SSStttooonnnyyy BBBrrrooooookkk UUUnnniiivvveeerrrsssiiitttyyy The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook University. ©©© AAAllllll RRRiiiggghhhtttsss RRReeessseeerrrvvveeeddd bbbyyy AAAuuuttthhhooorrr... Understanding the dielectric properties of water A Dissertation Presented by Daniel Christopher Elton to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics Stony Brook University December 2016 Stony Brook University The Graduate School Daniel Christopher Elton We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation Marivi Fern´andez-Serra- Dissertation Advisor Professor, Department of Physics and Astronomy Phillip B. Allen - Chairperson of Defense Professor, Department of Physics and Astronomy Matthew Dawber Associate Professor, Department of Physics and Astronomy Alan Calder Professor, Department of Physics and Astronomy Matthew Reuter Assistant Professor, Department of Applied Mathematics and Statistics This dissertation is accepted by the Graduate School Charles Taber Dean of the Graduate School ii Abstract of the Dissertation Understanding the dielectric properties of water by Daniel Christopher Elton Doctor of Philosophy in Physics Stony Brook University 2016 Liquid water is a complex material with many anomalous prop- erties. Three of these anomalies are an abnormally high dielec- tric constant, an abnormally high boiling point, and a solid phase which is less dense than the liquid phase. Each of these anomalies is known to have been critically important in the development of life on Earth. All of water's special properties can be linked to water's unique ability to form hydrogen bonds. Water's hydrogen bonds form a transient network. Understanding the average struc- ture of this network and how it changes through the phase diagram remains the focus of intense research. In this thesis we focus on understanding dielectric and infrared measurements, which measure the absorption and refraction of electromagnetic waves at different frequencies. Computer simu- lation is a necessary tool for correctly interpreting these measure- ments in terms of the microscopic dynamics of molecules. In the first part of this thesis we compare three classes of water molecule model that are used in molecular dynamics simulation iii rigid, flexible, and polarizable. We show how the inclusion of polar- ization is necessary to capture how water's properties change with pressure and temperature. This finding is relevant to biophysical simulation. In the next part, we conduct a detailed study of water's dielectric properties to discover vibrational modes that propagate through the hydrogen bond network. Parts of the absorption spec- trum of water are due to electromagnetic waves coupling to these modes. Previously, vibrational motions in water were thought to be confined to small clusters of perhaps five molecules. Our work upends this view by arguing that dynamics occur on the hydrogen bond network, resulting in modes that can propagate surprisingly long distances of up to two nanometers. These modes bear many similarities to optical phonon modes in ice. We show how the LO-TO splitting of these modes provides a new window into the structure of the hydrogen bond network. In the final part of this thesis we turn to the problems one en- counters when trying to simulate water from “first principles", ie. from the laws of quantum mechanics. The primary technique that physicists use to approximate the quantum mechanics of electrons, density functional theory, does not work well for water, and much work is being done to understand how to fix this problem. A usual assumption in first principles simulation is that only electrons need to be treated quantum mechanically. We argue that both electrons and nuclei need to be treated quantum mechanically and we present a new code to do this. The custom code presented in this thesis implements a novel algorithm which greatly speeds up the calcula- tion of nuclear quantum effects with only minor losses in accuracy. We hope that others will start using our technique to advance first principles simulation. Accurate first principles simulation of water is important for understanding and developing solar water splitting catalysts and batteries. First principles simulations are also being increasingly used to understand proteins and drug molecules, and this trend will continue with Moore's law. iv To my parents Contents List of Figures xii List of Tables xxii Acknowledgements xxv 1 Introduction 1 1.1 Some open questions about water ................ 3 1.2 The liquid-liquid phase transition hypothesis .......... 3 1.2.1 Issues regarding water structure ............. 4 1.3 Some questions that are explored in this thesis ......... 7 2 Introduction to dielectric properties 9 2.1 Key equations and conventions ................. 10 2.1.1 The polarizability ..................... 12 2.2 The dipole-dipole interaction ................... 12 2.3 A unified treatment of fluctuation formulas for the dielectric constant .............................. 16 2.3.1 The relation between P and E from electrostatics ... 17 2.3.2 The relation between P and E from statistical mechanics 18 2.3.3 The equation for the dielectric constant for different boundary conditions ................... 19 2.3.4 The infinite system .................... 22 2.3.5 Summary of formulas ................... 24 2.4 The Kirkwood g-factor ...................... 24 2.5 Systems of polarizable point dipoles ............... 25 2.5.1 Polarization catastrophe & smeared dipoles ...... 27 2.5.2 Dielectric constant and "(!) for polarizable dipoles .. 29 2.6 The frequency-dependent dielectric function .......... 32 2.6.1 The dielectric response function ............. 33 2.6.2 The “infinite frequency" dielectric constant ....... 35 vi 2.6.3 Kramers-Kronig relation ................. 35 2.6.4 Theory of absorption / loss ............... 36 2.7 Summary of important relations ................. 38 3 Simple mean field theories 39 3.1 Debye's theory .......................... 39 3.2 Onsager's theory ......................... 42 3.3 Kirkwood's theory ........................ 45 3.3.1 Application of the Kirkwood equation .......... 47 3.4 Conclusions ............................ 47 4 Computational methods 49 4.1 Integrating Newton's equations ................. 49 4.1.1 Velocity-Verlet method .................. 49 4.1.2 Multiple timestep (RESPA) method ........... 51 4.2 The TTM series of models .................... 52 4.3 Test for artifacts from thermostating .............. 56 4.4 Box size dependence ....................... 56 4.5 Comparison of three methods for calculating "(0) ....... 57 4.5.1 Method I: linear response equation ........... 58 4.5.2 Method II: Kirkwood g-factor .............. 60 4.5.3 Method III: applied electric field ............. 63 4.6 Method of calculating the dielectric function .......... 65 4.7 Conclusion ............................. 65 5 Comparison of rigid, flexible, & polarizable models 67 5.1 General overview of water models ................ 67 5.1.1 Sensitivity to water model geometry .......... 69 5.2 Details of the simulations that were run ............. 70 5.3 Dielectric constant ........................ 71 5.3.1 Dipole moments ...................... 74 5.3.2 Temperature derivative of "(0) .............. 75 5.4 Dielectric function ........................ 76 5.5 Conclusion ............................. 78 6 Dipole-dipole spatial correlation 79 6.1 Introduction ............................ 79 6.2 1D correlation functions ..................... 80 6.3 2D spatial correlation functions ................. 81 6.4 2D correlation functions for a point dipole in a homogeneous dielectric continuum ....................... 82 vii 6.5 Distance dependent Kirkwood function ............. 83 6.5.1 Axial and equatorial components ............ 84 6.6 The artificial enhancement of dipole correlation from periodic boundary conditions ....................... 85 6.6.1 Dependence of the artifact on box shape ........ 88 6.7 1D correlation functions for water ................ 90 6.8 Kirkwood function of water ................... 96 6.9 2D angular correlation function of water ............ 98 6.10 Conclusion ............................. 101 7 Dielectric relaxation of water 102 7.1 The most commonly used dielectric functions are all related . 102 7.2 More generalized lineshapes ................... 105 7.2.1 Power law relaxation ................... 109 7.3 The physical mechanism of Debye relaxation in water ..... 109 7.4 Wrong conceptions of the Debye relaxation ........... 111 7.4.1 Models based off the temperature dependence ..... 111 7.4.2 The Debye relaxation is purely due to rotational motions 112 7.5 Mean-field theories ........................ 112 7.6 The high frequency excess .................... 113 7.6.1 Hydrogen bond network modes ............. 114 7.6.2 Overlap of hydrogen bond network modes and inertial relaxation ......................... 115 7.7 Using the gLST relation as a novel constraint to fit the dielectric function of water ......................... 116 7.7.1 A new code for fitting dielectric spectra ......... 118 7.7.2 Results of fitting f-sum and gLST constraints ..... 118 7.7.3 Collective nature of the relaxation ............ 121 7.8 Appendix: Relaxation at ultra low frequencies ......... 122 7.9 Appendix: relevance to biology ................