Molecular modeling for functional materials design in chemical engineering and more

Transcription

1 Molecular modeling for functional materials design in chemical engineering and more Dr inż. Łukasz Radosiński Group of Bioprocess and Biomedical Engineering Wroclaw University of Science and Technology

2 Nanomaterials in technology Improved chemical resistance, Improved hardness and abrasion resistance, Increased tensile strength and flexibility, Favorable melting points, Favorable magnetic properties, Increased thermal and electrical conductivity, and Favorable surface-chemistry effects that improve the ability of a powder to be dispersed in a liquid.

3 CAD is now used in every part of industry Civil engineering, Chemical Engineering, Areonautics, Medical and pharmaceutical

4 Computer Aided Design Growing widely available computational power changed the product development process. We are able to predict properties of the system long before it was even created Golden management rule: if you want to optimize you have have numbers, if you have numbers you have to know how to calculate it, if you want to calculate you have to have tools computers + physics + chemistry + mathematics

5 Multi-scale view of the materials Buehler and Ackbarow, Materials Today, 2007

6 Molecular modelling Length [m] 10 3 DPD 10-3 Molecular Dynamics, Monte Carlo Coarse-Grained CFD 10-9 Ab Initio Time [s]

7 Ab initio methods Predicting of physical and chemical properties of materials from quantum theory Very accurate Without any empirical parameters Calculating parameters of forcefields for Molecular Dynamics and Monte Carlo methods Electronic properties of matter (charges, reactivity, dipole moments, IR spectrum and much more ) Very accurate geometrical parameters

8 Ab initio methods Size of a system is very limited (up to hundred atoms) Long computing time for even small systems We need high computing power Any movement of atoms or timedependent methods need much more computing time and power Theory is not so easy

9 Molecular dynamics Millions of atoms thousands of cores Solving of Newton equation of motion for all atoms Bigger systems (thousands of atoms) Phenomena depending on time (diffusion, cracking of materials, melting, solvation and so on ) Introduction of a temperature and pressure to the system

10 Molecular dynamics Size is still limited (up to billion of atoms) Simulated time is not enough long (bio-physical or chemical processes may take seconds) Everything depends on forcefield! Disregarding of quantum nature of matter (single atom has no energy no interaction with anything) Very large systems need high computing power and time

11 Monte Carlo method Averages over time (MD) are replaced by averages over states of a system States of a system are selected randomly

12 Monte Carlo method Cannot simulate time dependent properties Size is still limited Everything depends on forcefield! How to properly generate random state of a system, is it really random?

13 Coarse-Grained method Similar method to molecular dynamics - groups of atoms are replaced with superatoms Possibility to simulate very large systems Need to describe interactions between superatoms Disregarding of properties of separate atoms Everything depends on forcefield!

14 Macroscale Finite elements methods: CFD, FDM itp. tensions, flow of mass, temperature, etc.

15 When do we need molecular modeling? Beam deformation problem Question: displacement function d z x Governing equation: EI zz 4 d z,0 x 4 + q z = 0 Geometry: I zz = bh3 12 Boundary condition: q z = ρga Integration d z x = ρga 24EI zz x 4 d z x x How to obtain Young s modulus E?

16 How to obtain Young s modulus - experiment Laboratory measurements Molecular modeling A A L L + Δl F F = kδl E = Lk A

17 What is our goal? - To gain extensive knowledge about state-of-the-art methods of predicting and deisigning properties of material using molecular approach, - To be able to predict basic properties of the various physical systems and conntect the results with macroscopic values

18 Our way bottom up

19 Are Molecular Simulations part of Chemical Engineering?

20 Science

21 Science

22 Numerical methods numerical experiment

23 Can we simulate world?

24 Can we simulate world?

25 Can we simulate world?

26 Can we simulate world?

27 Why does it look real????

28 Why does it look real????

29 Why does it look real???? Newtonian motion!

30 Newton s Equation a F m

31 Newton s equation F dv F a v m dt m v dr dt r(t) Knowing force, initial velocities and position we can extract velocity and position in any moment of time!

32 How molecular dynamics works? V 1 1 (x1 p, y1 p,z1 p ) 2 (x2 p, y2 p,z2 p ) V 2

33 How molecular dynamics works? V 1 1 (x1 p, y1 p,z1 p ) F 12 (x1 p, y1 p,z1 p, x2 p, y2 p,z2 p ) F 21 (x1 p, y1 p,z1 p, x2 p, y2 p,z2 p ) 2 (x2 p, y2 p,z2 p ) V 2

34 How molecular dynamics works? V 1 1 (x1 p, y1 p,z1 p ) F 12 /m 1 =a 12 a = Dv Dt F 21 /m 2 =a 21 2 (x2 p, y2 p,z2 p ) V 2

35 How molecular dynamics works? ΔV 1 V 1 (t) 1 (x1 p, y1 p,z1 p ) F 12 /m 1 =a 12 a = Dv Dt V 1 (t+δt) Dv = adt V 2 (t+δt) F 21 /m 2 =a 21 2 ΔV 2 Dv = F m Dt (x2 p, y2 p,z2 p ) V 2 (t)

36 How molecular dynamics works? (x1 n, y1 n,z1 n ) 1 Dr 1 1 (x1 p, y1 p,z1 p ) Dr = v(t)dt V 1 (t+δt) (x2 p, y2 p,z2 p ) 2 Dr 2 2 (x2 n, y2 n,z2 n ) V 2 (t+δt)

37 How molecular dynamics works? (x1 n, y1 n,z1 n ) 1 V 1 (t+δt) F 21 (t + Dt) 2 V 2 (t+δt) (x2 n, y2 n,z2 n )

38 Mathematically Euler s scheme df f (t + Dt) - f (t) f (t + Dt) - f (t) dt = lim Dt 0 Dt» Dt df dt = f i+1 - f i Dt

39 Instead of solving function analytically step by step procedure f t + Δt = f t + f t Δt f Analytical solution t

40 Instead of solving function analytically step by step procedure f t + Δt = f t + f t Δt f Analytical solution f Point-by-point solution t t

41 How does it works we start from f(0) 1. Choose step Δt = 1 2. Compute next point using chain rule f t + Δt = f t + f t Δt Example f t = 2 t f t = 2

42 How does it works we start from f(0) f 0 + Δt = 1 = = 2 f f(1) 1 t

43 How does it works we start from f(0) f = = 4 f f(2) f(1) t 2 2

44 And so on point by point we reproduce solution f f(t) = 2t f(2) f(1) t 2 2

45 Problems? - Accuracy Example derivative varies (change of the function s shape) f Wrong value! t

46 Solution? More points, shorter interval Example derivative varies (change of the function s shape) f Problem? Computing efficiency! t

47 Application Solving Newton s Eq. 1. Assuming we know the Force acting on a body we write down Newton s Eq.: a = dv dt = F m

48 Application Solving Newton s Eq. 1. Assuming we know the Force acting on a body we write down Newton s Eq.: a = dv dt = F m 2. We use Euler s scheme to rewrite differentials: dv dt v t + Δt v t Δt

49 Application Solving Newton s Eq. 3. We solve our equation of motion v t + Δt v t Δt = F m v t + Δt = v t + F m Δt

50 Application Solving Newton s Eq. 3. Knowing initial condition say v(0) and setting Δt we reproduce point by point evolution of velocity using our finite difference scheme: v Δt = v 0 + F m Δt

51 Application Solving Newton s Eq. 3. Knowing initial condition say v(0) and setting Δt we reproduce point by point evolution of velocity using our finite difference scheme: v Δt = v 0 + F m Δt v 2Δt = v Δt + F m Δt

52 Application Solving Newton s Eq. 3. Knowing initial condition say v(0) and setting Δt we can reproduce point by point evolution of velocity using our finite difference scheme: v Δt = v 0 + F m Δt v 2Δt = v Δt + F m Δt v 0, v Δt, v 2Δt, v 3Δt,

53 What about position? Same scheme! 1. Having collection of v(t) we use definion of velocity as: v(t) = Δx Δt

54 What about position? Same scheme! 1. Having collection of v(t) we use definion of velocity as: v(t) = Δx Δt x t + Δt x t Δt = v(t)

55 What about position? Same scheme! 1. Having collection of v(t) we use definion of velocity as: v(t) = Δx Δt x t + Δt x t Δt = v(t) x t + Δt = x t + v t Δt

56 What about position? Same scheme! 1. Having collection of v(t) we use definion of velocity as: v(t) = Δx Δt x t + Δt x t Δt = v(t) x t + Δt = x t + v t Δt We already have this!

57 What about position? Same scheme! 1. Having collection of v(t) we use definion of velocity as: v(t) = Δx Δt x t + Δt x t Δt = v(t) x t + Δt = x t + v t Δt We already have this! x 0, x Δt, x 2Δt,

58 POWEER! Stupid pig.. Angry Birds

59 Newton`s equation

60

61 Newton`s equation algorithm

62 Newton`s equation algorithm

63 Newton`s equation algorithm example

64 Newton`s equation algorithm example

65 Harmonic motion why so important?

66 Harmonic motion why so important?

67 Harmonic motion why so important? Equilibrium length no motion Out of equilibrium - F = kx, E p = kx2 2 Hook s law Holds up to certain distance

68 Harmonic motion why so important? Equilibrium length no motion Out of equilibrium - F = kx, E p = kx2 2 Hook s law Holds up to certain distance

69 Harmonic motion why so important? Equilibrium length no motion Out of equilibrium - F = kx, E p = kx2 2 Hook s law Holds up to certain distance Equilibrium shape no motion Deformation F = EA ΔL, E L p = 0 Young s modulus Holds up to certain def. scale EA L0 ΔL2 2

70 Harmonic motion why so important? Equilibrium length no motion Out of equilibrium - F = kx, E p = kx2 2 Hook s law Holds up to certain distance Equilibrium shape no motion Deformation F = EA ΔL, E L p = 0 Young s modulus Holds up to certain def. scale EA L0 ΔL2 2

71 Harmonic motion why so important? Equilibrium length no motion Out of equilibrium - F = kx, E p = kx2 2 Hook s law Holds up to certain distance Equilibrium shape no motion Deformation F = EA ΔL, E L p = 0 Young s modulus Holds up to certain def. scale EA L0 ΔL2 2 Bond length no motion (zero temp., no quantum motion) Out of eq. F~Δl, E p ~ Δl2 2 Quantum mechanics harmonic approx.

72 Harmonic motion why so important? Why so many similarities? Physics??? Equilibrium length no motion Out of equilibrium - F = kx, E p = kx2 2 Hook s law Holds up to certain distance Equilibrium shape no motion Deformation F = EA ΔL, E L p = 0 Young s modulus Holds up to certain def. scale EA L0 ΔL2 2 Bond length no motion (zero temp., no quantum motion) Out of eq. F~Δl, E p ~ Δl2 2 Quantum mechanics harmonic approx.

73 Harmonic motion why so important? NO! Mathematics!

74 Two balls in equilibrium no motion E E x x

75 Two balls in equilibrium no motion E E No motion x x

76 Two balls in equilibrium no motion E E x No motion No forces x

77 Two balls in equilibrium no motion E E x No motion No forces Energy minimum x

78 Energy change due to movement E E E 0 + Δx E 0 + Δx x E 0 = 0 x Δx Δx E 0 = 0

79 Can we calculate energy E(Δx)? E E E 0 + Δx E 0 + Δx x E 0 = 0 x Δx Δx E 0 = 0

80 Can we calculate energy E(Δx)? E E E 0 + Δx E 0 + Δx x E 0 = 0 x Δx Δx E 0 = 0 de 0 E 0 + Δx = E 0 + dx Δx + 1 d 2 E 0 2 dx 2 Δx 2 +

81 Can we calculate energy E(Δx)? Minimum de 0 dx = 0! Minimum de 0 dx = 0! E E E 0 + Δx E 0 + Δx x E 0 = 0 x Δx Δx E 0 = 0 de 0 E 0 + Δx = E 0 + dx Δx + 1 d 2 E 0 2 dx 2 Δx 2 +

82 When Δx is small Minimum de 0 dx = 0! Minimum de 0 dx = 0! E E E 0 + Δx E 0 + Δx x E 0 = 0 x Δx Δx E 0 = 0 E Δx 1 2 k d 2 E 0 dx 2 Δx 2

83 When Δx is small Minimum de 0 dx = 0! Minimum de 0 dx = 0! E E E 0 + Δx E 0 + Δx x E 0 = 0 x Δx Δx E 0 = 0 E Δx 1 2 k d 2 E 0 dx 2 Δx 2 = kδx2 2

84 What about force? Minimum de 0 dx = 0! F = kx F = kx Minimum de 0 dx = 0! E E E 0 + Δx E 0 + Δx x E 0 = 0 x Δx Δx E 0 = 0 F = de dx = d kx2 2 dt = kx

85 Harmonic motion why so important? Chemical bonds! And every coupling with eq. distance

86 But there is sth extra! F = kx x = Asin ω 0 t ω 0 = k m

87 But there is sth extra! F = kx F ext = Asin(ωt)

88 But there is sth extra! F = kx F ext = Bsin(ωt) ma = Bsin ωt kx A = m B ω 0 ω 2 When ω ω 0 A is huge! Great response! Resonance

89 Where is chemistry? Light! F = kx F ext = qesin(ωt) Photon! When frequency = energy of photon is close to normal mode there is huge transfer of energy Adsorption!

90 And this we can calculate! O C O

91 Molecular modelling Length [m] 10 3 DPD 10-3 Molecular Dynamics, Monte Carlo Coarse-Grained CFD 10-9 Ab Initio Time [s]

92 Ab initio methods Predicting of physical and chemical properties of materials from quantum theory Very accurate Without any empirical parameters Calculating parameters of forcefields for Molecular Dynamics and Monte Carlo methods Electronic properties of matter (charges, reactivity, dipole moments, IR spectrum and much more ) Very accurate geometrical parameters

93 Ab initio methods Size of a system is very limited (up to hundred atoms) Long computing time for even small systems We need high computing power Any movement of atoms or timedependent methods need much more computing time and power Theory is not so easy

94 Molecular dynamics Millions of atoms thousands of cores Solving of Newton equation of motion for all atoms Bigger systems (thousands of atoms) Phenomena depending on time (diffusion, cracking of materials, melting, solvation and so on ) Introduction of a temperature and pressure to the system

95 Molecular dynamics Size is still limited (up to billion of atoms) Simulated time is not enough long (bio-physical or chemical processes may take seconds) Everything depends on forcefield! Disregarding of quantum nature of matter (single atom has no energy no interaction with anything) Very large systems need high computing power and time

96 Monte Carlo method Averages over time (MD) are replaced by averages over states of a system States of a system are selected randomly

97 Monte Carlo method Cannot simulate time dependent properties Size is still limited Everything depends on forcefield! How to properly generate random state of a system, is it really random?

98 Coarse-Grained method Similar method to molecular dynamics - groups of atoms are replaced with superatoms Possibility to simulate very large systems Need to describe interactions between superatoms Disregarding of properties of separate atoms Everything depends on forcefield!

99 Our way bottom up

100 What do we obtain? Trajectories Position and velocity in time

101 Molecular Dynamics Tasks Fundamental: to compute trajectories Minimazation of energy geometrical analysis of compouds To compute avarages and current physical quantities in in the investigated systems (kinetic energy, potential energy, interatomic distance, pressure, volume, atoms spatial correlation, diffusion, heat transfer, viscosity etc.)

102 Molecular Dynamics Tasks Fundamental: to compute trajectories Minimazation of energy geometrical analysis of compouds To compute avarages and current physical quantities in in the investigated systems (kinetic energy, potential energy, interatomic distance, pressure, volume, atoms spatial correlation, diffusion, heat transfer, viscosity etc.)

103 What is missing? 1. Intermolecular forces. 2. Initial positions. 3. Initial velocities.

104 What is missing? 1. Forces forcefields (quantum chemical computations), empirical potentials. 2. Initial positions crystal structure, fluids order, gas density etc. 3. Initial velocities zero, random, temperature.

105 Initial positions Initial positions of atoms should be as close to equilibrium positions (minimal energy) as possible scientific knowledge To big deviation from equilibrium positions results in unphysical behaviour large forces

106 Temperature How to introduce temperature to the system? Simplest solution is to draw velocities from Boltzmann distribution for fixed T:

107 Temperature of the system is its average kinetic energy

108 Temperature is not constant Unperfect solution of the Verlet s algorithm error cumulation

109 Problems? What is the average fluctuation in microcanonical ensamble?

110 Using Boltzmann distribution

111 Solution? Thermostat (heat bath) Typical MD simulation is in NVE regime -> transformation to NVT and NPT ensambles only in the thermodynamical limit (inifnite system size) Introduction of the heat bath separate system with fixed temperature System T(t) Heat bath T 0

112 Beredsen thermostat Weak coupling via Newton s law Coupling constant Reservouir s temperature

113 How to apply? Rescale velocities by step dependant rescalling factor

114 How to obtain lambda?

115 Beredsen s thermostat Pros: Straight forward code Robust Exponential decay to the desired temperature Cons: Results do not correspond to any ensamble, however small deviation from canonical only during equilibriation No-time reversible

116 Noose-Hoover Thermostat Introducing friction to the heat reservouir by including new variable to the equation of motion Kinetic energy of the hb Potential energy of the hb Fictious mass of the hb

117 Nose-Hoover scheme Apply Hamilton s equation of motion, Obtain coupled system of diff. eq. for r and s with coupling constant Q Solve dynamics

118 Nose-Hoover thermostat Pros: Gives canonical properties Deterministic, time reversible Conserves avarages p A, r A p, s r c Cons: In some cases extended system not ergodic entrapment in subspace slow Not implemented in NAMD!

119 Langevin s thermostat Introducing hypothetical fluid of much smaller particles each system s particle is subjected to a drag and a stochastic force Force Drag Noise

120 The drag and noise are balanced in order to obtain constant temperature Gamma is a friction coefficiant A is a random force uncorrelated in time and across particles

121 Langevin thermostat Pros: Reproduces canonical properties Ergodic Allows larger time steps as compared to non-stochastic thermostats Cons: Undeterministic Momentum trasfer not conserved not advisable to use in the system with diffusion Temperature fluctuations

122

123 Ekwilibracja Potential Energy as a Function of Time 0 Potential Energy (kcal/mol) time (ps) 123

124 Fluctuations in Total Energy for an NVE Simulation of 216 TIP3P Water Molecules (1fs time step) [the run was initiated at E = kcal/mol] -1730,2-1730,25 Total Energy (kcal/mol) -1730,3-1730, ,4-1730, , time (ps) 124

125 Thermostating & Barostating Simulation system is dynamical. In order to preserve the system specific temperature or pressure we employ in the calculation the concept of thermostat and barostat. Conceptionaly these are separate systems with defined pressure an temperature connected to our simulation balls. Mathematicaly we simply add two Newtons equations, additionaly equations defining temperature and pressure.

126 Pressure virial theorem Virial theorem: Pressure in the system:

127 Berendsen Thermostat The velocities are scaled at each step, such that the rate of change of temperature is proportional to the difference in temperature. Scaling factor:

128 Berendsen Barostat. In barostat the mechanism is similar: Scaling factor:

129 NPT V nt/p During simulation ensemble uses barostat and thermostat to equilibrate temperature and pressure. Varying, operating both: P and T results in changes of volume of the simulation box. In other words, our goal is to reach density of water about 1.0 g/cc, that is a property of our system in 298 K and 1 bar.

130 Dynamics Calculation stages to define temperature and density NPT NVT NVE equilibration of density (volume changes) equilibration of temperature final dynamics

131 NPT Box length running average

132 NPT Density

133 NVT High kinetic energy equilibrated temperature Running average Random Velocities High potential Energy Temperature Strong thermostat work

134 Real systems simulations Problems and solutions

135 Problems of simulating real systems Most if the systems contains huge number of atoms >> Typical atomic time scale is10-15 s human is living in seconds Most of interactions are nonbonded (electrostatic, dispersion etc.) infinite range of interactions

136 Periodic Boundary Conditions

137 Periodic Boundary Conditions Pros: Elimination of the boundary problem Cons: Neutral charge of the system requirement unbalanced charge creates multiple images due to PBC resulting unphysical behaviour Angular momentum in not conserved - no rotational symmetry Cannot observe oscillation of wavelength longer then PBC box gas-liquid critical point

138 Infinite system

139 Simplication to a single simulation box

140 Periodic Boundary Conditions examples Spherical Cubical

141 Goal: glucose in water no PBC Glucose PBC addition Water addition

142 Goal: glucose in water no PBC No problems with non-bonded interactions Huge number of atoms Expensive PBC Atom may interact with itself Small number of atoms Cheaper

143 Geometry optimization Each system must be optimized before your proper simulation. Every algorithm leads potential energy function to its minimum. You must be aweare that potential energy is rather multidimensional hypersurface than 3D function (seen on the pic.). Initial energy Global or local minimum

144 Geometry optimization Example of convergence criteria: 1. Convergence of the function (energy) f x k+1 f(x k ) < ε f 2. Convergence of the variable (position) x k+1 x k < ε x 3. Covergence of the gradient (force) f x k+1 f(x k ) < ε g Initial energy Global or local minimum

145 Geometry optimization Examples of most popular energy minimization algorithms: - conjugate gradients, - steepest descent. And also some damping dynamics algorithms such as: - quickmin, - fire. Initial energy Global or local minimum

146 Potential energy Geometry optimization 60º 109.5º 145º 180º 215º 215.5º 300º Angle

147 Problems of simulating real systems The results are supposed have relevance to real systems values Becouse time scale of nanoscopic systems is fs and ours time scale is s we measure AVARAGES temperature, pressure, energy, specific heat etc. Question: how long do we have to simulate the process so the avarages obtained in the simulation are thermodynamical avarages? Ergodic hypothesis.

148 Molecular dynamics protocol Initialization (positions) Energy Minimanizatiob (excess potential energy removal) Initialization of velocities (temperature) Equilibration ( smearing of kinetic energyj) Simulation Trajectory Analysis

149 Molecular dynamics protocol Initialization (positions) Energy Minimanizatiob (excess potential energy removal) Initialization of velocities (temperature) Equilibration ( smearing of kinetic energyj) Simulation Trajectory Analysis

150 Molecular dynamics protocol Initialization (positions) Energy Minimanizatiob (excess potential energy removal) Initialization of velocities (temperature) Equilibration ( smearing of kinetic energyj) Simulation Trajectory Analysis

151 Molecular dynamics protocol Initialization (positions) Energy Minimanizatiob (excess potential energy removal) Initialization of velocities (temperature) Equilibration ( smearing of kinetic energy) Simulation Trajectory Analysis

152 Molecular dynamics protocol Initialization (positions) Energy Minimanizatiob (excess potential energy removal) Initialization of velocities (temperature) Equilibration ( smearing of kinetic energy) Simulation Trajectory Analysis

153 Molecular Dynamics Forces

154 Protokół dynamiki molekularnej

155 Types of interactions Bonded interactions: Strong, Short-ranged, Typically up to the second nearest neighbour, Strongly depends on the geometry and electronic structure. Can be quantified using quantum mechanics.

156 Types of interactions nonbonded interactions: Long-ranged, Every atoms interacts with every atom in the system Weak (van der Waalsa) or strong (electrostatic)

157 Forcefield Forcefield (AMBER, CHARMM), Empirical (AIREBO, LCBOP), Quantum based (COMPASS)

158 Potential function V = V bonds +V angle +V torsion +V improper +V nonbonded

159 Funkcja potencjału empirycznego V = V bonds +V angle +V torsion +V improper +V nonbonded ( ) 2 å + k q q -q 0 V = k b b - b 0 bonds N å n=1 ( ) + K j n angles ( ) 2 å + å 1+ cos( nj -d ) + K w w - w 0 dihedrals + 4e ij s ij éë 12 æ ö ç è r - s 6 é æ ö ù å ij ê ç ij ø è r ú + i, j ë ê ij ø û ú ù û å i, j æ ç è å impropers q i q j Dr ij ö ø ( ) 2

160 Bonded interactions V = å k ( b r - r ) 2 0 bonds

161 Chemical bond Morse potential E V bonds = D e (1- exp(-a(l - l 0 ))) 2 a = w n / 2D e distance

162 Chemical bond Morse potential E V bonds = D e (1- exp(-a(l - l 0 ))) 2 a = w n / 2D e distance Pauli s exclusion principle

163 Chemical bond Morse potential E V bonds = D e (1- exp(-a(l - l 0 ))) 2 a = w n / 2D e Eq. distance distance Pauli s exclusion principle

164 Chemical bond Morse potential E V bonds = D e (1- exp(-a(l - l 0 ))) 2 a = w n / 2D e Eq. distance distance Pauli s exclusion principle Bond brakage

165 Chemical bond Morse potential E Harmonic potential V bonds = D e (1- exp(-a(l - l 0 ))) 2 a = w n / 2D e Pauli s exclusion principle Eq. distance distance Bond brakage V bonds = k 2 (l - l 0) 2 Computational simplicity

166 Bonded interaction bond angle V angle = k 2 (q -q 0) 2

167 Bonded interaction torsion angle

168 Bonded interaction torsion angle V angle = N å V n 2 n=0 (1+ cos(nw -g )) Most popular model MM2 V = ½ V tor,1 (1 cos ω ) + ½ V tor,2 (1 - cos 2 ω ) + ½ V tor,3 ( 1 - cos 3 ω ) Dipol-dipol interactions, differences in electronegativity Conjugations (alkanes) double bond Sterical effects Interactions between orbitals

169 Bonded interaction torsion angle V angle = N å V n 2 n=0 (1+ cos(nw -g ))

170 Bonded interaction improper torsion V angle = k(1- cos2w) k i l j ω Thomas W. Shattuck

171 Bonded interaction improper torsion V angle = k(1- cos2w) k i l j ω Thomas W. Shattuck Induce planar structure Example: cyclobutan

172 Bonded interaction conjugates ω ω ω Bond - bond Bond - angle V cross = k 2 (l 1 - l 10 )(l 2 - l 2 0 ) V cross = k 2 (l 1 - l 10 )(w -w 0 ) Angle - angle V cross = k 2 (w 1 -w 10 )(w 2 -w 20 )

173 Van der Waals interactions å i, j 4e ij éæ ê ç ê ë è r m r ij ö ø 12 æ - r m ç è r ij ö ø 6 ù ú ú û dipol dipol interaction type: Weak, long-ranged 1 r ij 6 dipol dipol interaction 1 r ij 12 Pauli exclusion principle two particles cannot occupy one point in space

174 Coulomb interactions

175 Coulomb interactions Each atom interacts with every atom in the system computational complexity O(n 2 )

176 Non bonded interactions ( ) 2 V = å k b b - b 0 + å k q q -q 0 + bonds N angles ( ) 2 ( n) + å K j é ë 1+ cos( nj -d ) ù û + K w w - w 0 dihedrals å n=1 12 æ s + 4e ij ö ij ç è r - s 6 é æ ö ù ij å ê ç ij ø è r ú + i, j ê ë ij ø ú û å i, j å impropers æ q i q j ö ç è ø Dr ij ( ) 2

177 Non bonded interactions ( ) 2 V = å k b b - b 0 + å k q q -q 0 + bonds N angles ( ) 2 ( n) + å K j é ë 1+ cos( nj -d ) ù û + K w w - w 0 dihedrals å n=1 12 æ s + 4e ij ö ij ç è r - s 6 é æ ö ù ij å ê ç ij ø è r ú + i, j ê ë ij ø ú û å i, j å impropers æ q i q j ö ç è ø Dr ij ( ) 2 Greatest computational complexity Every atoms interacts with every atom

178 Non bonded interactions ( ) 2 V = å k b b - b 0 + å k q q -q 0 + bonds N angles ( ) 2 ( n) + å K j é ë 1+ cos( nj -d ) ù û + K w w - w 0 dihedrals å n=1 12 æ s + 4e ij ö ij ç è r - s 6 é æ ö ù ij å ê ç ij ø è r ú + i, j ê ë ij ø ú û å i, j å impropers æ q i q j ö ç è ø Dr ij ( ) 2 Greatest computational complexity Every atoms interacts with every atom 1500 atoms

179 Non bonded interactions ( ) 2 V = å k b b - b 0 + å k q q -q 0 + bonds N angles ( ) 2 ( n) + å K j é ë 1+ cos( nj -d ) ù û + K w w - w 0 dihedrals å n=1 12 æ s + 4e ij ö ij ç è r - s 6 é æ ö ù ij å ê ç ij ø è r ú + i, j ê ë ij ø ú û å i, j å impropers æ q i q j ö ç è ø Dr ij ( ) 2 Greatest computational complexity Every atoms interacts with every atom 1500 atoms force pairs

180 Non bonded interactions Solution: Cut-off distance Ewald Summation ( ) 2 V = å k b b - b 0 + å k q q -q 0 + bonds N angles ( ) 2 ( n) + å K j é ë 1+ cos( nj -d ) ù û + K w w - w 0 dihedrals å n=1 12 æ s + 4e ij ö ij ç è r - s 6 é æ ö ù ij å ê ç ij ø è r ú + i, j ê ë ij ø ú û å i, j å impropers æ q i q j ö ç è ø Dr ij Greatest computational complexity Every atoms interacts with every atom ( ) atoms force pairs

181 Non bonded interactions cut off distance For atom A we calculate non-bonded interactions only in the sphere of radius δ Verifaction process required

182 Ewald summation method Ewald summation is an algorithm wich effciently calculates the interactions between periodic mirrors of atoms or molecules. Van der Waals interactions deacys like 1 r6 so there is no porblem usually periodic mirrors are over the cutoff. In case of Coulomb electrostatic interactions k q iq i_mirror potential r i i_mirror deacys slow. Ewald proposed 1 + which states for short range, rapidly r r r varying function and long range flat function. The f r = erfc r = 2 exp t 2 dt. π x = f r 1 f r

183 Ewald summation method Reciprocal space Real space r r Each charge is neutralised with neutralising charge Another contribution counteracts neutralisation Real space r

184 Particle-particle particle-mesh In the particle-particle particlemesh method (PPPM, P 3 M) some simplifications are made. Beyond some distance R c particles (atoms) are discetized charge density gives a contribution to neighbouring grid nodes. For close particels (atoms) interactions are calcualted by common Ewald summation. Mesh grid

185 Ewald vs. PPPM (P3M) Ewald summation Particle Particle Particle mesh Summation over every atom Summation over a discretized grid of charge denisty Uses Fourier transform Uses fast Fourier transform Faster for smaller systems Significantly faster for large systems (N > 10000) # Atoms Ewald (c. time) P3M (c. time)

186 What about large molecules? atoms