclusion of the diffuse functions in 6-311+G* inhibits the ability of the calculation to describe reality.  This anomaly is evidence that a larger basis set is not always a superior approximation.

                After choosing the appropriate model chemistry, the next step in the study is to use the model to perform studies done by Purser and Suidan et al. (1, 2, 6).  Comparing results produced by a empirically valid model, it is hoped that an appropriate Lewis structure for SO₂, SO₃, and SO₄^2- may be presented.

Acknowledgements

                I would like to thank the providers and committee of the WU-CSI grant who provided the software Spartan ‘04 for Windows and the summer stipend.  I would like to thank the Washburn Chemistry Department for all their help.  I would like to thank my advisor Dr. Kevin Charlwood for his interest in this project.  Lastly, I would also like to thank my advisor Dr. Stephen A. Angel for his patience and enthusiasm. 

References

1.  Purser, Gordon H., Journal of Chemical Education 78, 981-983 (2001).

2.  Suidan, L.; Badenhoop, J.K.; Glendening, E.D.; Weinhold, F., Journal of Chemical Education 72, 583-      586 (1995).

3.  Hehre, W.J.; Radom, L.; Schleyer, P.v.R; Pople, J.A., Ab Initio Molecular Orbital Theory, Wiley, New       York, 1985.

4.  Hehre, W.J., A Guide to Molecular Mechanics and Quantum Chemical Calculations,Wavefunction,        California, 2003.

5.  Mulliken, R.S., The Journal of Chemical Physics 23, 1833-1849, (1955).

6.  Purser, Gordon H., Journal of Chemical Education 76, 1013-1018 (1999).

7.  Chrysostom, E.T.H.; Vulpanovici, N.; Masiello, T.; Barber, J.; Nibler, J.W.; Weber, A.; Maki, A.; Blake,                      T.A., Journal of Molecular Spectroscopy 210, 233-239 (2001).