5 Energies of Atomization from Population Analysis on Hückel Wave Functions F O R R E S T S. M O R T I M E R
Downloaded by CORNELL UNIV on September 28, 2016 | http://pubs.acs.org Publication Date: January 1, 1966 | doi: 10.1021/ba-1966-0054.ch005
Shell Development Co., Emeryville, Calif.
Three-dimensional Hückel molecular orbital calculations have been performed on a series of molecules made from the atoms H, C , N, O, F, and Cl. It has been found that the sum of the Mulliken overlap populations is closely related to the energy of atomization. For 40 compounds not containing carbon, the observed energies of atomization can be reproduced with a mean deviation of 11.1 kcal./mole by a simple empirical equation which includes a term to account for the extra stability of polar molecules. A slightly more complicated expression is needed for compounds of carbon, and the fit to the observed data is not as good.
' " p h i s study was initiated to determine to what extent empirical molecular orbital ( M O ) theories of the Hûckel type can provide information on the thermodynamic stability of a hypothetical unknown compound. The test, of course, has to be made on known compounds. O u r interests have centered on compounds involving atoms such as N , O, F , and CI, but compounds with C and H have also been included. The results thus far have been encouraging. From the papers of Lipscomb, Lohr, Hoffmann, et al (7, 8, 9,10,11, 12) we first learned of their work on an "extended" Huckel theory for polyatomic molecules. W e also benefited from a visit to Harvard to discuss this work before their computer program became generally avail able. O u r computer program is based on what we learned from them at that time and on our experience since then i n applying it to our particular types of molecules. The other major influence i n the work has come from the papers of Mulliken and his co-workers, i n particular the series (14) on population analysis of L C A O - M O wave functions and their relation to energies of atomization. As Mulliken suggested (14), we have attempted to relate 39
Holzmann; Advanced Propellant Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1966.
40
ADVANCED PROPELLANT CHEMISTRY
the calculated overlap populations to the energy of atomization for the molecule with corrections for the polarity of the bonds.
Downloaded by CORNELL UNIV on September 28, 2016 | http://pubs.acs.org Publication Date: January 1, 1966 | doi: 10.1021/ba-1966-0054.ch005
Three-Dimensional
Huckel
Theory
The theory (7, 8, 9,10,11,12) w i l l be outlined for molecules having η atoms with a total of Ρ valence shell electrons. W e seek a set of molecu lar orbitals ( L C A O - M O ' s ) , ψ, that are linear combinations of atomic orbitals centered on the atoms i n the molecule. Since we shall not ignore overlap, the geometry of the molecule must be known, or one must guess it. The molecule is placed i n an arbitrary Cartesian coordinate system, and the coordinates of each atom are determined. Orbitals of the s and ρ Slater-type ( S T O ) make up the basis orbitals, and as indicated above we restrict ourselves to the valence-shell electrons for each of the atoms in the molecule. The S T O s have the following form for the radial part of the function (13,18): R(r)
Nr>* exp(-Çr/a )
=
H
(1
where Ν is a normalization factor, m = 0 for 1 s electrons, 1 for 2 s or 2 ρ electrons and 2 for 3 5 or 3 ρ electrons, ζ = orbital exponent, and a = Bohr radius = 0.529175 A . The mathematical representation of the basis is needed only for cal culating the overlap matrix, which is assumed to give a good representa tion of the tendency to form a bond. If Φ is a row vector of the atomic orbitals that make up the basis: Φι, Φ . . . $N> then the molecular orbitals are given by an Ν Χ Ν matrix, ψ. H
2
φ
=
φ€
(2)
C is a transformation matrix that satisfies the equations: HC
=
SCe
(3)
and CSC
=
1
(C'
