## Development of non-local density functionals

- Author(s): Krull, Brandon
- Advisor(s): Furche, Filipp
- et al.

## Abstract

This thesis focuses on the use and development of electronic structure methods in the density functional theory (DFT) framework. The first half is devoted to the implementation of a new integral algorithm and its use in fully non-local density functional approximations. While costly compared to simpler semi-local approximations, non-local functionals of the density are a way to intuitively incorporate physics into a real-space model. Two broad classes of methods are targeted: locally-range separated hybrids and non-local exchange kernels. These methods are tested on a variety of small systems. In both cases, proper scaling to the high- density limit is satisfied which turns out to hamper the approximations’ ability to describe static correlation. This leads us to conclude that proper scaling methods will inherently need static correlation corrections if good performance in thermochemical measures is desired. In the second half, DFT is applied to three drastically different problems which highlights its ability to reasonably predict experimental structural parameters of compounds containing f- and d- block elements while simultaneously describing excited state properties of main group and transition metal compounds. Using time-dependent DFT, UV-vis spectra of lanthanides, transition metals, and the small molecule luciferin were simulated and found to be in good agreement with experiments, providing photophysical insight to the nature of their excited states.

Chapter 2 describes a new recursion scheme that was derived in order to target the evaluation of non-local interaction kernels. The integrals that arise from such kernels are first trans- formed into relative and center of mass coordinates and then rewritten in terms of an integral over the relative coordinate that is parametrically dependent on the center of mass. Using the shared-memory parallel progamming API, OpenMP, a multithreaded semi-analytical al- gorithm is presented where integrals over the relative coordinate are evaluated for blocks of grid points of the numerical integration. The algorithm scales as O(N5), where N is a mea- sure of the system size, but prescreening based on density and integral prefactors are used to reduce the cost. In its present state, the algorithm is still costly and has been limited to only smaller molecules. Benchmark tests show that the recursion scheme can precisely recover the Coulomb interaction and that screening reduces the overall computational cost by 10-15% for small systems. The prescreening which is based on the sparsity of the exchange density matrix, should lead to asymptotic linear scaling with increasing system size; however, this is not realized in practice and is not fully understood at this point. Despite efforts to efficiently parallelize the numerical integration, the total wall time as a function of number of CPUs, does not behave as 1/Ncpu, leading to the conclusion that there is a bottleneck executed on only a single CPU that reduces the efficiency. Testing of the implementation on larger systems is necessary to see if the screening behaves as expected and to further analyze the parallel efficiency.

Chapters 3 and 4 are focused on building non-local models of the density functional exchange- correlation energy using the integral scheme developed in chapter 2. Chapter 3 focuses on locally range-separated hybrids. Here, the exchange interaction is partitioned into long-range and short-range contributions via a range-separation function. We model this function using a density-dependent functional that is evaluted at the center of mass. Our models are made to satisfy exact constraints such as correct scaling in the high-density limit and good description of one-electron and iso-orbital regions. We propose and test a variety of forms that fall into two classes: ones that use long-range exact exchange with short-range approximate exchange and ones that use short-range exact exchange with long-range exchange. Overall, with scaling to the high-density limit being enforced, atomization energies tend to be too low, though self-interaction error is significantly reduced. These proper scaling methods hint at the need to reintroduce the static correlation effects not described by Hartree-Fock to achieve a pragmatic balance. In chapter 4, corrections to the random-phase approximation are explored by way of a short-ranged non-local static exchange kernel. Our first model constructed consisted of a short-ranged Gaussian function. Use of this type of exchange kernel was plagued with instabilities which can be traced to negative values of the Fourier transform of the Hartree-exchange-correlation kernel. A second similar model based on an error function was also constructed and was better behaved. In places where instabilities occured, the first-order RPA-renormalization framework was used and shown to alleviate instabilities in both cases. The fact that a direct modeling of exchange kernels often leads to instabilities indicates that this route to beyond-RPA corrections may be more difficult than once assumed. Additionally, despite the fact that the implementation is purely proof of concept, the high O(N5) cost of computing the non-local integrals is challenging to manage and considerable effort would be required to reduce it.

Chapters 5-7 are based on collaborative work with other research groups at UCI. In chapter 5, a joint computational and experimental study on the photophysical properties of the light-emitting molecule luciferin was undertaken. Using time-dependent DFT, the excited states of a plethora of luciferin analogues were studied in order to find novel, robust light emitting species. While computed oscillator strengths corresponded well with previously known luciferins, the simple picture of non-adiabatic emission from the first singlet excited state was only moderately reliable in predicting the chemiluminescent properties of newly synthesized luciferins. This work implies the need for explicit consideration of environmental effects and the inclusion of other nearby electronic states in order to have a reliably predictive methodology. In chapter 6, a comprehensive study of lanthanides recently discovered to have a new +2 oxidation state was undertaken. Time-dependent DFT was used to simulate the UV-vis spectra of new complexes of Eu, Sm, Tm, Yb, Dy, and Nd which were found to be in good agreement with experiments. Nd and Dy were found to be “crossover” elements in that their electronic ground state could be modulated easily by its ligand environment, in contrast to previous paradigms. In chapter 7, a comparison of complexes of Group V metals (vanadium, niobium, and tantalum) coordinated to redox active ligands is presented. The main challenge in using computational tools to study these complexes lies in the fact that they are coordinated to redox-active ligands which significantly complicate their electronic structure. It was discovered that while Nb and Ta are well described by a closed-shell singlet in their +5 oxidation state, V differs dramatically and has a more stable solution in the +4 oxidation state with open-shell singlet diradical character. This points to subtelties in the electronic structure of transition metals that can be exploited in catalytic applications.