DOCSLIB.ORG

Coupled Electron Ion Monte Carlo Calculations of Atomic Hydrogen Markus Holzmann, Carlo Pierleoni, David Ceperley

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Coupled Electron Ion Monte Carlo Calculations of Atomic Hydrogen Markus Holzmann, Carlo Pierleoni, David Ceperley Coupled Electron Ion Monte Carlo Calculations of Atomic Hydrogen Markus Holzmann, Carlo Pierleoni, David Ceperley To cite this version: Markus Holzmann, Carlo Pierleoni, David Ceperley. Coupled Electron Ion Monte Carlo Calculations of Atomic Hydrogen. 2004. hal-00003115 HAL Id: hal-00003115 https://hal.archives-ouvertes.fr/hal-00003115 Preprint submitted on 21 Oct 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Coupled Electron Ion Monte Carlo Calculationsof Atomic Hydrogen Markus Holzmann a,∗ Carlo Pierleoni b David M. Ceperley c aLPTL, UMR 7600 of CNRS, Universit´eP. et M. Curie, Paris, France bINFM and Department of Physics, University of L’Aquila, Via Vetoio, I-67010 L’Aquila, Italy cPhysics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Abstract We present a new Monte Carlo method which couples Path Integral for finite temperature protons with Quantum Monte Carlo for ground state electrons, and we apply it to metallic hydrogen for pressures beyond molecular dissociation. This method fills the gap between high temperature electron-proton Path Integral and ground state Diffusion Monte Carlo methods. Our data exhibit more structure and higher melting temperatures of the proton crystal than Car-Parrinello Molecular Dynamics results using LDA. We further discuss the quantum motion of the protons and the zero temperature limit. Key words: Quantum Monte Carlo, metallic hydrogen PACS: 05.30.Lq, 71.10.+x, 64.30.+t, 02.70.Lq 1. Introduction as liquid hydrogen within QMC. For this system, VMC/DMC and PIMC are computationally too inef- Quantum Monte Carlo (QMC) methods have been ficient to provide definite answers, e.g. regarding the developed for accurately solving the many-body nature of the melting transition from liquid to solid or Schr¨odinger equation. Zero temperature Variational metal to insulator. Whereas PIMC calculations have Monte Carlo (VMC) and Diffusion Monte Carlo been done at comparatively high temperatures [6], (DMC), and finite temperature Path Integral Monte this method becomes computationally inefficient at Carlo (PIMC) are currently the most accurate and temperatures lower than roughly 1/20 of the Fermi general methods for computing static properties of a temperature. Zero temperature calculations (VMC, quantum system [1,2]. They have been successfully DMC) have been used for ground state calculations applied to simple quantum many-body systems, such where both electronic and protonic degrees of free- as the electron gas, hydrogen, and helium. dom are treated quantum mechanically [7,8]. How- Recently, there have been new attempts[3,4,5] ever, the convergence of these calculations suffers to calculate properties of disordered systems such from the different masses of protons and electrons which introduce two time scales differing by three ∗ Corresponding Author: orders of magnitude, and, more important, low tem- Email address: [email protected] (Markus perature properties are inaccessible by these ground ccsd-00003115, version 1 - 21 Oct 2004 Holzmann). Preprint submitted to Elsevier Science 21 October 2004 state methods. To fill this gap, the Coupled Electron- moves (between 0.01Aand˚ 0.5Afor˚ classical protons) Ion Monte Carlo (CEIMC) has been developed [3,4] this method is much more efficient than performing to combine a classical or quantum Monte Carlo sim- two independent electronic calculations [3,4]. ulation of the nuclei at non-zero temperature with a Anessential partofthe CEIMC methodis the choice QMC calculation for the electronic energies where the of the trial wavefunction needed to calculate the Born- Born-Oppenheimer approximation helps to overcome Oppenheimer energies. Variational Monte Carlo de- the time scale problem. pends crucially on the quality of the trial wavefunc- In Ref. [5], the CEIMC method has been applied tion. To go beyond VMC, we implemented a Reptation to determine the equation of state of hydrogen for Quantum Monte Carlo algorithm (RQMC)[12] to sam- temperatures across the melting of the proton crystal. ple more accurately the electronic ground state. Simi- More structure and higher melting temperatures of the lar to DMC, RQMC projects the trial wavefunction on proton crystal compared to Car-Parrinello Molecular to the ground state within the Fixed-Node approxima- Dynamics (CPMD) results using LDA[9] have been tion. A high quality trial wave functions is important found. In this paper, we shortly summarize the method to relax to the ground state with a very limited number and the results as reported in Ref. [5] and discuss in of time slices and to provide accurate nodes. RQMC, more detail the quantum effects of the protons [10,11]. being a Metropolis based method, is more easily used to compute energy differences; conversely, the corre- lated sampling method within DMC is more involved because of the branching step. 2. Method To reduce finite size effects in metallic systems, we average over twisted boundary conditions (TABC) In the CEIMC method, the proton degrees of free- when computing electronic energies within CEIMC dom are advanced by a Metropolis algorithm in which (i.e. we integrate over the Brillouin zone of the super the energy difference between the actual state S and cell)[15,4]. the trial state S′ is computed by a Quantum Monte Quantum effects for protons are relevant at high Carlo calculation The energies of the states are cal- pressure. We represent protons by imaginary time path culated within the Born-Oppenheimer approximation, integrals without considering the statistics of the pro- where the electrons are assumed to remain in the tons. (those effects are negligible in this temperature- ground state with respect to the actual protonic po- density range.) For efficiency, it is important to mini- sitions. Since the Born-Oppenheimer energies E(S) mize the number of protonic time slices. We have used and E(S′) have to be sampled by a QMC calculation, the pair action of an effective proton-proton poten- they are affected by statistical noise which would bias tial and treated the difference between the true Born- the Monte Carlo sampling of the protons. At first sight Oppenheimer energy and the effective potential with one might expect that for an unbiased calculation one the primitive approximation[2]. When coupled with will need to reduce the accuracy of the energy differ- TABC, rather than using all the k-points for each pro- ′ ence E(S) − E(S ) much below kBT . However, it tonic time slice, we can, randomly assign a subset of has been shown that unbiased sampling of the proton k-points to each protonic slice without introducing a configurations can be efficiently achieved by using the detectable systematic effect. penalty method[13], a generalization of the Metropo- lis algorithm, where detailed balance is satisfied on average. Since only differences of electronic energies are 3. Results needed, we sample the electronic degrees of free- dom according to the sum of the electronic distribu- tion functions (e. g. the square of the trial wave func- tion in VMC) for the S and S′ states, and we com- Comparison of CEIMC with CPMD We first con- pute the energies for the two states as correlated sam- sider classical protons. For classical protons it is pos- pling averages[3,4]. For the typical size of the proton sible to compare the CEIMC results with previous 2 3.5 4 N =32 CEIMC-TABC 3 p Γ Np=54 CEIMC- point N =162 CPMD-Γ point 2.5 p 3 Np=32 CEIMC-TABC T=2000K 2 (r) melting pp 2 g 1.5 (hartree/atom) class. 2 rs=0.8 1 x10 p rs=1.0 K 1 rs=1.2 0.5 0 1 2 3 4 0 r (a.u) 0 2 4 6 3 T/10 (K) Fig. 1. Pair correlation function at rs = 1, T = 1000K. Comparison between CEIMC-VMC-TABC with 32 protons, Fig. 2. CEIMC-VMC-TABC with 54 protons. Protonic kinetic CEIMC-VMC-PBC with 54 protons and CPMD-LDA with 162 energy per particle at various densities versus temperature. The red protons (simulation with Np = 54 provides identical correlation). line estimates the melting of the bcc crystal from the Lindemann Data from CEIMC-VMC-TABC at T=2000K (stars) are also re- ratios. ported. on VMC calculation and RQMC is only exploited to Car-Parrinello molecular dynamics (CPMD) [9], the estimate the systematic error of VMC. For the sys- only difference is the method to calculate the poten- tem with 54 protons at the Γ point, we have found tial energy surface. Whereas in CEIMC the Born- no detectable differences in the correlation function Oppenheimer energies are calculated by QMC meth- between VMC and RQMC. ods, CPMD uses density functional theory (DFT) to calculate electronicenergies.Both methodsare in prin- ciple exact but rely on approximations of the unknown nodes of the trial wavefunctions in QMC and on the The Quantum effects of the protons The quantum approximation of the unknown exchange-correlation effects of the protons are summarized in Fig. 2 which energy functional in DFT. In Fig. 1 we compare the shows the kinetic energy of the protons versus temper- proton correlation function gpp(r) of both methods. ature for three different densities (rs = 1.2, 1.0, 0.8) The CEIMC results show more structure than CPMD and the deviation from its classical value 3kBT/2. of Ref.
Load more

Recommended publications
  • Annual-Report-BW Ceperley
    Annual Report for Blue Waters Allocation January 2016 Project Information o Title: Quantum Simulations o PI: David Ceperley (Blue Waters professor), Department of Physics, University of Illinois Urbana-Champaign o Norm Tubman University of Illinois Urbana-Champaign (former postdoc), Carlo Pierleoni (Rome, Italy), Markus Holzmann(Grenoble, France) o Corresponding author: David Ceperley, [email protected] Executive summary (150 words) Much of our research on Blue Waters is related to the “Materials Genome Initiative,” the federally supported cross-agency program to develop computational tools to design materials. We employ Quantum Monte Carlo calculations that provide nearly exact information on quantum many-body systems and are also able to use Blue Waters effectively. This is the most accurate general method capable of treating electron correlation, thus it needs to be in the kernel of any materials design initiative. Ceperley’s group has a number of funded and proposed projects to use Blue Waters as discussed below. In the past year, we have been running calculations for dense hydrogen in order to make predictions that can be tested experimentally. We have also been testing a new method that can be used to solve the fermion sign problem and to find dynamical properties of quantum systems. Description of research activities and results During the past year, the following 4 grants of which Ceperley is a PI or CoPI, and that involve Blue Waters usage, have had their funding renewed. Access to Blue Waters is crucial for success of these projects. • “Warm dense matter”DE-NA0001789. Computation of properties of hydrogen and helium under extreme conditions of temperature and pressure.
    [Show full text]
  • The Early Years of Quantum Monte Carlo (1): the Ground State
    The Early Years of Quantum Monte Carlo (1): the Ground State Michel Mareschal1,2, Physics Department, ULB , Bruxelles, Belgium Introduction In this article we shall relate the history of the implementation of the quantum many-body problem on computers, and, more precisely, the usage of random numbers to that effect, known as the Quantum Monte Carlo method. The probabilistic nature of quantum mechanics should have made it very natural to rely on the usage of (pseudo-) random numbers to solve problems in quantum mechanics, whenever an analytical solution is out of range. And indeed, very rapidly after the appearance of electronic machines in the late forties, several suggestions were made by the leading scientists of the time , - like Fermi, Von Neumann, Ulam, Feynman,..etc- which would reduce the solution of the Schrödinger equation to a stochastic or statistical problem which , in turn, could be amenable to a direct modelling on a computer. More than 70 years have now passed and it has been witnessed that, despite an enormous increase of the computing power available, quantum Monte Carlo has needed a long time and much technical progresses to succeed while numerical quantum dynamics mostly remains out of range at the present time. Using traditional methods for the implementation of quantum mechanics on computers has often proven inefficient, so that new algorithms needed to be developed. This is very much in contrast with what happened for classical systems. At the end of the fifties, the two main methods of classical molecular simulation, Monte Carlo and Molecular Dynamics, had been invented and an impressively rapid development was going to take place in the following years: this has been described in previous works [Mareschal,2018] [Battimelli,2018].
    [Show full text]
  • The Early Years of Quantum Monte Carlo (2): Finite- Temperature Simulations
    The Early Years of Quantum Monte Carlo (2): Finite- Temperature Simulations Michel Mareschal1,2, Physics Department, ULB , Bruxelles, Belgium Introduction This is the second part of the historical survey of the development of the use of random numbers to solve quantum mechanical problems in computational physics and chemistry. In the first part, noted as QMC1, we reported on the development of various methods which allowed to project trial wave functions onto the ground state of a many-body system. The generic name for all those techniques is Projection Monte Carlo: they are also known by more explicit names like Variational Monte Carlo (VMC), Green Function Monte Carlo (GFMC) or Diffusion Monte Carlo (DMC). Their different names obviously refer to the peculiar technique used but they share the common feature of solving a quantum many-body problem at zero temperature, i.e. computing the ground state wave function. In this second part, we will report on the technique known as Path Integral Monte Carlo (PIMC) and which extends Quantum Monte Carlo to problems with finite (i.e. non-zero) temperature. The technique is based on Feynman’s theory of liquid Helium and in particular his qualitative explanation of superfluidity [Feynman,1953]. To elaborate his theory, Feynman used a formalism he had himself developed a few years earlier: the space-time approach to quantum mechanics [Feynman,1948] was based on his thesis work made at Princeton under John Wheeler before joining the Manhattan’s project. Feynman’s Path Integral formalism can be cast into a form well adapted for computer simulation. In particular it transforms propagators in imaginary time into Boltzmann weight factors, allowing the use of Monte Carlo sampling.
    [Show full text]
  • Path Integrals Calculations of Helium Droplets
    Imaginary Time Path Integral Calculations of Supersolid Helium David Ceperley Dept. of Physics and NCSA University of Illinois Urbana-Champaign, Urbana, IL, 61801, USA In 1953 Feynman, introduced imaginary-time path integrals to understand superfluid 4He. Path integrals are an exact "isomorphism" between quantum systems and the classical statistical mechanics of ring "polymers" allowing one to understand Bose condensation from the point of view of classical statistical mechanics and to calculate many of its properties. Bose symmetry of the wave function implies that the polymers are allowed to “cross-link'' or exchange. Superfluidity (coupling to the boundaries) is proportional to the mean squared flux of polymers through a surface. Bose condensation is equivalent to the delocalization of the end-end distance of a cut polymer. We have developed specialized simulation methods (Path Integral Monte Carlo) based on the Metropolis Monte Carlo method, to simulate boson systems[1]. Andreev, Lifshitz, Chester, and Leggett suggested in about 1970 that a quantum crystal such as bulk helium-4 under pressure might show both crystallinity and superfluid behavior. Experiments by Kim and Chan within the last two years have found indications for such a supersolid phase. The theoretical explanation of superflow in a crystal assumed vacancies, however, they have not been seen experimentally and computer simulations do not find stable vacancies. The path integral picture allows one to address the question[2] of whether a highly fluctuating quantum crystal is “insulating” or “metallic”. We and others have done Path Integral Monte Carlo calculations of exchange frequencies[3], superfluid density and the condensate fraction[4].
    [Show full text]
  • Variational Density Matrix Method for Warm Condensed Matter And
    Variational Density Matrix Method for Warm Condensed Matter and Application to Dense Hydrogen Burkhard Militzera) and E. L. Pollockb) a)Department of Physics University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 b)Physics Department, Lawrence Livermore National Laboratory, University of California, Livermore, California 94550 (August 8, 2018) A new variational principle for optimizing thermal density matrices is intro- duced. As a first application, the variational many body density matrix is written as a determinant of one body density matrices, which are approximated by Gaus- sians with the mean, width and amplitude as variational parameters. The method is illustrated for the particle in an external field problem, the hydrogen molecule and dense hydrogen where the molecular, the dissociated and the plasma regime are described. Structural and thermodynamic properties (energy, equation of state and shock Hugoniot) are presented. I. INTRODUCTION Considerable effort has been devoted to systems where finite temperature ions (treated either classically or quantum mechanically by path integral methods) are coupled to degenerate electrons on the Born-Oppenheimer surface. In contrast, the theory for similar systems with non-degenerate electrons (T a significant fraction of TF ermi) is relatively underdeveloped except at the extreme high T limit where Thomas-Fermi and similar theories apply. In this paper we present a computational approach for systems with non-degenerate electrons analogous to the methods used for ground state many body computations. Although an oversimplification, we may usefully view the ground state computations as consisting of three levels of increasing accuracy [1]. At the first level, the ground state wave function consists of determinants, for both spin species, of single particle orbitals often taken from local density functional theory Φ1(r1) ..
    [Show full text]
  • 2014 Blue Waters Update
    2014 Blue Waters Update Bill Kramer Blue Waters Director Announcements • Today, Ed Seidel has invited the PIs to lunch in the Alma Mater Room. • The PI's have blue tickets in the back of their badges. • We will take a group photo of all attendees at the first break. • #BWsymp2014 for another chance BW Symposium - May 2014 2 Joint Dinner at Memorial Stadium – Tonight Joint with the Private Sector Program Workshop Attendees BW Symposium - May 2014 3 SETAC NSF PRAC • Paul Woodward, Physics and Astrophysics, University of Minnesota • Tom Cheatham, Chemistry, University of Utah • Patrick Reed, Civil and Environmental Engineering – Systems Optimization, Cornell • Klaus Schulten, Physics and Molecular Dynamic, University of Illinois Urbana-Champaign • David Ceperley, Physics and Material Science, University of Illinois Urbana-Champaign • Tiziana Di Matteo, Physics and Cosmology, Carnegie Mellon University • Dave Randall, Atmospheric Sciences and Climate Colorado State University GLCPC Chair • Joe Paris, Academic & Research Technologies in Information Technology, Northwestern University (Chair for 2013/2014, followed by Jorge Vinals, Structural Mechanics and Biophysics, University of Minnesota, Chair for 2014/2015) University of Illinois at Urbana-Champaign Allocation Chair • Athol Kemball, Atmospheric Sciences, University of Illinois at Urbana-Champaign Industry • Rick Authur, General Electric Global Research, Computer and Software Engineering BW Symposium - May 2014 4 Blue Waters Fellows • 6 Awards (so far) • Substantial Stipend + Blue Waters allocations • 10 other very deserving nominees are being offered Blue Waters allocations • Kenza Arraki, New Mexico State University • Jon Calhoun, University of Illinois at Urbana-Champaign • Sara Kokkila, Stanford University, • Edwin Mathews, University of Notre Dame • Ariana Minot, Harvard University • Derek Vigil-Fowler, University of California, Berkeley BW Symposium - May 2014 5 Blue Waters Usage 2/11/14 – Largest 10 Jobs-Torus View Each dot is a Gemini router and represents 64 AMD integer cores.
    [Show full text]
  • The Phases of Warm-Dense Hydrogen and Helium As Seen by Quantum Monte Carlo M
    The Phases of Warm-Dense Hydrogen and Helium as seen by Quantum Monte Carlo M. Morales, DMC: University of Illinois C. Pier leon i: L’Aqu ila, ITALY Eric Schwegler: Livermore • The coupled electron-ion Monte Carlo method • Hydrogen and Helium at High Pressure Supported by DOE DE-FG52-06NA26170 Computer time from NCSA and ORNL (INCITE grant) David Ceperley, UIUC March 4, 2010 KITP Materials Design • Giant Planets – Primary components are H and He –P(ρ,T,xi) closes set of hydrostatic equations – Interior models depend very sensitively on EOS and phase diagram • Saturn’s Luminosity – Homogeneous evolutionary models do not work for Saturn – Additional energy source in planet’s interior is needed – Does it come from Helium Taken from: Fortney J. J., Science 305, 1414 (2004). segregation (rain)? 2 EOS does matter how big is Jit’Jupiter’s core? Planet modeling needs P(ρ,TxT,x) accurate to 1%! Also entropy, compressibility,…. 3 QMC methods for Dense Hydrogen Path Integral MC for T > EF/10 Coupled-electron Ion MC Path Integral MC with an effective potential Diffusion MC T=0 4 MD and MC Simulations –Hard sphere MD/MC ~1953 (Metropolis, Alder) –Empirical potentials (e.g. Lennard-Jones) ~1960 (Verlet, Rahman) –Local density functional theory ~1985 (Car-Parrinello) –Quantum Monte Carlo (CEIMC) ~2000 • Initial simulations used semi-empirical potentials. • Much progress with “ab initio” molecular dynamics simulations where the effects of electrons are solved for each step. • However, the potential surface as determined by density functi ona l theory is not always accurate enough • QMC+MD =CEIMC/MD is a candidate for petascale computing 5 QuantumMonte Carlo • Premise: we need to use simulation techniques to “solve” many-body quantum problems just as you need them classically.
    [Show full text]
  • Lo^ Wc-Rangf Forces Computer Simulation
    ' ^, .*£*' % LO^WC-RANG F FORCES COMPUTER SIMULATION CONDENSED MEDIA i.i.r'.bl-'.v.j (..%-•[-! i..T i' • i '.i rk . C .\jiti irtiM \'. i.:. r ••. c- i i. i ° t-' 0 Hi r* PROCEEDINGS OF THE WORKSHOP THE PROBLEM OF LONG-RANGE FORCES IN THE COMPUTER SIMULATION OF CONDENSED MEDIA Sponsored by the National Resource for Computation in Chemistry Lawrence Berkeley Laboratory Berkeley, California 94720 Held at Vallambrosa Center Menlo Park, California January 8-11, 1980 NRCC Proceedings No. 9 Edited by: David Ceperely -m- CONTENTS Workshop Participants vii Foreword xi Introduction xii SESSION I. CONTROLLED STUDIES OF LONG-RANGE FORCE PROBLEM IN SIMULATION OF IONIC SYSTEMS Review Talk--The Problem of Coulombic Forces in Computer Simulation J. P. Valleau 3 Perturbation Theory for Hard Spheres B. Larsen and S. A. Rogde 9 Periodic Boundary Conditions in Simulation of Ionic Solids E. R. Smith 10 An Alternative to the Ewald Summation? D. J. Adams 11 Periodic, Truncated-Octahedral Boundary Conditions D. J. Adams 13 Summary of Session I B. Larsen 14 SESSION II. CONTROLLED STUDIES OF LONG-RANGE FORCE PROBLEM IN SIMULATION OF DIPOLAR SYSTEMS Review Talk—Dipolar Fluids G. N. Patey 19 Calculation of (k, ) by Computer Simulations of Permanent Dipolar Systems E. L. Pollock 23 Periodic Boundary Conditions for Dipolar Systems E. R. Smith 24 Computer Simulations of Hydrogen-bonded Liquids W. L. Jorgensen 25 -Tv- Dielectric Theory for Polar Molecules with Fluctuating Polarizability G. Stell 27 Summary of Session II J. J. Weis 28 SESSION III. EFFECT OF LONG-RANGE FORCES ON THE SIMULATION OF SOLVATION AND ON BIOPOLYMER HYDRATION: COULOMB AND HYDRODYNAMIC FORCES Review Talk P.
    [Show full text]
  • Novel Quantum Monte Carlo Approaches for Quantum Liquids
    Novel Quantum Monte Carlo Approaches for Quantum Liquids Brenda M. Rubenstein Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2013 c 2013 Brenda M. Rubenstein All Rights Reserved ABSTRACT Novel Quantum Monte Carlo Approaches for Quantum Liquids Brenda M. Rubenstein Quantum Monte Carlo methods are a powerful suite of techniques for solving the quantum many-body problem. By using random numbers to stochastically sample quantum proper- ties, QMC methods are capable of studying low-temperature quantum systems well beyond the reach of conventional deterministic techniques. QMC techniques have likewise been indispensible tools for augmenting our current knowledge of superfluidity and superconduc- tivity. In this thesis, I present two new quantum Monte Carlo techniques, the Monte Carlo Power Method and Bose-Fermi Auxiliary-Field Quantum Monte Carlo, and apply previ- ously developed Path Integral Monte Carlo methods to explore two new phases of quantum hard spheres and hydrogen. I lay the foundation for a subsequent description of my re- search by first reviewing the physics of quantum liquids in Chapter 2 and the mathematics behind Quantum Monte Carlo algorithms in Chapter 3. I then discuss the Monte Carlo Power Method, a stochastic way of computing the first several extremal eigenvalues of a matrix too memory-intensive to be stored and therefore diagonalized. As an illustration of the technique, I demonstrate how it can be used to determine the second eigenvalues of the transition matrices of several popular Monte Carlo algorithms. This information may be used to quantify how rapidly a Monte Carlo algorithm is converging to the equilibrium probability distribution it is sampling.
    [Show full text]
  • Division of Computational Physics
    DIVISION OF COMPUTATIONAL PHYSICS LETTER FROM THE PAST CHAIR Dear Colleagues, The Newsletter provides the traditional vehicle for Outgoing Chair to comment on the previous year of activities for the Division of Computational Physics. Several important events and initiatives come to mind. First, we began the year (March 2005–March 2006) having just completed a highly successful joint DCOMP-CCP international conference in conjunction with the regular March Meeting, which included an all-day plenary session on Sunday combined with specially-tailored invited symposia and focus sessions. The DCOMP Executive Committee at its Baltimore meeting agreed to organize another such DCOMP-CCP collaboration for the 2008 March Meeting, scheduled for New Orleans. Although this date appears distant, planning will soon commence so that comments on the 2005 conference and suggestions for 2008 should be sent to the new Officers. Second, DCOMP plans an educational initiative for this year to coincide with similar efforts across APS. Some ideas floated this past year, all associated in some degree with the March Meeting, included a special session along the lines of the DCOMP-CCP plenary session but aimed at students, an interactive session on Summer 2006 computational methods in the vein of an expanded tutorial, and IN THIS ISSUE: a “conference experience for students” with some initial back- I) LETTER FROM THE ground sessions that would guide the students through a selected PAST CHAIR set of invited and focus sessions. Upper-level undergraduates and II) OFFICERS AND beginning graduate students might form the main target group. COMMITTEES More information will come in future broadcasts, but the floor III) FELLOWS/PRIZES has opened for comments and suggestions ([email protected]).
    [Show full text]
  • Challenges in Path Integral Monte Carlo David Ceperley
    Challenges in Path Integral Monte Carlo David Ceperley Physics UIUC • Imaginary time path integrals • Exchange in quantum crystals • The fermion “sign problem” • Restricted paths • The Penalty method • Quantum dynamics? Imaginary Time Path Integrals PIMC Simulations • We do Classical Monte Carlo simulations to evaluate averages such as: 1 < V >= dRV (R)e−βV ( R) Z ∫ β = 1/ (k T ) where R = r ,r ,... B { 1 2 } • Quantum mechanically for T>0, we need both to generate the distribution and do the average: 1 <>=V dRV() Rρβ( R ; ) Z ∫ ρβ(R;) = diagonal density matrix • Simulation is possible since the density matrix is positive. The thermal density matrix ˆ • Find exact many-body HEφφααα= eigenstates of H. 2 −β E • Probability of (;RRe ) ()α β = 1/kT ρβ= ∑ φα occupying state ϕ is α exp(-βE ) α -β Hˆ ρˆ β = e operator notation • All equilibrium properties can be off-diagonal density matrix: calculated in terms of * E (,RR ';) (')() R Re−β α thermal o-d density ρβφφ= ∑ αα matrix α • Convolution theorem ρβ(RR , '; )≥ 0 (without statistics) relates high temperature to lower ρββ(,RR121 ;+= 2 ) temperature. = dR'ρβρ ( R , R '; ) (RR', ;β ) ∫ 11 22 ˆˆˆ or with operators: eee-(ββ12+ )HHH= - β 1 - β 2 Trotter’s formula (1959) • We can use the effects of operators ρˆ = e−β (T +V ) separately as long as we take small n enough time steps. ρˆ = lim ⎡e−τ T e−τ V ⎤ n→∞ ⎣ ⎦ • n is number of time slices. τ = β / n • τ is the “time-step” • We now have to evaluate the density matrix for potential and kinetic matrices by themselves: −3/2 2 −τ T −(r−r ') /4λτ • Do by FT’s r e r ' = (4πλτ ) e • V is “diagonal” r e−τ V r ' = δ (r − r ')e−τV (r ) • Error at finite n comes from commutator τ 2 − ⎡⎤TVˆˆ, e 2 ⎣⎦ Using this for the density matrix.
    [Show full text]
  • Annual Report for Blue Waters Professor Project January 2018
    Annual Report for Blue Waters Professor Project January 2018 Project Information o Title: Quantum Simulations o PI: David Ceperley (Blue Waters Professor), Department of Physics, University of Illinois Urbana-Champaign o Collaborators: Carlo Pierleoni (Rome, Italy), Markus Holzmann (Grenoble, France) o Corresponding author: David Ceperley, [email protected] Executive summary (150 words) Much of our research on Blue Waters is related to the “Materials Genome Initiative,” the federally supported cross-agency program to develop computational tools to design materials. We employ Quantum Monte Carlo calculations that provide nearly exact information on quantum many-body systems. This is the most accurate general method capable of treating electron correlation, thus it needs to be in the kernel of any materials design initiative. It is able to use Blue Waters effectively because there are several pathways to find parallel performance. Ceperley’s group has projects to use Blue Waters as listed below. In the past year, we have been running calculations for dense hydrogen in order to make predictions that can be tested experimentally. We have also been testing a new method that can be used to solve the fermion sign problem and to find dynamical properties of quantum systems. Description of research activities and results • Key Challenges: description of the science/engineering problem being addressed • Why it Matters: description of the potential impact of solving this research problem, and, if appropriate, the educational outcomes. For exploratory allocations, indicate whether the results will be used to substantiate a future proposal. • Why Blue Waters: explanation of why you need the unique scale and attributes of Blue Waters to address these challenges; if relevant, provide an assessment of code(s) performance on Blue Waters.
    [Show full text]