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Ab Initio Molecular Dynamics of Water by Quantum Monte Carlo

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Ab Initio Molecular Dynamics of Water by Quantum Monte Carlo Ab initio molecular dynamics of water by quantum Monte Carlo Author: Ye Luo Supervisor: Prof. Sandro Sorella A thesis submitted for the degree of Doctor of Philosophy October 2014 Acknowledgments I would like to acknowledge first my supervisor Prof. Sandro Sorella. During the years in SISSA, he guided me in exploring the world of QMC with great patience and enthusiasm. He's not only a master in Physics but also an expert in high performance computing. He never exhausts new ideas and is always advancing the research at the light speed. For four years, I had an amazing journey with him. I would like to appreciate Dr. Andrea Zen, Prof. Leonardo Guidoni and my classmate Guglielmo Mazzola. I really learned a lot from the collaboration on various projects we did. I would like to thank Michele Casula and W.M.C. Foulkes who read and correct my thesis and also give many suggestive comments. I would like to express the gratitude to my parents who give me unlimited support even though they are very far from me. Without their unconditional love, I can't pass through the hardest time. I feel very sorry for them because I went home so few times in the previous years. Therefore, I would like to dedicate this thesis to them. I remember the pleasure with my friends in Trieste who are from all over the world. Through them, I have access to so many kinds of food and culture. They help me when I meet difficulties and they wipe out my loneliness by sharing the joys and tears of my life. Especially, I would like to thank all my Italian friends because they often have to bear my poor spoken Italian. I would like to thank all my colleagues. They are always very kind to spare their precious time to share their wealth of knowledge with me in order to resolve my doubts. Last but not the least, I would like to thank SISSA and Trieste. SISSA is by far the best place I have ever worked or studied. Besides its academic prestige, it's attractive because of its wonderful location and facilities. The beautiful garden and the whole landscape view of Trieste gulf alleviate a lot my stress during the working hours. Trieste city is charming and the local people (in general, Italian people) are warmhearted. I had so many unforgettable memories of Trieste. For this reason, every time I see the lights on the Trieste coast when my train runs across the historic bridge in Barcola, I'm filled with happiness since my beloved city is approaching and my home is there. 2 Contents 1 Introduction5 2 Quantum Monte Carlo 10 2.1 Introduction . 10 2.2 Variational Monte Carlo . 13 2.3 Metropolis algorithm . 15 2.4 Wavefunction optimization . 16 2.4.1 Stochastic reconfiguration method . 17 2.4.2 Stochastic reconfiguration with Hessian accelerator . 20 2.4.3 Stochastic reconfiguration with conjugate gradients . 21 2.4.4 Stochastic reconfiguration with signal noise filter . 21 3 Jastrow correlated antisymmetrized geminal power wavefunc- tion 23 3.1 Introduction . 23 3.2 Antisymmetrized geminal power wavefunction . 24 3.3 Hybrid orbitals . 25 3.4 Molecular orbitals . 27 3.5 Jastrow factor . 29 3.5.1 One-body and two-body Jastrow . 30 3.5.2 Three/four-body Jastrow . 32 4 Molecular dynamics with quantum Monte Carlo 35 4.1 Introduction . 35 4.2 Force evaluation with quantum Monte Carlo . 36 4.3 Solution of the infinite variance problem for the forces . 39 4.4 Molecular dynamics with noisy forces . 42 4.5 Role of the force covariance matrix . 45 5 Vibrational frequencies 51 5.1 Introduction . 51 5.2 Calculation of vibrational properties . 53 5.2.1 Simple fitting method . 55 5.2.2 Fitting method with molecular dynamics . 57 5.2.3 Covariance matrix method . 58 3 5.3 Simulation setup . 61 5.4 Control of all the sources of systematic errors in QMC dynamics 62 5.4.1 Time step error . 62 5.4.2 Error in sampling the BO energy surface . 63 5.4.3 Residual QMC error due to a finite number of samples 65 5.5 Results . 66 5.6 Conclusions . 70 6 Liquid water 73 6.1 Introduction . 73 6.2 Simulation specifications . 76 6.3 Radial distribution function . 78 6.4 Many water molecules' interaction . 81 6.5 Role of the Hydrogen bond . 83 6.6 Discussion and conclusion . 86 7 Conclusion 88 A Solutions to the second order Langevin dynamics 91 A.1 Integration of second order Langevin equations . 91 A.2 Better integration scheme . 92 B Efficient calculation of S^2 95 B.1 Determinant part . 96 B.2 Jastrow part . 97 List of abbreviations 98 List of publications 99 4 Chapter 1 Introduction \The Blue Planet" is how we describe our beloved homeland planet | Earth. When we see its pictures taken from the satellites, the planet is wrapped by a beautiful blue color under the sunshine with a black background of the dark universe. The blue color comes from the oceans which cover 71% of the surface of Earth and all the oceans contain 97.1% of the total amount of the water on Earth in a liquid form. The rest part (less than 3%) is found in groundwater, glaciers or ice caps and other large water bodies like lakes and rivers. Water circulation is one of the most important circulations on Earth besides the carbon circulation. Water evaporates from the sea and forms clouds in the sky. Some of those clouds stay still and some others are carried to the lands by the wind and later turn into rainfalls. Rain drops accumulate and join rivers or groundwater and finally return back to the sea. During this circulation, the flowing rivers cut the ground and valleys form. Meanwhile, the mud and sand from the erosion of river banks are carried by the river to the lower course and plains form. In this way the landscape of Earth was reshaped gradually in the past billions of years. This circulation also causes the weather changes. Having clouds, rains, snows and fogs totally depends on the amount of water in the air and sky. Moreover, the abundance of water strongly affects the climate of certain areas. For instance, rain forests around the amazon river have plenty of rain while the Sahara deserts have almost zero precipitation. The other contribution of the water to Earth, even more important than that to the geographic evolution, is that it brings life to Earth. Even though there are many models of abiogenesis, the earliest life is discovered in the oceans without any doubts. From bacteria, later algae to marine plants and fishes, the forms of lives evolve their sizes and complexity during the billions of years. On the land, insects, plants and animals started to appear a little bit later than in the oceans. They no longer live in an aquatic environment but they need acquiring water for sustaining their lives since a very large portion of the body weights consists of water. For instance, the human body contains from 55% to 78% water. If one stops drinking water for a week, he will die soon because of organ failures due to the dehydration. Many 5 biological processes happen with the participation of water. As one of the most important processes in cyanobacteria and plants, the photosynthesis 2nCO2 + 2nH2O ! 2nCH2O + 2nO2 which generates the oxygen in the atmosphere needs water as one of the raw materials. In another case, when leaves fall off the trees and get dry, their color turns from green into yellow. According to our current knowledge about those lives we are familiar with, the existence of liquid water is a key prerequisite for any kind of lives. Therefore, whenever we send robot rovers to other planets, looking for the signs of water is always a very important subject in order to discover alien lives. When the Cassini-Huygens arrived Enceladus, a satellite of Saturn, we were so encouraged by this piece of inspiring news that water vapor was discovered there. Since water plays such an important role in lives on Earth, it deserves a very deep understanding. Our journey on understanding water started very early. In 1783, Cavendish published a paper on the production of water [1] by burning inflammable air (hydrogen) in dephlogisticated air (oxygen). Later physicists found that atoms could be divided into electrons and nuclei. As the positive charges of nuclei increase, all the elements can be sorted in the periodic table. We know that oxygen is the richest element on Earth and hydrogen is the lightest element and also the first element appeared at the original stage of the Universe ac- cording to the Big Bang theory. Not only in chemistry but also in physics, water has left so many footprints because of its easy accessibility. In the early days, the SI unit kilogram was chosen to be as heavy as 1 dm3 of wa- ter. In the Celsius measurement of temperature, zero and a hundred Celsius are the frozen and boiling points of water at the atmosphere. Besides, the lifting power, buoyancy, of liquids was first explained by Archimedes with an experiment in a bathtub filled with water. All our knowledge about water in physics and chemistry came from exper- iments till 1925 when quantum mechanics was developed to describe how the quantum states of a physical system evolve with time. It laid the foundation of studying materials by a theoretical method. By solving the Schr¨odingerequa- tion [2], in principle, all the electronic properties of materials can be studied theoretically.
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