DFT of many-electron systems in Cartesian coordinate grid

Ghosal, Abhisek (2021) DFT of many-electron systems in Cartesian coordinate grid. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Abhisek Ghosal (14RS068)) 14RS068.pdf - Submitted Version Restricted to Repository staff only Download (2MB)

Abstract

Throughout the past several decades, density functional theory (DFT) has been invoked for electronic structure calculations across an unusually wide variety of fields, fromorganic chemistry to condensedmatter physics, as it allows for an accurate quantum mechanical description at a relatively modest computational cost. Practical applications of DFT, to a large extent relies on the solution of Kohn-Sham (KS) equation or its generalization, (G)KS equation. Within this framework, the interactingmany-electron systemismapped onto an effective single-particle one through a local one-body potential, keeping the ground-state density fixed. In principle, the theory is exact and has the ability to capture many-body effects completely and uniquely. The aim of the thesis is to establish a viable alternative for DFT calculations, at least, for small to medium systems through a Cartesian coordinate grid (CCG) based pseudopotential KS-DFT framework using LCAO-MO ansatz. The important aspects of using pseudopotential are two folds; firstly, one can work only with the valence electrons. This is particularly important for heavy elements, where each atom involves several tens of electrons, while the number of valence electrons can be much smaller. Secondly, it takes into account relativistic effects, which are non-negligible for heavy elements. Further, the pseudo-valence orbitals which are typically smoother than the core electron orbitals, have fewer nodes near the nuclei. Henceforth, it enhances the smoothness of the orbitals at inner region that requires a less number of grid points. Thus, the pseudopotential approximation has certain advantages for our purpose. Accordingly, various quantities like localized basis functions, MOs, single-particle electron density, two-electron potential are directly set up in CCG. While one-electron contributions of KS-Fock matrix are evaluated through well-established recursion relations, the two-electron matrix elements are computed via direct numerical integration in CCG. Furthermore, to perform a quantum mechanical calculation for any given system, we need a systematic way of computation which should be universal. In this scenario, we have placed a simple grid-optimization procedure that brings a paramount of flexibility to perform the calculations routinely irrespective of systems of interest. To establish the viability and suitability of this framework, a decent number of species (atoms/molecules) have been tested using different XC functionals, grid parameters, basis sets within a desired level of accuracy, and it is found to be quite encouraging for its further development along with practical applications. Next, we present a purely numerical approach in CCG, for efficient computation of exact exchange density contribution in certain types of orbitaldependent density functionals. This takes inspiration from a semi-numerical algorithm, where the rate-determining step is the accurate evaluation of electrostatic potential integral. This introduces the Fourier convolution theorem in conjunction with a range-separated Coulomb interaction kernel. The latter is efficiently mapped into CCG through the above mentioned simple grid optimization procedure, giving rise to a constraint in the range-separated parameter. The overall process apart from pre-factors offers logarithmic scaling with respect to themolecular size. Again, a parallel semi-numerical approach has also been worked out that exploits the familiar Obara-Saika recursion algorithm without any additional technique. A critical analysis of these two algorithms reveals that the proposed numerical scheme in CCG could lead to very attractive and competitive scaling having an excellent agreement between these two routes. The success of this approach enables us for the development of optimally tuned (OT) range-separated (RS) functionals from first principles. We make use of the size dependency based ansatz i.e., RS parameter, γ, is a functional of density, ofwhich very little is known. To be consistent with this ansatz, a novel procedure is presented that relates the characteristic length of a given system (where ρ(r) exponentially decays to zero) with γ self-consistently via a simple mathematical constraint. In practice, γOT is obtained through an optimization of total energy as follows: γOT ≡ optγ. Etot,γ. It is found that the parameter γOT, estimated as above can show better performance in predicting properties (especially fromfrontier orbital energies) than conventional respective RSH functionals, of a given system. We have examined the nature of highest fractionally occupied orbital from exact piece-wise linearity behavior, which reveals that this approach is sufficient to maintain this condition. A careful statistical analysis then illustrates the viability and suitability of the current approach. Next, we report a simple, alternative time-independent DFT procedure, for computation of single-particle excitation energies, in particular, the lower bound excited singlet states, which are of primary interest in photochemistry. This originates froma recently published Becke’s excitonmodel, where a key step constitutes the accurate evaluation of correlated singlet-triplet splitting energy. It introduces a non-empiricalmodel, both from"adiabatic connection theorem" and "virial theorem" to analyze the role of two-electron integral in such calculations. The latter quantity is efficiently mapped onto CCG and computed accurately using a purely numerical strategy. The triplet and singlet excitation energies corresponding to first singly excited configuration, are found to be in excellent agreement with the theoretical best estimates. This demonstrates the viability and suitability of this approach in determining optical gaps, combining predictive accuracy with moderate computational cost. The last part of the thesis is dedicated to deal with a new approach towards real-time time-dependent density functional theory (RT-TDDFT) through time-dependent KS (TDKS) equations. This, however, is not used widely as the method of choice in the literature due to its huge computational burden as it requires smaller time-step to propagate the density for a period of time. On the other hand, there are not many alternatives which can accurately describe the non-linear electron dynamics of a many-electron system. Besides, high-lying excitations are attractive targets for RT-TDDFT which are very expensive for linear response algorithm due to the inherent iterative nature of implementation. In recent years, most of the developments have been engaged to reduce the computational cost while maintaining the accuracy and numerical stability of the approximate time-dependent propagator (ATDP). In this direction, we have proposed a new propagation scheme based on adiabatic eigenstate subspace (AES) approach which has a promise of larger time-step in RT-TDDFT calculations. This introduces a secondorder split operator technique in energy representation to implement the ATDP in AES. Further, we have also used density predictor/corrector algorithm to maintain the self-consistency in ρ(r, t) at each time step. Most of the elements in TDKS matrix are directly computed in CCG. The relevant time-dependent (TD) properties are computed through TD density matrix. To demonstrate the internal consistency of our proposed scheme, we have considered a generic situation where two representative molecules (H2 and N2 ) would interact with strong laser field. We computed the timedependent dipole moment and high harmonic generation spectra using adiabatic approximation on exchange-correlation functionals. The comparison with the available theoretical results ensures the feasibility of our proposed RT-TDDFT of many-electron systems in CCG. The thesis is outlined as follows: Chapter 1 gives a general introduction in the field of electronic structure theory and dynamics, in particularly focused on density functional theory and its time-dependent variant along with its present challenges. Chapter 2 presents the theoretical basis for the works developed in Chapters 3 to 6. It starts with a review of DFT, and then introduces pseudopotential CCG based KS-DFT formalism. We provide some general notions about pseudopotential, LCAO-MO ansatz, and CCG consideration. Finally, there is a brief introduction of Runge-Gross van Leeuwen theorems along with TDKS equations and adiabatic local density approximation. Chapter 3 presents a purely numerical approach in CCG, for efficient computation of exact exchange energy contribution in the Hartree-Fock and DFT models. In Chapter 4, we report a systematic self-consistent optimization of RS functionals from first principles. Then, we reveal the effectiveness of our pseudopotential based CCG framework in optical gap calculations through time-independent KS-DFT approach in Chapter 5. In Chapter 6, we present a general framework of RT-TDDFT of many-electron systems in CCG. Finally, Chapter 7 summarizes the important results of this thesis and presents possible future work and outlook.

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