1. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 Ch121a Atomic Level Simulations of Materials and Molecules William A. Goddard III, wag@wag.caltech.edu Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology 1 Lecture 5b and 61, April 14 and 16, 2014 MD2: dynamics Room BI 115 Lecture: Monday, Wednesday Friday 2-3pm TA’s Jason Crowley and Jialiu Wang

2. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 Homework and Research Project 2 First 5 weeks: The homework each week uses generally available computer software implementing the basic methods on applications aimed at exposing the students to understanding how to use atomistic simulations to solve problems. Each calculation requires making decisions on the specific approaches and parameters relevant and how to analyze the results. Midterm: each student submits proposal for a project using the methods of Ch121a to solve a research problem that can be completed in the final 5 weeks. The homework for the last 5 weeks is to turn in a one page report on progress with the project The final is a research report describing the calculations and conclusions

3. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 3 Previously we develop the force fields and we minimized the geometry now we do dynamics

4. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 4 Classical mechanics – Newton’s equations Newton’s equations are M k (d 2 R k /dt 2 ) = F k = -  Rk E where R k, F k, and  are 3D vectors and where goes over every particle k Here F k =  n≠k F nk where F nk is the force acting on k due to particle n From Newton’s 3 rd law F nk = - F kn Solving Newton’s equations as a function time gives a trajectory with the positions and velocities of all atoms Assuming the system is ergotic, we can calculate the properties using the appropriate thermodynamic average over the coordinates and momenta.

5. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 5 Consider 1D – the Verlet algorithm Newton’s Equation in 1D is M(d 2 x/dt 2 ) t = F t = - (dE/dx) t We will solve this numerically for some time step . First we consider how velocity is related to distance v(t+  /2) = [x(t+  ) – x(t)]/  that is the velocity at point t+  /2 is the difference in position at t+  and at t divided by the time increment . Also v(t-  /2) = [x(t) – x(t-  )]/  Next consider how acceleration is related to velocities a(t) = [v(t+  /2) - v(t-  /2)]/  = that is the acceleration at point t is the difference in velocity at t+  and at t-  divided by the time increment . Now combine to get a(t)=F(t)/M= {[x(t+  ) – x(t)]/  - [x(t) – x(t-  )]/  )}/  = {{[x(t+  ) – 2x(t) + x(t-  )]}/   Thus x(t+  ) = 2x(t) - x(t-  ) +   F(t)/M This is called the Verlet (pronounced verlay) algorithm, the error is proportional to  4. At each time t we calculate F(t) and combine with the previous x(t-  ) to predict the next x(t+  )

6. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 6 Initial condition for the Verlet algorithm The Verlet algorithm, x(t+  ) = 2x(t) - x(t-  ) +   F(t)/M starts with x(t) and x(t-  ) and uses the forces at t to predict x(t+  ) and all subsequent positions. But typically the initial conditions are x(0) and v(0) and then we calculate F(0) so we need to do something special for the first point. Here we can write, v(  /2) = v(0) + ½  a(0) and then x(  ) = x(0) +  v(  /2) = x(0) + v(0) + ½  F(0)/M As the special form for getting x(  ) Then we can use the Verlet algorithm for all subsequent points.

7. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 7 Consider 1D – the Leap Frog Verlet algorithm Since we need both velocities and coordinates to calculate the various properties, I prefer an alternative formulation of the numerical dynamics. Here we write v(t+  /2) = v(t-  /2) +  a(t) and x(t+  ) = x(t) + v(t+  /2) This is called leapfrog because the velocities leap over positions and the positions leap over the velocities Here also there is a problem at the first point where we have v(0) not v(  /2). Here we write v(0) = {v(  /2) + v(-  /2)]/2 and v(  /2) - v(-  /2) =  a(0) So that 2v(  /2) =  a(0) + 2 v(0) or v(  /2) = v(0) + ½  F(0)/M

8. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 8 Usually we start with a structure, maybe a minimized geometry, but where do we get velocities? For an ideal gas the distribution of velocities follows the Maxwell-Boltzmann distribution, For N particles at temperature T, this leads to an average kinetic energy of KE=  k=1,3N (M k v k 2 /2)= (3N)(k B T/2) where k B is Boltzmann’s constant. In fact our systems will be far from ideal gases, but it is convenient to start with this distribution. Thus we pick a distribution of random numbers {  1,…,  3N } that has a Gaussian distribution and write v i =  i sqrt[(2kT/M) This will lead to KE ~ (3N/2)(k B T), but usually we want to start the initial KE to be exactly the target bath T B. Thus we scale all velocities by  such that KE = (3N/2)(k B T B )

9. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 9 Equilibration Although our simulation system may be far from an ideal gas, most dense systems will equilibrate rapidly, say 50 picoseconds, so our choice of Maxwell-Boltzmann velocities will not bias our result Often we may start the MD with a minimized structure. If so some programs allow you to start with double T bath. The reason is that for a harmonic system in equilibrium the Virial theorem states that =. Thus if we start with =0 and the KE corresponding to some T initial, the final temperature, after equilibration will be T ~½ T initial.

10. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 10 The next 4 slides are background material The Virial Theorem - 1 For a system of N particles, the Virial (a scalar) is defined as G =  k (p k ∙ r k ) where p and r are vectors and we take the dot or inner product. Then dG/dt =  k (p k ∙ dr k /dt) +  k (dp k /dt ∙ r k ) =  k M k (dr k /dt ∙ dr k /dt) +  k (F k ∙ r k ) = 2 KE +  k (F k ∙ r k ) Where we used Newton’s equation: F k = dp k /dt Here F k =  n ’ F nk is the sum over all forces from the other atoms acting on k, and the prime on the sum indicates that n≠k

11. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 11 The Virial Theorem - 2 Thus  k (F k ∙ r k ) =  k  n ’ (F n,k ∙ r k ) =  k  n k (F n,k ∙ r k ) =  k  n n (F k,n ∙ r n ) =  k  n n (-F n,k ∙ r k ) =  k  n<k [F n,k ∙ (r k – r n )] Where line 3 relabeled k,n in 2 nd term and line 4 used Newton’s 3 rd law F k,n = - F n,k. But F n,k = -  rk PE = - [d(PE)dr] [(r k – r n )/r nk ]  k (F k ∙ r k ) = -  k  n<k [d(PE)dr] [(r k – r n )] 2 /r nk = -  k  n<k [d(PE)dr] r nk

12. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 12 The Virial Theorem - 3 Using  k (F k ∙ r k ) = -  k  n<k [d(PE)dr] r nk we get dG/dt = 2 KE -  k  n<k [d(PE)dr] r nk Now consider a system in which the PE between any 2 particles has the power law form, PE(r nk ) =  (r nk ) p for example: p=2 for a harmonic system and p=-1 for a Coulomb system. Then dG/dt = 2KE – p PE tot which applies to every time t. Averaging over time we get = 2 – p

13. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 13 The Virial Theorem - 4 = 2 – p Integrating over time ʃ 0  (dG/dt) dt = G(  G(  For a stable bound system G(∞)=G(0) Hence = 0, leading to the Virial Theorem 2 = p Thus for a harmonic system = = ½ E total And for a Coulomb system = -1/2 So that E total = ½ = -

14. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 microcanonical Dynamics (NVE) 14 Solving Newton’s equations for N particles in some fixed volume, V, with no external forces, leads to conservation of energy, E. This is referred to as microcanonical Dynamics (since the energy is fixed) and denoted as (NVE). Since no external forces are acting on this system, there is no heat bath to define the temperature.

15. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 canonical Dynamics (NVT) – isokinetic energy 15 We will generally be interested in systems in contact with a heat bath at Temperature T B. In this case the states of the system should have (q,p) such that P(p,q) = exp[- H (p,q)/k B T B ]/Z is the probability of the system having coordinates p,q. This is called a canonical distributions of states. However, though we started with KE=  k=1,3N (M k v k 2 /2)= (3N/2)(k B T B ) there is no guarantee that the KE will correspond to this T B at a later time. A simple fix to this is velocity scaling, where all velocities are scaled by a factor,, such that the KE remains fixed (isokinetic energy dynamics). Thus on each iteration we calculate the temperature corresponding to the new velocities, T new = (2k B /3N)  k=1,3N (M k v k 2 /2), then we multiply each v by, so that (2k B /3N)  k=1,3N (M k ( v k ) 2 /2) = T B where =sqrt(T B /T new ) Thus if T new is too high, we slow down the velocities.

16. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 Why scale the velocity rather than say, KE (ie v 2 ) 16 We will take the center of mass of our system to be fixed, so that the sum of the momenta must be zero P =  k m k v k = 0 where P and v (and 0) are vectors. Thus scaling the velocities we have P=  k m k ( v k ) = P = 0 If instead we used some other transformation on the velocities this would not be true.

17. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 17 Advanced Classical Mechanics Lagrangian Formulation The Lagrangian equation of motion become ∂L ( q, q)/∂q = (d/dt) [∂ L ( q, q)/∂ q ] Where the momentum is p = [∂ L ( q, q)/∂ q ] and dp/dt = [∂ L ( q, q)/∂q] Letting KE=1/2 M q 2 this leads to p = M q and dp/dt = -  PE = F or F = m a, Newton’s equation The Frenchman Lagrange (~1795) developed a formalism to describe complex motions with noncartesian generalized coordinates, q and velocities q =(dq/dt) (usually this is q with a dot over it) L ( q, q) = KE( q ) – PE (q)

18. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 18 Advanced Classical Mechanics Hamiltonia Formulation The Hamiltonian equation of motion become ∂H (p, q)/∂p = q ∂H (p, q)/∂q = - dp/dt Letting KE=p 2 /2M this leads to q = p/M dp/dt = -  PE = F or F = m a, Newton’s equation The irishman Hamilton (~1825) developed an alternate formalism to describe complex motions with noncartesian generalized coordinates, q and momenta p. Here the Hamiltonian H is defined in terms of the Lagrangian L as H (p, q) = pq– L ( q, q)

19. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 19 Statistical Ensembles - Review There are 4 main classes of systems for which we may want to calculate properties depending on whether there is a heat bath or a pressure bath. Microcanonical Ensemble (NVE) Canonical Ensemble (NVT) Isothermal-isobaric Ensemble (NPT) Isoenthalpic-isobaric Ensemble (NPH) Solve Newton’s Equations Normal experimental conditions Usual way to solve MD Equations

20. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 20 Calculation of thermodynamics properties Depending on the nature of the heat and pressure baths we may want to evaluate the ensemble of states accessible by a system appropriate for NVE, NVT, NPH, or NPT ensembles Here NVE corresponds to solving Newton’s equations and NPT describes the normal conditions for experiments We can do this two ways using our force fields Monte Carlo sampling considers a sequence of geometries in such a way that a Boltzmann ensemble, say for NVT is generated. We will discuss this later. Molecular dynamics sampling follows a trajectory with the idea that a long enough trajectory will eventually sample close enough to every relevant part of phase space (ergotic hypothesis)

21. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 21 Now we have a minimized structure, let’s do Molecular Dynamics This is just solving Newton’s Equation with time M k (∂ 2 R k )/∂t 2 ) = F k = - -(∂E/∂ R k ) for k=1,..,3N Here we start with 3N coordinates { R k } t0 and 3N velocities { V k } t0 at time, t 0, then we calculate the forces { F k } t0 and we use this to predict the 3N { R k } t1 and velocities { V k } t1 at some later time step t 1 = t 0 +  t

22. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 22 Ergodic Hypothesis Time/Trajectory average MD Ensemble average MC Thermodynamics Kinetics

23. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 23 Canonical Dyanamics (NVT) - Langevin Dynamics Consider a system coupling to a heat bath with fixed reference temperature, T B.. The Langevin formalism writes M k dv k /dt = F k –  k M k v k + R k (t) The damping constants  k determine the strength of the coupling to the heat bath and where R i is a Gaussian stochastic variable with zero mean and with intensity The Langevin equation corresponds physically to frequent collisions with light particles that form an ideal gas at temperature T B. Through the Langevin equation the system couples both globally to a heat bath and also subjected locally to random noise.

24. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 24 canonical Dynamics (NVT)-Berendsen Thermostat The problem with isokinetic energy dynamics is that the KE is constant, whereas a system of finite size N, in contact with a heat bath at temperature T B, should exhibit fluctuations in temperature about T B, with a distribution scaling as 1/sqrt(N) A simple fix to this problem is the Berendsen Thermostat. Here the velocities are scaled at each step so that the rate of change of the T is proportional to the difference between the instantaneous T and T B, dT(t)/dt = [T B – T(t)]/  where the damping constant  determines the relaxation time This leads to a change in T between successive steps spaced by  is  T = (  [T B -T(t)], which leads to 2 =1+ (  [T B /T(t-  /2) – 1], since we use leap frog for the velocities. (Note that  leads to isokinetic energy dynamics) The Berendsen Equation of motion is M k dv k /dt = F k – (M k /  )(T B /T – 1)v k

25. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 25 Canonical dynamics (NVT): Andersen Method where is the collision frequency (default 10 fs -1 in CMDF). It is reasonable to require the stochastic collision frequency to be the actual collision frequency for a particle. Andersen NVT does produce a Canonical distribution. Indeed Andersen NVT generates a Markov chain in phase space, so it is irreducible and aperiodoic. Moreover, Andersen NVT does not generate continuous (real) dynamics due to stochastic collision, unless the collision rate is chosen so that the time scale for the energy fluctuation in the simulation equal to correct values for the real system Hans Christian Andersen (the one at Stanford, not the one that wrote Fairy tales) suggested using Stochastic collisions with the heat bath at the probability

26. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 26 Size of Heat bath damping constant  The damping of the instantaneous temperature of our system due to interaction with the heat bath should depend on the nature of the interactions (heat capacity,thermal conductivity,..) but we need a general guideline. One criteria is that the coupling with the heat bath be slow compared to the fastest vibrations. For most systems of interest this might be CH vibrations (3000 cm -1 ) or OH vibrations (3500 cm -1 ). We will take (1/ )=3333 cm -1 Since =c the speed of light, the Period of this vibration is T = 1/ = (  c) = 1/[c(1/ )] = 1/[(3*10 10 cm/sec)(3333)] = =1/[10 14 ] = 10 -14 sec = 10 fs (femtoseconds) Thus we want  >> 10 fs = 0.01 ps. A good value in practice is  = 100 fs = 0.1 ps. Better would be 1 ps, but this means we must wait ~ 20  = 20 ps to get equilibrated properties.

27. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 27 Canonical Dynamics (NVT) – Nose-Hoover The methods described above do not lead properly to a canonical ensemble of states, which for a Hamiltonian H(p,q) has a Boltzmann distribution of states. That is the probability for p,q scales like exp[- H(p,q)/k B T B ]. Thus these methods do not guarantee that the system will evolve over time to describe a canonical distribution of states, which might invalidate the calculation of thermodynamic and other properties. S. Nose formulated a more rigorous MD in which the trajectory does lead to a Boltzmann distribution of states. S. Nosé, J. Chem. Phys. 81, 511 (1984); S. Nosé, Mol. Phys. 52, 255 (1984) Nosé introduced a fictitious bath coordinate s, with a KE scaling like Q(ds/dt) 2 where Q is a mass and PE energy scaling like [gk B T B ln s] where g=N dof + 1. Nose showed that microcanonical dynamics over g dof leads to a probability of exp[- H(p,q)/k B T B ] over the normal 3N dof.

28. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 28 Canonical Dynamics (NVT) – Nose-Hoover Bill Hoover W. G. Hoover, Phys. Rev. A 31, 1695 (1985)) modified the Nose formalism to simplify applications, leading to d 2 R k /dt 2 = F k /M k –  (dR k /dt ) (d  /dt) = (k B N dof /Q)T(t){(g/N dof ) [1 - T B /T(t))]} where  derives from the bath coordinate. The 1 st term shows that  serves as a friction term, slowing down the particles when  >0 and accelerating them when  <0. The 2 nd term shows that T B /T(t) 0 so that  starts changing toward positive friction that will eventually start to cool the system. However the instantaneous  might be negative so that the system still heats up (but at a slower rate) for a while. Similarly T B /T(t) > 1 (too cold) leads to (d  /dt) > 0 so that  starts changing toward negative friction.

29. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 29 Canonical Dynamics (NVT) – discussion This evolution of states from Nose-Hoover has a damping variable  that follows a trajectory in which it may be positive for a while even though the system is too cold or negative for a while even though the system is to hot. This is what allows the ensemble of states over the trajectory to describe a canonical distribution. But this necessarily takes must longer to converge than Berendsen which always has positive friction, moving T toward T B at every step I discovered the Nose and Hoover papers in 1987 and immediately programmed it into Biograf/Polygraf which evolved into the Cerius and Discover packages from Accelrys because I considered it a correct and elegant way to do dynamics. However my view now is that for most of the time we just want to get rapidly a distribution of states appropriate for a given T, and that the distribution from Berendsen is accurate enough.

30. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 The next 3 slides are background material details about Nose-Hoover - I 30 S. Nose introduced (1984) an additional degree of freedom s to a N particle system: The Lagrangian of the extended system of particles and s is postulated to be: Equation of motion: For particles: For s variable: or

31. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 31 details about Nose-Hoover - II The averaged kinetic energy coincides with the external T B : Momenta of particle: Momenta of s: The Hamiltonian of the extended system (conserved quantity, useful for error checking):

32. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 32 details about Nose-Hoover - III An interpretation of the variable s: a scaling factor for the time step  t The real time step  t ’ is now unequal because s is a variable. Q has the unit of “mass”, which indicates the strength of coupling. Large Q means strong coupling. Nose NVT yields rigorous Canonical distribution both in coordinate and momentum space. Hoover found Nose equations can be further simplified by introducing a thermodynamics friction coefficient  : Nose-Hoover formulation is most rigorous and widely used thermostat. S. Nose, Molecular Physics, 100, 191 (2002) reprint; Original paper was published in 1983.

33. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 33 All 3 thermostats yield an average temperature matching the target T B. All 3 allow similar instantaneous fluctuations in T around target T B (300 K). Nose should be most accurate 216 waters at 300K Instantaneous Temperature Fluctuations

34. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 Stopped April 14, 2014 34

35. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 What about volume and pressure? 35 For MD on a finite molecule, we consider that the molecule is in some box, but if the CM of the molecule is zero the CM stays fixed so that we can ignore the box. To describe the bulk states of a gas, liquid, or solid, we usually want to consider an infinite volume with no surfaces, but we want the number of independent molecules to be limited, (say 1000 independent molecules) In this case we imagine a box (the unit cell), which for a gas or liquid could be cubic, repeated in the x, y, and z directions through all space. If the cell has fixed sizes then there will be a pressure (or stress) on the cell that fluctuates with time (NVE, NVT) We can also adjust the cell sizes from step to step to keep the pressure constant (NPH or NPT)

36. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 36 Periodic Boundary Conditions for solids Consider a periodic system with unit cell vectors, {a, b, c} not necessarily orthogonal but noncoplanar We define the atom positions within a unit cell in terms of scaled coordinates, where the position along the a, b, c axes is (0,1). We define the transformation tensor h ={a,b,c} that converts from scaled coordinates S to cartesian coordinate r, r= h S, In the MD, we expect that the atomic coordinates will adjust rapidly compared to the cell coordinates. Thus for each time step the forces on the atoms lead to changes in the particle velocities and positions, which are expressed in terms of scaled coordinates, then the stresses on the cell parameters are used to predict new values for the cell parameters {a, b, c} and their time derivatives. Here the magnitude of r k is r i 2 = (s i |G|s i ) where G=h T h is the metric tensor for the nonorthogonal coordinate system and T indicates a transpose so that G is a 3 by 3 metric tensor

37. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 37 NPH Dynamics: Berendsen Method Berendsen NPT is similar to Berendsen NVT, but the scaling factor scales atom coordinates and cell length by a factor each time step Note that Berendsen NPH does NOT allow the change of cell shape. This is appropriate for a liquid, where the unit cell is usually a cube. The cell length and atom coordinates will gradually adjust so that the average pressure is P 0. The time scale for reaching equilibrium is controlled by  P, the coupling strength.  P is related to the sound speed and heat capacity. larger  P  weaker coupling  slower relaxation I recommend  P = 20  T = 400  (time step) Thus  = 1 fs (common default)   T = 0.02 ps   P = 0.4 ps.

38. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 38 The next 3 slides are background material NPH: Andersen Method I Use scaled coordinates for the particles. For a cubic cell V 1/3 = L the cell length,  i =(0,1) Andersen defined (1980) a Lagrangian by introducing a new variable Q, related to the cell volume Particle momentum: Volume momentum:  Q ~ PV Cell KE atom KE u ~ atom PE Replaces p = mv Similar to atom momentum but for the cell

39. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 39 NPH: Andersen Method II Equations of motion: Hamiltonian of the extended system: Internal stress scalar External stress scalar Replaces v = p/m Replaces dp/dt = F = -  E High M  slow Cell changes  Q ~ PV Cell KE atom KE atom PE

40. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 40 NPH: Andersen Method III  The variable M can be interpreted as the “mass” of a piston whose motion expands or compresses the studied system.  M controlls the time scale of volume fluctuations. We expect M ~ L/c where c is the speed of sound in the sample and L the cell length  Andersen NPH dynamics yields isoenthalpic-isobaric ensemble. In 1980 Andersen suggested “It might be possible to do this (constant temperature MD) by inventing one or more additional degree of freedom, as we did in the constant pressure case. We have not been able to do this in a practical way.” ~ 3 years later, Nose figured out how to implement Andersen’s idea! Hans C. Andersen, J. Chem. Phys. 72, 2384 (1980)

41. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 41 NPH: Parrinello-Rahman Method I Parrinello and Rahman (1980) defined a Lagrangian by involving the time dependence of the cell matrix h: Leading to the equations of motion: PV Cell KE atom KE atom PE φ =  (PE) Replaces v = p/m Rate of change in cell is proportional to difference in external stress and the internal stress Proportional to v, thus is friction P ext could be  ext

42. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 42 NPH: Parrinello-Rahman Method II RP Equations of motion: where the internal stress tensor. Cell KE Dynamics of cell changes Newton’s Eqn Friction from cell changes

43. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 43 NPH: Parrinello-Rahman Method III Hamiltonian of the extended system:  The Parrinello-Rahman method allows the variation of both size and shape of the periodic cell because it uses the full cell matrix (extension of Andersen method).  An appropriate choice for the value of W (Andersen) is such that the relaxation time is of the same order of magnitude as the time L/c, where L is the MD cell size and c is sound velocity. Actually this is much longer than defaults (  P = 0.4 to 2 ps)  Static averages are insensitive to the choice of W as in classical statistical mechanics (the equilibrium properties of a system are independent of the masses of its constituent parts). PVCell KE atom KE atom PE φ =  (PE) dh T /dt More general: p  stress and V  h dependent term

44. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 44 NPT: combine thermostat and barostat Barostat-Thermostat choices  Berendsen-Berendsen  Andersen-Andersen  Andersen-(Nose-Hoover)  (Parinello-Rahman)-(Nose-Hoover)  Other hybrid methods Thermostats and barostats in CMDF are fully decoupled functionality modules and user can choose different thermostats and barostats to create a customized hybrid NPT method.

45. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 45 Martyna-Tuckerman-Klein Methods  The original Nose-Hoover method generates a correct Canonical distribution for molecular systems using a single coupling degree of freedom. This is appropriate if there is only one conserved quantity or if there are no external forces and the center of mass remains fixed. (This is normal case.) Martyna, Tuckerman and Klein [G. J. Martyna, M. E. Tuckerman, D. J. Tobias and M. L. Klein "Explicit reversible integrators for extended systems dynamics", Molecular Physics 87 pp. 1117-1157 (1996) extended the Nose- Hoover thermostat to use Nose-Hoover chains, where multiple thermostats couple each other linearly.G. J. Martyna, M. E. Tuckerman, D. J. Tobias and M. L. Klein "Explicit reversible integrators for extended systems dynamics", Molecular Physics 87 pp. 1117-1157 (1996)  They also developed a reversible multiple time step integrator which can solve NPH dynamics explicitly without iteration (1996).

46. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 46 NPT dynamics at different temperatures (P= 0 GPa, T= 25K to 300K) 25K 300K

47. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 47 Compression becomes harder when pressure increases, as indicated by the smaller volume reduction. (A 3 ) 588.7938 525.7325 489.6471 469.1264 452.1472 439.3684 432.4970 NPT dynamics under different pressures (300K, 0 to 12 GPa) for PETN

48. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 48 New ReaxFF can predict compressed structures well even without further fitting (same parameter as that in 100 K, 0 GPa fitting). Exp. (300 K) ReaxFF+Disp How well can we predict volume as a function of pressure?

49. © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 49 Conclusion  Most rigorous thermostat: Nose-Hoover (chain), but in practice often use Berendsen  Most rigorous barostat: Parrinello-Rahman, but in practice often use Berendsen Stress calculations usually need higher accuracy than for normal fixed volume MD. Therefore, we often use choose NVT instead of NPT, but we use various sized boxes and calculate the free energy for various boxes to determine the optimum box size for a given external pressure and temperature (even though this may require more calculations).