Méthodes de simulation et modélisation

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1 Méthodes de simulation et modélisation Caroline Mellot-Draznieks avec la participation de Frédérik Tielens

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3 Computational Chemistry Chemistry? Theoretical Chemistry Software Hardware Quantum Chemical Packages QM/MM, MM approaches, Coarse grain Computational Chemistry Properties of molecules and solids Complements to experiments Predictions of never observed phenomena! Design of new molecules and materials Nobel Prize Chemistry 1998 Nobel Prize Chemistry 2013 Several Approaches Choose the appropriate approaches 3

4 Introduction Computational Chemistry? Each system its approach! Different groups of software

5 Computational Chemistry QM/MM DFT Gaussian physics, astrophysics, biochemistry, material sciences

6 Computational Chemistry QM/MM DFT Gaussian physics, astrophysics, biochemistry, material sciences

7 Which system for which method/software? Is there a particular category of computations that is of most interest? Structure: Geometry optimizations based on model chemistry Comparison of computational results to experimental results Transition state geometries Property: Electrical, optical, magnetic, etc Determination of spectra, from NMR to X-Ray Calculation of quantum descriptors Quantitative structure-property relationship (QSPR) (Re)activity: Reaction mechanisms in chemistry and biochemistry QSAR-types of problems Quantitative structure-activity relationship (QSAR) is the process by which chemical structure is quantitatively correlated with a well defined process 7

8 Each System its approach Approaches involve different approximations: Simplified forms = easier or faster to solve Approximations by limiting the size of the system For example, most ab initio calculations make the Born- Oppenheimer approximation, which greatly simplifies the underlying Schrödinger Equation by freezing the nuclei in place during the calculation. Ab Initio Methods In practice impossible Exact solution The goal of computational chemistry is to minimize this residual error while keeping the calculations tractable. 8

9 Which system for which method/software? Systems Atoms & molecules Clusters Molecular complexes Large molecular systems Biomolecules Solvated systems Biological membranes External Perturbations Electric/magnetic Fields Solids Bulk/Surfaces Crystals/Amorphous Metals/oxides Methods MM Parameters Exp./QC HF/Post-HF MPn/CC/CI Localized basis functions Slater/Gaussian Pseudo-potentials DFT Functionals Pure DFT/Hybrid Local Basis/Plane waves Pseudopotentials Cluster/Periodic 9

10 Methodologies Several Approaches No mathematical solution Too computationally demanding Specificity of the system Calculation Complexity Exact Only for H-atom Post HF Only for relatively small systems HF Finite and Periodic, etc. DFT System Complexity Semi Empir. MM Large & complex systems 10 Price to pay Accuracy!

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12 Limits of Computational Chemistry Exact Calculation Level Post-HF DFT HF Semi-Empirical MM Atomistic Methods Atoms Poly atomic molecules Solids Solids + Solvents Diatomic Molecules Metallic Clusters # Atoms Amorphous Solids 13

13 Intermolecular Interactions Interaction between atoms and molecules «Chemical» Forces 1. E covalent share of electrons short range (1-2 Å) kj/mol «Physical» Forces 2. E electrostatic charge - charge charge - dipole dipole - dipole charge - non polar atom long range (10 Å) NaCl, LiF, Rbi kj/mol 3. E repulsion short range 4. E dispersion dipole inst / dipole inst short to long range Ar liquid: U ~ 8 kj/mol 5. E hydrogen bond Short range kj/mol

14 Intermolecular Interaction between Interactions atoms and molecules Dipole moment q r i i V 1q2 r) 4 r ion dip ( 2 0 cos V dip dip ( r) (1 3cos ) 3 4 r 0 ind.e Dipoles in motion (Temp.) C V 2 3(4 ) k T dip dip C ( r) 6 r 0 B V dip dip. ind ( r) 2 1 ' r 0 2 6

15 Molecular Mechanics Model based on classical mechanics (not quantum) Molecules are treated like an ensemble of atoms in space linked to each other with bonds described by functions of elastic potentials Very big systems F=-kx Results not always reliable 16

16 Forcefield Methods: Forcefield methods They are based on a simple description of the potential energy between atoms using empirical equations Total potential energy: E(r N ) = E bond + E angles + E dihedral + E van der Waals + E electrostatic + E polarisation Bonded interactions INTRA-molecular only Non bonded interactions INTER-molecular only

17 FORCEFIELD = a parametrized function of the potential energy E ij E ij ijk E ijk ijkl E ijkl E ie covalent Q * e r* F 1 k ij (r ij -r* ij ) k ijk (Q ijk -Q* ijk ) 2 + k ijkl (1±cosnF ijkl ) 2 2 liaison angle torsion E ie électrost q i q j q i q j. 4 o r ij E ie répuls disp Potentiel de Lennard-Jones E ij = ij r* ij r* ij r ij r ij ij r*ij Potentiel de Buckingham E ij = A ij exp (-r ij / ij ) - B ij /r ij 6

18 Explore the Potential Energy Surfaces Initial Crystal Structure + ForceField Unit cell symmetry Atomic coordinates k ij r* ij k ijk Q* ijk k ijkl F ijkl q i q j ij k core-shell q core q shell SIMULATIONS = Exploration of the HYPERSURFACE OF ENERGY

Explore the Potential Energy Surfaces The minima of the total potential energy corresponds to - conformers of a single organic molecule - polymorphs of a solid (SiO 2, TiO 2 ) - adsorption sites on a

19 Explore the Potential Energy Surfaces The minima of the total potential energy corresponds to - conformers of a single organic molecule - polymorphs of a solid (SiO 2, TiO 2 ) - adsorption sites on a surface The PES possesses many minima and there is no general mathematical approach to find the global minimum. One uses numerical approaches that allow to find local minima. There are different strategies to explore the PES, depending on the kind of information wanted. STATISTICAL METHODS ENERGY MINIMISATION DYNAMICS MONTE CARLO

20 Molecular Mechanics Ball and spring description of molecules Able to compute relative strain energies Cheap to compute Lots of empirical parameters that have to be carefully tested and calibrated equilibrium geometries No electronic interactions into account No information on reactivity Cannot readily handle reactions involving the making and breaking of bonds ReaxFF(William A Goddard III) for hydrocarbons reactions, transition metals catalysed nanotube formation, zeolites, silica surfaces, benchmarking DFT.

MSI/Biosym Molecular Modelling Software NAMD - Scalable Molecular Dynamics TINKER package for molecular

21 Molecular Mechanics AMBER CHARMM VMD - Visual Molecular Dynamics MOLDY - Free MD program GROMACS Molecular Dynamics on Parallel Computers GROMOS Dynamic Modelling of Molecular Systems MacroModel - Molecular Modelling MSI/Biosym Molecular Modelling Software NAMD - Scalable Molecular Dynamics TINKER package for molecular mechanics and dynamics SYBYL - software from Tripos X-PLOR- MM program free for Academics DNAtools-Web tools to analyze DNA 22

22 Molecular Mechanics 23

23 Molecular Mechanics 24

24 Molecular Mechanics 25

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26 Molecular Dynamics Principle of Molecular Dynamics Molecular Dynamics/ Energy Minimisation Move atoms in the direction of the force which is acting on this atom: Minimisation: one conformation: problem of local minimum Dynamics: Trajectory with time calculations of averages comparison with macroscopic measurements The force F is derived from the potential energy, which is evaluted using the empirical forcefield or ab initio Dynamics can pass energy barriers Sampling of configurations Simulation of time-dependant events

27 Molecular Dynamics E ie T Equation de Newton: F = m. a = m. dv/dt = m. d 2 x/dt 2 a = dv/dt a = -1/m de/dr equations de Newton X i ( t ) v i ( t ) X i ( t + Dt ) v i ( t + Dt ) Long Rich in informations on dynamics and structures Cross energy barriers v = at + v o v = dx/dt x = v.t + x o x = a.t 2 + v o.t + x o Activation energies Diffusion coefficients Trajectories

Simulations Based on Statistics Molecular Dynamics Equations of movement (Newton) Monte Carlo Different configurations of a system are generated ad random, and a selected is made on the basis of the

28 Simulations Based on Statistics Molecular Dynamics Equations of movement (Newton) Monte Carlo Different configurations of a system are generated ad random, and a selected is made on the basis of the Boltzmann distribution *Grands systèmes *Effet solvant *Propriétés macroscopique (capacité calorifique, const. dielectr., diffusion) *Souvent parametrisé *Sinon long temps de calculs *Propriétés électroniques Start conditions known: Atom Positions Forces Masses Temperature Etc. Conditions after Dt 1 Conditions after Dt n Properties are calculated as mean values after a certain time (when equilibrium is reached) 29

Statistical ensembles Microcanonical ensemble (NVE):The thermodynamic state characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed energy, E.

29 Statistical ensembles Microcanonical ensemble (NVE):The thermodynamic state characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed energy, E. This corresponds to an isolated system. Canonical Ensemble (NVT): it is a collection of all systems whose thermodynamic state is characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed temperature, T. Isobaric Isothermal Ensemble (NPT):This ensemble is characterized by a fixed number of atoms, N, a fixed pressure, P, and a fixed temperature, T. Grand canonical Ensemble(μVT):The thermodynamic state for this e ensemble is characterized by a fixed chemical potential, μ, a fixed volume, V, and a fixed temperature, T.

30 Monte Carlo Methods Metropolis algorithm T (N,V,T): Rich in information Adsorption Heats (N,V,T) Isotherms (,V,T) Density of states Radiale distribution functions 1. Calculate the energy E(i) of the initial configuration of the N atomes 2. Random move of atoms (translation and rotation), with new energy E (i+1) DE (i, i+1) < 0 p acc = 1 DE (i, i+1) > 0 p acc exp (- DE (i, i+1) /kt) 3. Statistical average: <E (i acc) > NB: importance choice of the move parameters, acceptance/rejectance ratio of 50% for a good sampling

31 Simulated Annealing T Decrease the temperature T T Identifies various local minima

32 Adsorption of cyclohexane in zeolite HY NEUTRON DIFFRACTION low température (5 K) 2.74 Å Rietveld Refinement Location of adsorbed molecules In nanoporous frameworks 50 % of cyclohexane located In 12-ring windows

33 Adsorption of cyclohexane in zeolite HY ENERGY MINIMISATION (zéro K) Monte Carlo docking: Random generation of 20 initial configurations of cyclohexane 50 % 3 Å Energy Minimisation de of each of the 20 configurations 50 % 2.9 Å 2.8 Å 2.8 Å 2.8 Å 2.8 Å Vitale, Mellot and Cheetham, J. Phys. Chem. 1997, 101, 9886.

(a.u.) 16 methanol 32 methanol 48 methanol 96 methanol Intensity (a.u.)

34 CH 3 OH distribution through pair distribution functions Dominant Na + (II) - Om interactions Hydrogen bond between methanol molecules Intensity (a.u.) 16 methanol 32 methanol 48 methanol 96 methanol Intensity (a.u.) Liquid 1,2 1,6 2,0 2,4 2,8 3,2 3,6 4,0 Distance (Å) Distance (Å)

35 Quantum Chemistry 1926 Schrödinger: finds the solution for the hydrogen atom using quantum mechanics HY = EY systems No solution nor operational method, nor computation power for multi-electronic Hartree-Fock Approach (Each e - is described in the field of the other e -, No electron correlation, SCF method) systems! Poly-electronic Unpaired electrons & correlation of electron movements Looking for new methods for the calculation of systems in which electron correlation is important. 36

36 Quantum Chemistry 37

37 Post-HF methods 38

38 Post-HF methods 39

39 Density Functional Theory 40

40 The VASP approach Periodic models + DFT + plane waves + pseudopotentials All electrons Pseudopotentials meta-gga, hybrid GGA LDA periodic not periodic Atomic orbital Plane waves Numeric

41 Density Functional Density Functional Theory Theory 42

42 Self-consistent calculation procedure 43

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44 Exchange-correlation functional 45

45 Accuracy of DFT 46

46 Accuracy of DFT 47

47 Accuracy of DFT 48

48 Limitations of DFT 49

49 Limitations of DFT 50

50 Strategies Type of interactions in the system matters but also the Size! Methods Calculation Strategies 51

51 Calculation Strategies Finite Size vs. Periodic Simulations based on statistics Approaches for systems with a large number of atoms Tools Calculation of electronic properties Calculation of macroscopic properties 52

crossing points Pure DFT Heavy calcs for Hybrid methods & localized basis sets Finite Size Hybrid methods: B3LYP

52 Finite vs. Periodic? Periodic No edge effects Larger models Plane wave basis set IR and Raman frequencies Specific calculations: TSs, crossing points Pure DFT Heavy calcs for Hybrid methods & localized basis sets Finite Size Hybrid methods: B3LYP Localized basis sets IR and Raman intensities PBE/plane waves VASP 4.6 program Specific calculations: TSs, crossing points B3LYP 6-311G(2d,p) Gaussian03 program Smaller model Edge effects No coverage effect 53

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56 Approximations for Systems with a Large Number of Atoms QM/MM & ONIOM Ex. Bio Systems (Proteins, Enzymes, etc.) Ex. Zeolites Level 1: Ab Initio Level 2: Semi-Empirical/MM *Large systems *Very versatile *long calculation times *difficult description border zones between levels *MM Potential not always available Level 3: MM/charges 57

57 «The DFT» from the physicists used by chemists DFT not Computational but Conceptual The mathematical framework of DFT permits de precise definition of chemistry concepts (link to reactivity) Electronegativity Hardness and Softness Chemical Potential Variations of the electron density (Fukui) Chemical Reactivity Theory 58

58 Molecular Modeling Software Molecular Mechanics Quantum Chemistry Molecular visualization and editing Other Companies Academics Freeware Web Applications 59

59 Quantum Chemistry

60 Molecular visualization and editing Molecules Periodic Systems Molden Materials Studio Jmol Crystal Maker GaussView VMD ECCE ModelView Arguslab MOLDRAW (Molecules and crystals) VMD Molekel (Molecules and crystals) VegaZZ DeepView Discovery Studio MolView and Molview Lite - Macintosh Selection!

Example questions for Molecular modelling (Level 4) Dr. Adrian Mulholland

Example questions for Molecular modelling (Level 4) Dr. Adrian Mulholland 1) Question. Two methods which are widely used for the optimization of molecular geometies are the Steepest descents and Newton-Raphson