Molecular Orbital calculations of Dinitrogen, Hydrogen Cyanide, and Phenol


Ryan Bednarski and Larry Vue

Abstract:

    Quantum mechanic calculations become very complicated when the number of electrons and atoms get bigger than one. Computational approximation methods have been developed to reduce the time spent calculating and give rough estimates of large molecules. The geometry optimization used come from the Ab Initio theory. The Ab Initio levels of theory include: 621G, 631G, and DZV. Using these optimization methods were use on the molecules, Dinitrogen, Hydrogen Cyanide, and Phenol. The best optimization for each molecule had their own level of theory that work the best. For dinitrogen, 621G was the best for the best bond length, but DZV gave the more negative potential. HCN used DZV for the bond lengths and 621G for the dipole moment. Phenol used DZV for the bond lengths and 621G for the dipole moment.

Introduction:

    The electronic activity, especially the valence electrons, is what gives a molecule or an atom its reactive properties. The dipole moment, polarizability, vibrational frequencies, probability of absorption of visible light, and tendency to donate electrons in a reaction, can be determined if the structure of the molecule and the location of its electrons are known. To determine the structure of a molecule and where its electrons can be found, a wavefunction can be used to represent the electrons which are known as molecular orbitals. Unfortunately, this model of assigning a wavefunction to electrons or group of electrons can only give an exact value for a single atom with one electron. As the number of atoms and electrons increase, the more complicated the potential energy (eigenvalue) portion of the Hamiltonian operator becomes. No exact solutions can be found for two or more atoms and electrons.
    With this problem of not being able to obtain an exact value, an approximation method was implemented. The variational principle allows us to approximate wavefunctions with a sum of simpler wavefunctions. To put it in other words, a linear combination of atomic orbitals (LCAO).  The wavefunctions found with the lowest energy, or the most negative, have been proven to be the best approximation to the actual wavefunction. This energy found is the expectation value of the Hamiltonian wavefunction divided by the normalized wavefunction.
    First, a little background on the computation geometry optimization methods is needed. MOPAC3 or molecular orbital package combines these semi-empirical methods that provide estimation of two electrons overlap integrals needed for calculating Hamiltonians.
    The ab initio molecular orbital theory first states that nuclei are stationary relative to electrons due to the Borne-Oppenheimer approximation, which are equilibrated to the molecular geometry. Electrons also move independently of each other, and how it is affected by the average electric field created by all other electrons and nuclei in the molecule. The molecular orbital will then be constructed as a linear combination of atomic orbitals and calculated. The smallest calculation in the basis set is 621G. Then increasing in geometry theory is 631G, and DZV. Delta Zeta Valence (DZV) being the high level of theory for geometry optimization.
    First, the program Avogadro was used to construct each molecule to be optimized. Then the progam MacMolPlt was utilized to perform the MOPAC3 AM1 and PM3 optimizations. Those optimized files were then optimized again in MacMolPlt using the Ab Initio daisy chain of 621G to 631G to DZV. After the optimizations of all three molecules were complete, the molecules were ran through the GamessQ package to be able to view the .log files of each molecule. These .log files of each molecule were then search for dipole moments, UV-Vis wavelengths, and IR-speectrum data. All other values for each molecule were determined through the Jmol software. These values include, bond length, bond angles, electrostatic potential, partial atomic charge, and HOMO and LUMO orbitals.

Results:
The results from the experiment for each molecule can be found within these hyperlinks. These hyperlinks contain interactive diagrams along with figures and tables to represent all data collected:    Dinitrogen    Hydrogen Cyanide    Phenol

Conclusion:
    Overall, Quantum Mechanical geometry optimizations gave a rough estimate of values for molecules. The levels of theory used for the calculations were 621G, 631G, and DZV. Generally, the smaller the molecule is, the close the calculated values will be to the literature values. Dinitrogen was a simple diatomic molecule and it did not require a high level of theory to calculate the bond lengths. 631G was best for the bond lengths. However, DZV gave the lowest potential energy versus bond length which has been proven to be the best optimization according to the Hardtree-Fock self consistent field theory. For the smallest polar molecule, hydrogen cyanide, the opposite was true compared to dinitrogen. The best optimization for the bond length used DZV level of theory, where as the best optimization for the dipole moment only needed to use the 621G level of theory. Phenol was the last molecule to be optimized. Phenol used the highest level of geometry optimization for calculating the bond length, but only required the 621G theory for the dipole moment. HCN and phenol were similar with the theories used to obtain values. This might have to do with them being polar molecules.
    Since the quantum mechanical calculations of large molecules get increasingly complex to the point that it would take days to solve, computer programs and approximation methods have been implemented. Using these geometry optimization methods, HF-SCF, MOPAC3 and Ab Initio, can be very useful when determine the values or data of structures that are enormous, for example proteins, virus structures, and ribosomes.  These approximations do have their draw backs. For example, over approximating for IR spectrum values, dipole moments, and UV-Vis wavelengths. This was shown for the molecules.
 
References:

1. Mohrig R. Jerry, Hammond Noring Christina, Schatz F. Paul, Techniques in Organic Chemistry, Third Edition.; W.H. Freeman and Company: New York 2010; pp IR index.

2. Listing experimental Data for N2, http://cccbdb.nist.gov/exp2.asp?casno=7727379 accessed March 10, 2015.

3. Listing experimental Data for HCN (Hydrogen Cyianide), http://cccbdb.nist.gov/exp2.asp?casno=74908 accessed March 10, 2015.

4. Listing experimental Data for C6H5OH (Phenol), http://cccbdb.nist.gov/exp2.asp?casno=108952 accessed March 10, 2015.

5. McClellan L. A., Table of Experimental Dipole Moments, W.H. Freeman and Company: San Francisco and London 1963. pp 38.

6. The constants in Category "Universal constants", http://physics.nist.gov/cgi-bin/cuu/Value?c|search_for=universal_in! accessed March 10, 2015.