Representing The World Around Us: Applications of Group Representation Theory to Molecular Orbital (MO) Theory

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Molecular orbital (MO) theory is a theory at the forefront of modern chemistry, allowing for accurate descriptions of reactivity of molecules by using quantum mechanics to predict the location of electrons within a molecule, and their corresponding energies. The equations which govern their behavior, the Schrodinger equation, are often difficult to solve. Often, we can only approximate a solution using numerical methods. This paper discusses a method which exploits a molecule’s internal symmetry. Specifically, we use Group representation theory to help analyze and break down the molecular symmetry, and then use the analysis to help us find the MO’s. First, we establish key results about irreducible representations and characters. We then establish a correspondence between MO’s and irreducible representations. We then use the results we obtained to perform MO calculations on water (H2O). We then compare the results obtained via our MO theory calculations and Valance Bond Theory (VBT). We conclude by showing these calculations are best used for rough work, being most useful for deciding which atomic orbitals they arose from, and each MO’s energies relative to each other.