All-Electron Ground-State and Time-Dependent Density Functional Theory: Fast Algorithms and Better Approximations

Abstract

Density functional theory (DFT), in its ground-state as well as time-dependent variant, have enjoyed incredible success in predicting a range of physical, chemical and materials properties. Although a formally exact theory, in practice DFT entails two key approximations---(a) the pseudopotential approximation, and (b) the exchange-correlation approximation. The pseudopotential approximation models the effect of sharply varying core-electrons along with the singular nuclear potential into a smooth effective potential called the pseudopotential, thereby mitigating the need for a highly refined spatial discretization. The exchange-correlation approximation, on the other hand, models the quantum many-electron interactions into an effective mean-field of the electron density ($rho(mathbf{r})$), and, remains an unavoidable approximation in DFT.

The overarching goal of this dissertation work is ---(a) to develop efficient numerical methods for all-electron DFT and TDDFT calculations which can dispense with the pseudopotentials without incurring huge computational cost, and (b) to provide key insights into the nature of the exchange-correlation potential that can later constitute a route to systematic improvement of the exchange-correlation approximation through machine learning algorithms (i.e., which can learn these functionals using training data from wavefunction-based methods). This, in turn, involves---(a) obtaining training data mapping $rho(mathbf{r})$ to $v_text{xc}(mathbf{r})$, and (b) using machine learning on the training data ($rho(mathbf{r}) Leftrightarrow v_text{xc}(mathbf{r})$ maps) to obtain the functional form of $v_text{xc}[rho(mathbf{r})]$, with conformity to the known exact conditions.

The research efforts, in this thesis, constitute significant steps towards both the aforementioned goals. To begin with, we have developed a computationally efficient approach to perform large-scale all-electron DFT calculations by augmenting the classical finite element basis with compactly supported atom-centered numerical basis functions. We term the resultant basis as enriched finite element basis. Our numerical investigations show an extraordinary $50-300$-fold and $5-8$-fold speedup afforded by the enriched finite element basis over classical finite element and Gaussian basis, respectively. In the case of TDDFT, we have developed an efficient emph{a priori} spatio-temporal discretization scheme guided by rigorous error estimates based on the time-dependent Kohn-Sham equations. Our numerical studies show a staggering $100$-fold speedup afforded by higher-order finite elements over linear finite elements. Furthermore, for pseudopotential calculations, our approach achieve a $3-60$-fold speedup over finite difference based approaches. The aforementioned emph{a priori} spatio-temporal discretization strategy forms an important foundation for extending the key ideas of the enriched finite element basis to TDDFT. Lastly, as a first step towards the goal of machine-learned exchange-correlation functionals, we have addressed the challenge of obtaining the training data mapping $rho(mathbf{r})$ to $v_text{xc}(mathbf{r})$. This constitute generating accurate ground-state density, $rho(mathbf{r})$, from wavefunction-based calculations, and then inverting the Kohn-Sham eigenvalue problem to obtain the $v_text{xc}(mathbf{r})$ that yields the same $rho(mathbf{r})$. This is otherwise known as the emph{inverse} DFT problem. Heretofore, this remained an open challenge owing lack of accurate and systematically convergent numerical techniques. To this end, we have provided a robust and systematically convergent scheme to solve the inverse DFT problem, employing finite element basis. We obtained the exact $v_text{xc}$ corresponding to ground-state densities obtained from configuration interaction calculations, to unprecedented accuracy, for both weak and strongly correlated polyatomic systems ranging up to 40 electrons. This ability to evaluate exact $v_text{xc}$'s from ground-state densities provides a powerful tool in the future testing and development of approximate exchange-correlation functionals.