Méthodes de simulation et modélisation Caroline Mellot-Draznieks [email protected] avec la participation de Frédérik Tielens 2 Computational Chemistry Chemistry Theoretical Chemistry Software Hardware Quantum Chemical Packages ? QM/MM, MM approaches, Coarse grain… Computational Chemistry Complements to experiments Properties • Predictions of never observed phenomena! of molecules and solids • • Design of new molecules and materials • Nobel Prize Chemistry 1998 Choose the • Nobel Prize Chemistry 2013 Several appropriate Approaches approaches 3 Introduction • Computational Chemistry? • Each system its approach! • Different groups of software Computational Chemistry QM/MM DFT – Gaussian physics, astrophysics, biochemistry, material sciences… Computational Chemistry QM/MM DFT – Gaussian physics, astrophysics, biochemistry, material sciences… 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 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 Exact solution In practice impossible The goal of computational chemistry is to minimize this residual error while keeping the calculations tractable. 8 Which system for which method/software? • Systems • Methods • Atoms & molecules •MM • Clusters • Parameters Exp./QC • Molecular complexes •HF/Post-HF • Large molecular systems • MPn/CC/CI • Biomolecules • Localized basis functions • Solvated systems • Slater/Gaussian •Biological membranes • Pseudo-potentials • External Perturbations • DFT • Electric/magnetic Fields • Functionals • Solids • Pure DFT/Hybrid • Bulk/Surfaces • Local Basis/Plane waves • Crystals/Amorphous • Pseudopotentials • Metals/oxides • Cluster/Periodic 9 Methodologies No mathematical solution Several Too computationally demanding Approaches Specificity of the system Calculation Complexity Semi Exact Post HF HF DFT Empir. MM Only for Only for Finite and Large & complex H-atom relatively Periodic, systems small etc. systems System Complexity Price to pay … Accuracy!10 11 Limits of Computational Chemistry Exact l Post-HF e v e L DFT on HF ati l u Semi-Empiricalc l a C MM Atomistic Methods Atoms Poly atomic molecules Solids Solids + Solvents Diatomic Molecules Metallic Clusters Amorphous Solids13 # Atoms IntermolecularInteraction between Interactions atoms and molecules « Chemical » Forces short range (1-2 Å) 200-800 kJ/mol 1. E covalent share of electrons « Physical » Forces 2. E electrostatic charge - charge long range (10 Å) charge - dipole NaCl, LiF, Rbi… dipole - dipole 600-1000 kJ/mol charge - non polar atom 3. E repulsion short range 4. E dispersion dipole inst / dipole inst short to long range Ar liquid: U ~ 8 kJ/mol Short range 5. E hydrogen bond 10-40 kJ/mol IntermolecularInteraction between Interactions atoms and molecules Dipole moment qi r i q 1 2 2 .E 1 2 Vdipdip (r) 3 1( 3cos ) ind Viondip (r) cos 2 40r 40r Dipoles in motion (Temp.) 2 2 2 C C 1 2 V (r) 2 2 dipdip 6 4(3 0 ) kBT r 1 '2 Vdipdip.ind (r) 6 0r 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 Forcefield Methods: Forcefield methods They are based on a simple description of the potential energy between atoms using empirical equations Total potential energy: N E(r ) = Ebond + Eangles + Edihedral + E van der Waals + E electrostatic + E polarisation Bonded interactions Non bonded interactions INTRA-molecular only INTER-molecular only FORCEFIELD = a parametrized function of E Eij Eijk Eijkl the potential energy ij ijk ijkl F Q* 1 k (r -r* )2 + 1 k (Q -Q* )2 + k (1 cosnF ) ie ij ij ij ijk ijk ijk ijkl ± ijkl E covalent 2 2 liaison angle torsion e r* qi qj qi qj ie E électrost 4o rij . ie Potentiel de Lennard-Jones E répuls°- 12 6 disp ° r*ij r*ij Eij = ij -2 rij rij r*ij Potentiel de Buckingham ij 6 Eij = Aij exp (-rij/ij) - Bij/rij Explore the Potential Energy Surfaces Initial Crystal Structure + ForceField Unit cell kij r*ij kijk Q*ijk kijkl Fijkl symmetry qi qj ij Atomic coordinates kcore-shell qcore qshell 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 (SiO2, TiO2) - 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 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. 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 Molecular Mechanics 23 Molecular Mechanics 24 Molecular Mechanics 25 26 Molecular Dynamics Principle of Molecular Dynamics Molecular Dynamics/ Energy Minimisation • Move atoms in the direction of the force • Minimisation: one conformation: which is acting on this atom: problem of local minimum • Dynamics: Trajectory with time calculations of averages comparison with macroscopic measurements • The force F is derived from the potential • Dynamics can pass energy barriers energy, which is evaluted using the empirical forcefield or ab initio • Sampling of configurations • Simulation of time-dependant events Molecular Dynamics T Eie Equation de Newton: F = m . a = m . dv/dt = m . d2x/dt2 a = dv/dt a = -1/m dE/dr equations de Newton v = at + v o X i ( t ) X i ( t + Dt ) v i ( t ) v i ( t + Dt ) v = dx/dt • Long x = v.t + x • Rich in informations on o dynamics and structures 2 • Cross energy barriers x = a.t + vo.t + xo Activation energies Diffusion coefficients Trajectories Simulations Based on Statistics • Molecular Dynamics *Grands systèmes *Effet solvant Equations of movement (Newton) *Propriétés macroscopique (capacité calorifique, const. dielectr., diffusion) • Monte Carlo *Souvent parametrisé Different configurations of a system are *Sinon long temps de calculs generated ad random, and a selected is *Propriétés électroniques made on the basis of the Boltzmann distribution Start conditions known: Atom Positions Conditions Conditions Forces … after Dt1 after Dtn Masses Temperature Etc. Properties are calculated as mean values after a certain time (when equilibrium is reached) 29 Statistical ensembles • Microcanonical ensemble (NVE):The thermodynamic state characte- rized 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. Monte Carlo Methods Metropolis algorithm T (N,V,T): 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) Rich in information DE(i, i+1) < 0 p acc = 1 DE(i, i+1) > 0 p acc exp (- DE(i, i+1) /kT) Adsorption Heats (N,V,T) Isotherms (,V,T) Density of states 3. Statistical average: <E(i acc) > Radiale distribution functions NB: importance choice of the move parameters , acceptance/rejectance ratio of 50% for a good sampling Simulated Annealing Decrease the temperature