Very often, chemical reactivity can be thought of in terms of ionization energies. For example, the lower an atom or molecule's ionization energy, the higher its reactivity. For an atom or molecule, X, its ionization energy, IE, can be defined as
X(g) --> X+(g) + 1e- ΔE = IE
Of course, ionization energies will depend on from which molecular orbital an electron is removed. The lowest ionization energy would correspond to removing an electron from the highest occupied molecular orbital (HOMO). So, given its importance, it is often useful to calculate an atom or molecule's ionization energy to predict or explain its chemical reactivity. In this activity, we explore two approaches. One approach is to calculate explicitly the energy of X, and X+ and take energy differences:
IE = E(X+) − E(X) (1)
For smaller molecules, this is a straightforward approach. However, as the size of the molecule increases, this approach becomes increasingly difficult. Another more approximate method, known as Koopman's Theorem, requires the calculation of the electronic structure of X only:
Koopman's Theorem: The lowest (vertical) ionization energy of a molecule can be approximated as the negative of the highest occupied molecular orbital (HOMO) energy:
IE_Koopman = − e(HOMO) (2)
More generally, ionization from the ith molecular orbital (MO) is approximately the negative of the MO energy. In this exercise, we will explore to some degree the quality, or lack thereof, of Koopman's theorem for approximating ionization energies.