Lecture March30
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Transcript of Lecture March30
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Module-3
Ab Initio Molecular Dynamics
March 25
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Bulk vs Finite System
~1023 atoms cannot be treated
computationally
~10 - 106 atoms can be usually treated computationallysuch boundary
effects should be avoided!
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Periodic Boundary Conditions
if an atom goesout of the simulationbox, the same atom
should come into the box from the opposite
side
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⨉
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
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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:
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Home Work
• Write a working MD code for a two dimensional harmonic oscillator using Velocity Verlet Integrator (in any programing language)
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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
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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
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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
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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)
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• 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
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• 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
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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
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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
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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
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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