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Méthodes de simulation et modélisation
Caroline Mellot-Draznieks
Caroline.mellot-draznieks@college-de-france.fr
avec la participation de Frédérik Tielens
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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
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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
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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
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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
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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
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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
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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
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•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
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Molecular Mechanics
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Molecular Mechanics
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Molecular Mechanics
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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
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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
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Quantum Chemistry
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Post-HF methods
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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
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Self-consistent calculation procedure
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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
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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!
Recommended