7 To solve numerically the equation of motion, we use the velocity Verlet or leap frog algorithm. _ V i n = F i n m i (F.5) For time step, we approxim

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1 69 Appendix F Molecular Dynamics F. Introduction In this chapter, we deal with the theories and techniques used in molecular dynamics simulation. The fundamental dynamics equations of any system is the Lagrangian equation of motion d dt ( L )= L _q k q k (F.) where the Lagrangian function L(q; _q) is dened in terms of kinetic and potential energies L = K, V (F.) F. NVE Dynamics For an isolated system, we have Lagrangian as L = X i m i _r i, V (r ;r ;::: ;r i ;:::) (F.3) and dynamics equations r i = F i m i (F.4)

2 7 To solve numerically the equation of motion, we use the velocity Verlet or leap frog algorithm. _ V i n = F i n m i (F.5) For time step, we approximate _ V i n = V i n+, V i n, (F.6) leading to V i n+ = V i n, + F n i m i (F.7) and X n+ = X n + V i n+ (F.8) The problem with these two equations is that the velocities and coordinates are not from the same time step. Thus the KE and PE are available only for dierent time steps. To obtain equations involving V n and X n at the same time step, we start with Eq. (F.7) and estimate V n by V i n = (V i + V i n+ n, ) (F.9) Then we have V i n+ = V i n + F i n m i (F.)

3 7 and V i n = V i n, + F n i m i (F.) F.3 NVT Dynamics F.3. Theory There are many ways of doing molecular dynamics simulations under constant temperature. Among them, the extended system method originally formulated by Nose and Hoover is most important. An additional degree of freedom corresponding to a heat bath is introduced. The total energy of the physical system is allowed to uctuate by a thermal contact with a heat bath. In Nose's original formulation, two frames of variables, real variables corresponding to realistic motion of particles and virtual variables, are introduced. The relations between these two kinds of variables are derived from an assumption of time scaling dt = dt=s, t is a real time, t is a virtual time, and the scaling factor s corresponds to a heat bath variable. The canonical distribution is realized in a physical system if we choose the Lagrangian as the following L = X i m i s r_ i, V (r ;r ;::: ;r i ;:::)+ Q s _s,gk B T B ln s (F.) Equation for particles d dt (m i s _r i )=m i s_s_r i +m i s r i (F.3) Convert the equation to real time by _ A = s da dt (F.4)

4 m i ds dt dr i dt + m is( s 7 d r i dt, s ds dt dr i dt )=Fi (F.5) Dene s = s ds dt (F.6) we have d r i dt = F i, s dri m i dt (F.7) Equation for s d dt (Q s _s) = X i m i s_r i,gk B T B s (F.8) Convert to real time we have Since s Q d X s s dt = s m i _r i, gk B T B s i X i m i s _r i =KE = fk B T (F.9) (F.) where f is the degree of freedom. If we choose g as degree of freedom also, we have d s dt = gk BT B Q s ( T T B, ) (F.)

5 F.3. Numerical Integration 73 Let's consider a general form of dynamics equation with velocity dependent forces, or friction forces V t = F M, fv (F.) where f indicates a frictional term. The numerical equation thus becomes V n+ = V n, + F n M, f nv n (F.3) approximate V n by V n = V n+ + V n, (F.4) Then we have V n+ =, f n + f V n, + n + f n F n M (F.5) It is often convenient to break the leap frog algorithm into two half steps V n,,! V n which is V n+ =(, f n)v n + F n M (F.6) and V n,! V n+

6 74 which is Replacing f by s,wehave V n = + f n V n, + F n + f n M (F.7) V i n+ =(, n)v i n + F i n m i (F.8) V i n = + nv i n, + F n i + n m i (F.9) and n = n, + gk BT B ( T n,, ) (F.3) Q s T B F.4 NPT(Gibbs) Dynamics Formulation F.4. Constant Temperature and Constant Stress Ensemble Constant temperature and constant stress ensemble is one of the most important ensemble being studied, because most of the experiments are done under such condition. Results from molecular dynamics simulation of constant temperature and constant constant stress ensemble can be directly related to experimental results. Andersen 3 has shown how MD calculations can be modied to study systems under constant pressure by introducing the volume of the system as an additional variable. Later on, Parrinello and Rahman 4 extended the variable to simulation cell parameters which results the constant stress ensemble. By combining the Hamiltonian of Nose-Hoover formulation and that of Parrinello-Rahman, we get a constant temperature and constant stress ensemble. Parrinello-Rahman's original formulation used cell transformation matrix H = (a; b; c) which is not invariant under rotation. This

7 75 may introduce articial eects into the dynamics systems, lead to instable systems. In our implementation, we take the metric tensor G = H T H as our variable, which may gaveus better stability. F.4. Coordinates Transformation Let e x ;e y ;e z =fe gbe a set of orthogonal unit vectors and write the primitivevectors for the unit cell as a; b; c = H. Then we can write the coordinates of particle i as R i = H i (F.3) where R i is the coordinate of particle i, i From equation (F.3), we have is the scaled coordinate of particle i. _ R i = H _ i (F.3) The inner product becomes (R i R j )=R i Rj =H H i j = G i j (F.33) where the metric tensor G = H H = H T H =(H T H) (F.34) or G = B H + H + H 3 H H + H 3 H 3 H 3 H 33 H H + H 3 H 3 H + H 3 H 3 H 33 C A (F.35) H 3 H 33 H 3 H 33 H 33

8 76 while H = B p G, H, H 3 (G, H 3 H 3 )=H p G, H 3 C A (F.36) G 3 =H 33 G 3 =H 33 p G33 The H matrix is calculated in the order of H 33! H 3! H 3! H! H! H F.4.3 Strain Tensor Consider a homogeneous distortion H,! H (F.37) in which the unit cell change size but the scaled particle coordinates remain xed. The new particle coordinates are given by R i = H H, ; Ri ; (F.38) Thus the displacement of each particle is given by ~u = ~ R, ~ R =(HH,, )R (F.39) Using the Landau-Lifschitz denition of nite strain = (u x + u x + u x u x ) (F.4) leads then to = ( ~ H, GH,, ) (F.4)

9 77 where the tilde indicates transpose of a matrix and H refers to the reference state of the cell in terms of which the external stress is dened. F.4.4 Lagrangian for Particles and Cell Kinetic Energy For particle kinetic energy, we have KE particle = s X i M i _R i _ R i = s X i M i _ i G _ i (F.4) For nose parameter s KE nose = Q _s (F.43) For cell kinetic energy, we have KE cell = Ws _G _G (F.44) Potential Energy Particles potential energy PE particle includes all valence and nonbond interactions. while the nose potential energy is dened as PE nose = gk B T B log s (F.45) The total potential energy for the stresses system is PE cell = p(, )+ (,S+p) (F.46) where istheunit cell volume, p = T race(s) isthepressure and S is the external 3 stress tensor.

10 78 Dening the transformed stress tensor as = H, (,S+p)~ H, (F.47) we have PE cell = p(, )+ G (F.48) Lagrangian The above results leads to a Lagrangian L = s X i m i _ i G _ i, V (~r ;~r ;::: ;~r i ;:::) + Qs, gk B T B log s + W _G _G s, p(, ), G (F.49) F.4.5 Dynamics Equations Internal Coordinates Since L i = Ri i L R i = H F i (F.5) ( L )= (m t s _ i i s G _ i t )=m i s_sg _ i + m i s G i + m i s _G _ i s (F.5) By using ( L )= L t s _ i i (F.5)

11 79 we have m i s G i = H F i, m i s _sg _ i, m i s _G _ i (F.53) m i s G, G i = G, H F i, m i s _sg, G _ i, m i s G, G _ i i = m i s H, F i, _s s _i, G, G _ i (F.54) (F.55) where we have used G, G = (F.56) G, H = H, (F.57) Convert nose time to real time _A = A t s = s A t (F.58) then s t ( s i t )= i s t, s s 3 t i t (F.59) Dene s = s s t P = G, G t (F.6) (F.6) Finally we have i t = H, F i m i, si t, P i t (F.6)

12 Nose Parameter 8 L s L _s = X i m i s i G i, gk B T B s + Ws _ G _G (F.63) = Q _s (F.64) s = s t ( s Put them together, and we have t t )= s s t (F.65) Q s t = X i m i s i G i, gk B T B + Ws _G _G =(KE particles + KE cell ), gk B T B = gk B (T, T B ) (F.66) where we have used KE particles + KE cell = gk B T with T as the system temperature. Dening Q = gk B T B s (F.67) we have s t = ( T, ) (F.68) s T B

13 8 Metric G Since, V = P pe V = V G G (F.69) (F.7) = H G ~ G, H, G = ~ H, [ ( + )]H, = 4 ( ~ H, H, + ~ H, H, ) (F.7) So V =, G 4 ( H ~, H, + ~ H, H, pe )P =, H, P pe ~ H, (F.7) where we have used the symmetry property of P pe. Similarly KE particle G = H, P ke ~ H, (F.73) Since det H = ijk H i H j H k (F.74) det H H i = ijk H j H k (F.75) H i det H H i = ijk H i H j H k = det H (F.76)

14 8 det H H = ~ H, det H (F.77) = ~a ( ~ b ~c) = ijk H i H j H k3 = det H (F.78) det G = det ~ H det H = (det H) = (F.79) = G G ~, = G, (F.8) By using the above equations, we have L G = H, (P pe + P ke ) ~ H, P ext G,, = H (P, internal, S external ) H ~, P ext G, (F.8) L _G = Ws _G (F.8) G = h i H, W (P internal, S external ) H ~, P ext G,, s _s _G (F.83) In Hoover formulation (real time) G t = h i H, W (P internal, S external ) H ~, P ext G,, s _G (F.84)

15 F.4.6 About Cell Mass 83 From the above derivation, W _G has unit of ML. We propose T has unit of WL4 T while the kinetic energy usually W = fm cell, 3 (F.85) with f a scale factor and M cell = nx i m i (F.86) F.4.7 Numeric Integration Two Step Leap-frog for TPN Following are the equations used in our integration i t = H, F i m i, si t, P i t (F.87) G t = h i H, W (P internal, S external ) H ~,, P ext G,, s _G (F.88) and The n,,! n, s t = ( T, ) (F.89) s T B equation for internal velocites of particles V n, =[, (P + s )]V n, + F atm n, M (F.9)

16 The n,,! n, 84 equation for velocity of metric parameters _G n, =(, s ) _G n, + F cell n, W (F.9) Nose parameter Internal coordinates of particles s n = s n, + ( T n,, ) (F.9) s T B X n = X n, + V n, (F.93) Metric parameters G n = G n, + _G n, (F.94) The n,,! n equation for internal velocity of particles V n = + (P n + s n) V n, + + (P n +n) s F atm n M (F.95) The n,,! n equation for velocity of metric parameters _G n = + s n _G n, + + s n F cell n W (F.96) since P n and F cell n depends on V n and _G n, we have to solve these two equations self-consistently. Following is the owchart of an integration circle.

17 85 Figure F.: NPT dynamics owchart

18 F.5 Rigid Dynamics 86 In general, atomistic molecular dynamics of various ensemble can handle most of the molecular systems; however treating some of the molecules as rigid can decrease the degree of freedoms and increase dynamics time-step, and thus, can simulate larger systems longer. F.5. Partition of dynamics equations Consider the motions of a rigid molecule M interacting with other atoms or molecules. For a nonlinear molecule there are 6 degrees of freedom (3 translation, 3 rotation). The forces on the atoms of M are calculated and combined to obtain the net forces and torques on the body F M =, X im V r i (F.97) T M =, X im (r i, R cm ) V r i (F.98) These calculations are carried out in normal space-xed coordinates. We introduce a coordinate system attached to the center of mass with the same orientation as the space-xed system at the initial time. The particle position r(t) i =R cm (t)+a (t)[r() i, R cm ()] (F.99) where A (t) is the rotation matrix. The particle velocities are then given by v(t) i =V cm (t)+! (t)[r(t) i, R cm (t)] (F.) where! is the angular velocity

19 87 Dene r i cm(t) =r i (t),r cm (t) (F.) Using equation (F.) in KE, we have KE = X im m i v i v i = MV cm V cm + X im i m i h!(t) rcm(t) i = KE cm + KE rot (F.) So we can partition equation of motion into center of mass translation and rotation with respect to center of mass coordinates. It's the angular motion which needs further study. F.5. Dynamics Equation for Angular Motion We can write the rotational kinetic energy as KE rot = X im m i! r i cm;! r i cm; (F.3) By using =, (F.4) we have KE rot = = X im X im m i [!! r i cm; ri cm;,!! r i cm; ri cm;] m i! ( r i cm, r i cm; ri cm;)! = I!! (F.5)

20 88 where I = X im m i ( r i cm, r i cm; ri cm;) (F.6) is dened as the moment of inertia tensor. The angular momentum We can rewrite KE rot as J = KE rot! = I! (F.7) KE rot = I, J J (F.8) Then for micro-canonical ensemble L = L NV E + X M M cm V i cm V i cm + and for canonical ensemble we have Lagrangian as L = L NV T + X M M cm s V i cm V i cm + and for Gibbs ensemble we have Lagrangian as L = L NPT + X M M cm s V i cm V i cm + X M X M X M I M!M!M s I M!M!M s I M!M!M (F.9) (F.) (F.) noted, there are no coupling between cell deformation and rotation of rigid molecules. For rotational degree of freedom, the dynamics equation takes the form of _ J = T, fj (F.) where T =, V, f = s for canonical and Gibbs ensembles, f = for micro-

21 canonical ensemble. 89 The two-step numerical integration equations are: J n, =(, f n,)j n, + T n, (F.3) and J n = + f n, J n, + + f n, T n (F.4) The above equation is in space-xed coordinates. In order to solve for the new angular coordinates, we must transform from space-xed (J) to body-xed (J B ) coordinates by applying J B = A (t)j (F.5) where A (t) takes the current space-xed coordinates transform back to the body- xed coordinates at t =. Since J = I! (F.6) where I is evaluated in the original body-xed coordinate system at t =. Thus! B (t) =I, J B (t)=i, A(t)J (F.7) F.5.3 Solving the Equation Using Quaternions Since the potential energy and forces (thus torques) are evaluated in space-xed coordinate systems, while the moment ofinternia is most easily expressed in a body- xed coordinate system, we have to relating the coordinates of a moving-body to a space xed coordinate system. The most common approach is in terms of Euler angles:; ;, where and give the orientation of a body axis relative to the space-

22 9 z ζ η y z x φ ξ z ζ θ η ζ θ η x φ ξ y x φ ψ ξ y Figure F.: The rotations dening the Euler angles xed frame and is the rotation of the body about this axis. The transformation tensor is A = B cos cos, cos sin sin cos sin + cos cos sin sin sin, sin cos, cos sin cos, sin sin + cos cos cos cos sin sin sin, sin cos cos (F.8) C A However, Euler angles lead to singularities whenever the equation of motion takes too close to o or 8 o. The using of four variable quaternions rather than the three variable Euler angles can by-pass the singularity problems.

23 9 The quaternions are dened in terms of Euler angles as Q x = sin cos, (F.9) Q y = sin sin, (F.) Q z = cos sin + (F.) where the normalization condition is Q 4 = cos cos + 4X = Q = (F.) (F.3) The relation between A and Q is A ii = Q 4 + Q i, X j6=i Q j (F.4) A ij =Q i Q j + ijk Q 4 Q k if i 6= j (F.5) where i =;;3. In matrix form A = B Q x, Q y, Q z + Q 4 Q x Q y +Q 4 Q z Q x Q z,q 4 Q y Q x Q y,q 4 Q z,q x+q y,q z+q 4 Q y Q z +Q 4 Q x Q x Q z +Q 4 Q y Q y Q z,q 4 Q x,q x,q y+q z+q 4 C A (F.6)

24 9 the quaternions change smoothly in time with no singualr points. The relation between _Q and! is _Q i = (Q 4! i + ijk Q j! k ), i =;;3 (F.7) _Q 4 =, Q i! i (F.8) If we dene! 4 =,in matrix form _Q = S! (F.9) we have S = B Q 4,Q z Q y Q x Q z Q 4,Q x Q y,q y Q x Q 4 Q z C A (F.3),Q x,q y,q z Q 4 Note that ~ SS= Let's partition the equation Q n = Q n, + _ Q n, (F.3) into Q n, = Q n, + _ Q n, (F.3) Q n = Q n, + _Q n, (F.33) Then following is the owchart of one dynamics step

25 93 Figure F.3: Rigid dynamics owchart

26 F.5.4 Linear Molecule 94 Figure F.4: Orientation of linear molecule Let's assume the bond axis of a linear molecule as ~r as in the above gure; we can diagnolize the inertia tensor by transformation coordinate system (x; y; z) into (x ;y ;z ) I =AIA T = B a a C A (F.34) Since the inverse is singular, we haveto pay special attention to linear molecule. We require angular momentum J be orthogonal to the bond axis. Denoting the unit vector along the molecule as e a, we must have J e a = (F.35)

27 95 Then rewrite I as proportional to the unit matrix I = B C a a A = a a (F.36) The kinetic energy becomes KE rot = J J I (F.37) To ensure angular momentum J be orthogonal to linear molecule bond axis, we Schmit orthogonalize each iteration by applying J new = J old, (e a J old )e a (F.38) The inverse of inertia tensor I, can be calculated as I, = A B T racei T racei T racei C A ~ A (F.39) where transformation matrix A can be calculated with Euler angle (; ; ) as = arccos r z r r y =, arccos p r x + ry = (F.4) The quantities specied are dened in Fig. F.4

28 F.6 References 96. S. Nose, J. Chem. Phys. 8, 984, 5.. W.H. Hoover, Phys. Rev. A 3, 985, H.C. Andersen, J. Chem. Phys. 7, 98, M. Parrinello and A. Rahman, Phys. Rev. Lett. 45(4), 98,

Temperature and Pressure Controls

Ensembles Temperature and Pressure Controls 1. (E, V, N) microcanonical (constant energy) 2. (T, V, N) canonical, constant volume 3. (T, P N) constant pressure 4. (T, V, µ) grand canonical #2, 3 or 4 are