3.320: Lecture 13 (Mar 21 2005) FIRST-PRINCIPLES MOLECULAR DYNAMICSSimulated AnnealingMicro- to macro- : diffusion coefficientMean Square DisplacementsMean Square DisplacementsVelocity Autocorrelation FunctionGreen-Kubo relationsVelocity Autocorrelation FunctionMore Green-KuboDynamics, Lagrangian styleNewton’s second law, tooHamiltonianThermostats, barostats…Nose’ extended LagrangianErgodicity issuesClassical MD BibliographyFirst-principles molecular dynamicsPlane waves basis setIt’s really kinetic + potentialKinetic energyTotal energy (non-SCF)Dynamical evolution of c’sWe need the forceSkiing down a valleyConjugate-gradients minimizationHellmann-Feynman theoremProof of Hellmann-FeynmanBorn-Oppenheimer Molecular DynamicsThe extended Car-Parrinello LagrangianEquations of motionConstant of MotionBorn-Oppenheimer vs Car-ParrinelloBO vs CP forcesKolmogorov-Arnold-Moser invariant toriQuantum MD BibliographyMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari3.320: Lecture 15 (Mar 31 2005) FIRSTFIRST--PRINCIPLES MOLECULAR DYNAMICSPRINCIPLES MOLECULAR DYNAMICS…and let us, as nature directs, begin first with first principles.Aristotle (Poetics, I)Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariSimulated Annealing-3 -2-2-1 0 1 2 3121086420Figure by MIT OCW.Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMicro- to macro- : diffusion coefficient• From Fick to Einstein:),(),(2trcDttrc∇=∂∂∫∫∇=∂∂),(),(222trcrrdDtrcrrdtrrdDtrt2)(2=∂∂Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMean Square Displacements∑=∆=∆NiitrNtr122)(1)(Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMean Square DisplacementsMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariVelocity Autocorrelation Function∑=∆=∆NiitxNtx122)(1)(∫′′=∆txtvtdtx0)()(∫∫∫′′′′′′=⎟⎟⎠⎞⎜⎜⎝⎛′′=∆txxttxtvtvtdtdtvtdtx00202)()()()(∫∫′′′′′′′=txxttvtvtdtd00)()(2Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariGreen-Kubo relations20()2lim lim2 ()()tttxxxtDdtvtvtt′′→∞ →∞∂′′′′′==∂∫)0()()()(xxxxvttvtvtv′′−′=′′′00lim ( ) (0) ( ) (0)ttxx xxDdtvttvdvvττ′∞′→∞′′ ′ ′′=−=∫∫Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariVelocity Autocorrelation FunctionMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMore Green-Kubo• Other transport coefficients:– Shear viscosity, from the stress– Electrical conductivity, from the charge current– IR adsorption, from the polarizationMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariDynamics, Lagrangian style• First construct L=T-V• Then, the equations of motion are given by• Why ? We can use generalized coordinates. Also, we only need to think at the two scalar functions T and V0=∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂jjqLqLdtd&(the dot is a time derivative)Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariNewton’s second law, too• 1-d, 1 particle: T=1/2 mv2, V=V(x)0=∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂jjqLqLdtd&2120mxdVdt x x⎛⎞⎛⎞∂⎜⎟⎜⎟∂⎝⎠⎜⎟+=∂∂⎜⎟⎜⎟⎝⎠&&()dVmxdt x∂=−∂&Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariHamiltonian• We could use it to derive Hamiltonian dynamics (twice the number of differential equations, but all first order). We introduce a Legendre transformationiiqLp&∂∂=∑−=iiitqqLpqtpqH ),,(),,(&&iipHq∂∂=&iiqHp∂∂=−&Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThermostats, barostats…• We might want to sample a constant-temperature ensemble, or constants pressure– Stochastic approach– Extended system– Constraint methodMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariNose’ extended LagrangiansNsQVrsmLiiiNOSEln)13(2121222β+−+−=∑&&Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariErgodicity issuesVery harmonic solids (e.g. 1 harmonic oscillator !)Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariClassical MD Bibliography• Allen and Tildesley, Computer Simulations of Liquids (Oxford)• Frenkel and Smit, Understanding Molecular Simulations (Academic)• Ercolessi, A Molecular Dynamics Primer (http://www.fisica.uniud.it/~ercolessi/md)Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFirst-principles molecular dynamicsGraph removed for copyright reasons.Shows dramatic increase in number of citations per year of “CP PRL 1985” and “AIMD” beginning around 1990.Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariPlane waves basis set2ij ijGaπδ⋅=rrMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariIt’s really kinetic + potential21ˆ()2HVr=− ∇ +r)exp()( rGicrGnGnrrrrr⋅=∑ψ∑∑==nnnnnHEψψεˆMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariKinetic energy212kin n nnEψψ=−∇∑)exp()( rGicrGnGnrrrrr⋅=∑ψ2212nkinGnGEcG=∑∑rr∫′=′⎥⎦⎤⎢⎣⎡∇−−=′∇−GGGrGiiGrdrGG,22221)exp(21)exp(21δMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariTotal energy (non-SCF)()pot n nnEVrψψ=∑r)exp()( rGicrGnGnrrrrr⋅=∑ψ∫′−=′−=′)()exp()()exp()( GGVrGirViGrdrGrVG22,1()2nnntotGGGnGGGEcGccVGG∗′′⎛⎞′=+−⎜⎟⎝⎠∑∑ ∑rrrrrrrrMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariDynamical evolution of c’s22,1()2nnntotGGGnGGGEcGccVGG∗′′⎛⎞′=+−⎜⎟⎝⎠∑∑ ∑rrrrrrrrMar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariWe need the forceiiiEFδψψδ}][{−=}][{iEEψ=iHψˆ−=Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariSkiing down a valleyiiHψψµ−=&&iiHψψ−=&Mar 31 2005