Transcription
Ab-initio molecular dynamics:from the basics up to quantum effectsRoberto CarPrinceton UniversityHands-on Tutorial Workshop on Ab-Initio Molecular Simulations, FritzHaber-Institut, Berlin, July 12-21, 2011
Outline Basics Why MD with a DFT potential energy surface CP/BO variations Path-integral AIMD: quantal nuclei Open issues
Basic conceptsMolecular Dynamics!! #MIRI!" ({R})( I 1, N )!RI Nuclei are supposed to obey classical mechanics Trajectories are integrated numerically on a computer Under ergodic evolution temporal averages are equivalent to ensemble averages Correlation functions describing both static and dynamic equilibrium properties areevaluated as temporal averages (equilibrium statistical mechanics)!1C AB (t ) # A(t ) B(0) A(t ) B(0) % A(t t ") B(t ")dt "! 0
Potential energy surface withoutempirical fittingEGS ({R}) Minn ( r ) EV [n ]EV %'{! },{! }&( 2*)i 1, N e / 2! EV &{" *},{" }%'!" (r )*in( r ) 2"i 1, N e 2! i (r )2# ! 2 21 n(r )n(r ")e 2!i! i * V (r )n(r )dr **drdr " E XC [n ]"2m2r#r( # i" i (r ) H KS" i (r ) ( # i" i (r ) 0Kohn-Sham equations1Z I Z J e2" ({R}) EGS ({R}) 2 I ! J RI # RJW. Kohn and L.J. Sham, Phys Rev 140, A1133 (1965)
Ab-initio Molecular Dynamics Parameter-free molecular dynamics: the potential energysurface is obtained from density functional theory The price to pay is that the Kohn-Sham equations needto be solved at all the nuclear configurations in atrajectory The problem is greatly simplified (conceptually and inpractice) by calculating the electronic structure on the flyduring nuclear dynamics The task is achieved with a generalized moleculardynamics scheme in the extended space of nuclear {R}and electronic {! } coordinatesR. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985)
Why MD with a DFT potentialenergy surface?An example: Hydrogen production from afunctionalized pyrite surface immersed inacidified waterF. Zipoli, R.C., M.H. Cohen, A. Selloni,JACS 132, 8593 (2010)
Water splitting in PEC cellH2O H2 ½ O2OxidationReductionH2O 2h ½O2 2H 2H 2e- H2N-typesemiconductor, e.g.TiO2Pt typically used ascathode
Functionalization of the pyrite surface witha modified [FeFe]H cluster88
Hydrogen production is easy when the first proton is inplace!
Free energy barrier of 1st protonationReaction coordinates:Leung et al JACS 200910
What did we learn?a) importance of treating explicitly nuclei andelectrons (AIMD)b) crucial role of the solvatingmedium: the Grotthuss mechanismc) rare events are important and require specialtreatment
CP/BO variations(1LCP * M I R! I2 2 µ * !! i !! i & ' CP ({R};{! *},{! }%) 2* "ij ! j ! i & # ji2 Iii, jv v 21ZZ e% CP #&{R};{! *},{! } ' EV ({ R}) #&{! *},{! } ' ) I J2 I " J RI ( RJ!! *MIRIµ !!!i *&' CP ({R};{! *},{! }%)&RI"' CP ({R};{! *},{! }%)2" ! iHCP K Kf " CP ! j # ji * H KS ! i ! j # jijj1 # M I R! I2 2 µ # !! i !! i " CP const2 Ii)
quasi-BO schemes!! FMRBO!FQBO ! FBO " !E R!! F! "MR!!RQBOFor Boltzmann sampling:!is arbitrary!E!! F ! (! !)R! "MRBOE!(0)!(t) 6("E ")MkBT #(t)fixed by1 !2MR2 3k T2 BT.D. Kuhne, M. Krack, F.R. Mohamed, M. Parrinello, PRL 98, 066401 (2007)
Path Integral AIMD:quantal nucleiFeynman paths: closed and open
Protons are light nuclei:how important are quantum effects intheir behavior?
Quantal protons in water and iceThe momentum distribution differs considerablyfrom the classical equilibrium distributionPath integral simulations: J. Morrone, RC, PRL 2008Experiment: deep inelastic neutron scattering (DINS)Quantum delocalization of the nuclei affects structural and spectroscopic properties
Quantum (path integral) ab-initio simulations improve substantially thestructureFrom J. Morrone and RC, Phys. Rev. Lett. 101, 107801(2008)Experiment: Joint structure refinement (x-ray andneutron) A. Soper, J. Phys.: Condens. Matter 19,335206 (2007). but systematic deviations remain.Quantum effect seems to be roughlymimicked by an increase of T
Effect of functional approximations and quantum nuclei
Effect of functional approximations and quantum nuclei
Effect of functional approximations and quantum nuclei
Effect of functional approximations and quantum nuclei
Functional approximations and experimentSoper2007:JointstructurerefinementofX- ‐rayandneutrondiffrac;ondatafrom:1. Hartetal,PRL2005;2.Huraetal,PCCP2003;3.Narten- ‐Levy,JCP1971VdWuseTkatchenko- ‐Schefflertheory,PRL2009
In the end comparison to experiment is goodFrom Z. Li, R. Di Stasio, X. Wu, and RC (preliminary results)The approach is not entirely consistent: quantum corrections have beencalculated at the BLYP level of DFT, VdW corrections to the PBE0 energyhave been used to reweigh the g(r)’s, a procedure that is affected by largestatistical errors.However, fully consistent calculations should be available soon. Based onthe current results a promising outcome should be expected.
Effect of quantum nuclei on XAS spectra:liquid water at 300KExpt (Nilsson): Wernet et al., Science (2004)Theory: L. Kong, X. Wu, R.C., (work inprogress, 2011)COHSEX self-energy expression as inW. Chen, X. Wu, R.C., Phys. Rev. Lett.(2010)nuclear configurations from J. Morrone, R.C.,Phys. Rev. Lett. (2008)The residual deviation from experiment at high energy calls for improvements in theself-energy approximation
Zero-point motion in ice IhL. Lin, J. Morrone, RC, M. Parrinello, PRB 2011Quasi-harmonic model fits the data well: zero-pointmotion is large but still semi-classical in nature
New physics occurs when tunnel drivenphase transitions happenCartoon from J. Texeira, NatureN&V commentingBenoit, Marx, and Parrinello,Nature 1998Picture is suggestive but rootedin Mean Field Theoryice VIII - ice VII: antiferroelectric to disordered proton sublatticeAnother transition driven by quantum fluctuations occurs in KDP: ferroelectric to disordered
Reduced longitudinal modelMotions along and orthogonal to the bond are uncorrelated in all ice structures consideredin this study (Ih, VIII, VII, X)!(x, x !)The longitudinal density matrixcontains the interesting physicsL. Lin, J.A. Morrone, RC (2011)
Entangled protonsVon Neumann entanglement entropyFrom Lin, Morrone, RC (preprint 2011)What is the origin of the entanglement? Why does it occur in ice VII and X?
Spectrum of the longitudinal density matrixEffective potential and tunnel splittingTunnel splitting547KT of simulationThe first 2 eigenstates in ice VII: these arethe natural orbitals of the proton100KEntanglement is due to correlations
Ring tunneling does not violate ice rulesMapping to a spin model: spin 1/2 or spin 1?A quantum spin liquid
Simulation versus Mean FieldThe distribution of local charge fluctuations in a 3state (spin 1) model with N,C, and F statesMFA leads to a ionization catastrophe thatseverely violates local charge neutrality(from Reiter, Mayers, Platzman, PRL (2002)Is there a node in the proton momentum distributionof KDP in the paraeletric phase?Ring tunneling does occur but is not the only process as some ionized configurations do occurConsider, however: small cell size, T of the simulation, chosen lattice parameter (applied pressure)
Open issues 1-particle density matrix vs density: apossible way beyond DFT Quantum dynamics vs equilibrium statisticalsampling
Acknowledgement Funding from NSF and DOE
Ab-initio molecular dynamics: from the basics up to quantum effects. Outline Basics Why MD with a DFT potential energy surface CP/BO variations Path-integral AIMD: quantal nuclei Open issues. Basic concepts ({}) II (1,) I R MR IN R!" # !!! Nuclei are supposed to obey classical mechanics Trajectories are integrated numerically on a computer Under ergodic .
