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Ab initio molecular dynamics via the Car-Parrinello method: Basic ideas, theory and algorithms

Ab initio molecular dynamics via the Car-Parrinello method: Basic ideas, theory and algorithms. Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences New York University, 100 Washington Sq. East New York, NY 10003. 1808:

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Ab initio molecular dynamics via the Car-Parrinello method: Basic ideas, theory and algorithms

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  1. Ab initio molecular dynamics via the Car-Parrinello method: Basic ideas, theory and algorithms Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences New York University, 100 Washington Sq. East New York, NY 10003

  2. 1808: “We are perhaps not far removed from the time when we shall be able to submit the bulk of chemical phenomena to calculation.” Joseph Louis Gay-Lussac (1778-1850)

  3. “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact solution of these laws leads to equations much to complicated to be soluble.” Paul Dirac on Quantum Mechanics (1929).

  4. “Every attempt to refer chemical questions to mathematical doctrines must be considered, now and always, profoundly irrational, as being contrary to the nature of the phenomena.” August Comte, 1830

  5. Motivation • Car-Parrinello is a method for performing molecular dynamics with forces obtained from electronic structure calculations performed “on the fly” as the simulation proceeds. This is known as ab initio molecular dynamics (AIMD). • As a result, AIMD calculations are considerably more expensive than force-field calculations, which only involve evaluation of simple functions of position. • Force fields, although useful, are, with notable exceptions, unable to treat chemical bond breaking and forming events. • Force fields often lack transferability to thermodynamic situations in which they are not designed to work. • Polarization and manybody interactions included implicitly.

  6. From ISI Citation Report R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985) Total Cites = 4,812

  7. The “Universal” Hamiltonian M Electrons N Nuclei Operator Definitions: Electronic: Nuclear: Coupling:

  8. Molecular energy levels Electron coordinates Nuclear coordinates Notation: Complete energy level spectrum:

  9. Born-Oppenheimer Approximation à la W. H. Flygare, Molecular Structure and Dynamics Quasi adiabatic separability ansatz for wave function: . . . Schrödinger equation separates if Electrons in fixed back- ground nuclear geometry R Nuclei on each electronic hypersurface

  10. Born-Oppenheimer (electronic) surfaces and nuclear energy levels ε2 ε1 (no bound levels) Vibrations Rotations ε0

  11. Classical nuclear motion on an electronic surface Consider the ground-state electronic surface Nuclear Hamiltonian: “Demote” to a classical Hamiltonian: Nuclear motion now given by Hamilton’s equations:

  12. Classical nuclei (R,P) Quantum electrons

  13. Hellman-Feynman Theorem Ground-state electronic surface as expectation value: Because

  14. Kohn-Sham density functional theory Except for very small systems, we cannot solve for the exact Density functional theory represents a compromise between accuracy and computational cost. Wave function ansatz: Single-particle orbitals: Electron density:

  15. Kohn-Sham density functional theory Total energy functional: Energy definitions: Ground-state energy via constrained minimization

  16. The Born-Oppenheimer Algorithm Electrons Nuclei Start with nuclei Add electrons Compute Propagate nuclei a short time Δtwith F Add electrons e.g. Verlet:

  17. The Car-Parrinello scheme Avoid explicit minimization with a fictitious adiabatic dynamics for electronic orbitals: Lagrangian (note μnot a mass! It has units of energy x time2): Equations of motion: Conditions: 1) “Near” Born-Oppenheimer “Seed” the CP equations of motion with initially minimized orbitals.

  18. Energy Conservation in Born-Oppenheimer and Car-Parrinello dynamics CP 5 a.u CP 10 a.u. BO 10-6, 10 a.u. BO 10-6, 100 a.u. CP 10 a.u. BO 10-5, 100 a.u. BO 10-6, 100 a.u. CP 10 a.u. BO 10-4, 100 a.u. System: 8 Silicon atoms Marx and Hutter, Modern Methods and Algorithms of Quantum Chemistry (NIC Series)1, J. Grotendorst, ed. (Forschungszentrum, Jülich, 2000)

  19. Energy conservation timing comparison Marx and Hutter, Modern Methods and Algorithms of Quantum Chemistry (NIC Series)1, J. Grotendorst, ed. (Forschungszentrum, Jülich, 2000) System: 8 Silicon atoms

  20. Adiabatic Dynamics Consider a simple 2 degree-of-freedom system: Adiabatic conditions:

  21. Analysis of the dynamics Full phase-space vector evolves according to Liouville operator: Subdivision of Liouville operator:

  22. Analysis of dynamics (cont’d) Evolution of phase space over a time Δt characteristic of nuclear motion: Trotter factorization: Exact Trotter theorem: Evolution of momentum:

  23. Analysis of dynamics (cont’d) Time-average equated to phase-space average: Partition function for slow variable: Adiabatic method for free-energy profiles: [L. Rosso, et al. JCP 116, 4389 (2002)] Annealing property:

  24. Model Problem:

  25. Methods: Plane-wave basis sets (periodic box, FFTs) orbitals density Car-Parrinello

  26. l = 0 Eliminating core electrons l = 1 l = 2

  27. Why a real-space basis? • Plane-waves are elegant but scale as N 2M • Slow convergence of plane waves to the basis set limit. • Ease of localizing orbitals. • Ease of representing position-dependent operators. • Exact representation of • Common choice – Gaussians

  28. Selecting a real-space basis (why not Gaussians?) • Retain simplicity of plane waves. • Systematic convergence to the basis-set limit. • Spatially localized for possible linear-scaling. • Position independence and orthonormality. • No BSSE • For flexibility of use, seek noncompact support. • Choice: Discrete variable representations (DVRs). J. C. Light, et al. J. Chem. Phys. 82, 1400 (1985); Edwards, Tuckerman, Friesner, Sorensen, J. Comp. Phys. 110, 82 (1994).R. A. Friesner, Chem. Phys. Lett. 116, 39 (1985); Bacic and Light, Ann. Rev. Phys. Chem. 40, 469 (1989); J. T. Muckerman, Chem. Phys. Lett. 173, 200 (1990); Colbert and Miller, J. Chem. Phys. 96, 1982 (1992); Light and Carrington, Adv. Chem. Phys. 114, 263 (2000); Littlejohn and Cargo, J. Chem. Phys. 117, 27, 37, 59 (2002); Varga, et al. Phys. Rev. Lett. 93, 176403 (2004).

  29. Definition of a DVR Plane-waves (at the Γ (k=0)-point) -- momentum eigenfunctions: Discrete-variable representations (position eigenfunctions): Begin with a set of N square-integrable orthonormal functions φi(x) On an appropriately chosen quadrature grid {x1,…,xN} Expand orbitals as: Y. Liu, D. Yarne and MET, PRB68, 125110 (2003); H. –S. Lee and MET, JPCA110, 5549 (2006)

  30. DVR convergence for a 32 water box vs. plane-waves with TM PPs Force measure: DVR basis sets allow the complete basis set limit to be reached with the ease of plane waves

  31. Is Exc = BLYP water overstructured? Plane-wave basis (70-85 Ry cutoff) Grossman, et. al. JCP120, 300 (2004) Pseudopotentials: Hamann (1989) 85 Ry cutoff Mantz, et. al. JPCB 110, 3540 (2006) Pseudopotentials: Troullier-Martins 70 Ry cutoff 292 K 318 K Morrone and Car, PRL101, 017801 (2008) Pseudopotentials: Troullier-Martins 70 ry cutoff Gaussians: TZV2P VandeVondele, et. al. JCP 122, 014515 (2005)

  32. Radial distribution functions for BLYP Water Grid = 753,t =60 ps Ensemble: NVT, 300 K, μ= 500 au DVR Neutron X-ray Grossman, et. al. JCP120, 300 (2004) r(Å) From Akin-Ojo, et al. JCP129, 064108 (2008) H. –S. Lee and MET, JPCA 110, 549 (2006) H. –S. Lee and MET JCP 125, 154507 (2006). H. –S. Lee and MET JCP 126, 164501 (2007). Neutron: Soper, et. al. JCP106, 247 (1997) X-ray: Hura, et. al. Chem. Phys. 113, 9140 (2000) When basis sets are too small! from C. J. Mundy (2008)

  33. Selected References • R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985) • D. K. Remler and P. Madden, Mol. Phys. 70, 921 (1990) • G. Galli and M. Parrinello in Computer Simulations in Chemical Physics • (NATO ASI Series C)397, 261 (1993) • 4. M. Parrinello, Solid State Commun. 102, 107 (1997) • 5. D. Marx and J. Hutter, Modern Methods and Algorithms of Quantum Chemistry • (NIC Series)1, J. Grotendorst, ed. (Forschungszentrum, Jülich, 2000) • M. E. Tuckerman, J. Phys. Condens. Matter, 14, R1297 (2002) • F. Krajewski and M. Parrinello, Phys. Rev. B73, 041105 (2006) • T. D. Kunhe, M. Krack, F. R. Mohamed and M. Parrinello, • Phys. Rev. Lett. 98, 066401 (2007) • 9. H. –S. Lee and M. E. Tuckerman, J. Phys. Chem. A 110, 5549 (2006); • J. Chem. Phys. 125, 154507 (2006); J. Chem. Phys. 126, 164501 (2007). • 10. E. Bohm, et. al. IBM J. Res. Devel. 52, 159 (2008) Ab initio molecular dynamics codes: CPMD: http://www.cpmd.org CP2K: http://cp2k.berlios.de VASP: http://cms.mpi.univie.ac.at/vasp PINY_MD: http://www.nyu.edu/PINY_MD/PINY.html OpenAtom: http://charm.cs.uiuc.edu/OpenAtom NWChem: http://www.emsl.pnl.gov/docs/nwchem/nwchem.html SIESTA: http://www.lrz-muenchen.de/services/software/chemie/siesta

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