DOCSLIB.ORG

Computational Physics Lectures: Variational Monte Carlo Methods

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computational Physics Lectures: Variational Monte Carlo Methods Quantum Monte Carlo Motivation Computational Physics Lectures: Variational Monte Given a hamiltonian H and a trial wave function ΨT , the variational Carlo methods principle states that the expectation value of H , defined through h i dRΨ∗ (R)H(R)Ψ (R) E[H] = H = T T , h i d Ψ ( )Ψ ( ) Morten Hjorth-Jensen1,2 R R T∗ R T R R is an upper bound to the ground state energy E0 of the hamiltonian Department of Physics, University of Oslo1 H, that is E H . Department of Physics and Astronomy and National Superconducting Cyclotron 0 ≤ h i 2 Laboratory, Michigan State University In general, the integrals involved in the calculation of various expectation values are multi-dimensional ones. Traditional Aug 23, 2017 integration methods such as the Gauss-Legendre will not be c 1999-2017, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license adequate for say the computation of the energy of a many-body system. Quantum Monte Carlo Motivation Quantum Monte Carlo Motivation The trial wave function can be expanded in the eigenstates of the hamiltonian since they form a complete set, viz., In most cases, a wave function has only small values in large parts ΨT (R) = ai Ψi (R), of configuration space, and a straightforward procedure which uses Xi homogenously distributed random points in configuration space will most likely lead to poor results. This may suggest that some kind and assuming the set of eigenfunctions to be normalized one obtains of importance sampling combined with e.g., the Metropolis 2 a∗ an dRΨ∗ (R)H(R)Ψn(R) a E algorithm may be a more efficient way of obtaining the ground nm m m = n n n E , 2 0 state energy. The hope is then that those regions of configurations nm am∗ an dRΨm∗ (R)Ψn(R) n an ≥ P R P space where the wave function assumes appreciable values are where weP used that RH(R)Ψn(R) = EnΨn(R). InP general, the sampled more efficiently. integrals involved in the calculation of various expectation values are multi-dimensional ones. The variational principle yields the lowest state of a given symmetry. Quantum Monte Carlo Motivation Quantum Monte Carlo Motivation Construct first a trial wave function ψT (R, α), for a The tedious part in a VMC calculation is the search for the many-body system consisting of N particles located at variational minimum. A good knowledge of the system is required in positions order to carry out reasonable VMC calculations. This is not always R = (R1,..., RN ). The trial wave function depends on α the case, and often VMC calculations serve rather as the starting variational parameters α = (α1, . , αM ). point for so-called diffusion Monte Carlo calculations (DMC). DMC Then we evaluate the expectation value of the hamiltonian H is a way of solving exactly the many-body Schroedinger equation by dRΨ∗ (R, α)H(R)Ψ (R, α) means of a stochastic procedure. A good guess on the binding E[H] = H = T T . h i d Ψ ( , α)Ψ ( , α) energy and its wave function is however necessary. A carefully R R T∗ R T R performed VMC calculation can aid in this context. R Thereafter we vary α according to some minimization algorithm and return to the first step. Quantum Monte Carlo Motivation Quantum Monte Carlo Motivation Basic steps Define a new quantity Choose a trial wave function ψT (R). 1 2 EL(R, α) = HψT (R, α), ψT (R) ψT (R, α) P(R) = | | . ψ (R) 2 dR | T | called the local energy, which, together with our trial PDF yields R This is our new probability distribution function (PDF). The N 1 approximation to the expectation value of the Hamiltonian is now E[H(α)] = P(R)E (R)dR P(R , α)E (R , α) L ≈ N i L i Z i=1 dRΨ∗ (R, α)H(R)Ψ (R, α) X E[H(α)] = T T . d Ψ ( , α)Ψ ( , α) with N being the number of Monte Carlo samples. R R T∗ R T R R Quantum Monte Carlo Quantum Monte Carlo: hydrogen atom The Algorithm for performing a variational Monte Carlo The radial Schroedinger equation for the hydrogen atom can be calculations runs thus as this written as Initialisation: Fix the number of Monte Carlo steps. Choose an ~2 ∂2u(r) ke2 ~2l(l + 1) α 2 u(r) = Eu(r), initial R and variational parameters α and calculate ψT (R) . −2m ∂r 2 − r − 2mr 2 | | Initialise the energy and the variance and start the Monte Carlo calculation. or with dimensionless variables Calculate a trial position Rp = R + r step where r is a ∗ 1 ∂2u(ρ) u(ρ) l(l + 1) random variable r [0, 1]. + u(ρ) λu(ρ) = 0, ∈ 2 2 Metropolis algorithm to accept or reject this move −2 ∂ρ − ρ 2ρ − w = P(Rp)/P(R). with the hamiltonian If the step is accepted, then we set R = Rp. Update averages 1 ∂2 1 l(l + 1) H = + . Finish and compute final averages. −2 ∂ρ2 − ρ 2ρ2 Observe that the jumping in space is governed by the variable step. Use variational parameter α in the trial wave function This is Called brute-force sampling. Need importance sampling to α αρ get more relevant sampling, see lectures below. uT (ρ) = αρe− . Quantum Monte Carlo: hydrogen atom Quantum Monte Carlo: hydrogen atom We note that at α = 1 we obtain the exact result, and the variance Inserting this wave function into the expression for the local energy is zero, as it should. The reason is that we then have the exact EL gives wave function, and the action of the hamiltionan on the wave 1 α 2 function EL(ρ) = α . −ρ − 2 − ρ Hψ = constant ψ, × A simple variational Monte Carlo calculation results in yields just a constant. The integral which defines various α H σ2 σ/√N h i expectation values involving moments of the hamiltonian becomes 7.00000E-01 -4.57759E-01 4.51201E-02 6.71715E-04 then 8.00000E-01 -4.81461E-01 3.05736E-02 5.52934E-04 9.00000E-01 -4.95899E-01 8.20497E-03 2.86443E-04 n n dRΨT∗ (R)H (R)ΨT (R) dRΨT∗ (R)ΨT (R) 1.00000E-00 -5.00000E-01 0.00000E+00 0.00000E+00 H = = constant = constant. h i d Ψ ( )Ψ ( ) × d Ψ ( )Ψ ( ) 1.10000E+00 -4.93738E-01 1.16989E-02 3.42036E-04 R R T∗ R T R R R T∗ R T R 1.20000E+00 -4.75563E-01 8.85899E-02 9.41222E-04 R R 1.30000E+00 -4.54341E-01 1.45171E-01 1.20487E-03 This gives an important information: the exact wave function leads to zero variance! Variation is then performed by minimizing both the energy and the variance. Quantum Monte Carlo: the helium atom Quantum Monte Carlo: the helium atom The hamiltonian becomes then The helium atom consists of two electrons and a nucleus with ~2 2 ~2 2 2ke2 2ke2 ke2 charge Z = 2. The contribution to the potential energy due to the Hˆ = ∇1 ∇2 + , attraction from the nucleus is − 2m − 2m − r1 − r2 r12 2ke2 2ke2 and Schroedingers equation reads , − r − r 1 2 Hˆψ = Eψ. and if we add the repulsion arising from the two interacting electrons, we obtain the potential energy All observables are evaluated with respect to the probability distribution 2 2 2 2 ψ (R) 2ke 2ke ke P( ) = | T | . V (r1, r2) = + , R 2 − r − r r ψT (R) dR 1 2 12 | | generated by the trial wave function.R The trial wave function must with the electrons separated at a distance r12 = r1 r2 . | − | approximate an exact eigenstate in order that accurate results are to be obtained. Quantum Monte Carlo: the helium atom Quantum Monte Carlo: the helium atom This results in 1 d (r ) RT 1 = Z, Choice of trial wave function for Helium: Assume r 0. T (r1) dr1 − 1 → R and 1 1 1 2 Z Zr EL(R) = HψT (R) = ψT (R)+finite terms. (r ) e 1 . ψ (R) ψ (R) −2∇1 − r T 1 − T T 1 R ∝ A similar condition applies to electron 2 as well. For orbital 2 1 1 d 1 d Z momenta l > 0 we have EL(R) = 2 T (r1) + finite terms T (r1) −2 dr1 − r1 dr1 − r1 R R 1 d T (r) Z r R = . For small values of 1, the terms which dominate are (r) dr −l + 1 RT 1 1 d Z lim EL(R) = T (r1), Similarly, studying the case r12 0 we can write a possible trial r1 0 (r ) −r dr − r R → → RT 1 1 1 1 wave function as since the second derivative does not diverge due to the finiteness of α(r1+r2) βr12 ψT (R) = e− e . Ψ at the origin. The last equation can be generalized to ψT (R) = φ(r1)φ(r2) . φ(rN ) f (rij ), Yi<j for a system with N electrons or particles. The first attempt at solving the helium atom The first attempt at solving the Helium atom During the development of our code we need to make several checks. It is also very instructive to compute a closed form With closed form formulae we can speed up the computation of the expression for the local energy. Since our wave function is rather correlation. In our case we write it as simple it is straightforward to find an analytic expressions. Consider first the case of the simple helium function arij ΨC = exp , α(r1+r2) + βr ΨT (r1, r2) = e− 1 ij Xi<j The local energy is for this case which means that the gradient needed for the so-called quantum force and local energy can be calculated analytically.
Load more

Recommended publications
  • An Overview of Quantum Monte Carlo Methods David M. Ceperley
    An Overview of Quantum Monte Carlo Methods David M. Ceperley Department of Physics and National Center for Supercomputing Applications University of Illinois Urbana-Champaign Urbana, Illinois 61801 In this brief article, I introduce the various types of quantum Monte Carlo (QMC) methods, in particular, those that are applicable to systems in extreme regimes of temperature and pressure. We give references to longer articles where detailed discussion of applications and algorithms appear. MOTIVATION One does QMC for the same reason as one does classical simulations; there is no other method able to treat exactly the quantum many-body problem aside from the direct simulation method where electrons and ions are directly represented as particles, instead of as a “fluid” as is done in mean-field based methods. However, quantum systems are more difficult than classical systems because one does not know the distribution to be sampled, it must be solved for. In fact, it is not known today which quantum problems can be “solved” with simulation on a classical computer in a reasonable amount of computer time. One knows that certain systems, such as most quantum many-body systems in 1D and most bosonic systems are amenable to solution with Monte Carlo methods, however, the “sign problem” prevents making the same statement for systems with electrons in 3D. Some limitations or approximations are needed in practice. On the other hand, in contrast to simulation of classical systems, one does know the Hamiltonian exactly: namely charged particles interacting with a Coulomb potential. Given sufficient computer resources, the results can be of quite high quality and for systems where there is little reliable experimental data.
    [Show full text]
  • Variational Monte Carlo Technique Ground State Energies of Quantum Mechanical Systems
    GENERAL ⎜ ARTICLE Variational Monte Carlo Technique Ground State Energies of Quantum Mechanical Systems Sukanta Deb In this article, I discuss the use of the Variational Monte Carlo technique in the determination of ground state energies of quantum harmonic os- cillators in one and three dimensions. In order to provide a better understanding of this technique, a pre-requisite concept of Monte Carlo integra- Sukanta Deb is an tion is also discussed along with numerical ex- Assistant Professor in the amples. The technique can be easily extended Department of Physics at Acharya Narendra Dev to study the ground state energies of hydrogen College (University of atom, particle in a box, helium atom and other Delhi). His research complex microscopic systems involving N parti- interests include astro- cles in d-dimensions. nomical photometry and spectroscopy, light curve 1. Introduction modeling in astronomy and astrophysics, Monte Carlo There exist many problems in science and engineering methods and simulations whose exact solution either does not exist or is difficult and automated data to find. For the solutions of those problems, one has to analysis. resort to approximate methods. In the context of quan- tum mechanics, the exact solutions of the Schr¨odinger’s equation exist only for a few idealized systems. There are varieties of systems for which the Schr¨odinger’s equa- tion either cannot be solved exactly or it involves very lengthy and cumbersome calculations [1]. For example, solving the Schr¨odinger equation to study the proper- ties of a system with hundreds or thousands of particles often turns out to be impractical [2].
    [Show full text]
  • Unifying Cps Nqs
    Citation for published version: Clark, SR 2018, 'Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks', Journal of Physics A: Mathematical and General, vol. 51, no. 13, 135301. https://doi.org/10.1088/1751- 8121/aaaaf2 DOI: 10.1088/1751-8121/aaaaf2 Publication date: 2018 Document Version Peer reviewed version Link to publication This is an author-created, in-copyedited version of an article published in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/article/10.1088/1751-8121/aaaaf2/meta University of Bath Alternative formats If you require this document in an alternative format, please contact: [email protected] General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 04. Oct. 2021 Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks Stephen R Clarkyz yDepartment of Physics, University of Bath, Claverton Down, Bath BA2 7AY, U.K. zMax Planck Institute for the Structure and Dynamics of Matter, University of Hamburg CFEL, Hamburg, Germany E-mail: [email protected] Abstract.
    [Show full text]
  • Arxiv:1902.04057V3 [Cond-Mat.Dis-Nn] 19 Jan 2020 Statistical Mechanics Applications [16], As Well As Den- Spin Systems [13]
    Deep autoregressive models for the efficient variational simulation of many-body quantum systems 1, 1, 1, 2, 1, Or Sharir, ∗ Yoav Levine, † Noam Wies, ‡ Giuseppe Carleo, § and Amnon Shashua ¶ 1The Hebrew University of Jerusalem, Jerusalem, 9190401, Israel 2Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA Artificial Neural Networks were recently shown to be an efficient representation of highly-entangled many-body quantum states. In practical applications, neural-network states inherit numerical schemes used in Variational Monte Carlo, most notably the use of Markov-Chain Monte-Carlo (MCMC) sampling to estimate quantum expectations. The local stochastic sampling in MCMC caps the potential advantages of neural networks in two ways: (i) Its intrinsic computational cost sets stringent practical limits on the width and depth of the networks, and therefore limits their expressive capacity; (ii) Its difficulty in generating precise and uncorrelated samples can result in estimations of observables that are very far from their true value. Inspired by the state-of-the-art generative models used in machine learning, we propose a specialized Neural Network architecture that supports efficient and exact sampling, completely circumventing the need for Markov Chain sampling. We demonstrate our approach for two-dimensional interacting spin models, showcas- ing the ability to obtain accurate results on larger system sizes than those currently accessible to neural-network quantum states. Introduction.– The theoretical understanding and been limited to relatively shallow architectures, far from modeling of interacting many-body quantum matter rep- the very deep networks used in modern machine learn- resents an outstanding challenge since the early days of ing applications.
    [Show full text]
  • An Introduction to Quantum Monte Carlo
    An introduction to Quantum Monte Carlo Jose Guilherme Vilhena Laboratoire de Physique de la Matière Condensée et Nanostructures, Université Claude Bernard Lyon 1 Outline 1) Quantum many body problem; 2) Metropolis Algorithm; 3) Statistical foundations of Monte Carlo methods; 4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation; Outline 1) Quantum many body problem; 2) Metropolis Algorithm; 3) Statistical foundations of Monte Carlo methods; 4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation; 1 - Quantum many body problem Consider of N electrons. Since m <<M , to a good e N approximation, the electronic dynamics is governed by : Z Z 1 2 1 H =− ∑ ∇ i −∑∑ ∑∑ 2 i i ∣r i−d∣ 2 i j ≠i ∣ri−r j∣ which is the Born Oppenheimer Hamiltonian. We want to find the eigenvalues and eigenvectos of this hamiltonian, i.e. : H r1, r 2,. ..,r N =E r 1, r2,. ..,r N QMC allow us to solve numerically this, thus providing E0 and Ψ . 0 1 - Quantum many body problem Consider of N electrons. Since m <<M , to a good e N approximation, the elotonic dynamics is governed by : Z Z 1 2 1 H =− ∑ ∇ i −∑∑ ∑∑ 2 i i ∣r i−d∣ 2 i j ≠i ∣ri−r j∣ which is the Born Oppenheimer Hamiltonian. We want to find the eigenvalues and eigenvectos of this hamiltonian, i.e. : H r1, r 2,.
    [Show full text]
  • University of Bath
    Citation for published version: Clark, SR 2018, 'Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks', Journal of Physics A: Mathematical and General, vol. 51, no. 13, 135301. https://doi.org/10.1088/1751- 8121/aaaaf2 DOI: 10.1088/1751-8121/aaaaf2 Publication date: 2018 Document Version Peer reviewed version Link to publication This is an author-created, in-copyedited version of an article published in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/article/10.1088/1751-8121/aaaaf2/meta University of Bath General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 23. May. 2019 Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks Stephen R Clarkyz yDepartment of Physics, University of Bath, Claverton Down, Bath BA2 7AY, U.K. zMax Planck Institute for the Structure and Dynamics of Matter, University of Hamburg CFEL, Hamburg, Germany E-mail: [email protected] Abstract. Correlator product states (CPS) are a powerful and very broad class of states for quantum lattice systems whose (unnormalised) amplitudes in a fixed basis can be sampled exactly and efficiently.
    [Show full text]
  • Variational Monte Carlo Method for Fermions
    Variational Monte Carlo Method for Fermions Vijay B. Shenoy [email protected] Indian Institute of Science, Bangalore December 26, 2009 1/91 Acknowledgements Sandeep Pathak Department of Science and Technology 2/91 Plan of the Lectures on Variational Monte Carlo Method Motivation, Overview and Key Ideas Details of VMC Application to Superconductivity 3/91 The Strongly Correlated Panorama... Fig. 1. Phase diagrams ofrepresentativemateri- als of the strongly cor- related electron family (notations are standard and details can be found in the original refer- ences). (A) Temperature versus hole density phase diagram of bilayer man- ganites (74), including several types of antiferro- magnetic (AF) phases, a ferromagnetic (FM) phase, and even a glob- ally disordered region at x 0 0.75. (B) Generic phase diagram for HTSC. SG stands for spin glass. (C) Phase diagram of single layered ruthenates (75, 76), evolving from a superconducting (SC) state at x 0 2 to an AF insulator at x 0 0(x controls the bandwidth rather than the carrier density). Ruthenates are believed to be clean metals at least at large x, thus providing a family of oxides where competition and com- plexity can be studied with less quenched dis- order than in Mn ox- ides. (D) Phase diagram of Co oxides (77), with SC, charge-ordered (CO), and magnetic regimes. (E) Phase diagram of the organic k-(BEDT- TTF)2Cu[N(CN)2]Cl salt (57). The hatched re- gion denotes the co- existence of metal and insulator phases. (F) Schematic phase dia- gram of the Ce-based heavy fermion materials (51).
    [Show full text]
  • Quantum Monte Carlo Simulations: from Few to Many Species
    Quantum Monte Carlo Simulations: From Few to Many Species by William Gary Dawkins A Thesis Presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Physics Guelph, Ontario, Canada c William Gary Dawkins, May, 2019 ABSTRACT QUANTUM MONTE CARLO SIMULATIONS: FROM FEW TO MANY SPECIES William Gary Dawkins Advisor: University of Guelph, 2019 Professor Alexandros Gezerlis We study systems of particles with varying numbers of distinguishable species ranging from 4-species to the N-species limit, where N is the number of particles in the system itself. We examine these systems using various powerful simulation techniques that belong to the quantum Monte Carlo (QMC) family. The N-species limit is studied via Path Integral Monte Carlo (PIMC), which is a finite temperature, non- perturbative technique. At this limit, we explore the viability of two different thermal density matrices for systems that interact via hard-sphere, hard-cavity potentials. We then study the impact of finite-size effects on calculations of the energy, pressure and specific heat of the system when reaching the thermodynamic limit under periodic boundary conditions. We then move to studying a 4-species fermionic system using Variational Monte Carlo (VMC) and Diffusion Monte Carlo (DMC), which are ground state techniques. We apply these methods to the clustering problem of 4-particle and 8-particle systems. While the 4-particle, 4-species system is relatively simple to calculate as it is physically identical to a bosonic system of 4-particles, the 8-particle fermi system is more complicated. We detail the process of simulating a bound 8- particle state with respect to decay into two independent 4-particle clusters with DMC for the first time.
    [Show full text]
  • Deep Learning-Enhanced Variational Monte Carlo Method for Quantum Many-Body Physics
    PHYSICAL REVIEW RESEARCH 2, 012039(R) (2020) Rapid Communications Deep learning-enhanced variational Monte Carlo method for quantum many-body physics Li Yang ,1,* Zhaoqi Leng,2 Guangyuan Yu,3 Ankit Patel,4,5 Wen-Jun Hu,1,† and Han Pu1,‡ 1Department of Physics and Astronomy & Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA 2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 3Department of Physics and Astronomy & The Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA 4Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA 5Department of Neuroscience, Baylor College of Medicine, Houston, Texas 77030, USA (Received 26 May 2019; accepted 12 December 2019; published 14 February 2020) Artificial neural networks have been successfully incorporated into the variational Monte Carlo method (VMC) to study quantum many-body systems. However, there have been few systematic studies exploring quantum many-body physics using deep neural networks (DNNs), despite the tremendous success enjoyed by DNNs in many other areas in recent years. One main challenge of implementing DNNs in VMC is the inefficiency of optimizing such networks with a large number of parameters. We introduce an importance sampling gradient optimization (ISGO) algorithm, which significantly improves the computational speed of training DNNs by VMC. We design an efficient convolutional DNN architecture to compute the ground state of a one-dimensional SU(N) spin chain. Our numerical results of the ground-state energies with up to 16 layers of DNNs show excellent agreement with the Bethe ansatz exact solution. Furthermore, we also calculate the loop correlation function using the wave function obtained.
    [Show full text]
  • Variational Quantum Monte Carlo Abstract Introduction
    Variational Quantum Monte Carlo Josh Jiang, The University of Chicago > (1.1) Abstract The many electron system is one that quantum mechanics have no exact solution for because of the correlation between each electron with every other electron. Many methods have been developed to account for this correlation. One family of methods known as quantum Monte Carlo methods takes a numerical approach to solving the many-body problem. By employing a stochastic process to estimate the energy of a system, this method can be used to calculate the ground state energy of a system using the variational principle. The core of this method involves choosing an ansatz with a set of variational parameters that can be optimized to minimize the energy of the system. Recent research in this field is focused on developing methods to faster and more efficiently optimize this trial ansatz and also to make the ansatz more flexible by allowing more parameters. These topics along with the theory behind variational Monte Carlo (VMC) will be the subject of this Maple worksheet. Introduction Quantum mechanics is the study of particles at a very small scale. At this scale, particles can be characterized by wave functions and behave much differently than classical systems. By understanding how molecules behave at the quantum level, researchers can apply that knowledge to gain insights into how macroscopic systems work. While the applications are promising, quantum mechanics still struggles to understand the behavior of large atomic systems. For small, simple systems like particle in a box, harmonic oscillator, and the hydrogen atom, the Schrodinger equation: can be solved exactly, revealing the energy levels and exact wave functions for those systems.
    [Show full text]
  • Variational Monte Carlo Method for Fermions
    Variational Monte Carlo Method for Fermions Vijay B. Shenoy [email protected] Indian Institute of Science, Bangalore Mahabaleswar, December 2009 1/36 Acknowledgements Sandeep Pathak 2/36 Acknowledgements Sandeep Pathak Department of Science and Technology 2/36 Plan of the Lectures on Variational Monte Carlo Method Motivation, Overview and Key Ideas (Lecture 1, VBS) 3/36 Plan of the Lectures on Variational Monte Carlo Method Motivation, Overview and Key Ideas (Lecture 1, VBS) Details of VMC (Lecture 2, VBS) 3/36 Plan of the Lectures on Variational Monte Carlo Method Motivation, Overview and Key Ideas (Lecture 1, VBS) Details of VMC (Lecture 2, VBS) Application to Superconductivity (Lecture 3, VBS) 3/36 Plan of the Lectures on Variational Monte Carlo Method Motivation, Overview and Key Ideas (Lecture 1, VBS) Details of VMC (Lecture 2, VBS) Application to Superconductivity (Lecture 3, VBS) Application to Bosons (Lecture 4, AP) 3/36 Plan of the Lectures on Variational Monte Carlo Method Motivation, Overview and Key Ideas (Lecture 1, VBS) Details of VMC (Lecture 2, VBS) Application to Superconductivity (Lecture 3, VBS) Application to Bosons (Lecture 4, AP) Application to Spin Liquids (Lecture 5, AP) 3/36 The Strongly Correlated Panorama... Fig. 1. Phase diagrams ofrepresentativemateri- als of the strongly cor- related electron family (notations are standard and details can be found in the original refer- ences). (A) Temperature versus hole density phase diagram of bilayer man- ganites (74), including several types of antiferro- magnetic (AF) phases, a ferromagnetic (FM) phase, and even a glob- ally disordered region at x 0 0.75.
    [Show full text]
  • Arxiv:1807.10633V1 [Physics.Chem-Ph] 27 Jul 2018
    Improved speed and scaling in orbital space variational monte carlo Iliya Sabzevari1 and Sandeep Sharma1, ∗ 1Department of Chemistry and Biochemistry, University of Colorado Boulder, Boulder, CO 80302, USA In this work, we introduce three algorithmic improvements to reduce the cost and improve the scal- ing of orbital space variational Monte Carlo (VMC). First, we show that by appropriately screening the one- and two-electron integrals of the Hamiltonian one can improve the efficiency of the al- gorithm by several orders of magnitude. This improved efficiency comes with the added benefit that the resulting algorithm scales as the second power of the system size O(N 2), down from the fourth power O(N 4). Using numerical results, we demonstrate that the practical scaling obtained is in fact O(N 1:5) for a chain of Hydrogen atoms, and O(N 1:2) for the Hubbard model. Second, we introduce the use of the rejection-free continuous time Monte Carlo (CTMC) to sample the determinants. CTMC is usually prohibitively expensive because of the need to calculate a large number of intermediates. Here, we take advantage of the fact that these intermediates are already calculated during the evaluation of the local energy and consequently, just by storing them one can use the CTCM algorithm with virtually no overhead. Third, we show that by using the adaptive stochastic gradient descent algorithm called AMSGrad one can optimize the wavefunction energies robustly and efficiently. The combination of these three improvements allows us to calculate the ground state energy of a chain of 160 hydrogen atoms using a wavefunction containing ∼ 2 × 105 variational parameters with an accuracy of 1 mEh/particle at a cost of just 25 CPU hours, which when split over 2 nodes of 24 processors each amounts to only about half hour of wall time.
    [Show full text]