Méthodes de simulation et modélisation

Caroline Mellot-Draznieks

Caroline.mellot-draznieks@college-de-france.fr

avec la participation de Frédérik Tielens

2

3

Several

Approaches

• Nobel Prize Chemistry 1998

• Nobel Prize Chemistry 2013

Computational Chemistry

Chemistry

Computational Chemistry

Theoretical Chemistry Software

Quantum Chemical Packages QM/MM, MM approaches, Coarse grain…

Properties of molecules and solids

• Complements to experiments

• Predictions of never observed phenomena!

• Design of new molecules and materials

Hardware

Choose the

appropriate

approaches

?

Introduction

• Computational Chemistry?

• Each system its approach!

• Different groups of software

Computational Chemistry

DFT – Gaussian physics, astrophysics, biochemistry,

material sciences…

QM/MM

Computational Chemistry

DFT – Gaussian physics, astrophysics, biochemistry,

material sciences…

QM/MM

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

8

Each System its approach …

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.

Approaches involve different approximations:

• Simplified forms = easier or faster to solve

• Approximations by limiting the size of the system

The goal of computational chemistry is to minimize this

residual error while keeping the calculations tractable.

Ab Initio Methods Exact solution

In practice impossible

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

10

Several

Approaches

Exact Post HF

Calculation Complexity

No mathematical solution

Too computationally demanding

Specificity of the system

Only for

H-atom

Only for

relatively

small

systems

Finite and

Periodic,

etc.

Large & complex

systems

Price to pay … Accuracy!

Methodologies

DFT Semi Empir. MM HF

System Complexity

11

13

Limits of Computational Chemistry

# Atoms

Ca

lcu

lation

Le

ve

l

Atoms

Diatomic Molecules

Poly atomic molecules

Metallic Clusters

Solids

Amorphous Solids

Solids + Solvents

MM

Semi-Empirical

HF

DFT

Post-HF

Exact

Atomistic Methods

Interaction between atoms and molecules

« Chemical » Forces 1. E covalent share of electrons

« Physical » Forces

2. E electrostatic charge - charge

charge - dipole

dipole - dipole

charge - non polar atom

3. E repulsion short range 4. E dispersion dipole inst / dipole inst

5. E hydrogen bond

short range (1-2 Å) 200-800 kJ/mol

long range (10 Å) NaCl, LiF, Rbi… 600-1000 kJ/mol

short to long range Ar liquid: U ~ 8 kJ/mol

Short range 10-40 kJ/mol

Intermolecular Interactions

Interaction between atoms and molecules Intermolecular Interactions

ii rq

Dipole moment

cos4

)(2

0

21

r

qrV dipion

)cos31(4

)( 23

0

21

rrV dipdip

6)(

r

CrV dipdip Tk

CB

20

22

21

)4(32

Dipoles in motion (Temp.)

Eind

.

60

221

.

')(

rrV inddipdip

16

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

F=-kx Results not always

reliable

Very big systems

They are based on a simple description of the potential energy

between atoms using empirical equations

Total potential energy:

E(rN) = Ebond + Eangles + Edihedral + E van der Waals + E electrostatic + E polarisation

Bonded interactions

INTRA-molecular only Non bonded interactions

INTER-molecular only

Forcefield Methods: Forcefield methods

Eieélectrost

.

qi qj

4o rij

qi qj

Eiecovalent

e r*

Q*

F

1 kij (rij-r*ij)2 + 1 kijk (Qijk-Q*ijk)

2 + kijkl(1±cosnFijkl) 2 2

liaison angle torsion

Eierépuls°-

disp°

Potentiel de Lennard-Jones

ij

Potentiel de Buckingham

Eij = Aij exp (-rij/ij) - Bij/rij6

r*ij

rij

r*ij

rij -2

12 6

Eij = ij

ijkl

ijkl

ijk

ijk

ij

ij EEEEFORCEFIELD = a parametrized function of the potential energy

r*ij

Initial Crystal Structure ForceField kij r*ij kijk Q*ijk kijkl Fijkl qi qj ij

kcore-shell qcore qshell

Unit cell symmetry Atomic coordinates

+

SIMULATIONS

=

Exploration of the HYPERSURFACE

OF ENERGY

Explore the Potential Energy Surfaces

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.

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

There are different strategies to explore the PES,

depending on the kind of information wanted.

ENERGY MINIMISATION DYNAMICS

STATISTICAL METHODS

MONTE CARLO

Explore the Potential Energy Surfaces

• 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

22

•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

Molecular Mechanics

23

Molecular Mechanics

24

Molecular Mechanics

25

Molecular Mechanics

26

Molecular Dynamics

Principle of Molecular Dynamics

• Move atoms in the direction of the force which is acting on this atom:

• The force F is derived from the potential energy, which is evaluted using the empirical forcefield or ab initio

Molecular Dynamics/ Energy Minimisation

• Minimisation: one conformation:

problem of local minimum

• Dynamics: Trajectory with time

calculations of averages

comparison with macroscopic

measurements

• Dynamics can pass energy barriers

• Sampling of configurations

• Simulation of time-dependant events

• Long • Rich in informations on dynamics and structures • Cross energy barriers

Activation energies Diffusion coefficients Trajectories

equations de Newton

Eie

X i ( t ) v i ( t )

X i ( t + Dt ) v i ( t + Dt )

T

Equation de Newton:

F = m . a = m . dv/dt = m . d2x/dt2 a = dv/dt a = -1/m dE/dr

v = at + vo

v = dx/dt

x = v.t + xo

x = a.t2 + vo.t + xo

Molecular Dynamics

29

Simulations Based on Statistics

• Molecular Dynamics

• Monte Carlo

Equations of movement (Newton)

Start conditions known:

Atom Positions

Forces

Masses

Temperature

Etc.

Conditions

after Dt1

Conditions

after Dtn …

Properties are calculated as mean values after a certain time (when equilibrium

is reached)

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

• 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 eensemble is characterized by a fixed chemical potential, μ, a fixed volume, V, and a fixed temperature, T.

Statistical ensembles

Metropolis algorithm

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) >

(N,V,T):

NB: importance choice of the move parameters , acceptance/rejectance ratio of

50% for a good sampling

Rich in information Adsorption Heats (N,V,T) Isotherms (,V,T) Density of states Radiale distribution functions

Monte Carlo Methods

T

Simulated Annealing

T

T

T

Identifies various local minima

Decrease the

temperature

NEUTRON DIFFRACTION «@ low température (5 K)

2.74 Å

50 % of cyclohexane located In 12-ring windows

Rietveld Refinement

Location of adsorbed molecules In nanoporous frameworks

Adsorption of cyclohexane in zeolite HY

Adsorption of cyclohexane in zeolite HY

Vitale, Mellot and Cheetham, J. Phys. Chem. 1997, 101, 9886.

ENERGY MINIMISATION (zéro K)

3 Å

2.9 Å

2.8 Å

2.8 Å 2.8 Å

2.8 Å

50 % Monte Carlo docking: Random generation of

20 initial configurations of cyclohexane

Energy Minimisation de of each of the 20 configurations

50 %

CH3OH distribution through pair distribution functions

Dominant Na+(II) - Om

interactions

1,2 1,6 2,0 2,4 2,8 3,2 3,6 4,0

16 methanol 32 methanol 48 methanol 96 methanol

Inte

nsity

(a.

u.)

Distance (Å)

Hydrogen bond between methanol molecules

0 1 2 3 4 5 6 7 8

Liquid

Inte

nsity

(a.

u.)

Distance (Å)

36

Quantum Chemistry

• 1926 Schrödinger: finds the solution for the hydrogen atom using quantum mechanics

• Hartree-Fock Approach 1930 … 1960 (Each e- is described in the field of the other e-, No electron correlation, SCF method)

Unpaired electrons &

correlation of electron

movements

No solution nor operational

method, nor computation

power for multi-electronic

systems

Poly-electronic

systems! Looking for new methods for the calculation of systems in

which electron correlation is important.

HY = EY

37

Quantum Chemistry

38

Post-HF methods

39

Post-HF methods

40

Density Functional Theory

The VASP approach

Periodic models + DFT + plane waves + pseudopotentials

All electrons

Pseudopotentials

periodic

not periodic

Atomic orbital Plane waves

Numeric

LDA GGA

meta-GGA, hybrid

42

Density Functional Theory Density Functional Theory

43

Self-consistent calculation procedure

44

45

Exchange-correlation functional

46

Accuracy of DFT

47

Accuracy of DFT

48

Accuracy of DFT

49

Limitations of DFT

50

Limitations of DFT

51

Strategies

Type of interactions in the system

matters … but also the Size!

Methods Calculation

Strategies

52

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

53

• Finite Size – – Hybrid methods: B3LYP

– Localized basis sets

– IR and Raman intensities

– Specific calculations: TSs, crossing points

– – Smaller model

– Edge effects

– No coverage effect

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

B3LYP 6-311G(2d,p)

Gaussian03 program

PBE/plane waves

VASP 4.6 program

54

55

56

57

Level 3:

MM/charges

Approximations for Systems with a

Large Number of Atoms

• QM/MM & ONIOM • Ex. Bio Systems (Proteins, Enzymes, etc.)

• Ex. Zeolites

Level 2:

Semi-Empirical/MM

Level 1:

Ab Initio

*Large systems

*Very versatile

*long calculation times

*difficult description border zones between levels

*MM Potential not always

available

58

« 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

59

Molecular Modeling Software

• Molecular Mechanics

• Quantum Chemistry

• Molecular visualization and editing

• Other …

Companies – Academics

Freeware – Web Applications

Quantum Chemistry

Molecular visualization and editing

Molecules

•Molden

•Jmol

•GaussView

•ECCE

•Arguslab

•VMD

•VegaZZ

•DeepView

•Discovery Studio

•MolView and Molview Lite - Macintosh

Periodic Systems

Materials Studio

Crystal Maker

VMD

ModelView

MOLDRAW (Molecules and crystals)

Molekel (Molecules and crystals)

Selection!