From wavefunctions to chemical reactions: New mathematical tools for predicting the reactivity of atomic sites from quantum mechanics

Solving the electronic Schrodinger equation for the molecular wavefunction is the central problem in theoretical chemistry. From these wavefunctions (possibly with relativistic corrections), one may completely characterise the chemical reactivity and physical properties of atoms, molecules, and materials. Unfortunately, there are very few systematic approaches for obtaining highly-accurate molecular wavefunctions. The approaches that do exist suffer from the so-called curse of dimensionality: their computational cost grows exponentially as the number of particles increases. Furthermore, even after obtaining an accurate wavefunction, partitioning the molecule into atoms is not straightforward. This is because the kinetic energy operator is a differential operator in spatial coordinates. This is a source of ambiguity in the definition of an atom-in-a-molecule and the associated atomic properties. Even after selecting an appropriate definition of an atom and obtaining the atoms from the wavefunction, the atom's intrinsic reactivity cannot be completely characterised without considering every possible reaction partner. This is because each set of two molecules produces a new wavefunction that is more complicated than the products of the wavefunctions of the separate molecules.;Partitioning a molecule into atoms is not straightforward because of the inherent nonlocality of quantum mechanics. In the context of molecular electronic structure, this nonlocality arises from the nature of the kinetic energy operator. The quantum theory of atoms in molecules (QTAIM) is a popular method that partitions molecules into atoms. QTAIM resolves the problem of ambiguity for all permissible forms of the kinetic energy operator. In this thesis the characterisation of an atom provided by QTAIM is extended to include relativistic contributions in the zero-order regular approximation (ZORA). The intrinsic ambiguity arising from the kinetic energy operator is also examined in detail.;Computing atomic or molecular properties (including computing the general-purpose reactivity indicator) almost always requires a wavefunction. For this reason, obtaining accurate wavefunctions is the central hurdle of quantum chemistry. This thesis proposes algorithms for finding high-accuracy molecular wavefunctions without exponentially exploding computational cost. To do this, tools for exploiting the smoothness of electronic wavefunctions are crafted. Computational methods that use these tools can break the curse of exponential scaling without sacrificing accuracy. Specifically, the computation cost of these new methods grows only as some polynomial of the electron number. The wavefunctions obtained from these methods are much simpler than those from conventional approaches of similar accuracy, and are therefore ideal for computing the electron density and atomic properties.;This thesis presents methods for addressing the three challenges raised in the previous paragraph: computing atomic properties (e.g. chemical reactivity), partitioning molecules into atoms, and computing accurate molecular wavefunctions. The first challenge is addressed by developing a general-purpose reactivity indicator to quantify the reactivity of an atom within a molecule. This indicator quantifies the reactivity of any point of the molecule using only the electrostatic potential and Fukui potential at that point. The key idea is to include only a vague description of an incoming molecule and compute an approximate interaction with the incoming object; this ensures that the general-purpose reactivity indicator is simple enough to be useful. Practically, this indicator is most useful when it is used to compute the reactivity of the atomic sites in the molecule of interest.