Quantum Calculations of Hydrogen Chloride, Sulfur Dioxide, and m-dichlorobenzene
Sean Boulanger & Erica Kirinovic
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)