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.