Lecture March30

Module-3

Ab Initio Molecular Dynamics

March 25

Bulk vs Finite System

~1023 atoms cannot be treated

computationally

~10 - 106 atoms can be usually treated computationallysuch boundary

effects should be avoided!

Periodic Boundary Conditions

if an atom goesout of the simulationbox, the same atom

should come into the box from the opposite

side

⨉

Minimum image convention:

longest distance

should not be larger than L/2

L

dx = xI � xJ

If |dx| > L/2, then

dx = dx� L sgn(dx)

dx

I

J

dx

O

If(x � L) thenx = x� L

If(x < 0) thenx = x+ L

wrapping the coordinates:correct distance:

J

Simulation of NVE Ensemble

Constant energy simulation (i.e. by solving the Hamilton’s equations of motion) in a constant volume

closed box (periodic/non-periodic)

A = hai =RdX a(X) �(H(X)� E)R

dX �(H(X)� E)

Ensemble average:

Home Work

• Write a working MD code for a two dimensional harmonic oscillator using Velocity Verlet Integrator (in any programing language)

Simulation of NVT Ensemble

Constant temperature simulation in a constant volume closed box (periodic/non-periodic)

systemsystem

Bath at T

• Hamilton’s equations of motion for the system, but with their momentum coupled to “bath

variables”• Total energy of the system will

no longer conserved• Bath+system energy is

conserved

A = hai =RdX a(X) exp(��H(X))R

dX exp(��H(X))

Fluctuations of a canonical ensemble should be captured in the simulations. For e.g. fluctuation in total energy

�2(E) = kBT2CV

Thermostat for NVT simulations

• Direct scaling of velocities:

T / R2I

Tt

T0=

R2I(t)

R20,I(t)

R0,I(t) = RI(t)

rT0

Tt

Usually, velocity scaling is used only in helping to equilibrate the system.

Scaling is often done if temperature goes beyond a window, or at some frequency (say every 50 MD steps); scaling

every time step doesn’t allow fluctuations, and thus leads to wrong ensemble!

req. velocity

current velocity

current temperature

req. temperature

Velocities are replaced by that from Maxwell-Boltzmann distribution (generated through Random numbers).

In a “Single particle” Andersen thermostat mode, thermostat is applied to a randomly picked single particle.

In “massive” Andersen thermostat mode every particle is coupled to the thermostat.

Thermostat is applied at a certain frequency (“collision frequency”) and not every MD step.

It can be proven that the correct canonical ensemble can be obtained by this thermostat; the thermostat disturbs

the dynamics (not good for computing diffusion const. etc.)

• Andersen Thermostat (system coupled to a stochastic bath)

Hans C. Andersen. J. Chem. Phys. 72, 2384 (1980)

• Langevin Thermostat

FI(t) = �rIU(RN )� �IMIRI(t) + gI

frictional coeff. Gaussian

random forcewith zero mean and

� =p2kBT0�IMI/�t

Brünger-Brooks-Karplus Integrator for the implementation of Lang. thermostat.

A. Brünger, C. L. Brooks III, M. Karplus, Chem. Phys. Letters, 1984, 105 (5) 495-500.

http://localscf.com/localscf.com/LangevinDynamics.aspx.html

• Berendsen Thermostat

Scaling velocity with �

�2 = 1 +�t

⌧

✓T0

T (t)� 1

◆

timescale of heat transfer

(0.1-0.4 ps)

Scaled every time stepProper fluctuations of a canonical ensemble is not well

captured

Nose Hoover Chain Thermostat

RI =PI

MI

PI = FI �p⌘1

Q1PI

⌘j =p⌘j

Qj, j = 1, ·,M

p⌘1 =

"X

I

P2I

MI� dNkBT

#� p⌘2

Q2p⌘1

p⌘M =

"p2⌘M�1

QM�1� kBT

#

p⌘j =

"p2⌘j�1

Qj�1� kBT

#�

p⌘j+1

Qj+1p⌘j

Ref: 1 Martyna, Kein, Tuckermann (1992),

J. Chem. Phys. 97 2635

Q1 = dNkBT ⌧2

Qj = kBT ⌧2, j = 2, · · · ,M

⌧ � 20�t

Results in canonical ensemble distribution

Widely used today in molecular simulations.

Special integration scheme is required: RESPA Integrator (Martyna 1996)

Martyna et al., Mol. Phys. 87 , 1117 (1996)

Usually

Ab Initio MD: Born-Oppenheimer MD

HBOMD({RI}, {PI}) =NX

I=1

P2I

2MI+ Etot({RI})

=NX

I=1

P2I

2MI+

min{ }

nD

({ri}, {RI})�

�

�

Hel

�

�

�

({ri}, {RI})Eo

+NX

J>I

ZIZJ

RIJ

rRI

D |Hel| |

E=DrRI |Hel| |

E+D |rRI Hel| |

E+D |Hel|rRI |

E

6=D |rRI Hel| |

E

Basis set should be large enough!

Convergence of wave function and energy conservation:

Time step (fs)

Convergence (a.u.)

conservation (a.u./ps)

CPU time (s) for 1 ps

trajectory

0.25 10-6 10-6 16590

1 10-6 10-6 4130

2 10-6 6 x 10-6 2250

2 10-4 1 x 10-3 1060