In 2022–23, I will be offering a Part II project in the Department of Chemistry, University of Oxford. This project can be based around any of the areas discussed here, or related topics that interest you. Please get in touch directly via email.
We are developing a new way to interpret molecular electronic energy levels as stationary points of an energy landscape, rather like a mountain range. The lowest-energy level corresponds to the valley bottom, while higher-energy levels are represented as high mountain passes and summit peaks. We can then investigate the fascinating pathways that connect different stationary points.
Excited electronic states in molecules have traditionally been predicted by considering the response of an approximate ground state to an external field, or by solving large matrix eigenvalue problems. These approaches are typically biased towards the ground-state properties, limiting the accuracy of the corresponding excited-state models. We are exploring an entirely new perspective whereby excited-state approximations exist as higher-energy stationary points of a high-dimensional electronic energy surface. These higher-energy stationary points correspond to individually optimised “state-specific” wave functions and can provide a more accurate representation of excited-state properties. By understanding the structure of this high-dimensional energy surface for different electronic structure approximations, we hope to develop new automated algorithms to locate state-specific representations of excited states.
Our energy landscape perspective can also help us understand where and why simple electronic structure models break down. For example, the failure of the mean-field Hartree–Fock approximation in the presence of strong electron coupling corresponds to the formation of symmetry-broken local minima on the energy surface. Directly comparing the simplistic model energy surface to the exact landscape can allow us to understand this failure and identify new ways to improve our computational models.
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Quantum mechanics is usually considered with mathematically real energies. This leads to the idea that electrons in a molecule can only take particular “discrete” energy levels. We explore the fascinating consequences of allowing the energy to be a complex number. Under this simple mathematical extension, discrete energy levels become unified as one continuous structure to create an entirely new perspective on quantum chemistry.
Electronic energy levels in molecules are conventionally considered as real-valued and discrete, giving “ground” and “excited” states. However, if we mathematically extend our equations to complex numbers — known as analytic continuation — we find that ground and excited states become unified as one continuous energy surface: a Riemann surface. In fact, discrete energy levels only arise by restricting our perspective to real numbers. These remarkable continuous energy surfaces allow a ground-state wave function to be continuously evolved into an excited-state wave function by following a well-defined pathway in the complex-plane. We have discovered the first examples of such pathways within electronic structure theory and we investigating how they can be used to locate state-specific representations of excited states.
We are also interested in the fundamental properties of non-Hermitian extensions to electronic structure methods and how this affects their behaviour on the real axis. For example, the convergence of perturbation theory is controlled by the location of non-Hermitian degeneracies in the complex plane. Furthermore, we have discovered that combined parity-time symmetry in electronic structure can enforce real-valued energies, even for complex non-Hermitian wave functions. We believe that understanding these fundamental properties will allow entirely new electronic structure approximations to be developed.
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The energy of molecules near their stable structure is often quite easy to predict by assuming that electron particles move independently of each other. However, when a bond is stretched or an electron is unpaired, the positions of different electrons become strongly correlated. This correlated motion can become frozen in multiple solutions to the independent particle model. We are developing new methods to predict such scenarios by combining these multiple snapshots of correlated electrons.
The ground-state of a molecule near its equilibrium structure can usually be predicted with a single mean-field electronic structure approximation such as Hartree–Fock theory. However, this approximation fails catastrophically in the presence of competing electronic configurations, for instance with two unpaired electrons or at the transition state for a reaction. We have found that, when many electronic configurations arise, there are generally multiple solutions to the mean-field approximation which represent the dominant configurations. Taking a linear combination of these multiple solutions in a Nonorthogonal Configuration Interaction (NOCI) expansion can recover qualitatively accurate potential energy surfaces for all molecular structures, allowing reaction profiles to be predicted.
We have recently developed a standardised approach to predicting molecular energy surfaces using NOCI. This approach has been implemented in the new LIBNOCI library available with the Q-Chem software package. We have also introduced a second-order perturbative correction “NOCI-PT2” that provides quantitative accuracy while retaining the computational efficiency and chemical intuition offered by NOCI.
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Simple models of molecules assume that electrons move independently in an effective field created by all other electrons. The algebraic equations for these models can produce multiple solutions that represent different important arrangements of the electrons in a molecule. Surprisingly, these solutions can vanish as atoms are moved, creating sudden jumps in the electronic energy. We have pioneered an algebraic extension that allows each solution to be smoothly continued over all molecular structures.
Mean-field approximations such as Hartree–Theory are the bedrock of modern electronic structure theory. These approximations often produce multiple solutions that represent different electron configurations, or ground and excited states. However, these multiple solutions can spontaneously disappear as the molecular structure changes, creating kinks and jumps in the energy surface. We have pioneered Holomorphic Hartree–Fock theory as a way of analytically continuing Hartree–Fock solutions beyond the points at which they vanish. This theory involves removing the complex-conjugate from the Hartree–Fock energy to construct a non-Hermitian complex-analytic function. Every holomorphic solution then exists over all molecular geometries to give a smooth and continuous energy. We have recently extended this approach to Kohn–Sham Density-Functional Theory, allowing the first investigation into how multiple solutions are connected between different exchange-correlation functionals.
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