Computational Chemistry for Dummies Svein Saebø Department of Chemistry Mississippi State University

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  • Computational Chemistry for DummiesSvein SaebDepartment of ChemistryMississippi State University

  • Computational Chemists / Theoretical ChemistsComputational Chemists use existing computer software (often commercial) to study problems from chemistryTheoretical Chemists develop new computational methods and algorithms.

  • Theoretical / Computational ChemistryTool: Modern ComputerApplication ofMathematicsPhysicsComputer Scienceto solve chemical problems

  • Chemistry Molecular ScienceStudies of moleculesLarge Molecules, macromolecules:Proteins, DNABiochemistry, Medicine, Molecular BiologyOther polymersMaterial Science (physics)

  • Computational ChemistryWHYdo theoretical calculations?

    WHATdo we calculate?

    HOWare the calculations carried out?

  • WHY?Evolution of Computational ChemistryConfirmation of experimental resultsInterpretation of experimental resultsassignmentPrediction of new resultsThe truth is experimental!AdvantagesAvoid experimental difficultiesSafetyCostWidely used by chemical and pharmaceutical industryVisualization

  • WHAT?Molecular SystemOne or several moleculesCollection of atomsStructure (geometry):3-dimensional arrangement of these atoms

  • WHAT?Molecular Potential SurfacesA molecular system with N atoms is described by 3N Cartesian (x,y,z) or 3N-6 internal coordinates (bond lengths, angles, dihedral angles)R = {q1 ,q2 ,q3 q4 ,.. q3N-6}Potential Energy Surface (PES) : E(R)the energy as a function of the three-dimensional arrangement of the atoms.

  • Diatomic MoleculeOnly one coordinate:R= bond lengthPotential Surface: E(R)

  • E(R)Morse Potential: E=D(1-exp(-F(R-R0))2Parabola: E=1/2 F (R-R0)2First derivative: dE/dR = F (R-R0)Second derivative d2E/dR2 =F (force-constant, Hooks Law)Vibrational Frequency n=1/(2p) (F/m)

  • Intercept through a PES.Stationary pointsMinimaSaddle points (transition states)

  • Potential SurfacesWe are normally interested in stationary pointsGlobal Minimum : Equilibrium StructureLocal Minima: Other (stable?) forms of the systemSaddle Points: Transition States

  • Stationary PointsMathematical ConceptdE / dqi = 0 for all iSlope of potential energy curve = 0Minimum: second derivatives positiveMaximum: second derivatives negativeSaddle Point: All second derivatives positive except one (negative)

  • WHAT do we calculate?Energy: E(q1,q2,q3,.q3N-6)Gradients:dE/dqiForce Constants:Fi,j = d2E/dqidqj+ Other second derivatives with respect to nuclear coordinateselectric , magnetic fields

  • Gradients (dE/dq)Needed for automatic determination of structureForce (f) in direction of coordinate qf = -(dE/dq) = -F (R-R0)Geometry relaxed until the forces vanishQuadratic surface:R+ f/F=R-F(R-R0)/F=R0

  • Force Relaxation:(1) start with an initial guess of the geometry Rn and the force constants F (a matrix) (n=1)(2) calculate the energy E(Rn), and the gradient g(Rn) at Rn (3) get an improved geometry:Rn+1 = Rn F-1 g(Rn) (4) Check the largest element of g(R)If larger than THR (e.g. 10-6) n=n+1 go to (2)If smaller than THR finishedThe final result does not depend on F

  • Optimization MethodsCalculation of the gradient at several geometries provide information about the force-constants FWidely used optimizations methods:Newton RaphsonSteepest DecentConjugate gradientVariable Metric (quasi Newton)

  • WHY Second Derivatives?provide many important molecular(spectroscopic) propertiesTwice with respect to the nuclear coordinatesF=d2E/dqidqj Force Constants Vibrational Frequencies Dipole moment derivatives IR intensitiesPolarizability Derivatives Raman IntensitiesOnce with respect to external magnetic field, once with respect to magnetic moment. Magnetic shielding- chemical shifts (NMR)

  • Summary (What?)Common for all Computational chemistry Methods:

    Potential Energy SurfacesNormally seeking a local minimum (or a saddle point)Get energy and structure

    Spectroscopic properties are normally only calculated by quantum mechanical methods

  • HOW?How do the computer programs work?Many different computational chemistry programsWide range of accuracyLow price - low accuracyUser InterfaceInput/outputMany modern programs are very user friendlymenu-driven point and click

  • HOW?Molecular MechanicsBased on classical mechanicsNo electrons or wave-functionInexpensive; can be applied to very large systems (e. g. proteins)Quantum Mechanical MethodsSeek approximate solutions of the Schrdinger equation for the system HY = E Y

  • Quantum Mechanical MethodsExact solutions only for the hydrogen atom s, p, d, f functionsMolecules: LCAO-approximation fi = S CimCm {C} H-like atomic functions: Basis set (AO) {f} Molecular orbitals (MO)

  • Quantum Mechanical MethodsSemiempirical MethodsUse parameters from experimentsInexpensive, can be applied to quite large systemsAb Initio MethodsLatin: From the beginningNo empirical data used [except the charge of the electron (e) and the value of Plancks constant (h)]

  • Semi empirical Methodsp-electrons onlyHckel, PPP (Pariser Parr Pople)Semi empirical MO methodsExtended HckelCNDO, INDO, NDDOMINDO, MNDO,AM1,PM3(Dewar)

  • Ab Initio MethodsHartree-Fock (SCF) Methodbased on orbital approximationSingle ConfigurationWave-Function:Y a single determinantEach electron is interacting with the average of the other electrons Absolute Error (in the energy) : ~1% Formal Scaling : N4 (N number of basis functions)

  • Ab Initio MethodsElectron CorrelationEcorr = E(exact) - EHFMany Configurations (or determinants)MP2-scale as N5CI, QCISD, CCSD, MP3, MP4(SDQ) - scale as N6

  • Power-Law ScalingAb Initio Methods: N4 N6N proportional to the size of the systemDouble the size: Price increases by a factor of ~60! (from 1 minute to 1 hour)Increase computational power with a factor of 10001000 ~ 3.56Could only do systems 3-4 time as big as today

  • The Future of Quantum ChemistryDirac (1929):The underlying physical laws necessary for the mathematical theory of the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble1950s it is wise to renounce at the outset any attempt at obtaining precise solutions of the Schrodinger equation for systems more complicated than the hydrogen molecule ionLevine: Quantum ChemistryFifth Edition

  • Future of Quantum ChemistryThe difficulties at least partly overcome by application of high speed computers1998 Nobel Price Committee: (Chemistry Price shared by Walter Kohn and John Pople):Quantum Chemistry is revolutionizing the field of chemistryWe are able to study chemically interesting systems, but not yet biologically interesting systems using quantum mechanical methods.

  • Low-Scaling Methods for Electron Correlation Low Scaling MP2 Near linear scaling for large systemsformal scaling N5Applied to polypeptides (polyglycines up to 50 glycine units, C100H151N50O51)

    Saebo, Pulay, J. Chem. Phys. 2001, 115, 3975.Saebo in: Computational Chemistry- Review of Current Trends Vol. 7, 2002.

  • Molecular MechanicsBased on classical mechanicsNo electrons or wave-functionInexpensive; can be applied to very large systems, macromolecules.PolymersProteinsDNA

  • Molecular MechanicsE=Estr + Ebend + Eoop + Etors + Ecross + EvdW + EesEstr bond stretchingEbend bond-angle bendingEoop out of planeEtors internal rotationEcross combinations of these distortions

    Non-bonded interactions:EvdW van der Waals interactionsEes electrostatic

  • Force FieldsThe explicit form used for each of these contributions is called the force field.

    Will consider bond stretching as an example

  • Bond StretchingE(R)=D (1-exp(-F(R-R0))2

    D - dissociation energyF - force-constantR0 - natural bond lengthThe parameters D, F, R0 are part of the so-called force-field.The values of these parameters are determined experimentally or by ab initio calculations

  • Force-fieldsSimilar formulas and parameters can be defined for:Bond angle bendingOut of plane bendingTwisting (torsion)Hydrogen bonding , etc.

  • Molecular Mechanics

  • Force-fieldsEach atom is assigned to an atom type based on: atomic number and molecular environmentExamples:Saturated carbon (sp3)Doubly bonded carbon (sp2)Aromatic carbonCarbonyl carbon..

  • Force-fieldsAn energy function and parameters (D, F, R0) are assigned to each bond in the molecule.In a similar fashion appropriate functions and parameters are assigned to each type of distortionHydrogen bonds and non-bonding interactions are also accounted for.

  • Commonly Used Force FieldsOrganic Molecules:MM2, MM3, MM4 (Allinger)Peptides,proteins, nucleic acids AMBER (Assisted Model Building with Energy Refinement) (Kollman) CHARMM (Chemistry at HARvard Molecular Modeling (Karplus)MMFF94 (Merck Molecular Force Field) (Halgren)

  • More Force Fields..CFF93, CFF95 (Consistent Force Field)Hagler (Biosym, Molecular Simulations)SYBYL or TRIPOS (Clark)

  • Computational Chemistry and NMRPowerful technique for protein structure determination competitive with X-ray crystallography.NOE: Nuclear Overhauser EffectProton-proton distancesStructure optimized under NOE constraintsChemical shifts are also used

  • Protein G Angela Gronenborn, NIH

  • 5-Enolpyruvylshuikimate-3-phosphate Synthase

  • AcknowledgementsDr. John K.Young, Washington State University