Modeling the Molecular Orbitals
of
Quantum Mechanics

Robert Hoppe and Kimberly R. Hopfensperger
University of Wisconsin -- Oshkosh
Physical Chemistry

Introduction

    With the inception of quantum theory in the 1920's, quantum mechanics has proven itself to be the most accurate model to date for describing the motion of elementary particles (electrons); the position of these particles in three dimensional space can be approximated using quantum mechanical models. This approximate area in space is known as an orbital. Utilizing a molecular modeling software known as GAMESS, the potential energy of the optimized geometrical configurations of molecular nitrogen and hydrofluoric acid were calculated using the 6-21G, 6-31G(d,p), and 6-311G(2d,p). Also, the spectra for UV-Vis at the 6-21G and 6-31G(d,p) and IR using the best ab initio level of theory were generated for nitrobenzene. This was done to test the accuracy of the software against known empirical values in the National Institute of Standards and Technology (NIST) database.


Nitrogen gas (N2)

Optimized geometry for nitrogen



Optimized geometry for nitrogen at the 6-31G level of theory.  A bond length of 1.083 Angstroms was compared against the experimental data on the NIST website.  The accuracy of this value to the experimental was what determined 6-31G to be the best level of theory. 


HOMO image for N2


This is the HOMO, or the highest occupied molecular orbital, for the molecule of nitrogen.  This is  level where electrons will react with another molecule.  Note, however, nitrogen HOMO is doubly occupied, which explains why this molecule is not exceptionally reactive.


This is a graph of the potenital energy of nitrogen at the three ab initio calculations
This is a graph of the of the potential energy vs. bond length distance (Angstroms) for the three ab initio.  As shown, the lowest energy of configuration was calculated by the 6-311g(2d,p), which makes this the most accurate.   Note, this level of theory used the highest number of basis sets.

An overlay of all of the potential energies
Note, when when placed on top of each other, all levels of calculation give the same shaped curvature for geometry. 



Hydrofluoric Acid (HF

image of hydrofluoric acid


Optimized geometry for hydrofluoric acid, where the bond length of 0.9004928 Angstroms was compared against the experimental data on the NIST website.  The accuracy of this value to the experimental was what determined 6-31G to be the best level of theory. 


The HOMO for HF


This is the HOMO, or the highest occupied molecular orbital, for the molecule of hydrofluoric acid.  This is  level where electrons will react with another molecule.  Note, however, hydrofluoric acid HOMO is singly occupied, which explains this molecule's acidity.


Potential Energy graphs of HF

This is a graph of the of the potential energies vs. bond length distance (Angstroms) for the three ab initio for hydrofluoric acid.  Again, as found in the nitrogen above, the lowest energy of configuration was calculated by the 6-311g(2d,p), which makes this the most accurate.   Note, this level of theory used the highest number of basis sets.

Overlay of all energies of HF
Again the energies are placed on top of each other, all levels of calculation give the same generally shaped curvature for geometry.



Nitrobenzene (C6H5NO2)
image of nitrobenzene


Optimized geometry for nitrobenzene.  When compared to the the experimental data on the NIST website, the most accurate level was the 6-311G(2d,p)  level of theory. 


The HOMO for nitrobenzene


This is the HOMO, or the highest occupied molecular orbital, for the molecule of nitrobenzene.  This is  level where electrons will react with another molecule.  These areas of reactivity are on the ring.



experimental IR of nitrobenzene

Above is the experimentally derived IR spectrum of nitrobenzene.  When compared with the NIST website's  IR spectrum for  this compound,  it can be seen that the peaks are shifted to a higher wavenumber and, in general, are not as intense.

Excited State
 Wavelength (nm)
 Oscillator Strength
              
Excited   State
Wavelength (nm)
 Oscillator Strength
1
 230.1
 0.000000
 1
 241.3 0.000000
2
 218.0
 0.000369
 2
 172.8 0.000604
3
 211.6
 0.034349
 3
 160.6 0.000117
4
 207.7
 0.105517
 4
 155.5 0.394369
5
 169.9
 0.000000
 5
 148.5 0.256104
6
 169.3
 0.0006224
 6
 139.2 0.000000
7
 164.8
 0.030938
 7
 125.3 0.12147
8
 163.7
 0.982578
 8
 117.7 1.015837
9
 159.6
 0.028036
 9
 115.7 0.016794
10
 154.9
 0.962891
 10
 113.8 0.000000
The UV-Vis data from 6-21G level of theory. The UV-Vis data from 6-31G(d,p) level of theory.


These values for both theory levels share the same trend as the spectrum on the NIST website but results have been deemed inconclusive as the range of values for the generated data do not fit the one of the standard experimental spectrum.  

A larger value in oscillator strength correlates to a greater dipole transition from the excited state to the ground state, making it a greater contributer to the overall dipole moment of the molecule.


Comparison of the potential bond stretching surfaces between heteronuclear and homonuclear diatomic atoms
This is a graph of the potenital energy of nitrogen at the three ab initio calculations
The potential energy graph of nitrogen
 Potential Engery graphs of HF
The potential energy graph of hydrofluoric acid.
Comparing the two graphs above for a homonuclear molecule (nitrogen) to a heteronuclear (hydrofluoric acid), the energy change between the different levels of theory has a greater shift in the latter's case.  Also, there is a steeper incline out of the well in the heteronuclear case.


Conclusion

    Computational results can be extremely useful in simple classroom usage to see characteristics and differences between different molecules.  The values that are calculated are close enough to experimental values that.  However, such results should be used carefully when under industrial or research purposes, for the calculations are not necessarily as accurate as can be obtained.