CHEM699.08

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CHEM699.08 Lecture 9 Calculating Time-dependent
Properties June 28, 2001 MM1stEd. Chapter
6 -- 333-342 MM2ndEd. Chapter 7.6 --
374-382
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Calculating Time-dependent Properties

An advantage of a molecular dynamics (MD)
simulation over a Monte Carlo simulation is that
each successive iteration of the system is
connected to the previous state(s) of the system
in time.

The evolution of a MD simulation over time allows
the data, or some property, at one time (t) to be
related to the same or different properties at
some other time (tdt).

A time correlation coefficient is a calculated
measurement of the degree of correlation for an
observed time-dependent property.
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Calculating Time-dependent Properties

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.
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Calculating Time-dependent Properties
y

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.
x
3
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Calculating Time-dependent Properties
y

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.

Is the movement of the sphere in the x direction
related to the motion in the y direction?
x
3
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Calculating Time-dependent Properties
y

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.

Is the movement of the sphere in the x direction
related to the motion in the y direction?
x
3
t 0
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Calculating Time-dependent Properties
y

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.

Is the movement of the sphere in the x direction
related to the motion in the y direction?
x
3
t 1
t 0
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Calculating Time-dependent Properties
y

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.

Is the movement of the sphere in the x direction
related to the motion in the y direction?
x
3
t 1
t 2
t 0
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Calculating Time-dependent Properties
y

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.

Is the movement of the sphere in the x direction
related to the motion in the y direction?
x
3
t 1
t 2
t 3
t 0
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Calculating Time-dependent Properties
y

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.

Is the movement of the sphere in the x direction
related to the motion in the y direction?
x
3
t 4
t 1
t 2
t 3
t 0
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Calculating Time-dependent Properties
y

Our simple 2D MD simulation of a single hard
sphere moving through an arbitrarily chosen plane.

Is the movement of the sphere in the x direction
related to the motion in the y direction?
x
3
t 4
t 5
t 1
t 2
t 3
t 0
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Calculating Time-dependent Properties

If there are two sets of data, x and y, the
correlation between them (Cxy) can be defined as
(1)
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Calculating Time-dependent Properties

If there are two sets of data, x and y, the
correlation between them (Cxy) can be defined as
(1)

This can also be normalized to a value between -1
and 1 by dividing by the rms of x and y
(2)
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Calculating Time-dependent Properties

A value of cxy 0 would indicate no correlation
between the values of x and y, while a value of 1
indicates a high degree of correlation.
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Calculating Time-dependent Properties

A value of cxy 0 would indicate no correlation
between the values of x and y, while a value of 1
indicates a high degree of correlation.

If x and y are found to only fluctuate around
some average value as would be the case for bond
lengths, for example, Equation 2 is commonly
expressed only as the fluctuating part of x and y.
(3)
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Calculating Time-dependent Properties

One drawback to Equation 3 is that the mean
values of x and y cant accurately be known until
the MD simulation has completed all M steps.
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Calculating Time-dependent Properties

One drawback to Equation 3 is that the mean
values of x and y cant accurately be known until
the MD simulation has completed all M steps.

Tired of waiting for those pesky MD simulations
to finish before generating your time-correlation
coefficients?

Well theres a way around this.
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Calculating Time-dependent Properties

Equation 3 can be re-written without the mean
values of x and y
(4)

This expression allows for the calculation of cxy
on the fly, as the MD simulation progresses!
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Calculating Time-dependent Properties

As the MD simulation proceeds the values of one
property can be compared to the same, or another
property at a later time
(5)
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Calculating Time-dependent Properties

As the MD simulation proceeds the values of one
property can be compared to the same, or another
property at a later time
(5)

If x and y are different properties, then Cxy is
referred to as a cross-correlation function. If
x and y are the same property, then this is
referred to as an autocorrelation function.

The autocorrelation function can be though of as
an indication of how long the system retains a
memory of its previous state.
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Calculating Time-dependent Properties

An example is the velocity autocorrelation
coefficient which gives an indication of how the
velocity at time (t) correlates with the velocity
at another time.
(6)

We can normalize the velocity autocorrelation
coefficient thusly
(7)
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Calculating Time-dependent Properties

For properties like velocities, the value of cvv
at time t 0 would be 1, while at loner times
cvv would be expected to go to 0.

The time required for the correlation to go to 0
is referred to as the correlation time, or the
relaxation time. The MD simulation must be at
least long enough to meet the relaxation time,
obviously.
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Calculating Time-dependent Properties

For properties like velocities, the value of cvv
at time t 0 would be 1, while at loner times
cvv would be expected to go to 0.

The time required for the correlation to go to 0
is referred to as the correlation time, or the
relaxation time. The MD simulation must be at
least long enough to meet the relaxation time,
obviously.

For long MD simulations the relaxation times can
be calculated relative to several starting points
in order to reduce the uncertainty.
Fig.1
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Calculating Time-dependent Properties
Shown here are the velocity autocorrelation
functions for the MD simulations of argon at two
different densities.

At time t 0 the velocity autocorrelation
function is highly correlated as expected, and
begins to decrease toward 0.

cvv(t)
Fig.2
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Time (ps)
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Calculating Time-dependent Properties
The long time tail of cvv(t) has been ascribed to
hydrodynamic vortices which form around the
moving particles, giving a small additive
contribution to their velocity.

Fig.3
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Calculating Time-dependent Properties
This slow decay of the time correlation toward 0
can be problematic when trying to establish a
time frame for the MD simulation, and also in the
derivation of some properties.

Transport coefficients require the correlation
function to be integrated between time t 0 and
t .

In cases where the time correlation has a long
time-tail there will be fewer blocks of data over
a sufficiently wide time span to reduce the
uncertainty in the correlation coefficients.

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Calculating Time-dependent Properties
Another example is the net dipole moment of the
system. This requires the summation of the
individual dipoles (vector quantities) of each
molecule in the system -- which will change over
time.

(8)
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Calculating Time-dependent Properties
Another example is the net dipole moment of the
system. This requires the summation of the
individual dipoles (vector quantities) of each
molecule in the system -- which will change over
time.

(8)
The total dipole correlation function is
expressed as

(9)
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Calculating Time-dependent Properties
Transport Properties

A mass or concentration gradient will give rise
to a flow of material from one region to another
until the concentration is even throughout.

The word transport suggests the system is at
non-equilibrium.

Here we will deal with calculating
non-equilibrium properties by considering local
fluctuations in a system already at equilibrium.

Examples temperature gradient, mass gradient,
velocity gradient, etc.

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Calculating Time-dependent Properties
The flux (transport of some quantity) can be
expressed by Ficks first law of diffusion thusly

Jz -D (dN / dz)
(10)
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Calculating Time-dependent Properties
The flux (transport of some quantity) can be
expressed by Ficks first law of diffusion thusly

Jz -D (dN / dz)
(10)
The time dependence (time-evolution of some
distribution) is expressed by Ficks second law

N (z,t)
2N (z,t)
(11)
D
t
z2
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Calculating Time-dependent Properties
Einstein showed that the diffusion coefficient
(D) is related to the mean square of the
distance, and in 3-dimensions this is given by

(12)
3D
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Calculating Time-dependent Properties
Einstein showed that the diffusion coefficient
(D) is related to the mean square of the
distance, and in 3-dimensions this is given by

(12)
3D
It is important to point out that Ficks law only
applies at long time durations, such as the case
above. To a good approximation some duration
where t effectively approaches infinity as far
as the simulation is concerned will be sufficient.

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Calculating Time-dependent Properties
fin
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