Development of Practical Non-Local Many-Body Polarization Functionals

Electronic structure calculations can now achieve the highly coveted chemical accuracy (less than 1 kcal/mol average error in energy differences) for molecules with a few dozens of atoms; however, extending these approaches to larger systems is an area of active research. A major challenge is devising methods that are widely applicable to both molecules and materials. To achieve this goal, computational advances must be coupled to a deep understanding of the related physical principles. Response functions and, in particular, polarizability, play a central role in our conceptual understanding of both electron correlation and polarization/dispersion interactions -- quantum mechanical effects that are notably hard to properly capture due to the underlying non-local nature of the quantities needed to compute these interactions.

To develop a practical formalism for non-local polarizability, one first needs to deeply elaborate the corresponding local and semi-local approaches. This can be achieved by studying model systems, atoms, and molecules, as the numerical results for molecular systems can be complemented by the physical understanding from the model results. By analyzing quantum systems ranging from model Hamiltonians to real molecules, in this work it is shown that polarizability can be factored into a spectrum-dependent and geometry-dependent part. Notably, the geometry-dependent part influences polarizability by a four-dimensional scaling law, enabling the proper description of response properties of individual atoms within molecules. A novel parametrization for representing the response of atoms by an effective harmonic oscillator model is also introduced, showing that spatially resolved polarization potentials can be predicted using just integrated dipolar properties of atoms.

Moving from model systems and atoms to molecules, it is found that the corresponding polarizability and HOMO-LUMO (highest occupied molecular orbital -- lowest unoccupied molecular orbital) gap are independent. In parallel, the theoretical foundations of non-local polarizability are examined, presenting expressions for a range of model systems via the polarization field correlation function. By following a rigorous derivation of this response function, it is shown that not only existing methods can be obtained from it as limiting cases, but the design of a general non-expanded many-body dispersion energy functional is also feasible.

Overall, this thesis aims to show that combining the fundamental physics of model systems, atoms, and molecules with a theory of non-local polarizability can lead to practical functionals for electronic structure calculations based on the advanced non-local description of response functions.