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A Density Functional Odyssey Beyond Ground State Energies


Density functional theory or DFT is presently the most popular route for computing the electronic structure of chemical systems. Although DFT is formally exact, the exact functional that maps the electron density to the energy remains unknown to date. A large number of density functional approximations (DFAs) have consequently been developed to compute the energies of molecules and extended materials. Use of exact constraints, large amounts of highly accurate benchmark data, and intelligent data-driven design schemes have resulted in modern functionals that provide an excellent balance between computational cost and predictive accuracy. However, nearly all the DFA development efforts in recent years had focused on improving chemically relevant energy differences in the ground state. Even the electron density, which is the central quantity of the theory, has been mostly neglected. This dissertation tries to explore usage of DFT beyond ground state energies through the investigation of electrical response properties and electronic excited states. Information from these regimes should prove helpful extending the applicability of DFT beyond computation of ground state energy differences, and also assist in designing more transferable DFAs that better approximate the exact functional.

The first half of the dissertation assesses the accuracy of modern DFAs in predicting molecular properties associated with the response of the energy to electric fields. The exact functional is formally capable of predicting exact energies even when the system is subjected to arbitrary electric fields. However, approximate functionals only model the electrical response well if the density is accurate. The ability of DFAs to compute electrical response properties thus indicate their accuracy in modeling densities. The dissertation therefore studies dipole moments (Chapter 4), second cumulants of the density (Chapter 5) and static dipole polarizabilities (Chapter 6). High level coupled cluster benchmarks at the complete basis set limit for ≥ 100 chemical species has been generated for all the three properties. These benchmark datasets are used to evaluate the performance of several popular and recent functionals, in order to gauge performance. This analysis reveals that some of the most accurate modern DFAs for ground state energies yield suboptimal predictions for electrical response properties. Future DFA development therefore should utilize these benchmark datasets for training and assessment purposes, in order to obtain functionals that simultaneously yield accurate energies and densities. In addition, we use the static dipole polarizability as a sensitive probe for electronic structure in Chapter 7, to identify qualitative problems in DFAs. This demonstrates that several modern DFAs are challenged by homolytic single bond dissociation, as they fail to completely unpair electrons over the right distance scales. The material in this half of the dissertation therefore provides information about how existing functionals struggle to model density, and should be helpful for the design of more accurate DFAs.

The second half of the dissertation examines behavior for electronic excited states, focusing on the popular linear-response time-dependent DFT (TDDFT) and the less well known orbital optimized DFT (OO-DFT) approaches. Chapter 10 shows that TDDFT methods cannot describe bond dissociations in the excited state, developing unphysical derivative discontinuities at the onset of spin unpairing in the ground state. The other chapters focus on OO-DFT, and applications to core spectroscopy. Chapter 11 presents a robust new algorithm for excited state OO, that ensures the optimization process remains on the chosen state and does not undergo ‘variational collapse’, to a lower energy state. This SquareGradient Minimization or SGM algorithm is used to model core-level excitations for closed-shell systems (Chapter 12) and radicals (Chapter 13), using OO-DFT. Chapter 13 also presents a scheme to recouple three unpaired electrons to obtain spin-pure doublets, which are relevant for core to unoccupied orbital transitions in radicals. The results of Chapter 12 and 13 demonstrate the OO-DFT with the SCAN DFA can model core-level spectra of second period elements to semiquantitative accuracy of ∼ 0.3 eV, against experimental values with ∼ 0.1 eV uncertainty. This is a dramatic improvement over the ∼ 15 eV errors observed from TDDFT, indicating that OO-DFT/SCAN is a cheap and reliable way to model core-level spectra. Indeed, OO-DFT/SCAN can be directly used to simulate experimental spectra, such as time-resolved X-ray transient absorption studies of chemical dynamics. The energies and densities of these core-excited states also provide new information for functional training beyond ground state energies. Incorporation of this very distinct form of data in the DFA development process thus can help better approximate the exact functional.

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