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# Lennard-Jones Fluid Reference Calculations

## Share

In this section, we provide sample configurations of Lennard-Jones atoms and report the internal energy and pair virial for those configurations. These sample configurations and reference calculations can be used to validate the energy and force routines for an existing or new molecular simulation code.

## 1. Sample Configurations of Lennard-Jones Atoms

Four sample configurations of Lennard-Jones atoms are available for download as a gzipped tarball archive. This archive contains five files: the four sample configuration files and one metadata file that explains the format of the sample configurations. These configurations should be converted to the configuration file format native to a user's simulation software.

## 2. Reference Calculations for the Internal Energy and Pair Virial

The following table contains calculations of the pair internal energy (Upair), the instantaneous pair virial (Wpair), and the appropriate tail correction to the internal energy (ULRC) for the four sample configurations in Section 1, where periodic boundary conditions [1,2] (also known as the minimum-image convention) were applied to all three Cartesian directions. The definition of these energetic terms are given below in Section 3. The reference calculations are for three truncation schemes: (1) potential truncation at a cutoff radius (rc) of 3σ (σ is the Lennard-Jones diameter), with long-range corrections (LRC), (2) potential truncation at rc=4σ, with long-range corrections, and (3) potential truncation at rc=3σ, with the linear-force shift applied at the cutoff radius (LFS). All sample energies and distances are given in appropriate Lennard-Jones reduced units, denoted by the superscript "*".

 LRC, rc*=3.0 LRC, rc*=4.0 LFS, rc*=3.0 Configuration Upair* Wpair* ULRC* Upair* Wpair* ULRC* Upair* Wpair* ULRC* 1 -4.3515E+03 -5.6867E+02 -1.9849E+02 -4.4675E+03 -1.2639E+03 -8.3769E+01 -3.8709E+03 3.1754E+02 0.0000E0 2 -6.9000E+02 -5.6846E+02 -2.4230E+01 -7.0460E+02 -6.5599E+02 -1.0226E+01 -6.2012E+02 -4.4533E+02 0.0000E0 3 -1.1467E+03 -1.1649E+03 -4.9622E+01 -1.1754E+03 -1.3371E+03 -2.0942E+01 -1.0210E+03 -9.3578E+02 0.0000E0 4 -1.6790E+01 -4.6249E+01 -5.4517E-01 -1.7060E+01 -4.7869E+01 -2.3008E-01 -1.5001E+01 -4.3096E+01 0.0000E0

## 3. Definitions

For the reference calculations given here, the following definitions are relevant:

A. The traditional Lennard-Jones Potential is given by:

$$\Large V_{LJ}\left(r\right)=4\epsilon\left[\left(\dfrac{\sigma}{r}\right)^{12}-\left(\dfrac{\sigma}{r}\right)^6\right]$$

B. When the "Long-Range Correction" (LRC) is applied to the tail of the Lennard-Jones Potential, the actual potential in a molecular simulation is given by:

$$\Large V\left(r\right) = \begin{cases} V_{LJ} \left( r \right) & r \leq r_c \\ 0 & r > r_c \end{cases}$$

C. When the "Linear-Force Shift" is applied to the tail of the Lennard-Jones Potential, the actual potential in a molecular simulation is given by:

$$\Large V\left( r \right) = \begin{cases} V_{LJ} \left( r \right) - V_{LJ} \left(r_c\right) - \left. \dfrac{\partial V_{LJ}}{\partial r}\right|_{r_c} \left(r-r_c\right) & r \leq r_c \\ 0 & r > r_c \end{cases}$$

D. The pair internal energy is given by the following nist-equation, where V(r) is the simulated pair potential:

$$\Large U_{pair} = \sum_{i=1}^{N-1} \sum_{j=i+1}^N V\left(r_{ij}\right)$$

E. The Long-Range correction to the Lennard-Jones potential, per particle, is given by [1,2]:

$$\Large U_{LRC} = \dfrac{1}{2} 4 \pi \rho \int _{r_c} ^{\infty} dr~r^2~V_{LJ} \left( r \right)$$

F. The instantaneous pair virial is given by:

$$\Large W_{pair} = -\sum_{i=1}^{N-1} \sum_{j=i+1}^N r_{ij} \left.\dfrac{\partial V}{\partial r}\right|_{r_{ij}}$$

References

1. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, New York, 1989).
2. D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd ed. (Academic, San Diego, 2002), pp.37-38.
Created November 29, 2012, Updated September 20, 2017