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# Lennard-Jones Fluid Reference Calculations: Non-cuboid Cell

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In this section, we provide sample configurations of Lennard-Jones atoms in non-cuboid cells 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. Two of the configurations are triclinic, i.e. none of the lattice angles α,β,γ are 90°, and two configurations are monoclinic, i.e., α=γ=90° and β \= 90°. The lattice angles are given in the header of each sample configuration as described in the metadata file; the user can use the cell side lengths and lattice angles to generate either a lattice matrix or the "tilt" factors often used molecular simulation.

## 2. Periodic Boundary Conditions in Non-cuboid Cells

Application of periodic boundary conditions (minimum image convention) for non-cuboid simulation cells differs from that used for cuboid simulation cells. For description of the technique, consult the classic tutorial by W.R. Smith [1] or documentation of the LAMMPS molecular dynamics software [2].

## 3. 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 [3,4] (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 Cell Upair* Wpair* ULRC* Upair* Wpair* ULRC* Upair* Wpair* ULRC* 1 Triclinic -4.23138E+02 1.8088E+04 -8.1589E+01 -4.7011E+02 1.7806E+04 -3.4433E+01 -2.2534E+02 1.8454E+04 0.0000E0 2 Monoclinic -2.1435E+02 4.3744E+01 -1.1892E+01 -2.2132E+02 1.9153E+00 -5.0188E+00 -1.8556E+02 9.7173E+01 0.0000E0 3 Triclinic -5.0579E+02 5.5753E+02 -2.9372E+01 -5.2276E+02 4.5575E+02 -1.2396E+01 -4.3502E+02 6.8970E+02 0.0000E0 4 Monoclinic -3.7943E+01 6.8656E+00 -2.0724E+00 -3.9126E+01 -2.2966E-01 -8.7464E-01 -3.3033E+01 1.5862E+01 0.0000E0

## 4. 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 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 [3,4]:

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

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. W. R. Smith, "The Minimum Image Convention in Non-Cubic MD Cells," (1989).
2. LAMMPS Documentation, Section 8.2.2. "Triclinic (non-orthogonal) simulation boxes." Accessed October 29, 2021.
3. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, New York, 1989).
4. D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd ed. (Academic, San Diego, 2002), pp.37-38.
Created October 28, 2021, Updated October 29, 2021