Introduction to Geometry Optimization

1 Introduction to Geometry Optimization Dr. Zaheer Ul-HaqAssociate Professor Dr. PanjwaniCenter for Molecular Medicine and Drug Research ICCBS, University of Karachi, PakistanDFT 2016, Isfahan, Iran. 6-5-2016 IntroductionInthelasttwodecadesanewfield inchemistr yhasopenedup;experim (SPM),atomicforcemicroscopy(AFM) :elasticpropertiesofpolymersconfor molecular :the equilibrium Geometry Reqand the transition state Geometry Rtsbothcorrespondtothestationarypointson thepotentialenergysurface(PES)(molecular energyE(R)asafunctionofnuclearpositionsR =(R1,R2,..))Theequilibriumgeometry localminimumThetransitionstate saddlepoint(I-storder)Thesepointsaredete rminedbytheconditionthatthefirstderivati vesoftheenergywithrespecttothenucleiposi tionsvanish(thetotalforceactingoneachnuc leusvanishes) molecular structuresItisproposetousethegeometryopt imizationprocedurealsotodetermineenforce dstructuralchangesinamolecule.

2 Inthefieldofcomputationalchemistry,energ yminimizationistheprocessoffindinganarra ngementinspaceofacollectionofatomswhere, thenetinter-atomicforceoneachatomisaccep tablyclosetozeroandthepositiononthepoten tialenergysurface(PES)isastationarypoint . Thecollectionofatomsmightbeasinglemolecu le,anion,acondensedphase,atransitionstat eorevenacollectionofanyofthese. Asanexample,whenoptimizingthegeometryofa watermolecule,oneaimstoobtaintheH-Hbondl engthsandtheH-OHbondanglewhichminimizeth eforces. Themotivationforperformingageometryoptim izationisthephysicalsignificanceoftheobt ainedstructure:optimizedstructuresoftenc orrespondtoasubstanceasitisfoundinnature andthegeometryofsuchastructurecanbeusedi navarietyofexperimentalandtheoreticalinv estigations. Typica l ly,butnotalways,theprocessseekstofindthe geometryofaparticulararrangementoftheato msthatrepresentsalocalorglobalenergymini mum. Insteadofsearchingforglobalenergyminimum ,itmightbedesirabletooptimizetoatransiti onstate,thatis, ,certaincoordinates(suchasachemicalbondl ength) OptimizationLe Chatliers PrincipleThe optimum Geometry is the Geometry which minimizes the strain on a given system.

3 Any perturbation from this Geometry will induce the system to change so as to reduce this perturbation unless prevented by external forcesMathematical Surface Reflects This Principle!!Features of Potential Energ y S urfacesCyclohexaneLocal maximaGlobal minimumGlobal maximaLocal minimaEx. PESS addle pointLocal minimumGlobal minimumPotential Energy Surface Terms Gradient-the first derivative of the energy with respect to Geometry (X, Y also termed the Force(strictly speaking, the negativeof the gradient is the force) Stationary Points-points on the PES where the gradient (or force) is zero; this includes Maxima, Minima, Transition States (which are first order saddle points),and higher orderSaddle In order to distinguish among the latter, one must examine the second derivativesof the PES with respect to Geometry ; the matrix of these is termed a Hessian(or force) matrix. Diagonalization of this matrix yields Eigenvectorswhich are normal modes of vibration; the Eigenvaluesare proportional tothe square of the vibrational frequency.)

4 (IR spectra can be derived from these)Sign of 2nd Derivatives The signof the second derivative can be used to distinguish between Maximaand Minimaon the PES Minimaon the PES have only positive eigenvalues (vibrational frequencies) Maximaor Saddle Points(maximum in one direction but minimum in other directions) have one or more negative(imaginary) frequencies. A frequency calculationmust be performed to determine the sign of the vibrational Geometry and mathematical interpretation Thegeometryofasetofatomsormoleculescanbe describedbyCartesiancoordinatesoftheatom sor,internalcoordinatesformedfromasetofb ondlengths,bondanglesanddihedralangles. Givenasetofatomsandavector,r,describingt heatoms'positions,onecanintroducetheconc eptoftheenergyasafunctionofthepositions, E(r). Geometryoptimizationisthenamathematicalo ptimizationproblem,inwhichitisdesiredtof indthevalueofrforwhichE(r)isatalocalmini mum,thatis,thederivativeoftheenergywithr especttothepositionoftheatoms, E/ r, , (r)couldbebasedonquantummechanics(usinge itherdensityfunctionaltheoryorsemi-empir icalmethods),forcefields, aspects of Optimization Somemethodsuchasquantummechanicscanbeuse dtocalculatetheenergy,E(r),thegradientof thePES,thatis,thederivativeoftheenergywi threspecttothepositionoftheatoms, E/ r.

5 Anoptimizationalgorithmcanusesomeorallof E(r), E/ rand E/ ri rjtotrytominimizetheforcesandthiscouldin theorybeanymethodsuchasgradientdescent,c onjugategradientorNewton'smethod. Formostsystemsofpracticalinterest,howeve r, ,forexample,areredundantsinceanon-linear moleculewithNatomshas3N ,Cartesiancoordinatesarehighlycorrelated ,thatis, ,because,forexample, , of freedom restriction Somedegreesoffreedomcanbeeliminatedfroma noptimization,forexample, state Optimization Transitionstatestructurescanbedetermined bysearchingforsaddlepointsonthePES. Afirst-ordersaddlepointisapositionontheP EScorrespondingtoaminimuminalldirections exceptone;asecond-ordersaddlepointisamin imuminalldirectionsexcepttwo,andsoon. Algorithmstolocatetransitionstategeometr iesfallintotwomaincategories:localmethod sandsemi-globalmethods. ,suchastheDimermethod, searches ,or,thenegativeEigenvaluecorrespondingto thereactioncoordinatemustbegreaterinmagn itudethantheothernegativeEigenvalues.

6 Giventheabovepre-requisites,alocaloptimi zationalgorithmcanthenmove"uphill"alongt heEigenvectorwiththemostnegativeEigenval ueand"downhill"alongallotherdegreesoffre edom, method dimer (ultimatelynegative).Activation Relaxation Technique (ART) TheActivationRelaxationTechnique(ART) (computedusingtheLanczosalgorithm)ontheP EStoreachthesaddlepoint,relaxingintheper pendicularhyperplanebetweeneach"jump"(ac tivation) methods Chain-of-statemethodsuseaseriesofvectors ,thatispointsonthePES,connectingthereact antandproductofthereactionofinterest,rre actantandrproduct, r ycommonly,thesepointsarereferredtoasbead sduetoananalogyofasetofbeadsconnectedbys tringsorsprings, , forexample,foraseriesofN+1beads,beadimig htbegivenby wherei 0,1,.., ,E(ri),andforces,- E/ ,spacingconstraintsmustbeappliedsothatea chbeadridoesnotsimplygetoptimizedtothere actantandproductgeometry. Oftenthisconstraintisachievedbyprojectin goutcomponentsoftheforceoneachbeadri,ora lternativelythemovementofeachbeadduringo ptimization, ,ifforconvenience,itisdefinedthatgi= E/ ri,thentheenergygradientateachbeadminust hecomponentoftheenergygradientthatistang entialtothereactionpathwayisgivenby whereIistheidentitymatrixand , transit Thesimplestchain-of-statemethodistheline arsynchronoustransit(LST) (QST)methodextendsLSTbyallowingaparaboli creactionpath, elastic bandInNudgedelasticbandmethod,thebeadsal ongthereactionpathwayhavesimulatedspring forcesinadditiontothechemicalforces,- E/ ri, ,theforcefioneachpointiisgivenbywhereist hespringforceparalleltothepathwayateachp ointri(kisaspringconstantand i,isaunitvectorrepresentingthereactionpa thtangentatri).

7 Inatraditionalimplementation, (nudgedelasticband)method,suchincludingt heclimbingimageNEB, method Thestringmethodusessplinesconnectingthep oints,ri, ,thepointsmightbemovedaccordingtotheforc eactingonthemperpendiculartothepath,andt hen,iftheequidistanceconstraintbetweenth epointsisno-longersatisfied,thepointscan beredistributed,usingthesplinerepresenta tionofthepathtogeneratenewvectorswiththe requiredspacing. Variationsonthestringmethodincludethegro wingstringmethod,inwhichtheguessofthepat hwayisgrowninfromtheendpoints(thatisther eactantandproducts) with other techniques ,subjecttotemperature,chemicalforces,ini tialvelocities,Brownianmotionofasolvent, andsoon,viatheapplicationofNewton' , ,bycontrast,doesnotproduceda"trajectory" withanyphysicalmeaning itisconcernedwithminimizationoftheforces actingoneachatominacollectionofatoms, , METHODSM ethods of Optimisation Energy only: simplex Energy and first derivatives (forces).

8 Steepest descents (poor convergence) conjugate gradients (retains information) approximate Hessian update Energy, first and second derivatives Newton-Raphson Broyden(BFGS)updating of Hessian (reduces inversions) Rational Function Optimisation(for transition states/and soft modes)Energy Only (Univariate) Method Simplest to implement Proceeds one direction until energy increases, then turns 90 , etc. Least efficient many steps steps are not guided Not used very Descents Simplest method in use Start at xo Minimize f(x) along the line defined by the gradient Follows most negative gradient (max. force) Fastest method from a poor starting Geometry Converges slowly near the energy minimum, start again until tolerance is reachedSteepest DescentsPerformance depends on Eigenvalues of Hessian ( max / min) Starting pointConjugate Gradients Same idea, butretaininginformationaboutpreviousstep s Search directions conjugate (orthogonal) to previous Convergence assured for N steps Variations on this procedure are the Fletcher-Reeves, the Davidon-Fletcher-Powell and the Derivative Methods The 2nd derivative of the energy with respect to X,Y,Z [the Hessian] determines the pathway.

9 Computationally more involved, but generally fast and reliable, esp. near the minimum. Geometry Optimization (Summary) Optimum structure gives useful information First Derivative is Zero -At minimum/maximum Use Second Derivative to establish minimum/maximum As N increases so does dimensionality/complexity/beauty/difficu lty0)(=rdrdV!!max,0)(min,0)(2222<>rdrVdrdrVd!!!! Geometry Optimization (Summary) Method used depends on System size 1-d (line search, bracketing, steepest descent) N-d local (Downhill) W/o derivatives Simplex Direction set methods (Powell s) W/ derivatives Conjugate gradient Newton or variable metric methods N-d Global Monte Carlo Simulated Annealing Genetic Algoritms For m of e ne r g y Analytic Not analytic