## Thesis Format

Monograph

## Degree

Master of Science

## Program

Chemistry

## Supervisor

Viktor N. Staroverov

## Abstract

It is a non-intuitive but well-established fact that the first and higher vertical ionization energies (VIE) of any N-electron system are encoded in the system's ground-state electronic wave function. This makes it possible to compute VIEs of any atom or molecule from its ground-state wave function directly, without performing calculations on the (N-1)-electron states. In practice, VIEs can be extracted from the wave function by using the (extended) Koopmans' theorem or by taking the asymptotic limit of certain wave-function-based quantities such as the ratio of kinetic energy density to the electron density. However, when the wave function is expanded in a Gaussian basis set, the latter method fails because the ratio diverges in the asymptotic limit. We show that, in such cases, the first VIE of any finite system can still be estimated by taking the asymptotic limit of the average local electron energy function. This function is constructed from an exact or approximate ground-state wave function of the system and approaches a system- and method-dependent constant in the asymptotic limit. For Hartree--Fock and density-functional theory, this limit reduces to the eigenvalue of the highest-occupied molecular orbital and hence the first VIE according to Koopmans' theorem. We also show that, in the finite-basis approximations of these theories, this constant will generally be more negative than the eigenvalue of the highest-occupied molecular orbital. The results are generalized to finite-basis-set post-Hartree--Fock theory.

## Summary for Lay Audience

Consider the amount of energy it takes to kick out an electron from a molecule --- the ionization energy. The ionization energy is a measure of how tightly a molecule holds on to its electrons. Molecules with low ionization energy tend to be more reactive as they lose electrons easily to other molecules. Because the ionization energy explains a lot about chemical reactivity, chemists have long tried to predict it from the quantum-mechanical model of atoms and molecules.

The quantum-mechanical model revolves around the molecule's wave function which encodes all the properties of the molecule, including its ionization energy. In fact, it turns out that the ionization energy determines the shape of the furthest region of the wave function. That means that just by knowing the shape of that region, quantum-chemists can predict the ionization energy or* vice versa*. However, in common approximations to the wave function, the outermost region suffers the most and predicts nonsensical ionization energies. We present the average local electron energy (ALEE). The ALEE is calculated from the wave function but shows more promising predictions of the ionization energy in its furthest region. In fact, we also show that its deviation from exact ionization energy follows a certain law when the ALEE function is expressed in the commonly used finite-basis set approximation.

## Recommended Citation

El-Samman, Amer Marwan, "Vertical Ionization Energies from the Average Local Electron Energy Function" (2019). *Electronic Thesis and Dissertation Repository*. 6633.

https://ir.lib.uwo.ca/etd/6633

#### Included in

Numerical Analysis and Scientific Computing Commons, Other Chemistry Commons, Physical Chemistry Commons