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On the Effect of Beating during Nonlinear Frequency Chirping

Spectroscopic analyses of energetic particle (EP) driven bursts of MHD fluctuations in magnetically confined plasmas often exhibit chirps that occur simultaneously in groups of two or more. While the superposition of oscillations at multiple frequencies necessarily causes beating in the signal acquired by a localized external probe, self-consistent hybrid simulations of chirping EP modes in a JT-60U tokamak plasma have demonstrated the possibility of global beating, where the mode's electromagnetic field vanishes globally between beats and reappears with opposite phase [Bierwage et al., Nucl. Fusion 57, 016036 (2017)]. This implies that there can be a single coherent field mode that oscillates at multiple frequencies simultaneously when it is resonantly driven by multiple density waves in EP phase space. Conversely, this means that the EP density waves are mutually coupled and interfere with each other via the jointly driven field, a mechanism ignored in some theories of chirping. In this thesis-style treatise, we study the role of field pulsations in general and beating in particular using the Hamiltonian guiding center orbit-following code ORBIT with a reduced wave-particle interaction model in realistic geometry. Beating is found to drive the evolution of EP phase space structures. Additionally, a key mechanism is the pulsation of effective phase space islands combined with the alternation of their effective O- and X-points due to phase jumps between each beat. Observations: (1) Beating causes density wave fronts to advance radially in a pulsed manner and the resulting chirps become staircase-like. (2) The pulsations facilitate convective transfer of material between neighboring layers of phase space density waves. On the one hand, this may inhibit the early detachment of solitary phase space vortices. On the other hand, it facilitates the accumulation of hole and clump fragments into larger structures. (3) Long-range chirping is observed when massive holes or clumps detach and drift away from the turbulent belt around the seed resonance. It is remarkable that the detached vortices remain robust and,on average, maintain their concentric nested layers while being visibly perturbed by the field's continued beating.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Testing the conservative character of particle simulations: I. Canonical and noncanonical guiding center model in Boozer coordinates

The guiding center (GC) Lagrangian in Boozer coordinates for toroidally confined plasmas can be cast into canonical form by eliminating terms containing the covariant component B Ψ P of the magnetic field vector with respect to the poloidal flux function Ψ P . In an unperturbed plasma, B Ψ P can be eliminated via exact coordinate transformations, but, in general, one relies on approximations, assuming that the effect of B Ψ P is small. In this work, we are interested in the question whether Hamiltonian conservation laws are still satisfied when B Ψ P is retained in the presence of fluctuations. Considering fast ions in the presence of a shear Alfvén wave field with fixed amplitude, fixed frequency, and a single toroidal mode number n, we show that simulations using the code ORBIT with and without B Ψ P yield practically the same resonant and nonresonant GC orbits. The numerical results are consistent with theoretical analyses (presented in the appendix), which show that the unabridged GC Lagrangian with B Ψ P retained yields equations of motion that possess two key properties of Hamiltonian flows: (i) phase space conservation and (ii) energy conservation. As counter-examples, we also show cases where energy conservation (ii) or both conservation laws (i) and (ii) are broken by omitting certain small terms. When testing the conservative character of the simulation code, it is found to be beneficial to apply perturbations that do not resemble normal (eigen) modes of the plasma. The deviations are enhanced and, thus, more easily spotted when one inspects wave-particle interactions using nonnormal modes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

ORNL_AISD-Ex: Quantum chemical prediction of UV/Vis absorption spectra for over 10 million organic molecules

We performed calculations of electronic excitation energies and associated oscillator strengths based on the time-dependent density-functional tight-binding (TD-DFTB) method [1]. The SMILES (Simplified molecular-input line-entry system) strings of the molecules from the AISD HOMO-LUMO database [2] were converted to a 3D atomistic structure and stored in a PDB file after preliminary geometry optimization using the Merck Molecular Force Field (MMFF94) in RDKit [3,4]. The primary information stored in the PDB file archive consists of Cartesian coordinates for each atom of the molecule in their 3D location in space, along with summary information about the structure, sequence, and experiment. We then performed molecular geometry optimization using the density-functional tight-binding (DFTB) method [5] in the electronic ground state, followed by single-point excited states calculations, as described below. We note that, since RDKit employs a random choice for the generation of molecular conformers, the molecular geometries obtained in this dataset could be different from the ones that were generated when the AISD HOMO-LUMO dataset was generated. The computed excitation energies and associated oscillator strengths can be converted to predict UV/Vis absorption spectra, where excitation energies correspond to absorption peak positions, and oscillator strengths are a good measure of the probability of absorption of visible or UV light in transitions between electronic ground and excited states. The conversion of SMILES strings to 3D Cartesian coordinates of fully DFTB-optimized molecules was successful for 10,502,904 out of 10,502,917 molecules. For these molecules, both geometry optimizations and excited states calculations were successful. The DFTB calculations did not complete for 13 molecules of the original AISD HOMO-LUMO dataset. We still provide information about the geometry of these molecules. The molecules are diverse for chemical compositions (which span 5 non-hydrogen elements: oxygen, carbon, nitrogen, fluorine, sulfur) and molecular size (the smallest molecule contains 5 non-hydrogen atoms, and the largest molecule contains 71 non-hydrogen atoms). The DFTB method [5] is an approximation to density functional theory (DFT), utilizing a minimal basis set in conjunction with a two-center approximation to the electronic Hamiltonian and overlap matrix elements. The DFTB total energy is the sum of an electronic and a repulsive energy contribution, and their calculation requires optimized electronic parameters and diatomic repulsive potential energy functions. All DFTB calculations were performed using the DFTB+ code [6] (version 21.2) and the wrapper for DFTB+ in the Atomic Simulation Environment (ASE) (version 3.22.1) [7], which performed an internal conversion of Cartesian coordinates from PDB to the .gen file format. For the geometry optimizations on the electronic ground state potential energy surface of the molecules, we have chosen the third-order DFTB (DFTB3) method [5c] and employed the matching 3ob set of electronic parameters and repulsive potentials [8]. The empirical γ-damping for hydrogen bond correction, and Grimme's D3 empirical dispersion correction with Becke-Johnson damping (D3(BJ)) [9] dispersion correction was included to improve the description of non-covalent interactions. For excited states single-point energy calculations, we employed the TD-DFTB method in conjunction with the DFTB2 method [5b] and the matching mio [5b,10] and halorg [11] parameter sets. We opted to request the simultaneous calculation of 50 excited states for singlet transition to investigate sufficient number of excited states, based on linear response theory using the Casida equation [Ref: T. A. Niehaus, S. Suhai, F. Della Sala, P Lugli, M. Elstner, G. Seifert, and Th. Frauenheim. Tight-binding approach to time-dependent density-functional response theory. Phys. Rev. B, 63:085108, 2001] and the ARPACK diagonalizer [R. B. Lehoucq, D. C. Sorensen, and C. Yang. Arpack users guide: Solution of large-scale eigenvalue problems by implicitly restarted arnoldi methods, 1997. 46, 51]. The dataset contains 1001 tar.gz files. Tar files are named as “ornl_aisd_ex_1.tar.gz†through “ornl_aisd_ex_1000.tar.gzâ€. Additionally, the 13 failed molecules are in “ornl_aisd_ex_unprocessed.tar.gzâ€. Except for the tar files listed below, each tar file contains 10,500 molecules. Tar files numbered 34, 121, 128, 352, 360, 429, 495, 509, 518, 627, 676, 668, and 862 contain 10,499 molecules each. The last tar file numbered 1000 contains 13,417 molecules. The total size of the uncompressed dataset is over 283 Gigabytes. The code for calculating the electronic excitation energies and statistical analysis of the dataset is provided at the following GitLab repository: https://github.com/ORNL/Analysis-of-Large-Scale-Molecular-Datasets-with-Python Calculating the UV spectrum of a molecule requires performing 3 main operations: 1. Converting the smiles string representation of a molecule into a geometric structure where each atom is assigned XYZ coordinates. The geometric structure is written to the file smiles.pdb. 2. Using smiles.pdb to compute the relaxed geometry of the molecule, which corresponds with the position of the atoms at the position of equilibrium at the ground state. This generates the files band.out, detailed.out, and geo_end.gen. 3. Using geo_end.gen to calculate the UV spectrum of the molecule which is written into the file EXC.DAT. Every molecule in the dataset has its own directory. The files contained in each molecule directory are as follows: 1. geo_end.gen 2. detailed.out 3. band.out 4. EXC.DAT 5. smiles.pdb REFERENCES [1] Niehaus, T. A.; Suhai, S.; Della Salla, F.; Lugli, P.; Elstner, M.; Seifert, G.; Frauenheim, Th. Tight-binding approach to time-dependent density-functional response theory. Phys. Rev. B, 2001, 63, 085108/1-9. [2] Blanchard, A.; Gounley, J.; Metha, K.; Yoo, P.; Irle, S. AISD HOMO-LUMO. DOI: 10.13139/ORNLNCCS/1869409 [3] RDKit: Cheminformatics and Machine Learning Software. 2013, [http://www.rdkit.org] [4] Tosco, P.; Stiefl, N. and Landrum, G. Bringing the MMFF force field to the RDKit: implementation and validation. J Cheminform. 2014, 6, 1–4. [5] a) Porezag, D.; Frauenheim, T.; Kohler, T.; Seifert, G.; Kaschner, Construction of tight-binding-like potentials on the basis of density-functional theory: Application to carbon, R. Phys. Rev. B 1995, 51, 12947-12957; b) Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, Th.; Suhai, S.; Seifert, G.; Phys. Rev. B 1998, 58, 7260-7268; c) Gaus, M.; Cui, Q.; Elstner, M. DFTB3: Extension of the Self-Consistent-Charge Density-Functional Tight-Binding Method (SCC-DFTB), J. Chem. Theory Comput. 2011, 7, 931-948; d) Cui, Q.; Elstner, M. Density functional tight binding: values of semi-empirical methods in an ab initio era, Phys. Chem. Chem. Phys. 2014, 16, 14368-14377. [6] Hourahine, B. et al. DFTB+, a software package for efficient approximate density functional theory based atomistic simulations, J. Chem. Phys. 2020, 152, 124101/1-19. [7] Larsen, A. H. et al. The atomic simulation environment—a Python library for working with atoms. J. Phys.: Cond. Matter 2017, 29, 273002. [8] Kubillus, M.; Kubar, T.; Gaus, M.; Rezac, J.; Elstner, M. Parameterization of the DFTB3 Method for Br, Ca, Cl, F, I, K, and Na in Organic and Biological Systems, J. Chem. Theory Comput. 2015, 11, 332-342. [9] Brandenburg, J. G.; Grimme, S. Accurate Modeling of Organic Molecular Crystals by Dispersion-Corrected Density Functional Tight Binding (DFTB), J. Phys. Chem. Lett. 2014, 5, 1785−1789. [10] a) Niehaus, T. A.; Elstner, M.; Frauenheim, Th.; Suhai, S. Application of an approximate density-functional method to sulfur containing compounds. J. Mol. Struct.: THEOCHEM 2001, 541, 185-94; b) Elstner, M.; Hobza, P.; Frauenheim, Th.; Suhai, S.; Kaxiras, E. Hydrogen bonding and stacking interactions of nucleic acid base pairs: A density-functional-theory based treatment. J. Chem. Phys. 2001, 114, 5149-55. [11] Kubar, T.; Bodrog, Z.; Gaus, M.; Köhler, C.; Aradi, B.; Frauenheim, Th.; Elstner, M. Parametrization of the SCC-DFTB Method for Halogens. J. Chem. Theory Comput. 2013, 9, 2939-49.

36 MATERIALS SCIENCE↗