Heat Capacity and Energy Fluctuations
Running an MD Simulation
Equilibration Phase
Before data-collection and results can be analyzed the system must be
prepared via equilibration
Minimize energy
Velocity/Pressure scaling (move T/P to desired value)
Heat cycles
Tempering of potential parameters
During equilibration monitor thermodynamic properties and structure
After achieving stability, perform “production” run
Equilibration of the 2D argon system
0.5 1 1.5 2 2.5 3 3.5 4
Time (ps)
-320
-340
-360
-380
-400
-420
-440
-460
90
Temperature (K)
200
100
0
-100
-200
-300
-400
-500
-600
-700
Temperature
Total Energy
Energy (kJ/mol)
Energy (kJ/mol)
Potential and Kinetic Energy
0.5 1 1.5 2 2.5 3 3.5 4
Time (ps)
85
80
75
70
65
60
0.5 1 1.5 2 2.5 3 3.5 4
Time (ps)
The properties (such as time averages) should not depend
on the initial conditions !
Compute averages from several simulations :
Initial
condition
time
1
2
3
Equilibration
Production
Compute block-averages :
time
Equilibration
#1 #2
#3
Production
#4
#5
#6
#7
Difficulty: Block-averages might be the same, because the
“equilibration” is very slow
Sometimes several simulations are performed with
di erent system sizes to check equilibration
Evolution of the radial distribution function of the 2D argon system
0-1 ps
1-2 ps
16
14
2-3 ps
6
6
5
5
4
4
g(r)
g(r)
10
8
6
g(r)
12
3
3
2
2
1
1
4
2
0
0
1
2
3
4
5
6
7
8
9
10
Distance (Ang)
0
0
1
2
3
4
5
6
7
8
Distance (Ang)
9
10
0
0
1
2
3
4
5
6
7
8
Distance (Ang)
Production phase
It is important to continue to check the properties monitored during the
equilibration phase during the production phase. They may still not be
stable, in which case the beginning or the whole of the simulation might
have to be discarded.
9
10
Properties from MD Simulations
Time averages
Thermodynamic properties (energies, pressure . . . )
Structural properties
...
Dynamic quantities
Time correlation functions (and their FT’s, related to
spectroscopic properties.)
Transport properties (di usion . . . )
Analyzing Simulation Results
Directly visualize the results using molecular graphics.
The results can (of course) be analyzed by inspection,
although this is not as trivial as it may sound !
Snapshot from a simulation containing
512 water molecules and one Na + ion
Local environment of Na+ (aq)
Structural Properties
The radial distribution function gives a measure of the
local structure. It corresponds to the local concentration
of particles in a (thin) spherical shell at the distance r
around a central particle, relative to a uniform distribution
of particles.
r
Examples
Liquid Argon
NaCl Melt
2
4.5
1.8
4
1.6
3.5
1.4
3
g(r)
g(r)
1.2
1
0.8
2.5
2
1.5
0.6
0.4
1
0.2
0.5
0
Na+ - Na+
Cl- - ClNa+ - Cl-
0
1
2
3
4
5
6
Distance (Angstrom)
7
8
9
10
0
0
1
2
3
4
5
6
Distance (Angstrom)
7
8
9
10
Radial Distribution Functions
Useful for molecular liquids. Site-site radial distribution functions for water :
Hydrogen-Hydrogen
Oxygen-Hydrogen
3.5
1.4
1.6
3
1.2
1.4
2.5
1
2
0.8
1.5
0.4
0.5
0.2
0
1
2
3
4
5
6
Distance (Ang)
7
8
9
10
1
0.6
1
0
1.2
g(r)
g(r)
g(r)
Oxygen-Oxygen
0
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
Distance (Ang)
7
8
9
10
0
0
1
2
3
4
5
6
Distance (Ang)
For large molecules the number of site-site distribution functions obviously
becomes very large and only a subset is usually computed
For molecules it is also possible to compute various angular dependent
functions, and/or compute so-called spatial distribution functions
7
8
9
10
Integrating the radial distribution function gives the number of particles
surrounding the central particle
30
25
g(r)
n(r)
g(r), n(r)
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
Distance (Ang)
Structure of water around an Al 3+ ion: g Al − O ( r ) and n Al − O (r)
Dynamical Properties
Time Correlation Functions
Correlations between two di erent quantities A and B are
measured using a time correlation function :
C AB ( t) = < A( t) · B ( 0) >
Such time correlation functions are interesting since :
They give a picture of the dynamics in the system
Their time integrals are often related to various transport
properties
Their Fourier transforms are often related to experimental
spectra
If A and B are di erent properties, C is called a cross
correlation function
If A and B are the same property, C is called an auto
correlation function . The auto correlation function is a
measure of the “memory” of the system for some property
Dynamical Properties: Time Correlation Functions (cont)
If A( t) is a property of many particles the correlation function is collective
If A( t) is a property of a single particle the function is a single particle
correlation function
The single particle velocity auto-correlation (VAC) function :
C vv ( t) = < v ( t) · v ( 0) >
Example : Hydrogen in liquid water
1
0.8
0.6
Cvv(t)
0.4
0.2
0
-0.2
-0.4
-0.6
0
0.1
0.2
0.3
Time (ps)
0.4
0.5
Dynamical Properties: Time Correlation Functions (cont)
The average in the velocity auto-correlation function is typically taken
over all particles in the system and for a number of di erent time origins
< v ( t) · v ( 0) >
=
< v i ( t) · v i ( 0) >
t=0
j=2
t=0
j=1
t=0
j=0
0
N
1
N
Σ
i
1
M
=
< v i ( t) · v i ( 0) >
M
Σ v (t + t) · v (t )
j
i
t=1
j=2
vi(2+t). vi(2)
t=1
j=1
vi(1+t). vi(1)
vi(0+t). vi(0)
2
j
j
t=1
j=0
1
i
3
4
5
6
7
8
9
time
Dynamical Properties: Time Correlation Functions (cont)
The normalized time correlation function is
< A( t) · B ( 0) >
< A( 0) · B ( 0) >
C AB ( t) =
Fourier transforming the correlation function
Ĉ AB ( ω) =
!
∞
−∞
C AB ( t) e −i2πωt dt
Fourier transform of the hydrogen in liquid water VAC
3
2.5
2
DOS
1.5
water intramolecular bend
1
water intramolecular stretch
0.5
0
-0.5
0
500
1000
1500
2000
2500
3000
Wavenumber (cm -1)
3500
4000
Transport Properties
Integrating the velocity auto-correlation function gives
the di sion coefficient
1 ∞
D =
< v ( t) · v ( 0) > dt
3 0
This is an expression of the general type
γ =
∞
0
< Ȧ( t) · Ȧ( 0) > dt
The corresponding “Einstein relation” is
2tγ = < ( A( t) − A( 0)) 2 >
An alternate way to compute the di
sion coefficient
< |r ( t) − r ( 0) | 2 >
2tD =
3
The Einstein relation holds at long times !
3
MSD (Angstrom^2)
2.5
MSD
FIT
2
1.5
1
0.5
0
0
1
2
3
4
5
Time (ps)
Examples of other dynamical properties that can be studied
using time correlation functions and/or Einstein relations :
Total dipole moment auto-correlation function :
Related to (infrared) absorption spectrum
Auto-correlation function of elements of the pressure tensor :
Related to the viscosity
Orientational correlation functions :
Related to various spectroscopic techniques (NMR, IR, Raman . . . )
Handling Fast Vibrational Motion
Energy
Vibrational motion with high frequencies ( ω ≈ k BT )
are really quantized
V(x)=kx 2/2
7 hω/2
5 hω/2
3 hω/2
hω/2
k B T/2?
Displacement (x)
Also, since the frequencies are very high, short timesteps
are required
Flow of energy might be slow, due to poor coupling
between the degrees of freedom. This can lead to
problems with equilibration.
Treatment of Rigid Molecules
One “solution” is to make molecules / bonds rigid !
Rigid molecules
Separate motion into translation and rotation ; Separate
equations of motion for the center of mass, and some
representation of the rotation of the molecule (use Euler
angles or quaternions)
Rigid bonds
Constraint dynamics (for ”holonomic constraints”)
Appropriate for molecules that are partially flexible, such
as a polymer
Rigid small molecules can also be handled by introducing
fixed “bonds”, three per atom (or, actually, 3N - 6 bonds per
molecule)
Constraint Dynamics
Relatively simple algorithms exist : SHAKE and RATTLE
SHAKE enforces the (for instance) distance between two atoms
to be constant
r
rHH=1.63298
Å
1.0
=
H
O
Å
H=
rO
1.0
Holonomic constraints :
f ( q1 , q2 , . . . , t)
= 0
rij2 − dij2
= 0
The SHAKE method is tightly connected to the integrator used,
the variant for the velocity Verlet integrator is termed RATTLE
The method introduces an extra force directed along the bond
between two atoms at time zero (i.e. before the integration)
First the integration step is completed as if there were no
constraint force
Then all constraint forces are solved for, one by one, iteratively
Long Range Interactions
A long range interaction decays no faster than r − d ,
where “d” is the dimensionality of the system
The problem : The interaction decay is so slow that we
cannot just truncate it at a reasonably short distance
Even worse : Conditionally convergent !
Important members of this class :
charge-charge (r − 1 )
charge-dipole (r − 2 )
dipole-dipole (r − 3 )
charge-quadrupole (r − 3 )
The Coulomb interaction :
V =
N
1
4
0
N
i= 1 j= i+ 1
qi qj
r ij
Di erent methods to treat this kind of interaction have been
devised (Ewald, reaction field, various multipole methods)
Here only the ordinary Ewald method is (briefly) considered
Sum over periodic images built up in spherical layers :
εs
The very large sphere is surrounded by a medium with relative
permittivity s
The potential energy can be written as :
V =
1
4
0
1
2
n
N
N
i
j
qi qj
|r ij + n|
where n = ( nx L, ny L, nz L) . nx , ny , nz are integers and L the
size of the central image. The ’ in the sum : i = j for |n| = 0
The Ewald method : Add screening charge distribution with
3 − 3/ 2 − α 2 r 2
)
e
opposite charge and equal magnitude ( α π
+
+
-
-
The interaction between the charges is now short ranged :
V real
=
erfc( x ) =
1
4
0
2π
1
2
n
∞
− 1/ 2
x
N
N
i
j
e
− t2
erfc ( α|r ij + n|)
qi qj
|r ij + n|
dt
For suitable values of the α parameter, |n| can be truncated to 0
The original potential is restored by adding a canceling charge
distribution :
The canceling distribution is summed in reciprocal (Fourier)
space :
1 1
V reciprocal =
3
0L 2
k=0
1 −
e
2
|k |
|k|2
4α 2
N
N
qi qj cos ( r ij · k )
i
j
The sum goes over reciprocal vectors , k = 2πn/ L
A correction term needs to be subtracted as the sum in
reciprocal space includes the interaction of the canceling
distribution at r i with itself :
V self = −
N
α
4π 3/ 2
0
qi2
i
The expression V = V real + V reciprocal + V self , corresponds to the
potential energy for the large sphere surrounded by a medium
with s = ∞