Stochastic density functional theory at finite temperatures

Yael Cytter, Eran Rabani, Daniel Neuhauser, and Roi Baer

Phys. Rev. B 97, 115207 – Published 27 March 2018

Abstract

Simulations in the warm dense matter regime using finite temperature Kohn-Sham density functional theory (FT-KS-DFT), while frequently used, are computationally expensive due to the partial occupation of a very large number of high-energy KS eigenstates which are obtained from subspace diagonalization. We have developed a stochastic method for applying FT-KS-DFT, that overcomes the bottleneck of calculating the occupied KS orbitals by directly obtaining the density from the KS Hamiltonian. The proposed algorithm scales as $O (N T^{- 1})$ and is compared with the high-temperature limit scaling $O (N^{3} T^{3})$ of the deterministic approach, where $N$ is the system size (number of electrons, volume, etc.) and $T$ is the temperature. The method has been implemented in a plane-waves code within the local density approximation (LDA); we demonstrate its efficiency, statistical errors, and bias in the estimation of the free energy per electron for a diamond structure silicon. The bias is small compared to the fluctuations and is independent of system size. In addition to calculating the free energy itself, one can also use the method to calculate its derivatives and obtain the equations of state.

Received 7 January 2018
Revised 14 February 2018

DOI:https://doi.org/10.1103/PhysRevB.97.115207

Physics Subject Headings (PhySH)

Warm-dense matter

Density functional theory First-principles calculations Monte Carlo methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yael Cytter

Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

Eran Rabani^*

Department of Chemistry, University of California and Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA and The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv, Israel 69978

Daniel Neuhauser^†

Department of Chemistry, University of California at Los Angeles, California 90095, USA

Roi Baer^‡

Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

^*eran.rabani@berkeley.edu
^†dxn@chem.ucla.edu
^‡roi.baer@huji.ac.il

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Issue

Vol. 97, Iss. 11 — 15 March 2018

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Images

Figure 1
CPU wall time for self-consistent KS-DFT calculations using the stochastic (sDFT, $I = 80$ stochastic orbitals) and deterministic (single thread dDFT, Quantum Espresso [58]) calculations on ${Si}_{64}$ having a lattice constant of $21 a_{0}$ . The dashed line is the expected $O (T^{3})$ extrapolation for the dDFT timings.
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Figure 2
Left panels: We show in the top left panel the estimated expected value of the Helmholtz free energy per electron (dots) and its square-root-variance $σ$ (half length of error bars $\pm σ$ which is also shown in the log plot of the bottom panel) as a function of the number of electrons $N_{e}$ in four Si supercell sizes (see text) using $I = 80$ stochastic orbitals. The dashed green line is a guide to the eye designating the free energy per electron of the largest system. Middle panels: In the top middle panel we show the estimated expected value of the Helmholtz free energy per electron (dots) and its square-root-variance $σ$ (which is also shown in the log plot of the bottom panel) as a function of the number of stochastic orbitals $I$ for ${Si}_{64}$ . The dashed green line designates the deterministic value of the free energy per electron for this system. Right panel: The 70% confidence intervals of the estimated Helmholtz free-energy per electron, for several system sizes, as a function of the inverse number of stochastic orbitals $I^{- 1}$ the solid lines are best-fit linear curves for the data (their equations are given in the legend).
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Figure 3
The estimated expected value of the Helmholtz free energy per electron $〈A〉 / N_{e}$ (dots) and its square-root-variance $σ$ (half length of error bars $\pm σ$ ) for ${Si}_{64}$ as a function of inverse temperature $β$ , representing a single run of $I = 80$ stochastic orbitals. The expected value and standard deviation were estimated from six independent such runs. The square symbols are the corresponding deterministic values of the free energy.
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Figure 4
Top panel: The calculated values of the Helmholtz free energy $A$ for the ${Si}_{216}$ system (with $N_{e} = 864$ electrons) at $β = 20 E_{h}^{- 1}$ , at several discrete values of the density $ρ = N_{e} / V$ (shown as points on plot) calculated for six independent seeds, i.e., each using six sets of $I = 80$ stochastic orbitals. The smooth lines express the free energy $A (β, N_{e}, ρ)$ as a function of $ρ$ using cubic polynomials which best fit the points. Middle and bottom panels: the pressure and bulk modulus isotherms derived from the six free energy curves of the top panel.
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Figure 5
Top two panels: Isobars of the density (top panel) and bulk modulus (middle panel) in the ${Si}_{216}$ system, as a function of temperature under different pressures. The stochastic calculations were done using $I = 80$ stochastic orbitals and the data was discerned from the free energy calculations discussed in the text. The darker squares are the results of a stochastic calculation for the ${Si}_{64}$ system. Bottom panel: The heat capacity for several values of the density as a function of temperature. The error bars are the errors per one seed and are the size of the markers. The lines are the polynomial fits for a specific seed.
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