Lecture March30
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Transcript of 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