Density functional theory embedding for correlated wavefunctions

Methods that exploit the intrinsic locality of molecular interactions show significant promise in making tractable the electronic structure calculation of large-scale systems. In particular, embedded density functional theory (e-DFT) offers a formally exact approach to electronic structure calculations in which the interactions between subsystems are evaluated in terms of their electronic density. In the following dissertation, methodological advances of embedded density functional theory are described, numerically tested, and applied to real chemical systems.

First, we describe an e-DFT protocol in which the non-additive kinetic energy component of the embedding potential is treated exactly. Then, we present a general implementation of the exact calculation of the non-additive kinetic potential (NAKP) and apply it to molecular systems. We demonstrate that the implementation using the exact NAKP is in excellent agreement with reference Kohn-Sham calculations, whereas the approximate functionals lead to qualitative failures in the calculated energies and equilibrium structures.

Next, we introduce density-embedding techniques to enable the accurate and stable calculation of correlated wavefunction (CW) in complex environments. Embedding potentials calculated using e-DFT introduce the effect of the environment on a subsystem for CW calculations (WFT-in-DFT). We demonstrate that WFT-in-DFT calculations are in good agreement with CW calculations performed on the full complex.

We significantly improve the numerics of the algorithm by enforcing orthogonality between subsystems by introduction of a projection operator. Utilizing the projection-based embedding scheme, we rigorously analyze the sources of error in quantum embedding calculations in which an active subsystem is treated using CWs, and the remainder using density functional theory. We show that the embedding potential felt by the electrons in the active subsystem makes only a small contribution to the error of the method, whereas the error in the nonadditive exchange-correlation energy dominates. We develop an algorithm which corrects this term and demonstrate the accuracy of this corrected embedding scheme.

A variational framework for spectral discretization of the density matrix in Kohn Sham density functional theory

Kohn-Sham density functional theory (KSDFT) is currently the main work-horse of quantum mechanical calculations in physics, chemistry, and materials science. From a mechanical engineering perspective, we are interested in studying the role of defects in the mechanical properties in materials. In real materials, defects are typically found at very small concentrations e.g., vacancies occur at parts per million, dislocation density in metals ranges from $10^{10} m^{-2}$ to $10^{15} m^{-2}$, and grain sizes vary from nanometers to micrometers in polycrystalline materials, etc. In order to model materials at realistic defect concentrations using DFT, we would need to work with system sizes beyond millions of atoms. Due to the cubic-scaling computational cost with respect to the number of atoms in conventional DFT implementations, such system sizes are unreachable. Since the early 1990s, there has been a huge interest in developing DFT implementations that have linear-scaling computational cost. A promising approach to achieving linear-scaling cost is to approximate the density matrix in KSDFT. The focus of this thesis is to provide a firm mathematical framework to study the convergence of these approximations. We reformulate the Kohn-Sham density functional theory as a nested variational problem in the density matrix, the electrostatic potential, and a field dual to the electron density. The corresponding functional is linear in the density matrix and thus amenable to spectral representation. Based on this reformulation, we introduce a new approximation scheme, called spectral binning, which does not require smoothing of the occupancy function and thus applies at arbitrarily low temperatures. We proof convergence of the approximate solutions with respect to spectral binning and with respect to an additional spatial discretization of the domain. For a standard one-dimensional benchmark problem, we present numerical experiments for which spectral binning exhibits excellent convergence characteristics and outperforms other linear-scaling methods.

Density-potential mapping in the standard and quantum electrodynamical time-dependent density functional theory

131 p.

A hybrid density functional view of native vacancies in gallium nitride.

We investigated the transition energy levels of the vacancy defects in gallium nitride by means of a hybrid density functional theory approach (DFT). We show that, in contrast to predictions from a recent study on the level of purely local DFT, the inclusion of screened exchange stabilizes the triply positive charge state of the nitrogen vacancy for Fermi energies close to the valence band. On the other hand, the defect levels associated with the negative charge states of the nitrogen vacancy hybridize with the conduction band and turn out to be energetically unfavorable, except for high n-doping. For the gallium vacancy, the increased magnetic splitting between up-spin and down-spin bands due to stronger exchange interactions in sX-LDA pushes the defect levels deeper into the band gap and significantly increases the associated charge transition levels. Based on these results, we propose the ϵ(0| - 1) transition level as an alternative candidate for the yellow luminescence in GaN.

Origin of antiferromagnetism in CoO: A density functional theory study

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Correcting the systematic error of the density functional theory calculation: the alternate combination approach of genetic algorithm and neural network

The alternate combinational approach of genetic algorithm and neural network (AGANN) has been presented to correct the systematic error of the density functional theory (DFT) calculation. It treats the DFT as a black box and models the error through external statistical information. As a demonstration, the AGANN method has been applied in the correction of the lattice energies from the DFT calculation for 72 metal halides and hydrides. Through the AGANN correction, the mean absolute value of the relative errors of the calculated lattice energies to the experimental values decreases from 4.93% to 1.20% in the testing set. For comparison, the neural network approach reduces the mean value to 2.56%. And for the common combinational approach of genetic algorithm and neural network, the value drops to 2.15%. The multiple linear regression method almost has no correction effect here.

Quantum master equation scheme of time-dependent density functional theory to time-dependent transport in nanoelectronic devices

In this work a practical scheme is developed for the first-principles study of time-dependent quantum transport. The basic idea is to combine the transport master equation with the well-known time-dependent density functional theory. The key ingredients of this paper include (i) the partitioning-free initial condition and the consideration of the time-dependent bias voltages which base our treatment on the Runge-Gross existence theorem; (ii) the non-Markovian master equation for the reduced (many-body) central system (i.e., the device); and (iii) the construction of Kohn-Sham master equations for the reduced single-particle density matrix, where a number of auxiliary functions are introduced and their equations of motion (EOMs) are established based on the technique of spectral decomposition. As a result, starting with a well-defined initial state, the time-dependent transport current can be calculated simultaneously along with the propagation of the Kohn-Sham master equation and the EOMs of the auxiliary functions.