International Journal of Computational and Theoretical Chemistry 2015; 3(1): 1-5 Published online April 13, 2015 (http://www.sciencepublishinggroup.com/j/ijctc) doi: 10.11648/j.ijctc.20150301.11 ISSN: 2376-7286 (Print); ISSN: 2376-7308 (Online) A Theoretical Method for Calculating the Bond Integral Parameter for Atomic Orbitals Dale J. Igram 1 , Jason W. Ribblett 2 , Eric R. Hedin 1 , Yong S. Joe 1, * 1 Center for Computational Nanoscience, Department of Physics and Astronomy, Ball State University, Muncie, USA 2 Department of Chemistry, Ball State University, Muncie, USA Email address: ysjoe@bsu.edu (Y. S. Joe) To cite this article: Dale J. Igram, Jason W. Ribblett, Eric R. Hedin, Yong S. Joe. A Theoretical Method for Calculating the Bond Integral Parameter for Atomic Orbitals. International Journal of Computational and Theoretical Chemistry. Vol. 3, No. 1, 2015, pp. 1-5. doi: 10.11648/j.ijctc.20150301.11 Abstract: In molecular orbital theory, the bond integral parameter k is used to calculate the bond integral β for different molecular structures. The bond integral parameter k, which represents the ratio of bond integrals between two atoms of a diatomic molecule, is a function of the bond length. This parameter is usually obtained empirically; however, it will be shown that k can be determined analytically by utilizing the overlap integral S. k will be calculated for different atomic orbital combinations (ss,pp) of σ and π interactions as a function of bond length for a carbon-carbon diatomic molecule. The results, which are represented graphically, indicate that different atomic orbitals in different interactions can have the same, or very close to the same, k values. The graphs reveal some significant features for the different atomic orbital combinations with respect to magnitude and profile, as well as illustrate good agreement with experimental results, which validates the utilization of the overlap integral calculation method for the determination of the bond integral parameter k. Keywords: Molecular Orbital Theory, Bond Integral, Overlap Integral, Atomic Orbitals, Carbon Dimer 1. Introduction Molecular Orbital Theory (MOT) has been used extensively in physical, inorganic, and organic chemistry to predict physical properties of a single molecule, such as orbital energies, bond length, bond order, bond energies, bond delocalization energies, and electron densities [1, 2, 3]. An important parameter that has been employed in MOT and has been used extensively in the Hückel method for molecular systems containing different atoms is the bond integral parameter k. This parameter is considered to be a scaling factor for bond integral calculations. The bond integral parameter k is very useful for large and complex molecular systems with many different types of atoms, such as DNA, and allows the general molecular orbital energy equation [1, 2, 3, 4] to be solved. One such application [4] pertains to the determination of the strength of the electronic coupling between two adjacent base pairs for a B-DNA molecule, where k values were only determined for 2pσ and 2pπ atomic orbital pairs, which resulted in the calculation of 342 atomic orbital combinations. These k values were utilized in a 361 element energy matrix for the calculation of the molecular orbital wave function coefficients, which were then used for the electronic coupling computation. What is new is a detailed analysis of k for different atomic orbital combinations and interactions resulting in a more comprehensive understanding of some important and interesting features, as well as a comparison between the calculated and empirical values of k for a carbon dimer. By definition, the bond integral β=kβ o , where β is the measured interaction energy between two atomic orbitals (which is difficult to obtain empirically), and β o represents a standard β defined at a specific bond length, such as the carbon-carbon bond distance in benzene (1.397Å). Now, β has been proposed [5] to also be proportional to the overlap integral S, which can be written as β=Sβ o /S o . The overlap integral S is a non-energy quantity [1], which can be determined theoretically, thus allowing for the calculation of k, where k=S/S o , and S o represents a standard overlap integral defined at a specific bond length. For this study, a homo-nuclear carbon dimer was considered, thus simplifying the calculations of the overlap integral S and the bond integral parameter k. The carbon dimer consists of two carbon atoms, where each carbon atom consists of a 1s orbital, 2s orbital, and