ofADMPisrstupdatedusingitsvelocityandacceleration.Theresultisusedasan - PDF document

ofADMPisrstupdatedusingitsvelocityandacceleration.Theresultisusedasaninitialguessforapuricationpro-cess,Eq.ofRef.4,thatconservesidempotencyuptogoodnumericalaccuracyandisnonlinear.Generally,idem-potencyismaintainedwithaboutteniterationsofthisschemeforlargesystems.3–6,9–11Furthermore,thereexistsadirectrelationbetweenADMPandcurvy-ELMDasdis-cussedintheAppendix.Finally,curvy-ELMDnecessitatesresettingtozeroateverytimestep,sincetheenergyderivativewithreferencetoisobtained=0.Thisimpliesthattheclassicaldy-namicsinthephasespaceisnotstrictlysmooth.ThisdoesnothavenumericalbearingsontheresultsinRef.1.However,thisalsoimpliesthatcurvy-ELMDdoesnotconservethe“total”energyofthesystem.InHerbertandHead-Gordonstatethatthisstatementis“correct,andalsowhollyirrelevant.”WenotethatthederivationofEuler–Lagrangeequationsinclassicalmechan-icsassumeaformallyconservedHamiltonianandcorre-spondingLagrangian.Hence,thetotalenergywithinanextended-Lagrangianframeworkdoeshaveapracticalvalue.InADMP,thisisnotaproblem,sincethedynamicalvari-ablesmoveonastrictlysmoothsurface,smoothnessbeinglimitedonlybynumericalprecisionandthetotalenergyisformallyconserved.APPENDIX:FORMALCONNECTIONSThedensity-matrixpropagationinADMPsatises 2
E beforeidempotencyisimposed.Incurvy-ELMD,propaga-tionisachievedthroughexponentialtransformation:.Theanti-Hermitianoperator 2
E Theenergyderivativesare.Theoccupied–occupiedandvirtual–virtualblocksofarezero.Theoccupied-virtualblocksarerelated: PRQ=E .Belowweassumethatattheinitialstep,ADMP,andcurvy-ELMDvelocities,,arezero.Wealsoassumethatthetimestepsandctitiousmassesareidentical.UsingEq.anditstranspose,theADMPandcurvy-ELMDdisplacementvectorssarerelated.HenceInADMP,idempotencyisenforcedthroughiterative,and.Thusaf-tertherstpuricationiterationSubsequentidempotencyiterationswillchangetheoccupied–occupiedandvirtual–virtualblocks.Forcurvy-ELMD,anidempotentdensitymatrixisobtainedfromBCH Thus,withthesameinitialdensitiesandcorrespondingve-locities,thetwoapproachesyieldthesamechangesinthedensityuptosecondorder.Theserelations,however,holdonlyforscalar-massADMP.Forthemasstensorialscheme,therelationsarecomplicatedbyidempotencycontributiontooccupied-virtualblocksofdensity.Electronicmail:iyengar@indiana.eduJ.M.HerbertandM.Head-Gordon,J.Chem.Phys.,11542R.CarandM.Parrinello,Phys.Rev.Lett.,2471H.B.Schlegel,J.M.Millam,S.S.Iyengar,G.A.Voth,A.D.Daniels,G.E.Scuseria,andM.J.Frisch,J.Chem.Phys.114,9758S.S.Iyengar,H.B.Schlegel,J.M.Millam,G.A.Voth,G.E.Scuseria,andM.J.Frisch,J.Chem.Phys.115,10291H.B.Schlegel,S.S.Iyengar,X.Li,J.M.Millam,G.A.Voth,G.E.Scuseria,andM.J.Frisch,J.Chem.Phys.117,8694S.S.Iyengar,H.B.Schlegel,G.A.Voth,J.M.Millam,G.E.Scuseria,andM.J.Frisch,Isr.J.Chem.,191J.M.HerbertandM.Head-Gordon,J.Chem.Phys.,027102followingpaper.P.TangneyandS.Scandolo,J.Chem.Phys.116,14N.Rega,S.S.Iyengar,G.A.Voth,H.B.Schlegel,T.Vreven,andM.J.Frisch,J.Phys.Chem.B,4210S.S.IyengarandM.J.Frisch,J.Chem.Phys.,5061S.S.Iyengar,T.J.F.Day,andG.A.Voth,Int.J.Mass.Spectrom.Briey,asetoforthonormalvectorsindimensionbelongstotheSteifelmanifoldandallelementsoftheStiefelmanifoldrelatedbyaunitarytransformformasinglepointontheGrassmannmanifold.H.Goldstein,ClassicalMechanicsAddison-Wesley,Cambridge,MA,027101-2Iyengaretal.J.Chem.Phys.,027101 Commenton“Curvy-stepsapproachtoconstraint-freeabinitiomoleculardynamics,usingatom-centeredbasisfunctions:ConvergencetowardBorn–Oppenheimertrajectories”J.Chem.Phys.121,11542SrinivasanS.IyengarDepartmentofChemistryandDepartmentofPhysics,IndianaUniversity,Bloomington,Indiana-47405H.BernhardSchlegelDepartmentofChemistry,WayneStateUniversity,Detroit,Michigan48202-3489GustavoE.ScuseriaDepartmentofChemistry,RiceUniversity,Houston,Texas77005-1892JohnM.MillamSemichem,Inc.,Mission,Kansas66222-0649MichaelJ.FrischGaussian,Inc.,Wallingford,Connecticut06492