Self-interaction error (SIE) is a commonly known problem that most Kohn-Sham density functional theory (KS-DFT) approximate functionals display to varying extents.
It originates from the incomplete cancellation of the Coulomb self-repulsion by the approximate exchange functionals.
This is one of the major challenges for DFT, and therefore increasing our understanding of it could have great benefits for future use of DFT.
Herein we advance techniques to dissect, understand, and textit{avoid} SIE in new ways.
Considering that KS-DFT requires solving the self-consistent field (SCF) equations, we first present a robust and economical SCF solver - the ``Quasi-Newton Unitary Optimization with Trust region" (QUOTR) solver.[Slattery, et. al. textit{Phys. Chem. Chem. Phys.}, textbf{2024}, 26, 6557-6573] To be robust, the solver is a direct-minimization solver equipped with a trust region (TR); to be economical, the solver uses an L-BFGS approximate Hessian and a physically-relevant preconditioner.
Coupling these two aspects together is a solver for the TR subproblem that exploits the low-rank structure of the L-BFGS Hessian.
We demonstrate that QUOTR is useful, not only for obtaining KS-DFT wave functions in difficult cases, but also for solving for Hartree-Fock (HF) orbitals in challenging chemical systems containing Cr or Fm.
Although not able to beat the low cost of traditional Roothaan-Hall (RH) solvers with acceleration, QUOTR is robust in its convergence at only a modest increase in computational cost.
The many examples of SCF convergence problems when using semi-local KS-DFT functionals are known to be the result of a vanishing HOMO-LUMO gap, which is further the result of SIE.
A major motivation for developing QUOTR came from our desire to understand the ``true" (albeit unphysical) ground state solutions in cases where KS-DFT could not be converged by a traditional diagonalization-based SCF solver.
We reinvestigate the relationship between the vanishing HOMO-LUMO gap and SCF non-convergence using our QUOTR solver.
A set of difficult biological systems that had previously been shown to display convergence problems [Rudberg, et. al. textit{J. Phys.: Condens. Matter}, textbf{2012}, 24, 072202] was selected for deeper analysis.
In addition to being able to obtain converged solutions, we analyze the resulting densities matrices in comparison to HF.
The source of the vanishing HOMO-LUMO gaps is demonstrated to be incompatible eigenspectrums of spatially distant fragments in the peptides.
We show that by using a local solver (QUOTR) with an appropriate initial guess, that a non-Aufbau filled stationary point can be found for vacuum-separated charged fragments.
A systematic scan of all 20 natural amino acids for some common DFAs is used to examine the prevalence of predicted non-Aufbau filling.
We find that hybrid functionals improve upon GGAs more than meta-generalized gradient approximations (GGAs) do, and range-separated functionals are much better - though not completely solving the problem.
Having addressed SIE in biomolecules in terms of where charges comes from and where it goes, we finally analyze SIE on an orbital-by-orbital basis.
We define the textit{genuine} exchange energy as the difference between the HF energy and the (self-interaction free) orthogonal Hartree product wave function energy.
We propose that the Edmiston-Ruedenberg [Edmiston, et. al. textit{Rev. Mod. Phys.}, textbf{1963}, 35, 457-464] localized HF orbitals are the most appropriate HF frame for this analysis, due to their connection with the Hartree product wave function.
Although the use of HF orbitals to quantify the genuine exchange energy is an approximation, we demonstrate that the error of total exchange energy is approximately 10-15% for a set of small molecules.
The good performance of two popular GGAs is shown to arise with considerable error cancellation between orbitals (particularly core and valence).
We also examine two orbital-dependent DFAs: the Perdew-Zunger self-interaction correction (PZ-SIC), and a generalization of the Hartree-Fock-Gopinathan (HFG) method. / Doctor of Philosophy / Much of theoretical chemistry is built on a formalism where electrons are described by single-particle functions, known as orbitals.
Because the orbitals influence each other through a variety of interactions, it is not trivial to determine accurate approximations to the best set of orbitals for any given chemical system.
The method for obtaining optimal mathematical forms of the orbitals is called the self-consistent field (SCF) procedure.
In essence solving the SCF equations is a nonlinear optimization problem, requiring iterative solution techniques.
Both of the major camps in the field of theoretical chemistry, density functional theory (DFT) and wave function methods, almost always start with an SCF calculation.
In wave function methods, SCF is typically only the first step, in this case the SCF is done with an approximation called Hartree-Fock (HF), and a further calculation will include many-body effects through other means which are more computationally costly.
In Kohn-Sham DFT (the most popular variety), the SCF equations contain approximations to part of the energy known as density functional approximations (DFAs) that in principle account for all many-body effects not included exactly.
Thus, for KS-DFT, solving the SCF equations is basically all that needs to be done for a complete description of the system.
For this reason DFT is attractive from a cost perspective, but the DFAs are only approximations and this can introduce unphysical errors.
In this work, we focus on providing new solutions and perspectives on a major contributor to poor performance of KS-DFT calculations, the self-interaction error (SIE), which is not present in HF.
The origin of the SIE in approximate KS-DFT is how the 2-body interactions of electrons are approximated.
Classical electrostatics provides a simple formula for the repulsion energy of like-charged particles.
When the quantum mechanical electron probability density is treated as a classical charge density then a single electron will unphysically interact with itself.
What is missing is the nonclassical ``exchange" interaction, which in the HF method cancels the Coulomb self-repulsion.
In KS-DFT this second contribution is approximated, and this causes the cancellation to be inexact.
Thus, it is the treatment of these two energy contributions in fundamentally different ways that causes SIE to appear in approximate KS-DFT, but not in wave function methods.
Problems with convergence of the SCF equations are more prevalent for approximate KS-DFT than for HF, and this has been attributed to SIE.
To address the issue of poor SCF convergence we developed the ``Quasi-Newton Unitary Optimization with Trust region" (QUOTR) SCF solver. [Slattery, et. al. textit{Phys. Chem. Chem. Phys.}, textbf{2024}, 26, 6557-6573] This solver is robust in converging to minima in the energy surface, while also being economical in computational cost.
In cref{ch:quotr} we demonstrate the usefulness of QUOTR for solving not only problems where KS-DFT SCF convergence is difficult, but also cases where even HF SCF is not simple.
With the QUOTR solver in hand, in cref{ch:sie_pep} we reexamine the SCF non-convergence problem for some approximate KS-DFT functionals when applied to biological systems in vacuum.
By comparing the spatial distribution of the electron density of our KS-DFT solutions to that of HF, we are able to pinpoint regions of the biological systems that donate and receive electron density relative to HF.
As is already known, how approximate KS-DFT treats charged groups often is generally the source of the error.
Therefore, we performed a systematic scan of all 20 naturally occurring amino acids to find combinations that might be problematic.
Finally, in cref{ch:orb_anatomy} we investigate the source of SIE in approximate KS-DFT functionals on an orbital-by-orbital basis, and attempt to remove it from the calculation a priori.
While the total density is unique, the orbital decomposition within KS-DFT is not.
We therefore, first justify our choice of a particular set of orbitals as the most physically reasonable for our analysis.
We find that two popular functionals display much error cancellation between orbitals to achieve good overall results.
Two generalizations of KS-DFT for orbital-dependent approaches, which attempt to be free of SIE, are examined.
Taken as a whole, this work examines SIE in KS-DFT both by improving convergence of the SCF procedure despite the presence of SIE (in order to understand what is really happening), and by dissecting the orbital structure of SIE, including for methods that attempt to be free of SIE.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/118949 |
| Date | 09 May 2024 |
| Creators | Slattery, Samuel Alexander |
| Contributors | Chemistry, Valeyev, Eduard Faritovich, Troya, Diego, Crawford, Daniel, Mayhall, Nicholas |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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