Parameterization of Lennard-Jones Equation in the qAIREBO Model via Tetrafluoromethane

1. Parameterization of Lennard-Jones Equation in the
qAIREBO Model via Tetrafluoromethane
MINH H. TRAN, STEVEN J. STUART *
Department of Chemistry, Clemson University, Clemson, South Carolina 29634
Abstract: The objective of this project is to determine a pair of Lennard-Jones (LJ) parameter values, σ
and ε, that describe van der Waals
interactions between two non-bonded
atoms in tetrafluoromethane (CF4).
Recently developed as an improvement of
the Reactive Empirical Bond-Order
(REBO) model,1
the AIREBO model
simulates chemical reactions more
realistically by introducing torsion,
dispersion and non-bonded repulsion terms
into the original potential function.2
The
AIREBO model has been parameterized
for use with systems of carbon and
hydrocarbon. In 2008, the qAIREBO,
which is under development, is promising
to be the most adaptive model by adding
electrostatic interaction. With this new
advantage, this project extends the application of the qAIREBO to fluorocarbon systems by
parameterization. Molecular dynamics simulations were performed on models of CF4 gas and CF4 liquid,
composed of 128 molecules. The LJ parameters were obtained by fitting them to reproduce CF4 liquid
with experimental properties of 90.792 Å3
/ molecule in density and 0.12415 eV in heat of vaporization at
the boiling temperature of 145.19 K and standard pressure of 1 bar. If successful, these parameter values
will be validated with larger systems of 256 and 512 CF4 molecules.
Introduction
Computational molecular modeling has a long history of development. To simulate molecular
interactions on computer, chemists have, at first, applied molecular dynamics and Monte Carlo method.
This classical molecular simulation made use of Newton’s second law to describe motion through time of
each individual atom in a chemical system. However, this technique successfully simulates physical
processes, but reflects no chemical reactions such as bond breaking and bond formation, which are
considered quantum mechanical processes. Not until 1988, Tersoff introduced a new model, which
modified the classical potential in order to empirically simulate the formation and dissociation of
saturated and unsaturated covalent bonds in chemical systems such as silicon and carbon. The most
successful and well-known example of Tersoff potential is the reactive empirical bond-order (REBO)
which was significantly modified by Brenner in 1990 to include radical and conjugated hydrocarbon
bonds.3
One feature that gives the REBO model an advantage over non-reactive molecular dynamics is
the contribution of both the repulsive and attractive potentials to the interacting potential between two
bonded atoms which is expressed as

2. Vij(r) = (r) + bij (r) (1)
where i, j represent two bonded atoms, r is the distance between them, and b is the bond order whose
terms vary as bonding environment changes. As a result, an atom is no longer covalently constrained to
its original neighbor or a specific hybridization state, but able to dissociate and make a new bond with
other atoms in an adaptive way during a chemical reaction. This REBO model was originally
parameterized to simulate properties of diamond. New development has extended the application of the
REBO potential to the simulation of the energetic, elastic and vibrational properties of solid carbon and
small hydrocarbons.4
The REBO model, however, still has some drawbacks. With the absence of
intermolecular (van der Waals) and torsional interaction potentials, the REBO model poorly reflects
chemical properties of condensed phase molecular systems or large hydrocarbons. The need for a more
complete model led to a new generation of the REBO’s family, called adaptive intermolecular REBO
(AIREBO), proposed by Stuart in 2000.5
There are still some disadvantages that make this model less
attractive. The Lennard-Jones (LJ) equation that describes the van der Waals interaction can adapt its
strength to the change in local chemical environment, but not the shape of the interaction. In other
words, the parameter values do not vary as the bonding environment changes. Another characteristic that
is included in classical molecular dynamics, but absent in the REBO potentials is the electrostatic
interaction. Therefore, the current model, called qAIREBO, is under development, and expected to
address these limitations. Table 1 provides a brief comparison of different models mentioned above.
Classical Molecular Dynamics REBO AIREBO qAIREBO
Covalent Bonding
   
Bond Stretching
   
Bond Angle
Bending
   
Torsion
   
van der Waals
   
Electrostatics
   
Table 1. Brief comparison of molecular modeling potentials
With more complete terms in the potential expression, the qAIREBO model is intended to not only
simulate hydrocarbon systems more accurately, but also simulate systems other than hydrocarbon such as
FC, NH, CHNO, which is the motivation of this paper. Experimental energetic property of
tetrafluoromethane (CF4) is used as a target to fit the LJ parameters. The pair of parameter values
obtained then can be used for all F atoms in any chemical environment. Table 2 lists the family of bond-
order potentials and their applications.

3. Model Applications
Tersoff Si and C systems
REBO Si, C and small HC systems
AIREBO C, HC and condensed phase molecular systems
qAIREBO AIREBO’s applications and, possibly, FC, NH, CHNO systems
Table 2. REBO potential family and their applications
The application of FC potential can be widely appreciated, including the fields of materials science
and biological research. One example is the simulation of C2F6 etching of silicon. Due to its favorable
kinetic properties and high selectivity, C2F6 plasmas are used in the etching industry. A potential model
will simulate the surface reaction mechanisms so that a future plan will be developed to optimize the
etching process. In the semiconductor manufacturing, a similar application is the simulation of
monatomic fluorine, removing the photoresist from an etched wafer during the plasma ashing process. In
pharmacy, many pharmaceutical products contain C-F bonds such as 5-fluorouracil, an anticancer drug.
As a result, the capability of simulating fluorocarbon systems proves itself useful in experimenting new
medical products. The list is long and an effort to improve the qAIREBO model to handle more types of
chemical systems is, therefore, a demand.
Model
A simplified equation for the qAIREBO potential has the form:
V = VREBO
+ VLJ
+ Vtors
(2)
in which the Lennard-Jones potential, VLJ
, is the expression that will be parameterized for the purpose of
this paper. VLJ
is a function of the distance rij between two non-bonded atoms i and j. Its dependence on
two parameters σij and εij for the description of the intermolecular interactions is expressed as the
following
(rij) = 4εij [( )12
_
( )6
] (3)
With its shortness and simpler terms, this 12-6 form proves itself to be more computationally
efficient than other functional forms, which may be more accurate, yet too complicated. The sum of the
intermolecular interactions acting on atoms i and j is then expressed as

4. Eij = α (rij) (4)
where the many-body term α varies according to changes in bonding environment.
There are three factors that determine whether long range interactions should take effect during a
simulation. These factors are covalent-bonding cutoff distance and bond order between atoms i and j, and
the relative positions of all atoms bonded to i and j. As a result, if two atoms are located beyond a cutoff
distance and not 1-4 neighbors on the same molecule, and have a low value of bond order, a full LJ
potential will exist between them. The strength of the LJ interactions, therefore, will vary smoothly with
different bonding environment and totally disappear if all three conditions above are not met. This
adaptability is where the REBO and the AIREBO models compromise. Two atoms, when approaching
close enough, will react or bond to each other and the REBO model is valid. In this case, the rij
-12
term in
equation (3) does not contribute. However, its contribution is taken into account when atoms are
chemically saturated.
The LJ potential is formulated around the variability of the parameters σ and ε. In the simulation
of soft, molecular systems, these parameters have an important role in determining material properties
such as cohesive energy, boiling point and density. The variability of σ and ε reflects the change in the
polarization and the radius of the electronic distribution on an atom, which invokes intermolecular
interactions. In the AIREBO model, each atom is assigned a pair of σ and ε values regardless of any
hybridization that occurs during a chemical reaction. As a result, the AIREBO model simulates a
chemical system that was used in the fitting process with greater accuracy than systems that were not. It
is the case of saturated and unsaturated hydrocarbons in which the AIREBO model is better suited to
saturated hydrocarbons. In addition, a single pair of σ and ε is used when an atom is bonded to other
different atoms. The type of covalent bonds, therefore, should not be ignored since different bond length
and polarity result in different intermolecular interactions. The most recent version of the AIREBO
model, however, resolves these problems. By considering the number of each type of neighbor that is
bonded to the target atom, the LJ parameters appear in the AIREBO model as functions, such as σF(NC
,
NF
) and εF(NC
, NF
), rather than fixed numbers. Again, the advantage of the AIREBO model over its
counterparts lies in the simplification of its functions to tackle several problems that, otherwise, require
complicated mathematical expressions. One limitation that may become a goal for future development is
that the AIREBO model uses the first-neighbor method to count the number of atoms bonded to the
target atom. When simulating many different chemical structures, it is sometimes necessary to consider
second or third neighbors. This functional form is applied to both the reactive and nonreactive, bond-
order potential. The difference is that in the latter, the values of N are integers whereas in the former, the
values of N are non-integers.
As mentioned in the previous section, fitting the values of σ and ε using CF4 is the topic of this
paper. However, due to the scope of this paper, only σ and ε values of F will be determined. Based on
the functional form that was developed for the LJ parameters, the mathematical expressions for σ and ε
are σF(1, 0) and εF(4, 0). This single pair of parameters is adequate for F since all F atoms in CF4 are
bonded to C in the same manner and none is bonded to each other.
Fitting Process
The objective of this fitting process was to find a pair of parameter values σF and εF that best
reproduce the energetic property of CF4 liquid, enthalpy of vaporization (∆Hvap) in this case, under the

5. experimental conditions. For consistency with the setting of the qAIREBO model, the units of the
experimental density and ∆Hvap are converted to Å3
/molecule and eV/molecule, respectively. Therefore,
∆Hvap of CF4, which was taken from a published paper,6
was recorded as 0.12425 eV/molecule. At the
boiling point of 145.19K, the density of liquid CF4 was taken from the same source as 90.792
Å3
/molecule under the pressure of 1 bar.
According to the definition of the enthalpy,
∆Hvap = ∆Uvap - ∆(PV) (5)
It is assumed that CF4 vapor behaves as an ideal gas and the molar volume of the liquid is
negligible compared to that of the gas. Then, equation (5) becomes
∆Hvap = ∆Uvap – (∆ngas)RT (6)
Since ∆ngas = 1, equation (6) is reduced to
∆Hvap = ∆Uvap – RT (7)
The change in internal energy ∆Uvap is calculated from a simulation by taking the difference in
internal energy of a gas-phase CF4 molecule and a CF4 liquid molecule at the boiling point.
Two separate simulations were run on two CF4 systems. The gas phase system included only one
single CF4 molecule under the non periodic boundary condition while the liquid phase included 128 CF4
molecules under periodic boundary conditions. The CF4 gas molecule was heated up from an initial
temperature of 2 K to a final temperature of 145.19 K with periodic velocity reassignment thermostat for
a period of 80 ps (Figure 1). Then the molecule was equilibrated over 10 ps from which its average
internal energy was obtained (Figure 2). Unlike the CF4 gas simulation, the simulation of CF4 liquid
required much more labor. In order to obtain a periodic liquid cell with each side length greater than 20.4
Å or twice of the LJ cutoff of 3σCC, a minimum number of 128 CF4 molecules were required. The side
length was calculated and adjusted based on the experimental liquid density of 90.792 Å3
/molecule so
that the total volume of the cell was exactly equal to that of the physical liquid. Figure 3 shows a 3-D
view of a CF4 liquid cell. After the cell was built, its internal energy at the boiling point was measured in a
similar manner as done with CF4 gas. The cell was heated up from 2 K to its boiling point of 145.19 K
with Langevin thermostat for a period of 20 ps (Figure 4). For molecular interactions beyond the LJ
cutoff, long range corrections were applied assuming a uniform density over that distance. The equations
of motion were integrated with the velocity Verlet integrator. Then the cell was equilibrated over 200 ps
from which its average internal energy and external pressure were measured (Figures 5 & 6). The
average internal energy of a single CF4 liquid molecule was calculated by dividing the cell’s average
energy by the number of CF4 molecules contained in the cell. The difference in the internal energy
between two phases was then substituted into equation (7) to calculate ∆Hvap at the boiling temperature
and with the universal gas constant R of 0.000086 . This result was compared with the
experimental ∆Hvap of 0.12425 eV / molecule. Also, the average external pressure measured from the
simulation graph was compared with the experimental pressure of 1 bar. If all values match, the values of
σ and ε, which were set to run the simulations, will be the LJ parameters for F. If they don’t, the
simulation for CF4 liquid cell must be performed again from the beginning with a new pair of parameter
values until the cell reaches the experimental energetic property under the experimental conditions.

6. Figure 1. Internal energy of CF4 gas molecule from Figure 2. Internal energy of CF4 gas molecule
2 K to 145.19 K over 80 ps. during the equilibrium state over 10 ps.
The line shows the average internal energy
Figure 3. A 3-D view of CF4 liquid with 128 Figure 4. Internal energy of CF4 liquid from 2 K
molecules to 145.19 K over 20 ps

7. Figure 5. Internal energy of CF4 liquid Figure 6. Pressure on CF4 liquid during the
during the equilibrium state over 200 ps. equilibrium state over 200 ps. The line
The line shows the average internal energy shows the average pressure.
Results and Discussions
Due to the substantial amount of time required to fit a pair of values for the LJ parameters, the
current work has not reached its goal yet. So far, the best pair of parameter values is σ = 2.13 Å and ε =
50.717 K. These values will reproduce the enthalpy of vaporization of CF4 liquid under the pressure of
-225 bar ± 63 bar. In figure 7, at σ = 2.13, each value of ε provides a ∆Hvap, which is then compared to
the experimental ∆Hvap to give the difference. As seen from the graph, at ε = 50.717 K, ∆Hvap from the
simulation is equal to that from the experiment. Figure 8 shows a graph of pressure corresponding to
each value of σ with the value of ε that reproduces the ∆Hvap from the experiment. According to this
graph, the pressure is getting closer to 1 bar as the value of σ decreases.

8. Figure 7. The difference between ∆Hvap from simulations and ∆Hvap from experiment corresponding to
each value of ε at σ = 2.13 Å
Figure 8. The value of pressure corresponding to each value of σ with the value of ε* that reproduces
the ∆Hvap from the experiment.
More research is needed to determine the final pair of parameter values that can reproduce ∆Hvap
of CF4 liquid under 1 bar pressure. As extrapolated from the graph in figure 8, a pressure close to 1 bar

9. can be achieved at σ = 1.36 Å. However, this approximation should be confirmed by simulations when
available.
When the size of the simulation is larger with more molecules, the values of σ and ε may require
adjustment. Only the pair of values that do not vary with larger models will be accepted. In other words,
a satisfied pair of parameters will provide a consistent internal energy for each individual CF4 liquid
molecule (U/N) over different large CF4 liquid systems. As a result, after successfully obtaining proper
values of σ and ε for CF4 liquid corresponding to 128 CF4 molecules, a series of simulations will be run
for 256 and 512 CF4 molecules with those same parameter values. If these systems continue to reproduce
exact energetic property of CF4 liquid under the experimental conditions, the parameter values obtained
are valid. If not, a new quest for new parameter values will resume. As explained in the previous sections,
with the adaptability of the LJ parameters that gives the current model an advantage, it is necessary to
direct future project to the transferability of the LJ parameters to other fluorocarbon systems.
References
1. Stuart, S. J.; Tutein, A. B.; Harrison, J. A. J Chem Phys 2000, 112, 6472
2. Liu, A.; Stuart, S. J. J Comput Chem 2008, 29, 601
3. Brenner, D. W.; Shenderova, O. A.; Harrison, J. A.; Stuart, S. J.; Ni, B.; Sinnott, S. J Phys Condens
Matter 2002, 14, 783.
4. Stuart, S. J.; Tutein, A. B.; Harrison, J. A. J Chem Phys 2000, 112, 6472
5. Stuart, S. J.; Tutein, A. B.; Harrison, J. A. J Chem Phys 2000, 112, 6472
6. Lobo, L. Q.; Staveley, L. A. K. J Chem Eng 1981, 26, 406

10. can be achieved at σ = 1.36 Å. However, this approximation should be confirmed by simulations when
available.
When the size of the simulation is larger with more molecules, the values of σ and ε may require
adjustment. Only the pair of values that do not vary with larger models will be accepted. In other words,
a satisfied pair of parameters will provide a consistent internal energy for each individual CF4 liquid
molecule (U/N) over different large CF4 liquid systems. As a result, after successfully obtaining proper
values of σ and ε for CF4 liquid corresponding to 128 CF4 molecules, a series of simulations will be run
for 256 and 512 CF4 molecules with those same parameter values. If these systems continue to reproduce
exact energetic property of CF4 liquid under the experimental conditions, the parameter values obtained
are valid. If not, a new quest for new parameter values will resume. As explained in the previous sections,
with the adaptability of the LJ parameters that gives the current model an advantage, it is necessary to
direct future project to the transferability of the LJ parameters to other fluorocarbon systems.
References
1. Stuart, S. J.; Tutein, A. B.; Harrison, J. A. J Chem Phys 2000, 112, 6472
2. Liu, A.; Stuart, S. J. J Comput Chem 2008, 29, 601
3. Brenner, D. W.; Shenderova, O. A.; Harrison, J. A.; Stuart, S. J.; Ni, B.; Sinnott, S. J Phys Condens
Matter 2002, 14, 783.
4. Stuart, S. J.; Tutein, A. B.; Harrison, J. A. J Chem Phys 2000, 112, 6472
5. Stuart, S. J.; Tutein, A. B.; Harrison, J. A. J Chem Phys 2000, 112, 6472
6. Lobo, L. Q.; Staveley, L. A. K. J Chem Eng 1981, 26, 406