MODELING MATTER AT
NANOSCALES
3. Empirical classical PES and typical
procedures of optimization
3.05. Molecular Dynamics
Second Newton’s Law in
Nanoworld
Second Newton’s law and
Hypersurfaces
For an isolated system, the translational motion of a given n
particle in a container of N particles is described by the second
Newton’s law:
2
d rn
mn 2  Fn r1 , r2 ,..., rN 
dt
being the force on the n particle expressed in terms of the
potential originated by all particles in the system:
Fn   nV r1 , r2 ,...rN 
where t is the time, mn the mass of n and rn the position vector
of any particle.
Second Newton’s law and
Hypersurfaces
For an isolated system, the translational motion of a given n
particle in a container of N particles is described by the second
Newton’s law:
2
d rn
mn 2  Fn r1 , r2 ,..., rN 
dt
being the force on the n particle expressed in terms of the
potential originated by all particles in the system:
Fn   nV r1 , r2 ,...rN 
where t is the time, mn the mass of n and rn the position vector
of any particle.
Thus, Fi is the net force acting on a given particle i, crated by a
potential V of interaction with all other particles of the system in
an instant dt.
Second Newton’s law and
Hypersurfaces
If we assume that:
1.- The potential can be computed from all pairs of particles and
that it is additive;
Second Newton’s law and
Hypersurfaces
If we assume that:
1.- The potential can be computed from all pairs of particles and
that it is additive;
2.- There are no external forces
Second Newton’s law and
Hypersurfaces
If we assume that:
1.- The potential can be computed from all pairs of particles and
that it is additive;
2.- There are no external forces
Then:
V r1 , r2 ,..., rN    unl (rnl )
n l
Second Newton’s law and
Hypersurfaces
If we assume that:
1.- The potential can be computed from all pairs of particles and
that it is additive;
2.- There are no external forces
Then:
 


V r1 , r2 ,..., rN    unl (rnl )
n l
where and the force acting on each particle is expressed as:
unl (rnl )
Fn   Fnl   Fln   
rnl
l
l
l
Second Newton’s law and
Hypersurfaces
Thus, the problem is reduced to integrate a differential equation
of motion as:
d 2 rn
unl (rnl )
mn 2   
dt
rnl
l
corresponding to each particle of mass mn and a position in space
given by the vector rn
9
Second Newton’s law and
Hypersurfaces
Thus, the problem is reduced to integrate a differential equation
of motion as:
d 2 rn
unl (rnl )
mn 2   
dt
rnl
l
corresponding to each particle of mass mn and a position in space
given by the vector rn that is developing a potential energy unl
with each one of the other particles m of the system, found at rnl
distances .
10
Second Newton’s law and
Hypersurfaces
Therefore, the requirements for carrying out a simulation of
molecular dynamics are:
- Selecting a hypersurface or potential energy function,
expressing reliable unl(rnl) energies between individual
particles.
11
Second Newton’s law and
Hypersurfaces
Therefore, the requirements for carrying out a simulation of
molecular dynamics are:
- Selecting a hypersurface or potential energy function,
expressing reliable unl(rnl) energies between individual
particles.
- Solving the equation of motion in a way that given positions
and speeds of all and each particles in a given time t, all
positions and speeds can be also obtained for a time t + Dt
where the interval Dt is small and limited.
12
Integration of the Equations of
Motion and Temporary
Parameters
13
Equation of motion
Requirements: Numerical integration of the equations of motion
for hundreds or thousands of atoms or molecules.
14
Equation of motion
Requirements: Numerical integration of the equations of motion
for hundreds or thousands of atoms or molecules.
Approximation: Exists a compromise between the quality of
results and the time interval size for successive evaluation of
integrals, called as Dt.
15
Equation of motion
A common method for integration is the steepest descent
procedure where the solution is represented by the values of a
finite number of points in the space of each system, named as
“grid points” of the equation:
d 2 rn 1
unl (rnl )


2
dt
mn l
rnl
16
Equation of motion
A common method for integration is the steepest descent
procedure where the solution is represented by the values of a
finite number of points in the space of each system, named as
“grid points” of the equation:
d 2 rn 1
unl (rnl )


2
dt
mn l
rnl
Such points are simulated by a polynomial and the complete
solution is obtained after the evaluation of the derivative:
2
d rn dt
2
with different values of variables obtained by the polynomial
in each one of the points of the “grid”.
17
Translational motion: The Gear’s
Predictor – Corrector Algorithm
Integration is carried out in two successive processes:
1.- The position of the center of mass and its derivatives is
“predicted” for time t + Dt from the actual position and the
derivatives in the previous step at t, according to a truncated
Taylor’s series of order m :
m  J !Dt J
 d kr 
 d J r (t ) 
 k   


J
 dt  pred J k  k! J  k ! dt 
18
Translational motion: The Gear’s
Predictor – Corrector Algorithm
2.- Derivatives of all orders are “corrected” by the forces F (or
negative gradients) that were computed in the predicted
positions of the step 1:

2
pred
2

D
t
F
(
r
)
d
r 
d r 
 d kr 

m
 2  
 k    k   Ck 
m
 dt corr  dt  pred
 dt  pred 
 2
k
where the values of the constants Ckm depend on the order and
can be found in literature.
19
Translational motion: The Verlet
Algorithm
It uses positions and accelerations at a time t and in a previous
moment t - Dt in order to predict positions at time t + Dt. The
scheme of integration is based in a Taylor expansion to the 3rd.
order:
2
d
rn (t )
2
rn t  Dt   2rn t   rn t  Dt   Dt
dt 2
and the speeds can be obtained by the basic rule of
differentials:
drn (t ) rn t  Dt   rn t  Dt 

dt
2Dt
20
Translational motion: The “Leap
Frog” Verlet Algorithm
It tries the Verlet algorithm for calculations of speeds more
precisely:
1
1
drn (t  Dt ) drn (t  Dt ) d 2 r (t )
n
2
2


Dt
2
dt
dt
dt
1
drn (t  Dt )
2
rn t  Dt   rn t  
Dt
2
dt
21
Translational motion: The “Leap
Frog” Verlet Algorithm
It tries the Verlet algorithm for calculations of speeds more
precisely:
1
1
drn (t  Dt ) drn (t  Dt ) d 2 r (t )
n
2
2


Dt
2
dt
dt
dt
1
drn (t  Dt )
2
rn t  Dt   rn t  
Dt
2
dt
 Accelerations at time t are calculated from coordinates in
that moment.
22
Translational motion: The “Leap
Frog” Verlet Algorithm
It tries the Verlet algorithm for calculations of speeds more
precisely:
1
1
drn (t  Dt ) drn (t  Dt ) d 2 r (t )
n
2
2


Dt
2
dt
dt
dt
1
drn (t  Dt )
2
rn t  Dt   rn t  
Dt
2
dt
 Accelerations at time t are calculated from coordinates in
that moment.
 Speeds are calculated at the mean positive interval of the
first equation and are substituted in the second to obtain
the positions at time t + Dt.
23
Translational motion: The “Leap
Frog” Verlet Algorithm
It tries the Verlet algorithm for calculations of speeds more
precisely:
1
1
drn (t  Dt ) drn (t  Dt ) d 2 r (t )
n
2
2


Dt
2
dt
dt
dt
1
drn (t  Dt )
2
rn t  Dt   rn t  
Dt
2
dt
 Accelerations at time t are calculated from coordinates in
that moment.
 Speeds are calculated at the mean positive interval of the
first equation and are substituted in the second to obtain
the positions at time t + Dt.
 Speeds at time t are calculated as the mean of the speed at
24
t - Dt and t + Dt.
Selection of time intervals for
the simulation
If the time intervals Dt are “large” (i.e. several femto seconds, or
fs), molecular dynamics simulations can be more comprehensive
in the exploration of the configuration’s space of the system.
25
Selection of time intervals for
the simulation
If the time intervals Dt are “large” (i.e. several femto seconds, or
fs), molecular dynamics simulations can be more comprehensive
in the exploration of the configuration’s space of the system.
However, the integrated equations for translation oblige forces
to be constant in each step. This fact compromises the lasting of
the simulation with the molecular internal motions originating
dynamics in the physical reality, which are mostly molecular
vibrations.
26
Selection of time intervals for
the simulation
The size of the time interval Dt is limited by the maximal frequency of
vibrations among the component particles, and must be:
1
Dt 
 max
27
Selection of time intervals for
the simulation
The size of the time interval Dt is limited by the maximal frequency of
vibrations among the component particles, and must be:
1
Dt 
 max
For a typical stretching vibrational mode corresponding to a non –
associated O-H bond [~ 3500 cm-1 = 1.05 (1014) s-1], Dt cannot be
larger than --1 = 9.5 (10-15) s, or 9.5 fs. In this case, Dt uses to be
selected of 1 fs.
28
Selection of time intervals for
the simulation
The size of the time interval Dt is limited by the maximal frequency of
vibrations among the component particles, and must be:
1
Dt 
 max
For a typical stretching vibrational mode corresponding to a non –
associated O-H bond [~ 3500 cm-1 = 1.05 (1014) s-1], Dt cannot be
larger than --1 = 9.5 (10-15) s, or 9.5 fs. In this case, Dt uses to be
selected of 1 fs.
Then, if the bond modes [3700 to 400 cm-1 or 1.11(1014) s-1 to
1.20(1013) s-1] are excluded and, for example, only the internal
rotational modes (torsional) are considered in a certain case [~ 200
cm-1 = 6.00(1012) s-1], Dt can easily be smaller than --1 = 166.8 (10-13)
s, or 166.8 fs. These are the cases of the so – called constrained
29
molecular dynamics and Dt could reach 2 fs and even more.
Comparisons of methods
 Small Dt values for each step favors the Gear’s algorithm
because presents smaller fluctuations of the total energy than
the Verlet’s.
30
Comparisons of methods
 Small Dt values for each step favors the Gear’s algorithm
because presents smaller fluctuations of the total energy than
the Verlet’s.
 Intermediate Dt values for each step give similar behaviors for
both methods.
31
Comparisons of methods
 Small Dt values for each step favors the Gear’s algorithm
because presents smaller fluctuations of the total energy than
the Verlet’s.
 Intermediate Dt values for each step give similar behaviors for
both methods.
 Large Dt values for each step favor the Verlet’s algorithm.
32
Integration of molecular rotation
Rotational molecular motion can be modeled with the rigid body
approximation, where rotation of the main axis is given by the Euler’s
equations. They are also integrated with the “leap frog” or Gear’s
routines:
d x
I xx
 I yy  I zz  y z  Tx
dt
d y
I yy
 I zz  I xx  z x  Ty
dt
d z
I zz
 I xx  I yy  x y  Tz
dt
where I’s are components of the inertia moment on coordinates x, y,
z, q is the angular speed and T the torque (depending on the
applied rotational force and the distance between the point of
application and the center) acting on the molecule that originates the
33
motion of such coordinate during the simulation.




Stages of the Simulation
34
Initialization
 Each participating center is initially placed according a trial or guess
geometry. The simulation begins with random speeds, although
corresponding to the input simulation temperature. Temperatures in
molecular dynamics are usually depending on the equipartition
principle. The calculated temperature of each step is used for the
energetic control of the system.
35
Initialization
 Each participating center is initially placed according a trial or guess
geometry. The simulation begins with random speeds, although
corresponding to the input simulation temperature. Temperatures in
molecular dynamics are usually depending on the equipartition
principle. The calculated temperature of each step is used for the
energetic control of the system.
 The speed of all and each particle is normalized initially to avoid a
resulting linear translation of the whole system, that is essentially
meaningless.
36
Initialization
 Each participating center is initially placed according a trial or guess
geometry. The simulation begins with random speeds, although
corresponding to the input simulation temperature. Temperatures in
molecular dynamics are usually depending on the equipartition
principle. The calculated temperature of each step is used for the
energetic control of the system.
 The speed of all and each particle is normalized initially to avoid a
resulting linear translation of the whole system, that is essentially
meaningless.
 During the initialization step of the simulation, renormalization of
speeds is repeated as much as needed to maintain the system at the
input temperature. Initialization is considered done when the
temperature becomes stable (the energy of the system becomes
stationary and equilibrium is reached) and, therefore,
37
renormalizations are no longer necessary.
Development
 The “development” stage means a simulation during a given
virtual time given by a significant number of Dt steps. This
time of simulation is virtual because is NOT the time elapsed
by the computer to carry it out, but an imposed parameter to
the molecular dynamics modeling.
38
Development
 The “development” stage means a simulation during a given
virtual time given by a significant number of Dt steps. This
time of simulation is virtual because is NOT the time elapsed
by the computer to carry it out, but an imposed parameter to
the molecular dynamics modeling.
 During this stage the position and speed of all particles in
selected time steps are registered and stored for further
processing and property calculations.
39
Analysis
The analysis stage means the calculation of all static and dynamic
properties of the system from data stored during the previous
step of “development”. The treatment of positions and speeds is
performed statistically from such information collected during
the simulation.
40
Thermodynamics and molecular
dynamics
 The common practice is performing molecular dynamics simulations
in isolated systems, although such system meant several interacting
molecules. It means that “temperature”, being a typical
macroscopic property, only got interest as a consequence and result
of the calculation of kinetic energies of the molecular set on the
grounds of the classical principle of equipartition of energy.
41
Thermodynamics and molecular
dynamics
 The common practice is performing molecular dynamics simulations
in isolated systems, although such system meant several interacting
molecules. It means that “temperature”, being a typical
macroscopic property, only got interest as a consequence and result
of the calculation of kinetic energies of the molecular set on the
grounds of the classical principle of equipartition of energy.
 Any kind of calculation of macroscopic termodynamic properties
will be on the grounds that the system’s energy is maintained as
constant during the development stage, in equilibrium conditions,
although such energy is expressed in terms of temperature.
42
Thermodynamics and molecular
dynamics
 The common practice is performing molecular dynamics simulations
in isolated systems, although such system meant several interacting
molecules. It means that “temperature”, being a typical
macroscopic property, only got interest as a consequence and result
of the calculation of kinetic energies of the molecular set on the
grounds of the classical principle of equipartition of energy.
 Any kind of calculation of macroscopic termodynamic properties
will be on the grounds that the system’s energy is maintained as
constant during the development stage, in equilibrium conditions,
although such energy is expressed in terms of temperature.
 Thus, molecular dynamics systems are usually treated as
microcanonical statistical ensembles (NVE).
43
Computing Properties
44
Phase space restrictions
As molecular dynamics is a simulation considering that the
behavior of each particle is classical, knowing positions and
instant speeds at such points must do possible the exact
calculation of any related collective property.
45
Phase space restrictions
As molecular dynamics is a simulation considering that the
behavior of each particle is classical, knowing positions and
instant speeds at such points must do possible the exact
calculation of any related collective property.
As it has been previously noticed, the main constraint is that the
phase space can no account very large numbers, being them of
the Avogadro’s number order.
46
Phase space restrictions
As molecular dynamics is a simulation considering that the
behavior of each particle is classical, knowing positions and
instant speeds at such points must do possible the exact
calculation of any related collective property.
As it has been previously noticed, the main constraint is that the
phase space can no account very large numbers, being them of
the Avogadro’s number order.
Therefore, only significant phases of one or several molecules in
the system are considered as they are selected to be at minima
of the hypersurface.
47
Types of Properties
 Properties are statical when they are not related with a time
dependent path (i.e. heat capacity at constant pressure), or
dynamical when they are (i.e. self diffusion coefficients).
48
Types of Properties
 Properties are statical when they are not related with a time
dependent path (i.e. heat capacity at constant pressure), or
dynamical when they are (i.e. self diffusion coefficients).
 Properties are unimolecular when they depend only from the
behavior of a single particle (i.e. self diffusion coefficients), or
collective when depend from an ensemble of particles (i.e.
dielectric constants).
49
Static Properties
Translational kinetic energy:
K transl
Potential energy:
1
drn 

  mn  
2 n  dt 
N
2
N
V   un
n
Where the expression between <...> means and average over all
Dt time steps of the simulation during the development stage, N
is the number of atoms or molecules in the system under study,
un is the energy calculated over each atom or molecule n in the
system, and mn is the atomic mass of such n body.
50
Static Properties
2 Ktransl
Ttransl 
3 Nk Boltzmann
It is the main contributing component to system’s calculated T.
Translational temperature:
51
Static Properties
2 Ktransl
Ttransl 
3 Nk Boltzmann
It is the main contributing component to system’s calculated T.
Rotational temperature can also be calculated by replacing
force by torque in the previous kinetic energy expressions and
provides the complementary value.
Translational temperature:
52
Static Properties
2 Ktransl
Ttransl 
3 Nk Boltzmann
It is the main contributing component to system’s calculated T.
Rotational temperature can also be calculated by replacing
force by torque in the previous kinetic energy expressions and
provides the complementary value.
Translational temperature:
If internal vibrations and torsions are to be accounted, their
temperature can be obtained from their corresponding
expressions of kinetic energy.
53
Static Properties
2 Ktransl
Ttransl 
3 Nk Boltzmann
It is the main contributing component to system’s calculated T.
Rotational temperature can also be calculated by replacing
force by torque in the previous kinetic energy expressions and
provides the complementary value.
Translational temperature:
If internal vibrations and torsions are to be accounted, their
temperature can be obtained from their corresponding
expressions of kinetic energy.
The total temperature of the system is the sum of all these
contributions from the equipartition principle:
2 ( Ktransl  K rot  K vib )
T
3
Nk Boltzmann
54
Static Properties
The heat capacity at constant volume can be obtained from the
fluctuations of energy during the simulation development:


2
2


K  K
fn
CV  R 1 

f
2   n 2
2
Nk
T
  Boltzmann 
  2 

1
where fn is the number of degrees of freedom per particle, R is
the ideal gas constant and N the number of atoms or molecules
of the system.
55
Static Properties
Pressure is calculated with a truncated virial ecuation, as is
normally managed in the case of gases at low pressure:
1
p  ( Nk BoltzmannT  w)
V
where V is the system’s volume and w is the second virial
coefficient as given by:
1 N  un
1 N N  unl
w    ri      rnl 
3 n rn
3 n l n rnl
56
Static Properties
The radial distribution function of pairs is the probability of
finding a pair of bodies a and b separated by a distance r
between them.
57
Static Properties
The radial distribution function of pairs is the probability of
finding a pair of bodies a and b separated by a distance r
between them.
Being n(r) the number of body pairs ab that are found separated
at distances between rn and rn + dr, then:
V n( r )
gab (r ) 
N 4r 2dr
where V is the volume of simulation.
58
Static Properties
The radial distribution function of pairs is the probability of
finding a pair of bodies a and b separated by a distance r
between them.
Being n(r) the number of body pairs ab that are found separated
at distances between rn and rn + dr, then:
V n( r )
gab (r ) 
N 4r 2dr
where V is the volume of simulation.
X ray or neutron diffraction occurs because irregularities of
radial distribution with respect to uniformity. Thanks to that,
the obtained values in simulations can be compared with
experiments to test the reliability of a potential or
hypersurface used in it.
59
Static Properties
As it is well known from elementary sciences, correlation
functions have been defined for establishing the relationship or
dependence between, i.e. any two variables, being x and y, over
M measurements.
60
Static Properties
As it is well known from elementary sciences, correlation
functions have been defined for establishing the relationship or
dependence between, i.e. any two variables, being x and y, over
M measurements.
A common expression is:
Cxy 
x 
x 
x
x
 y  y 
 y  y 
M
M
M
2
M
2
M
M
M
61
Static Properties
As it is well known from elementary sciences, correlation
functions have been defined for establishing the relationship or
dependence between, i.e. any two variables, being x and y, over
M measurements.
A common expression is:
Cxy 
x 
x 
x
x
 y  y 
 y  y 
M
M
M
2
M
2
M
M
M
The correlation function outputs a number between -1 and 1
from anti-correlated to fully correlated variables. A null value
means that they are independent (uncorrelated).
62
Static Properties
The so – called Liouville´s theorem establishes that the phasespace distribution function f0(xn,pn) of a given system in
equilibrium is constant along the trajectories of the system:

f0 ( xn , pn )  0
t
Static Properties
The so – called Liouville´s theorem establishes that the phasespace distribution function f0(xn,pn) of a given system in
equilibrium is constant along the trajectories of the system:

f0 ( xn , pn )  0
t
It is, itself, a statistical definition of equilibrium in a given
system, and can be applied to the stage of development in a
molecular dynamics simulation.
Static Properties
The so – called Liouville´s theorem establishes that the phasespace distribution function f0(xn,pn) of a given system in
equilibrium is constant along the trajectories of the system:

f0 ( xn , pn )  0
t
It is, itself, a statistical definition of equilibrium in a given
system, and can be applied to the stage of development in a
molecular dynamics simulation.
Then, when correlated variables of a system in equilibrium are
time – dependent, the correlation function also becomes time –
dependent.
Static Properties
Autocorrelation is the correlation of a signal or calculation result
with itself in different conditions, i.e. the similarity between a
function evaluated at different separated points of any
independent variable x among M samples:
1 M
C AA  x   A( x) A( x  Dx) 
 A( xi ) A( xi  Dx)
M i 1
Static Properties
Autocorrelation is the correlation of a signal or calculation result
with itself in different conditions, i.e. the similarity between a
function evaluated at different separated points of any
independent variable x among M samples:
1 M
C AA  x   A( x) A( x  Dx) 
 A( xi ) A( xi  Dx)
M i 1
If the variable is time, it is a mathematical tool for finding
repeating patterns, such as the presence of a periodic signal
which has been buried under noise.
Static Properties
Obviously, time autocorrelation relates values of the same
function A(t) calculated at times t apart and the convergence of
CAA(t) denotes equilibrium or stationary states.
Static Properties
Obviously, time autocorrelation relates values of the same
function A(t) calculated at times t apart and the convergence of
CAA(t) denotes equilibrium or stationary states.
The mathematical expression of time autocorrelation of a given
A(t) quantity in molecular dynamics is defined as:
1 M
C AA t   A(0) A(t ) 
 A(0) A(ti )
M i 1
being the ensemble average < …> of the product of the
initial value of a certain A property and the values of A at
several t times.
Static Properties
In just the same way we may define time-correlation functions of
different variables as cross-correlation:
CAB t   A(0) B(t )
Static Properties
In just the same way we may define time-correlation functions of
different variables as cross-correlation:
CAB t   A(0) B(t )
If the choice of origin of the time scale is entirely arbitrary and
the equilibrium ensemble distribution function is invariant to
changes of time, according the Liouville´s theorem, we have the
obvious general identity:
A(t0 ) B (t0  Dt )  A(0) B (t )
A(t0 ) A(t0  Dt )  A(0) A(t )
both for cross- and auto- correlation.
Static Properties
In just the same way we may define time-correlation functions of
different variables as cross-correlation:
CAB t   A(0) B(t )
If the choice of origin of the time scale is entirely arbitrary and
the equilibrium ensemble distribution function is invariant to
changes of time, according the Liouville´s theorem, we have the
obvious general identity:
A(t0 ) B (t0  Dt )  A(0) B (t )
A(t0 ) A(t0  Dt )  A(0) A(t )
both for cross- and auto- correlation.
This is a desirable ergodic condition of molecular dynamics
simulations.
Static Properties
In these conditions the time correlation formalism to the
trajectory in a molecular dynamics simulation implies that we
can use many time origins provided that they are sufficiently
separated that there is no correlation between them.
Static Properties
In these conditions the time correlation formalism to the
trajectory in a molecular dynamics simulation implies that we
can use many time origins provided that they are sufficiently
separated that there is no correlation between them.
It means that molecular dynamics can use many separate time
frames instead of many ensembles (as is normal in the usual
statistical mechanical approach) in order to obtain useful time
decays during separate simulation times t that can be analyzed
for given Dt time steps.
Static Properties
For the same variable (autocorrelation), also expressed as
a summation of steps:
1 N
A(0) A(t ) 
 A(ti ) A(ti  Dt ), ti  0,t ,2t ,...
Nt i 1
t
For samples in the sum to be independent. t should be
chosen so that:
A(0) A(t )  A(0) 2
Static Properties
The initial value of the time autocorrelation value in any starting
reference time is:
C AA (t , t )  A(0) A(0)  A(t ) A(t )  A2  0
Static Properties
The initial value of the time autocorrelation value in any starting
reference time is:
C AA (t , t )  A(0) A(0)  A(t ) A(t )  A2  0
If the evolution is random, very separated times t and t’
(the value of t) must give uncorrelated values of the
variable:
lim C AA (t ,t )  A(t ) A(t )  A
t 
2
Static Properties
Using <A(t)A(t)> as a normalization factor, a graphic of decay of
autocorrelation of the A quantity during a molecular dynamics
simulation is:
Static Properties
Using <A(t)A(t)> as a normalization factor, a graphic of decay of
autocorrelation of the A quantity during a molecular dynamics
simulation is:
The time required for a full relaxation of the autocorrelation is
called relaxation time (t) and simulations must be longer than
the one orresponding to the property of interest.
Static Properties
Using <A(t)A(t)> as a normalization factor, a graphic of decay of
autocorrelation of the A quantity during a molecular dynamics
simulation is:
The time required for a full relaxation of the autocorrelation is
called relaxation time (t) and simulations must be longer than
the one orresponding to the property of interest.
Resulting structures and energy values after each period of
correlation times is a “shot” or “frame” of the simulation.
Dynamic Properties
We consider as dynamic properties of the system those that
depend on time advance during the simulation.
Dynamic Properties
We consider as dynamic properties of the system those that
depend on time advance during the simulation.
Position, velocity, dipole moment, density, etc. are dynamic
variables.
Dynamic Properties
We consider as dynamic properties of the system those that
depend on time advance during the simulation.
Position, velocity, dipole moment, density, etc. are dynamic
variables.
In general, their sampling during the development stage in
a given simulation provides a statistical description of the
time-evolution of such variables or a correlated pair of
them for an ensemble at thermal equilibrium.
Dynamic Properties
A molecular dynamics simulation produces a substantial amount
of useful information, and it is normal to store:
• vectors of the positions of each particle,
• velocities (as angular velocities),
• forces (as torques) for each molecule,
as well as the instantaneous values of all the calculated
properties.
Dynamic Properties
A molecular dynamics simulation produces a substantial amount
of useful information, and it is normal to store:
• vectors of the positions of each particle,
• velocities (as angular velocities),
• forces (as torques) for each molecule,
as well as the instantaneous values of all the calculated
properties.
They use to be sampled after a certain number of steps, say 5
or 10, to save data storage.
Dynamic Properties
Let us consider a set of n particles (atoms or molecules) in a unit
of volume and n1 the number of certain labeled particles among
them in such volume. If Jz is the mean number of the labeled
particles crossing or diffusing through a unit area of a plane
dividing and inside such volume per unit of time in the positive z
direction, we can write:
n1
J z  D
z
Dynamic Properties
Let us consider a set of n particles (atoms or molecules) in a unit
of volume and n1 the number of certain labeled particles among
them in such volume. If Jz is the mean number of the labeled
particles crossing or diffusing through a unit area of a plane
dividing and inside such volume per unit of time in the positive z
direction, we can write:
n1
J z  D
z
This relation establishes that the diffusion is proportional to the
gradient of the number of labeled particles in the flow direction
if the n1 particles are not uniformly distributed.
Dynamic Properties
Let us consider a set of n particles (atoms or molecules) in a unit
of volume and n1 the number of certain labeled particles among
them in such volume. If Jz is the mean number of the labeled
particles crossing or diffusing through a unit area of a plane
dividing and inside such volume per unit of time in the positive z
direction, we can write:
n1
J z  D
z
This relation establishes that the diffusion is proportional to the
gradient of the number of labeled particles in the flow direction
if the n1 particles are not uniformly distributed.
If the n1 particles are uniformly distributed in the volume
there are no gradient and then Jz = 0.
Dynamic Properties
Let us consider a set of n particles (atoms or molecules) in a unit
of volume and n1 the number of certain labeled particles among
them in such volume. If Jz is the mean number of the labeled
particles crossing or diffusing through a unit area of a plane
dividing and inside such volume per unit of time in the positive z
direction, we can write:
n1
J z  D
z
This relation establishes that the diffusion is proportional to the
gradient of the number of labeled particles in the flow direction
if the n1 particles are not uniformly distributed.
If the n1 particles are uniformly distributed in the volume
there are no gradient and then Jz = 0.
D is called as the self – diffusion coefficient of the substance,
and can be considered as the particle flow number when the
gradient is unitary.
Dynamic Properties
A particular case is the time correlation function of particle
velocities, known as velocity autocorrelation function, defined
as before:


Cvv (t )  v n (t0 )  v n (t0  Dt )
t0
Dynamic Properties
A particular case is the time correlation function of particle
velocities, known as velocity autocorrelation function, defined
as before:


Cvv (t )  v n (t0 )  v n (t0  Dt )
t0
It can be deduced that the velocity autocorrelation function
describes how long velocities of the involved particles persist
until they averaged out by microscopic motions and
interactions with its surroundings.
Dynamic Properties
The integral value of Cvv in the total time of simulation serves to
find the self - diffusion coefficient:
1 
D  0 C vv (t )dt
3
Dynamic Properties
The integral value of Cvv in the total time of simulation serves to
find the self - diffusion coefficient:
1 
D  0 C vv (t )dt
3
It can be defined as the integral of all velocity self – correlation
functions in a given system during the whole simulation time. It
is a good measure of translational mobility.
Dynamic Properties
Vibrational spectra simulations are a natural result of molecular
dynamics as the motion in nanoscopic dimensions dealing with
them are directly modeled with this method.
Dynamic Properties
Vibrational spectra simulations are a natural result of molecular
dynamics as the motion in nanoscopic dimensions dealing with
them are directly modeled with this method.
Therefore, the Fourier transform of D self – difussion
coefficient is a measure of the spectral density of states.
1 m  it
1 m  it
fMD   
e
C
(
t
)
dt

e Ddt


vv
0
0
3N kT
N kT
Dynamic Properties
Vibrational spectra simulations are a natural result of molecular
dynamics as the motion in nanoscopic dimensions dealing with
them are directly modeled with this method.
Therefore, the Fourier transform of D self – diffution
coefficient is a measure of the spectral density of states.
1 m  it
1 m  it
fMD   
e
C
(
t
)
dt

e Ddt


vv
0
0
3N kT
N kT
Left: velocity autocorrelation functions for oxygen and hydrogen atoms in a water MD simulation;
Right: Corresponding spectral density for the liquid and the gas.
Dynamic Properties
Vibrational spectra simulations are a natural result of molecular
dynamics as the motion in nanoscopic dimensions dealing with
them are directly modeled with this method.
Therefore, the Fourier transform of D self – diffution
coefficient is a measure of the spectral density of states.
1 m  it
1 m  it
fMD   
e
C
(
t
)
dt

e Ddt


vv
0
0
3N kT
N kT
Left: velocity autocorrelation functions for oxygen and hydrogen atoms in a water MD simulation;
Right: Corresponding spectral density for the liquid and the gas.
Dynamic Properties
Vibrational spectra intensities depend on transition dipoles.
Dynamic Properties
Vibrational spectra intensities depend on transition dipoles.
They can be obtained for systems in which the time
evolution of the dipole moment and polarizability can be
described with the classical potentials used in the molecular
dynamics simulation.
Dynamic Properties
Vibrational spectra intensities depend on transition dipoles.
They can be obtained for systems in which the time
evolution of the dipole moment and polarizability can be
described with the classical potentials used in the molecular
dynamics simulation.
The absorption cross section (giving absorption intensities)
of IR spectra is proportional to the line shape obtained
after a Fourier transform of the dipole moment
autocorrelation function:

 kT
I  
a     1  e
I     C t e it dt

Dynamic Properties
Vibrational spectra intensities depend on transition dipoles.
They can be obtained for systems in which the time
evolution of the dipole moment and polarizability can be
described with the classical potentials used in the molecular
dynamics simulation.
The absorption cross section (giving absorption intensities)
of IR spectra is proportional to the line shape obtained
after a Fourier transform of the dipole moment
autocorrelation function:

 kT
I  
a     1  e
I     C t e it dt

where


C t   i (t0 )  i (t0  Dt )
is the dipole moment autocorrelation that can be averaged
over single molecule dipole moment autocorrelation
functions for localized vibrations.
More Common Potentials and
Computer Programs
AMBER
“Assisted Model Builder with Energy Refinement” (AMBER) by
the former Peter Kollman, in UC San Francisco, is used for
simulations of molecular dynamics. It uses only five bonding and
no bonding terms, as well as a special treatment for electrostatic
potentials. No cross terms are included. It gives good results with
proteins and nucleic acids although could be erratic with other
systems if the appropriate hypersurface is not selected.
Etotal   K r r  req 
2
bonds
  K   eq 
2
angles
  12 Kf 1  cosnf   
dihedrals
 Bnl Anl qn ql 
   12  6 

rnl rnl 
nl  rnl
103
AMBER
1. Weiner, P. K.; Kollman, P. A., AMBER: Assisted model building with energy refinement. A general program for modeling molecules
and their interactions. J. Comput. Chem. 1981, 2 (3), 287-303.
CHARMM
“Chemistry at HARvard Macromolecular Mechanics” (CHARMM)
by Martin Karplus, in Harvard, was originally designed for
proteins and nucleic acids, although now has been applied to
various types of biomolecules, to solvation energies and
structures, vibrational analyses, and QM/MM (quantum
mechanics / molecular mechanics) studies. The general terms
are:
Ebondstretch 
 Kb b  b0 
Ebondbond 
2
2
angles
1, 2 pairs
Erotatealongbond 
 K   0 
 Kf 1  cos(nf )
Evander Waals
1, 4 pairs
 Anl Cnl 
   12  6 
r 
nl
r
nl
Eelectrostatic
qn ql

nl Drnl
nl
105
GROMOS
“GROnigen MOlecular Simulations” (GROMOS) by van
Gunsteren, ETHZ, is popular for dynamics simulations of
molecules and pure liquids, as well as biomolecules.
V r1 , r2 ,..., rN    kb bn  b
Nb
n 1
1
2
n
  k  n  
N
n 1
1
2

0 2
n
n
bonding

0 2
n
  12 k  n   n0 
N
2
n
n 1
angular
improper
angles
Nf
  Kf 1  cos(nn 'fn '  d n ' 
n'
n '1
N AB
 [
n l

c12 n, l  c6 n, l 

12
rnl
rnl6
qn ql
4 0 r rnl
]S ( rnl )
torsions
non bonding
electrostatics
106
Some examples
107
Water dimer with the NCC
potential
The water dimer model is represented by O (centers 5 and 6, in
figure) and H nuclei (centers 1, 2, 3 and 4) although fiction
“centers” are also created to represent electric dipoles (centers 7
and 8).
108
Water dimer with the NCC
potential
The employed potential is called as Nieser-Corongiu-Clementi
(NCC) and consider two parts: one for two body interactions and
other for polarization:
VNCC   [Vtwo body (i, j )]  V pol
n ,l  n
109
Water dimer with the NCC
potential
The two body component is:
110
Water dimer with the NCC
potential
The polarization component is required for simulating quantum
effects related with interactions between charge densities
between molecules:
1 N N  ind  q
     n En
2 n 
m
V pol
d


where
Nm
is the number of molecules
Nd
is the number of dipoles in molecules
nind
is the induced dipole moment in molecule i by dipole .
Enq
is the electric field in the site of the induced dipole i
originated by the distribution of point charges in the
surroundings.


111
Water dimer with the NCC
potential
Results for water dimer:
112
Water dimer with the NCC
potential
Radial distribution:
113
Water dimer with the NCC
potential
Radial distribution:
114
Water dimer with the NCC
potential
Radial distribution:
115
Example of water simulation in a
myoglobin crystal
Here water is studied as solvent. Coordinates of 1261 heavy
atoms of myoglobin were obtained from X ray diffraction data of
the crystalized protein as the starting system. Hydrogen atoms
were absent because the experimental technique cannot resolve
them and were introduced by considering typical geometries, up
to 2532 atoms. 856 molecules of water were used in a virtual
unit cell, giving a crystal with a density of 1.2705 g cm-3.
116
Example of water simulation in a
myoglobin crystal
Water - protein interactions were calculated by a potential based
in ab initio quantum calculations and water – water were
potentials were standard.
117
Example of water simulation in a
myoglobin crystal
Water - protein interactions were calculated by a potential based
in ab initio quantum calculations and water – water were
potentials were standard.
The simulation was carried out in 50 ps, with an interval or time
step of 0.5 fs. After 20 ps equilibrium was reached and the
following 30 ps were used for the “development stage”. Average
temperature of simulation was 304.4 ± 8.15 K and the total
potential energy of –17.46 ± 0.03 kcal mol-1 with maximal
fluctuations of 0.001 kcal mol-1.
118
Example of water simulation in a
myoglobin crystal
Radial distribution function water – water and water – protein
119
Example of water simulation in a
myoglobin crystal
Comparison between velocity autocorrelation functions of bulk water
with respect to water in myoglobin
120
Molecular dynamics simulation
of salt dissolving in water
Chlorine ions are shown in yellow, sodium in blue. The large chlorine
dissociates from the salt crystal.
http://www.chem.ucl.ac.uk/ice
Group leader
Prof. Angelos Michaelides
University College London
London Centre for Nanotechnology
17-19 Gordron Street London, WC1H 0AH
Tel: +44 (0) 207 679 0647 (Ext: 30647) 121
Email: angelos.michaelides@ucl.ac.uk
References
Alder, B. J.; Wainwright, T. E., Studies in Molecular Dynamics. I. General
Method. J. Chem. Phys. 1959, 31 (2), 459-466.
Zwanzig, R., Time-Correlation Functions and Transport Coefficients in
Statistical Mechanics. Annu. Rev. Phys. Chem. 1965, 16 (1), 67-102.
Clementi, E.; Corongiu, G.; Bahattacharya, D.; Feuston, B.; Frye, D.;
Preiskorn, A.; Rizzo, A.; Xue, W., Selected topics in ab initio
computational chemistry in both very small and very large chemical
systems. Chem. Revs. 1991, 91 (5), 679-699.
Clementi, E.; Corongiu, G.; Aida, M.; Niesar, U.; Kneller, G., Monte Carlo
and Molecular Dynamics Simulations. In MOTECC. Modern Techniques
in Computational Chemistry, Clementi, E., Ed. ESCOM: Leiden, 1990; pp
805-888.
122