Extended Lagrangian Born–Oppenheimer molecular dynamics using a Krylov subspace approximation

Niklasson, Anders Mauritz N.

doi:10.1063/1.5143270

Title: Extended Lagrangian Born–Oppenheimer molecular dynamics using a Krylov subspace approximation

Journal Article · 11 March 2020 · Journal of Chemical Physics

DOI: https://doi.org/10.1063/1.5143270 · OSTI ID:1630868

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Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Uppsala Univ. (Sweden)

It is shown how the electronic equations of motion in extended Lagrangian Born–Oppenheimer molecular dynamics simulations can be integrated using low-rank approximations of the inverse Jacobian kernel. This kernel determines the metric tensor in the harmonic oscillator extension of the Lagrangian that drives the evolution of the electronic degrees of freedom. The proposed kernel approximation is derived from a pseudoinverse of a low-rank estimate of the Jacobian, which is expressed in terms of a generalized set of directional derivatives with directions that are given from a Krylov subspace approximation. The approach allows a tunable and adaptive approximation that can take advantage of efficient preconditioning techniques. The proposed kernel approximation for the integration of the electronic equations of motion makes it possible to apply extended Lagrangian first-principles molecular dynamics simulations to a broader range of problems, including reactive chemical systems with numerically sensitive and unsteady charge solutions. This can be achieved without requiring exact full calculations of the inverse Jacobian kernel in each time step or relying on iterative non-linear self-consistent field optimization of the electronic ground state prior to the force evaluations as in regular direct Born–Oppenheimer molecular dynamics. We note the low-rank approximation of the Jacobian is directly related to Broyden’s class of quasi-Newton algorithms and Jacobian-free Newton–Krylov methods and provides a complementary formulation for the solution of nonlinear systems of equations.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1630868

Report Number(s):: LA-UR--19-31455

Journal Information:: Journal of Chemical Physics, Journal Name: Journal of Chemical Physics Journal Issue: 10 Vol. 152; ISSN 0021-9606

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

References (82)

A Self-Consistent Charge Density-Functional Based Tight-Binding Method for Predictive Materials Simulations in Physics, Chemistry and Biology Frauenheim, Th.; Seifert, G.; Elsterner, M. physica status solidi (b), Vol. 217, Issue 1 https://doi.org/10.1002/(sici)1521-3951(200001)217:1<41::aid-pssb41>3.0.co;2-v	journal	January 2000
QM/MM Methods for Biomolecular Systems Senn, Hans Martin; Thiel, Walter Angewandte Chemie International Edition, Vol. 48, Issue 7 https://doi.org/10.1002/anie.200802019	journal	January 2009
Time-reversible always stable predictor-corrector method for molecular dynamics of polarizable molecules Kolafa, Ji?� Journal of Computational Chemistry, Vol. 25, Issue 3 https://doi.org/10.1002/jcc.10385	journal	January 2003
Car-Parrinello molecular dynamics: Car-Parrinello molecular dynamics Hutter, Jürg Wiley Interdisciplinary Reviews: Computational Molecular Science, Vol. 2, Issue 4 https://doi.org/10.1002/wcms.90	journal	September 2011
Real-World Predictions from Ab Initio Molecular Dynamics Simulations Kirchner, Barbara; di Dio, Philipp J.; Hutter, Jürg Multiscale Molecular Methods in Applied Chemistry https://doi.org/10.1007/128_2011_195	book	January 2011
Density Functional Theory Dreizler, Reiner M.; Gross, Eberhard K. U. Springer Berlin, Heidelberg https://doi.org/10.1007/978-3-642-86105-5	book	January 1990
A mathematical investigation of the Car-Parrinello method Bornemann, Folkmar A.; Schütte, Christof Numerische Mathematik, Vol. 78, Issue 3 https://doi.org/10.1007/s002110050316	journal	January 1998
Convergence acceleration of iterative sequences. the case of scf iteration Pulay, Péter Chemical Physics Letters, Vol. 73, Issue 2 https://doi.org/10.1016/0009-2614(80)80396-4	journal	July 1980
Spin eigenstate-dependent Hartree—Fock molecular dynamics Hartke, Bernd; Carter, Emily A. Chemical Physics Letters, Vol. 189, Issue 4-5 https://doi.org/10.1016/0009-2614(92)85215-v	journal	February 1992
An extended-Lagrangian scheme for charge equilibration in reactive molecular dynamics simulations Nomura, Ken-ichi; Small, Patrick E.; Kalia, Rajiv K. Computer Physics Communications, Vol. 192 https://doi.org/10.1016/j.cpc.2015.02.023	journal	July 2015
Fock matrix dynamics Pulay, Peter; Fogarasi, Géza Chemical Physics Letters, Vol. 386, Issue 4-6 https://doi.org/10.1016/j.cplett.2004.01.069	journal	March 2004
Jacobian-free Newton–Krylov methods: a survey of approaches and applications Knoll, D. A.; Keyes, D. E. Journal of Computational Physics, Vol. 193, Issue 2 https://doi.org/10.1016/j.jcp.2003.08.010	journal	January 2004
Extended Lagrangian Density Functional Tight-Binding Molecular Dynamics for Molecules and Solids Aradi, Bálint; Niklasson, Anders M. N.; Frauenheim, Thomas Journal of Chemical Theory and Computation, Vol. 11, Issue 7 https://doi.org/10.1021/acs.jctc.5b00324	journal	June 2015
Hybrid Extended Lagrangian, Post-Hartree–Fock Born–Oppenheimer ab Initio Molecular Dynamics Using Fragment-Based Electronic Structure Li, Junjie; Haycraft, Cody; Iyengar, Srinivasan S. Journal of Chemical Theory and Computation, Vol. 12, Issue 6 https://doi.org/10.1021/acs.jctc.6b00001	journal	May 2016
Recursive Factorization of the Inverse Overlap Matrix in Linear-Scaling Quantum Molecular Dynamics Simulations Negre, Christian F. A.; Mniszewski, Susan M.; Cawkwell, Marc J. Journal of Chemical Theory and Computation, Vol. 12, Issue 7 https://doi.org/10.1021/acs.jctc.6b00154	journal	June 2016
A New Method for Treating Drude Polarization in Classical Molecular Simulation Albaugh, Alex; Head-Gordon, Teresa Journal of Chemical Theory and Computation, Vol. 13, Issue 11 https://doi.org/10.1021/acs.jctc.7b00838	journal	October 2017
Extended Lagrangian Excited State Molecular Dynamics Bjorgaard, J. A.; Sheppard, D.; Tretiak, S. Journal of Chemical Theory and Computation, Vol. 14, Issue 2 https://doi.org/10.1021/acs.jctc.7b00857	journal	January 2018
Efficient and Accurate Born–Oppenheimer Molecular Dynamics for Large Molecular Systems Peters, Laurens D. M.; Kussmann, Jörg; Ochsenfeld, Christian Journal of Chemical Theory and Computation, Vol. 13, Issue 11 https://doi.org/10.1021/acs.jctc.7b00937	journal	October 2017
Higher-Order Extended Lagrangian Born–Oppenheimer Molecular Dynamics for Classical Polarizable Models Albaugh, Alex; Head-Gordon, Teresa; Niklasson, Anders M. N. Journal of Chemical Theory and Computation, Vol. 14, Issue 2 https://doi.org/10.1021/acs.jctc.7b01041	journal	January 2018
GFN2-xTB—An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions Bannwarth, Christoph; Ehlert, Sebastian; Grimme, Stefan Journal of Chemical Theory and Computation, Vol. 15, Issue 3 https://doi.org/10.1021/acs.jctc.8b01176	journal	January 2019
Accurate Classical Polarization Solution with No Self-Consistent Field Iterations Albaugh, Alex; Niklasson, Anders M. N.; Head-Gordon, Teresa The Journal of Physical Chemistry Letters, Vol. 8, Issue 8 https://doi.org/10.1021/acs.jpclett.7b00450	journal	April 2017
The Role and Perspective of Ab Initio Molecular Dynamics in the Study of Biological Systems Carloni, Paolo; Rothlisberger, Ursula; Parrinello, Michele Accounts of Chemical Research, Vol. 35, Issue 6 https://doi.org/10.1021/ar010018u	journal	June 2002
DFTB3: Extension of the Self-Consistent-Charge Density-Functional Tight-Binding Method (SCC-DFTB) Gaus, Michael; Cui, Qiang; Elstner, Marcus Journal of Chemical Theory and Computation, Vol. 7, Issue 4 https://doi.org/10.1021/ct100684s	journal	March 2011
Stable and Efficient Linear Scaling First-Principles Molecular Dynamics for 10000+ Atoms Arita, Michiaki; Bowler, David R.; Miyazaki, Tsuyoshi Journal of Chemical Theory and Computation, Vol. 10, Issue 12 https://doi.org/10.1021/ct500847y	journal	November 2014
Dyanmics of organic reactions Wang, I. S. Y.; Karplus, M. Journal of the American Chemical Society, Vol. 95, Issue 24 https://doi.org/10.1021/ja00805a033	journal	November 1973
Inertial extended-Lagrangian scheme for solving charge equilibration models Leven, Itai; Head-Gordon, Teresa Physical Chemistry Chemical Physics, Vol. 21, Issue 34 https://doi.org/10.1039/C9CP02979F	journal	January 2019
Accelerated, energy-conserving Born–Oppenheimer molecular dynamics via Fock matrix extrapolation Herbert, John M.; Head-Gordon, Martin Physical Chemistry Chemical Physics, Vol. 7, Issue 18 https://doi.org/10.1039/b509494a	journal	January 2005
Towards large scale hybrid QM/MM dynamics of complex systems with advanced point dipole polarizable embeddings Loco, Daniele; Lagardère, Louis; Cisneros, Gérardo A. Chemical Science, Vol. 10, Issue 30 https://doi.org/10.1039/c9sc01745c	journal	January 2019
Ab initio molecular dynamics: Propagating the density matrix with Gaussian orbitals Schlegel, H. Bernhard; Millam, John M.; Iyengar, Srinivasan S. The Journal of Chemical Physics, Vol. 114, Issue 22 https://doi.org/10.1063/1.1372182	journal	June 2001
Ab initio molecular dynamics: Propagating the density matrix with Gaussian orbitals. II. Generalizations based on mass-weighting, idempotency, energy conservation and choice of initial conditions Iyengar, Srinivasan S.; Schlegel, H. Bernhard; Millam, John M. The Journal of Chemical Physics, Vol. 115, Issue 22 https://doi.org/10.1063/1.1416876	journal	January 2001
How well do Car–Parrinello simulations reproduce the Born–Oppenheimer surface? Theory and examples Tangney, P.; Scandolo, S. The Journal of Chemical Physics, Vol. 116, Issue 1 https://doi.org/10.1063/1.1423331	journal	January 2002
Approximate Self‐Consistent Molecular Orbital Theory. I. Invariant Procedures Pople, J. A.; Santry, D. P.; Segal, G. A. The Journal of Chemical Physics, Vol. 43, Issue 10 https://doi.org/10.1063/1.1701475	journal	November 1965
Approximate Self‐Consistent Molecular‐Orbital Theory. V. Intermediate Neglect of Differential Overlap Pople, J. A.; Beveridge, D. L.; Dobosh, P. A. The Journal of Chemical Physics, Vol. 47, Issue 6 https://doi.org/10.1063/1.1712233	journal	September 1967
Curvy-steps approach to constraint-free extended-Lagrangian ab initio molecular dynamics, using atom-centered basis functions: Convergence toward Born–Oppenheimer trajectories Herbert, John M.; Head-Gordon, Martin The Journal of Chemical Physics, Vol. 121, Issue 23 https://doi.org/10.1063/1.1814934	journal	December 2004
On the theory underlying the Car-Parrinello method and the role of the fictitious mass parameter Tangney, Paul The Journal of Chemical Physics, Vol. 124, Issue 4 https://doi.org/10.1063/1.2162893	journal	January 2006
Time-reversible ab initio molecular dynamics Niklasson, Anders M. N.; Tymczak, C. J.; Challacombe, Matt The Journal of Chemical Physics, Vol. 126, Issue 14 https://doi.org/10.1063/1.2715556	journal	April 2007
A note on the Pulay force at finite electronic temperatures Niklasson, Anders M. N. The Journal of Chemical Physics, Vol. 129, Issue 24 https://doi.org/10.1063/1.3036203	journal	December 2008
Extended Lagrangian Born–Oppenheimer molecular dynamics with dissipation Niklasson, Anders M. N.; Steneteg, Peter; Odell, Anders The Journal of Chemical Physics, Vol. 130, Issue 21 https://doi.org/10.1063/1.3148075	journal	June 2009
Higher-order symplectic integration in Born–Oppenheimer molecular dynamics Odell, Anders; Delin, Anna; Johansson, Börje The Journal of Chemical Physics, Vol. 131, Issue 24 https://doi.org/10.1063/1.3268338	journal	December 2009
Lagrangian formulation with dissipation of Born-Oppenheimer molecular dynamics using the density-functional tight-binding method Zheng, Guishan; Niklasson, Anders M. N.; Karplus, Martin The Journal of Chemical Physics, Vol. 135, Issue 4 https://doi.org/10.1063/1.3605303	journal	July 2011
Geometric integration in Born-Oppenheimer molecular dynamics Odell, Anders; Delin, Anna; Johansson, Börje The Journal of Chemical Physics, Vol. 135, Issue 22 https://doi.org/10.1063/1.3660689	journal	December 2011
Classical trajectories using the full a b i n i t i o potential energy surface H ⁻ +CH ₄ →CH ₄ +H ⁻ Leforestier, Claude The Journal of Chemical Physics, Vol. 68, Issue 10 https://doi.org/10.1063/1.435520	journal	May 1978
Energy conserving, linear scaling Born-Oppenheimer molecular dynamics Cawkwell, M. J.; Niklasson, Anders M. N. The Journal of Chemical Physics, Vol. 137, Issue 13 https://doi.org/10.1063/1.4755991	journal	October 2012
First principles molecular dynamics without self-consistent field optimization Souvatzis, Petros; Niklasson, Anders M. N. The Journal of Chemical Physics, Vol. 140, Issue 4 https://doi.org/10.1063/1.4862907	journal	January 2014
Generalized extended Lagrangian Born-Oppenheimer molecular dynamics Niklasson, Anders M. N.; Cawkwell, Marc J. The Journal of Chemical Physics, Vol. 141, Issue 16 https://doi.org/10.1063/1.4898803	journal	October 2014
Thermostating extended Lagrangian Born-Oppenheimer molecular dynamics Martínez, Enrique; Cawkwell, Marc J.; Voter, Arthur F. The Journal of Chemical Physics, Vol. 142, Issue 15 https://doi.org/10.1063/1.4917546	journal	April 2015
An efficient and stable hybrid extended Lagrangian/self-consistent field scheme for solving classical mutual induction Albaugh, Alex; Demerdash, Omar; Head-Gordon, Teresa The Journal of Chemical Physics, Vol. 143, Issue 17 https://doi.org/10.1063/1.4933375	journal	November 2015
Graph-based linear scaling electronic structure theory Niklasson, Anders M. N.; Mniszewski, Susan M.; Negre, Christian F. A. The Journal of Chemical Physics, Vol. 144, Issue 23 https://doi.org/10.1063/1.4952650	journal	June 2016
Analytic hyperpolarizability and polarizability derivative with fractional occupation numbers for large extended systems Nishimoto, Yoshio The Journal of Chemical Physics, Vol. 146, Issue 8 https://doi.org/10.1063/1.4976551	journal	February 2017
Performance of extended Lagrangian schemes for molecular dynamics simulations with classical polarizable force fields and density functional theory Vitale, Valerio; Dziedzic, Jacek; Albaugh, Alex The Journal of Chemical Physics, Vol. 146, Issue 12 https://doi.org/10.1063/1.4978684	journal	March 2017
Next generation extended Lagrangian first principles molecular dynamics Niklasson, Anders M. N. The Journal of Chemical Physics, Vol. 147, Issue 5 https://doi.org/10.1063/1.4985893	journal	August 2017
Molecular dynamics without effective potentials via the Car-Parrinello approach Remler, Dahlia K.; Madden, Paul A. Molecular Physics, Vol. 70, Issue 6 https://doi.org/10.1080/00268979000101451	journal	August 1990
Broyden's method for self-consistent field convergence acceleration Srivastava, G. P. Journal of Physics A: Mathematical and General, Vol. 17, Issue 6 https://doi.org/10.1088/0305-4470/17/6/002	journal	April 1984
Ab initio molecular dynamics: basic concepts, current trends and novel applications Tuckerman, Mark E. Journal of Physics: Condensed Matter, Vol. 14, Issue 50 https://doi.org/10.1088/0953-8984/14/50/202	journal	December 2002
Canonical-ensemble extended Lagrangian Born–Oppenheimer molecular dynamics for the linear scaling density functional theory Hirakawa, Teruo; Suzuki, Teppei; Bowler, David R. Journal of Physics: Condensed Matter, Vol. 29, Issue 40 https://doi.org/10.1088/1361-648x/aa810d	journal	September 2017
A class of methods for solving nonlinear simultaneous equations Broyden, C. G. Mathematics of Computation, Vol. 19, Issue 92 https://doi.org/10.1090/s0025-5718-1965-0198670-6	journal	January 1965
Inhomogeneous Electron Gas Hohenberg, P.; Kohn, W. Physical Review, Vol. 136, Issue 3B, p. B864-B871 https://doi.org/10.1103/physrev.136.b864	journal	November 1964
Theory of ab initio molecular-dynamics calculations Pastore, G.; Smargiassi, E.; Buda, F. Physical Review A, Vol. 44, Issue 10 https://doi.org/10.1103/physreva.44.6334	journal	November 1991
Efficient iteration scheme for self-consistent pseudopotential calculations Kerker, G. P. Physical Review B, Vol. 23, Issue 6 https://doi.org/10.1103/physrevb.23.3082	journal	March 1981
Modified Broyden’s method for accelerating convergence in self-consistent calculations Johnson, D. D. Physical Review B, Vol. 38, Issue 18 https://doi.org/10.1103/physrevb.38.12807	journal	December 1988
Energy versus free-energy conservation in first-principles molecular dynamics Wentzcovitch, Renata M.; Martins, José Luís; Allen, Philip B. Physical Review B, Vol. 45, Issue 19 https://doi.org/10.1103/physrevb.45.11372	journal	May 1992
Fractional occupations and density-functional energies and forces Weinert, M.; Davenport, J. W. Physical Review B, Vol. 45, Issue 23 https://doi.org/10.1103/physrevb.45.13709	journal	June 1992
Ab initiomolecular dynamics for liquid metals Kresse, G.; Hafner, J. Physical Review B, Vol. 47, Issue 1, p. 558-561 https://doi.org/10.1103/physrevb.47.558	journal	January 1993
Born-Oppenheimer molecular-dynamics simulations of finite systems: Structure and dynamics of ( H 2 O ) 2 Barnett, Robert N.; Landman, Uzi Physical Review B, Vol. 48, Issue 4 https://doi.org/10.1103/physrevb.48.2081	journal	July 1993
Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties Elstner, M.; Porezag, D.; Jungnickel, G. Physical Review B, Vol. 58, Issue 11, p. 7260-7268 https://doi.org/10.1103/physrevb.58.7260	journal	September 1998
Implicit purification for temperature-dependent density matrices Niklasson, Anders M. N. Physical Review B, Vol. 68, Issue 23 https://doi.org/10.1103/physrevb.68.233104	journal	December 2003
Wave function extended Lagrangian Born-Oppenheimer molecular dynamics Steneteg, Peter; Abrikosov, Igor A.; Weber, Valery Physical Review B, Vol. 82, Issue 7 https://doi.org/10.1103/physrevb.82.075110	journal	August 2010
Canonical density matrix perturbation theory Niklasson, Anders M. N.; Cawkwell, M. J.; Rubensson, Emanuel H. Physical Review E, Vol. 92, Issue 6 https://doi.org/10.1103/physreve.92.063301	journal	December 2015
Extended Born-Oppenheimer Molecular Dynamics Niklasson, Anders M. N. Physical Review Letters, Vol. 100, Issue 12 https://doi.org/10.1103/physrevlett.100.123004	journal	March 2008
Unified Approach for Molecular Dynamics and Density-Functional Theory Car, R.; Parrinello, M. Physical Review Letters, Vol. 55, Issue 22 https://doi.org/10.1103/physrevlett.55.2471	journal	November 1985
Crystal Structures of Zirconia from First Principles and Self-Consistent Tight Binding Finnis, M. W.; Paxton, A. T.; Methfessel, M. Physical Review Letters, Vol. 81, Issue 23 https://doi.org/10.1103/physrevlett.81.5149	journal	December 1998
Density Matrix Perturbation Theory Niklasson, Anders M. N.; Challacombe, Matt Physical Review Letters, Vol. 92, Issue 19 https://doi.org/10.1103/physrevlett.92.193001	journal	May 2004
Efficient and Accurate Car-Parrinello-like Approach to Born-Oppenheimer Molecular Dynamics Kühne, Thomas D.; Krack, Matthias; Mohamed, Fawzi R. Physical Review Letters, Vol. 98, Issue 6 https://doi.org/10.1103/physrevlett.98.066401	journal	February 2007
New Developments in Molecular Orbital Theory Roothaan, C. C. J. Reviews of Modern Physics, Vol. 23, Issue 2 https://doi.org/10.1103/revmodphys.23.69	journal	April 1951
Some Recent Advances in Density Matrix Theory McWeeny, R. Reviews of Modern Physics, Vol. 32, Issue 2 https://doi.org/10.1103/revmodphys.32.335	journal	April 1960
Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients Payne, M. C.; Teter, M. P.; Allan, D. C. Reviews of Modern Physics, Vol. 64, Issue 4 https://doi.org/10.1103/revmodphys.64.1045	journal	October 1992
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems Saad, Youcef; Schultz, Martin H. SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3 https://doi.org/10.1137/0907058	journal	July 1986
Hybrid Krylov Methods for Nonlinear Systems of Equations Brown, Peter N.; Saad, Youcef SIAM Journal on Scientific and Statistical Computing, Vol. 11, Issue 3 https://doi.org/10.1137/0911026	journal	May 1990
Iterative Procedures for Nonlinear Integral Equations Anderson, Donald G. Journal of the ACM, Vol. 12, Issue 4 https://doi.org/10.1145/321296.321305	journal	October 1965
Structural and dynamical properties of hot dense matter by a Thomas-Fermi-Dirac molecular dynamics Lambert, F.; Clérouin, J.; Mazevet, S. Europhysics Letters (EPL), Vol. 75, Issue 5 https://doi.org/10.1209/epl/i2006-10184-7	journal	September 2006
Using Neighbor Diversity to Detect Fraudsters in Online Auctions Lin, Jun-Lin; Khomnotai, Laksamee Entropy, Vol. 16, Issue 5 https://doi.org/10.3390/e16052629	journal	May 2014
Linear-scaling first-principles molecular dynamics of complex biological systems with the Conquest code Otsuka, Takao; Taiji, Makoto; Bowler, David R. Japanese Journal of Applied Physics, Vol. 55, Issue 11 https://doi.org/10.7567/jjap.55.1102b1	journal	September 2016

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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
97 MATHEMATICS AND COMPUTING
Born-Oppenheimer molecular dynamics
Density-matrix
Integration methods
Krylov subspace approximation
Mathematics
Molecular dynamics
Newton-Krylov method
Non-linear systems of equations
Nonlinear systems
Numerical linear algebra
Optimization problems
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Title: Extended Lagrangian Born–Oppenheimer molecular dynamics using a Krylov subspace approximation

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