Abstract

    Hydrogen chloride, sulfur dioxide, and m-dichlorobenzene were analyzed computationally at multiple levels of molecular orbital theory to determine which basis set provided the best geometry optimization for each molecule.  The comparison of optimized geometries from five basis sets over two levels of theory are presented for each molecule.  Two MOPAC basis sets were used; AM1 and PM3, along with three Ab initio basis sets; 6-21G,  6-31G, and Double Zeta Valence (DZV) (in order of increasing size).  The best geometries for all three molecules came from the DZV basis set, and the optimized geometries were used to further calculate the dipole moments and visualize the highest occupied molecular orbital (HOMO), lowest unoccupied molecular orbital (LUMO), electrostatic potential, and partial atomic charges for all three molecules. Additionally for hydrogen chloride, the vibrational frequencies were calculated and the three levels of Ab initio theory were compared directly by mapping the potential energy of the molecule vs. bond stretching to determine which level had the lowest energy minimum.  Additionally for m-dichlorobenzene, the predicted UV-Visible absorbance peaks were extracted and compared to literature values.

Introduction

    The molecular orbitals that contain electrons in atoms and molecules are described by mathematical representations, called wavefunctions, that predict where an electron in a given orbital is likely to be found.  If a wavefunction for a given atom or molecule is an eigenfunction of the Hamiltonian operator, then the Schrodinger equation can be solved analytically and the eigenvalue reveals the energy of the species. This information is incredibly useful in predicting properties of the atom such as the vibrational frequencies, dipole moment, potential energies, bond lengths and angles, and the absorption of UV-Visible light.  Unfortunately, analytical solutions to the Schrodinger equation are only available for single electron systems, such as the Hydrogen atom and the Helium cation.  For more complicated systems with more than two charged particles, the variational principle can be employed to approximate a wavefunction by forming linear combinations of trial wavefunctions, each with a coefficient that can be adjusted to supply more or less of its respective character to the overall wavefunction.  Since the variational principle does not provide analytical solutions to the Schrodinger equation, the exact energies cannot be solved.  However, the expectation value of the energy can always be found by calculating the expectation value of the Hamiltonian.  Historically, these lengthy calculations were done on paper by Scientists with advanced degrees.  The advent of computers and specialized software has made it possible for the slightly-above-average-Joe to perform these quantum calculations using large basis sets and geometrically optimized molecules, improving the accuracy of the electronic structure and energies that can be predicted.[1]
    The initial geometry optimizations were done for each molecule using Molecular Mechanics in the program Avogadro to generate input (.inp) files to be run in the GAMESS computation package.  The software at this level of theory performs the optimization based on classical mechanics by treating the molecules as collections of harmonic oscillators, or balls on springs.   This level of theory is useful for modeling large systems and predicting geometries since the calculations are based on the number of atoms present as opposed to the number of electrons.  The relative energies of various conformations can also be compared at this level of theory.
    The secondary geometry optimizations were performed by inputting the optimized geometries that were completed in Avogadro for each molecule into the program MacMolPlot and selecting a semi-empirical level of theory referred to as MOPAC.  This method estimates the values of the electron overlap integrals that are required for calculating the expectation value of the Hamiltonian.  Two Hamiltonians were selected and all three molecules were optimized under both AM1 and PM3 basis sets, both of which generated .inp files that were run in GAMESS.  The geometries that are predicted by this level of theory are very good, since they are derived from empirical enthalpies of formation, ground state geometries, dipole moments, and ionization potentials.
    The third geometry optimizations were performed by inputting the AM1 optimizations back into MacMolPlot and selecting the 6-21G Gaussian basis set, which is the smallest basis set chosen at the Ab initio level of theory, to be run in GAMESS.  At this level of theory all integrals are calculated, though there are still approximations required to assemble a self-consistent field using a finite sized basis set.  The software at this level can calculate dipole moments, electron transition frequencies, geometries, vibrational frequencies and many other properties.
    For hydrogen chloride, the AM1 optimizations were input back into MacMolPlot a second and a third time to further optimize the geometry using the 6-31G basis set and the DZV (double zeta valence) to generate input files to be run in GAMESS.  For sulfur dioxide and m-dichlorobenzene, the 6-21G optimization results were used as the input in MacMolPlot to build the 6-31G optimizations to run in GAMESS, and in turn the 6-31G optimizations were used as the input in MacMolPlot to build the DZV optimizations to run in GAMESS.
        The dipole moments for each of the three molecules were extracted from the .log files from each level of theory and compared to literature values to determine which basis set gave the most accurate optimized geometries.  The vibrational frequencies for all three molecules, and the UV-Vis absorbance spectra for m-dichlorobenzene were obtained by running separate calculations on the best optimized geometry for each molecule.  Additionally for hydrogen chloride, a potential energy surface vs. bond length was plotted in IGOR Pro using the best optimized geometry from the Ab initio level of theory. 

    All of the geometries, along with live displays of the highest occupied molecular orbital (HOMO), the lowest unoccupied molecular orbital (LUMO), maps of the electrostatic potential, and partial atomic charges from each molecule were visualized in Jmol, and can be viewed by clicking on the following links for each respective molecule:  ********INSERT DAT SHIT HERE**********

Conclusion

    The software used for running these types of calculations is incredibly powerful, and can provide useful results that help to predict some of the properties and behaviors of molecules.  However, the accuracy of these results cannot always be established.  For example, much is already known about hydrogen chloride.  Literature values for the dipole moment, vibrational frequencies, bond lengths and angles could all be easily obtained from the National Institute of Standards and Technology (NIST) Chemistry Webbook [2] or the NIST Computational Chemistry Comparison and Benchmark DataBase (CCCBDB).[3]  Having these standards in hand allowed the geometry to be further optimized by the inclusion of diffuse functions in order to better match the literature dipole moment.  That option was not available for m-dichlorobenzene, which had minimal to no literature values available for comparison.  The lack of available data on m-dichlorobenzene means that the validity of the results obtained for this molecule are questionable.
    The DZV basis set provided the best geometry optimizations for all three molecules, and the geometry that best matched the dipole moment for hydrogen chloride came from inputting D=3, F=1, light=3.  Interestingly, the 6-31G basis set provided the lowest potential energy minimum for hydrogen chloride out of the three Ab inito sets.  This is surprising due to the fact that the best match to the literature dipole moment came from the DZV basis set for hydrogen chloride.
    These discrepancies show that computational results should not be taken as absolute, and further investigation is needed to determine the true accuracy of the values obtained computationally.

References


1. Gutow, J.; Mihalick, J; Quantum Calculations; Physical Chemistry Lab II; Oshkosh, WI, 2016

2. The National Institute of Standards and Technology (NIST) Chemistry WebBook. http://webbook.nist.gov/chemistry/ (accessed February 15, 2016).

3. NIST Computational Chemistry Comparison and Benchmark Database. NIST Standard Reference Database Number 101. Release 17b, September 2015, Editor: Russell D. Johnson III 
http://cccbdb.nist.gov/ (accessed February 15 , 2016)