Susi Lehtola COMPUTATIONAL MODELING OF THE ELECTRON MOMENTUM DENSITY COMPUTATIONAL Report Series in Physics HU-P-D202 COMPUTATIONAL MODELING OF THE ELECTRON MOMENTUM DENSITY Susi Lehtola ISBN 978-952-10-8090-6 (printed version) ISSN 0356-0961 ISBN 978-952-10-8091-3 (pdf version) http://ethesis.helsinki.fi Helsinki University Print Helsinki 2013 UNIVERSITY OF HELSINKI REPORT SERIES IN PHYSICS HU-P-D202 COMPUTATIONAL MODELING OF THE ELECTRON MOMENTUM DENSITY Susi Lehtola Laboratory of Materials Physics Department of Physics Faculty of Science University of Helsinki Helsinki, Finland ACADEMIC DISSERTATION To be presented, with the permission of the Faculty of Science of the University of Helsinki, for public criticism in Auditorium E204 of the Physicum building (Gustaf Häll- strömin katu 2a), on 22 February 2013, at 12 o’clock noon. Helsinki 2013 Supervisors Dr. Mikko Hakala Prof. Keijo Hämäläinen Department of Physics Department of Physics University of Helsinki University of Helsinki Helsinki, Finland Helsinki, Finland Pre-examiners Prof. Kirk Peterson Prof. Tapio Rantala Department of Chemistry Department of Physics Washington State University Tampere University of Technology Washington, USA Tampere, Finland Opponent Custos Prof. Lars Pettersson Prof. Keijo Hämäläinen Department of Physics Department of Physics Stockholm University University of Helsinki Stockholm, Sweden Helsinki, Finland This thesis has been written on Fedora Linux with LYX, using the TEXLive LATEX compiler. The figures have been prepared with xfig and GLE, with Octave for preprocessing the raw data. Report Series in Physics HU-P-D202 ISSN 0356-0961 ISBN 978-952-10-8090-6 (printed version) 978-952-10-8091-3 (pdf version) http://ethesis.helsinki.fi Helsinki University Print Helsinki 2013 ABSTRACT Abstract The properties and the functionality of materials are determined to a large extent by their electronic structure. The electronic structure can be examined through the electron momentum density, which is classically equivalent to the velocity distribution of the electrons. Changes in the structure of materials induce changes on their electronic structure, which in turn are reflected as changes in the electron momentum densities that can be routinely measured using, e.g., x-ray Compton scattering. The changes in the momentum density can be linked back to the structural changes the system has experienced through the extensive use of computational modeling. This procedure naturally requires using a model matching the accuracy of the experiment, which is constantly improving as the result of the ongoing devel- opment of synchrotron radiation sources and beam line instrumentation. However, the accuracies of the current computational methods have not been hitherto es- tablished. This thesis focuses on developing the methods used to compute the electron momentum density in order to achieve an accuracy comparable to that of the experiment. The accuracies of current quantum chemical methods that can be used to model the electron momentum density are established. The completeness- optimization scheme is used to develop computationally efficient basis sets for modeling the electron momentum density at the complete basis set limit. A novel, freely available software program that can be used to perform all of the necessary electronic structure calculations is also introduced. i PREFACE I wanted to study physics, because I was confused. Now I am still confused, but on a much higher level. Adapted from Earl C. Kelley Preface I initially came to study physics in order to learn about the general theory of relativity. I did take the course a few years ago, but as it often happens, during my studies before that I found my interests to lie elsewhere. Of the many people who have had influence over my choice of specialization I want to acknowledge especially Keijo Kajantie, whose insight and childlike enthusiasm towards physics – even now as a pensioner – inspired me already as a freshman, Jouni Niskanen for deeply topical courses in quantum mechanics, and Antti Kuronen for courses in scientific programming. I ended up in the field of materials physics somewhat by coincidence, having looked for a topic for my master’s thesis after four years of studies in theoretical physics. I was especially interested in computational science and quantum mechanics. My supervisor Keijo Hämäläinen hired thus me to work on numerical quantum mechanics, i.e., to compute inelastic x-ray scattering spectra. Although the beginning of my work as a theoretician in an experimental group was quite rough, after a few years I remembered why I initially came to the x-ray laboratory in the first place and started following this interest. The result of this decision is this thesis, completed in a time span of roughly three years. I am in gratitude to Keijo Hämäläinen and my advisor Mikko Hakala, who indefati- gably steered me into the subtleties of scientific writing. I am also very much grateful to the Jenny and Antti Wihuri foundation, whose financial support has been essential for the making of this thesis. I thank my coauthors Pekka Manninen, Arto Sakko, and Juha Vaara, who have provided me incisive comments on my manuscripts and thus helped me to hone the texts towards perfection. As can be seen from the composition of this thesis, collaboration with Pekka has been especially fruitful. Thanks to the pre-examiners of this thesis, Kirk Peterson and Tapio Rantala, are also in order. Last but not least I owe gratitude for my family for support. The PhD experiences of my parents and my brother have comforted me when facing seemingly gargantuan problems. Also the free time spent with my friends has been indispensable. Thanks to you all. Susi Lehtola Espoo, January 2013 ii TABLE OF CONTENTS Table of contents Table of contents iii List of figures iv List of Tables iv List of abbreviations vi 1 Introduction 1 2 Inelastic x-ray scattering 5 2.1 Experimental setup . .7 3 Electronic structure theory 9 3.1 Molecular structure . .9 3.2 Hartree–Fock theory . 11 3.3 Basis sets . 12 3.3.1 STO basis . 13 3.3.2 GTO basis . 14 3.3.3 Formation of basis sets . 16 3.3.3.1 Completeness profiles . 16 3.3.3.2 Completeness-optimization of exponents . 17 3.3.3.3 Completeness-optimization of basis sets . 18 3.4 Configuration interaction . 19 3.5 Møller–Plesset perturbation theory . 20 3.6 Coupled cluster theory . 21 3.7 Density-functional theory . 21 3.8 Electron momentum density . 22 3.8.1 STO basis . 24 3.8.2 GTO basis . 25 3.9 Illustrative calculation . 26 4 Summary of papers 31 4.1 Paper I: Calculation of isotropic Compton profiles with Gaussian basis sets 31 4.2 Paper II: Completeness-optimized basis sets: Application to ground- state electron momentum densities . 32 4.3 Paper III: Contraction of completeness-optimized basis sets: Application to ground-state electron momentum densities . 33 4.4 Paper IV: ERKALE – A Flexible Program Package for X-ray Properties of Atoms and Molecules . 33 5 Discussion and conclusions 35 iii LIST OF FIGURES Bibliography 37 List of figures 1 Schematic view of inelastic x-ray scattering. .5 2 The experimental setup of the xenon experiment. .7 3 The measured raw spectrum in the CS experiment. .8 4 Expansion of a Slater-type orbital in terms of Gaussian-type orbitals. 15 5 Completeness profile example . 17 6 Width of the completeness plateau plotted as a function of exponents. 18 7 Illustration of the completeness-optimization scheme. 19 8 Electron density of the fluorine dimer. 27 9 Electron momentum density of the fluorine dimer. 28 10 Isotropic Compton profile of the fluorine dimer. 28 11 Difference Compton profile of the fluorine dimer. 29 12 The strive towards the exact solution to the electronic problem. 35 List of Tables 1 Grouping of the elements with similar electronic structures. 32 iv LIST OF PUBLICATIONS List of publications Publications included in this thesis and the author’s contribution I J. Lehtola, M. Hakala, J. Vaara and K. Hämäläinen, Calculation of isotropic Compton profiles with Gaussian basis sets, Phys. Chem. Chem. Phys. 13, 5630 (2011), doi:10.1039/C0CP02269A, on page 47. II J. Lehtola, P. Manninen, M. Hakala and K. Hämäläinen, Completeness-optimized basis sets: Application to ground-state electron momentum densities, J. Chem. Phys. 137, 104105 (2012), doi:10.1063/1.4749272, on page 61. III S. Lehtola, P. Manninen, M. Hakala and K. Hämäläinen, Contraction of complete- ness-optimized basis sets: Application to ground-state electron momentum den- sities, J. Chem. Phys. 138, 044109 (2013), doi:10.1063/1.4788635, on page 71. IV J. Lehtola, M. Hakala, A. Sakko and K. Hämäläinen, ERKALE – A Flexible Program Package for X-ray Properties of Atoms and Molecules, J. Comp. Chem. 33, 1572 (2012), doi:10.1002/jcc.22987, on page 81. SL (formerly JL) wrote the computer programs, performed all of the calculations, carried out the analysis of the results and wrote the first versions of the manuscripts I-IV. Other publications by the author related to this thesis V J. Lehtola, M. Hakala and K. Hämäläinen, Structure of Liquid Linear Alcohols, J. Phys. Chem. B 114 (2010), 6426. VI T. Pylkkänen, J. Lehtola, M. Hakala, A. Sakko, G. Monaco, S. Huotari and K. Hämäläinen, Universal Signature of Hydrogen Bonding in the Oxygen K-Edge Spectrum of Alcohols, J. Phys. Chem. B 114 (2010), 13076. v LIST OF ABBREVIATIONS List of abbreviations BO Born–Oppenheimer approximation BSSE Basis set superposition error CBS Complete basis set CC Coupled cluster CCSD Coupled cluster theory with single and double excitations CI Configuration