8.2.3. Triclinic (non-orthogonal) simulation boxes

By default, LAMMPS uses an orthogonal simulation box to encompass the particles. The boundary command sets the boundary conditions of the box (periodic, non-periodic, etc). The orthogonal box has its “origin” at (xlo,ylo,zlo) and is defined by 3 edge vectors starting from the origin given by a = (xhi-xlo,0,0); b = (0,yhi-ylo,0); c = (0,0,zhi-zlo). The 6 parameters (xlo,xhi,ylo,yhi,zlo,zhi) are defined at the time the simulation box is created, e.g. by the create_box or read_data or read_restart commands. Additionally, LAMMPS defines box size parameters lx,ly,lz where lx = xhi-xlo, and similarly in the y and z dimensions. The 6 parameters, as well as lx,ly,lz, can be output via the thermo_style custom command.

LAMMPS also allows simulations to be performed in triclinic (non-orthogonal) simulation boxes shaped as a parallelepiped with triclinic symmetry. The parallelepiped has its “origin” at (xlo,ylo,zlo) and is defined by 3 edge vectors starting from the origin given by a = (xhi-xlo,0,0); b = (xy,yhi-ylo,0); c = (xz,yz,zhi-zlo). xy,xz,yz can be 0.0 or positive or negative values and are called “tilt factors” because they are the amount of displacement applied to faces of an originally orthogonal box to transform it into the parallelepiped. In LAMMPS the triclinic simulation box edge vectors a, b, and c cannot be arbitrary vectors. As indicated, a must lie on the positive x axis. b must lie in the xy plane, with strictly positive y component. c may have any orientation with strictly positive z component. The requirement that a, b, and c have strictly positive x, y, and z components, respectively, ensures that a, b, and c form a complete right-handed basis. These restrictions impose no loss of generality, since it is possible to rotate/invert any set of 3 crystal basis vectors so that they conform to the restrictions.

For example, assume that the 3 vectors A,B,C are the edge vectors of a general parallelepiped, where there is no restriction on A,B,C other than they form a complete right-handed basis i.e. A x B . C > 0. The equivalent LAMMPS a,b,c are a linear rotation of A, B, and C and can be computed as follows:

\[\begin{split}\begin{pmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{pmatrix} = & \begin{pmatrix} a_x & b_x & c_x \\ 0 & b_y & c_y \\ 0 & 0 & c_z \\ \end{pmatrix} \\ a_x = & A \\ b_x = & \mathbf{B} \cdot \mathbf{\hat{A}} \quad = \quad B \cos{\gamma} \\ b_y = & |\mathbf{\hat{A}} \times \mathbf{B}| \quad = \quad B \sin{\gamma} \quad = \quad \sqrt{B^2 - {b_x}^2} \\ c_x = & \mathbf{C} \cdot \mathbf{\hat{A}} \quad = \quad C \cos{\beta} \\ c_y = & \mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})} \times \mathbf{\hat{A}} \quad = \quad \frac{\mathbf{B} \cdot \mathbf{C} - b_x c_x}{b_y} \\ c_z = & |\mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})}|\quad = \quad \sqrt{C^2 - {c_x}^2 - {c_y}^2}\end{split}\]

where A = | A | indicates the scalar length of A. The hat symbol (^) indicates the corresponding unit vector. \(\beta\) and \(\gamma\) are angles between the vectors described below. Note that by construction, a, b, and c have strictly positive x, y, and z components, respectively. If it should happen that A, B, and C form a left-handed basis, then the above equations are not valid for c. In this case, it is necessary to first apply an inversion. This can be achieved by interchanging two basis vectors or by changing the sign of one of them.

For consistency, the same rotation/inversion applied to the basis vectors must also be applied to atom positions, velocities, and any other vector quantities. This can be conveniently achieved by first converting to fractional coordinates in the old basis and then converting to distance coordinates in the new basis. The transformation is given by the following equation:

\[\begin{split}\mathbf{x} = & \begin{pmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{pmatrix} \cdot \frac{1}{V} \begin{pmatrix} \mathbf{B \times C} \\ \mathbf{C \times A} \\ \mathbf{A \times B} \end{pmatrix} \cdot \mathbf{X}\end{split}\]

where V is the volume of the box, X is the original vector quantity and x is the vector in the LAMMPS basis.

There is no requirement that a triclinic box be periodic in any dimension, though it typically should be in at least the second dimension of the tilt (y in xy) if you want to enforce a shift in periodic boundary conditions across that boundary. Some commands that work with triclinic boxes, e.g. the fix deform and fix npt commands, require periodicity or non-shrink-wrap boundary conditions in specific dimensions. See the command doc pages for details.

The 9 parameters (xlo,xhi,ylo,yhi,zlo,zhi,xy,xz,yz) are defined at the time the simulation box is created. This happens in one of 3 ways. If the create_box command is used with a region of style prism, then a triclinic box is setup. See the region command for details. If the read_data command is used to define the simulation box, and the header of the data file contains a line with the “xy xz yz” keyword, then a triclinic box is setup. See the read_data command for details. Finally, if the read_restart command reads a restart file which was written from a simulation using a triclinic box, then a triclinic box will be setup for the restarted simulation.

Note that you can define a triclinic box with all 3 tilt factors = 0.0, so that it is initially orthogonal. This is necessary if the box will become non-orthogonal, e.g. due to the fix npt or fix deform commands. Alternatively, you can use the change_box command to convert a simulation box from orthogonal to triclinic and vice versa.

As with orthogonal boxes, LAMMPS defines triclinic box size parameters lx,ly,lz where lx = xhi-xlo, and similarly in the y and z dimensions. The 9 parameters, as well as lx,ly,lz, can be output via the thermo_style custom command.

To avoid extremely tilted boxes (which would be computationally inefficient), LAMMPS normally requires that no tilt factor can skew the box more than half the distance of the parallel box length, which is the first dimension in the tilt factor (x for xz). This is required both when the simulation box is created, e.g. via the create_box or read_data commands, as well as when the box shape changes dynamically during a simulation, e.g. via the fix deform or fix npt commands.

For example, if xlo = 2 and xhi = 12, then the x box length is 10 and the xy tilt factor must be between -5 and 5. Similarly, both xz and yz must be between -(xhi-xlo)/2 and +(yhi-ylo)/2. Note that this is not a limitation, since if the maximum tilt factor is 5 (as in this example), then configurations with tilt = …, -15, -5, 5, 15, 25, … are geometrically all equivalent. If the box tilt exceeds this limit during a dynamics run (e.g. via the fix deform command), then the box is “flipped” to an equivalent shape with a tilt factor within the bounds, so the run can continue. See the fix deform page for further details.

One exception to this rule is if the first dimension in the tilt factor (x for xy) is non-periodic. In that case, the limits on the tilt factor are not enforced, since flipping the box in that dimension does not change the atom positions due to non-periodicity. In this mode, if you tilt the system to extreme angles, the simulation will simply become inefficient, due to the highly skewed simulation box.

Box flips that may occur using the fix deform or fix npt commands can be turned off using the flip no option with either of the commands.

Note that if a simulation box has a large tilt factor, LAMMPS will run less efficiently, due to the large volume of communication needed to acquire ghost atoms around a processor’s irregular-shaped subdomain. For extreme values of tilt, LAMMPS may also lose atoms and generate an error.

Triclinic crystal structures are often defined using three lattice constants a, b, and c, and three angles \(\alpha\), \(\beta\), and \(\gamma\). Note that in this nomenclature, the a, b, and c lattice constants are the scalar lengths of the edge vectors a, b, and c defined above. The relationship between these 6 quantities (a, b, c, \(\alpha\), \(\beta\), \(\gamma\)) and the LAMMPS box sizes (lx,ly,lz) = (xhi-xlo,yhi-ylo,zhi-zlo) and tilt factors (xy,xz,yz) is as follows:

\[\begin{split}a = & {\rm lx} \\ b^2 = & {\rm ly}^2 + {\rm xy}^2 \\ c^2 = & {\rm lz}^2 + {\rm xz}^2 + {\rm yz}^2 \\ \cos{\alpha} = & \frac{{\rm xy}*{\rm xz} + {\rm ly}*{\rm yz}}{b*c} \\ \cos{\beta} = & \frac{\rm xz}{c} \\ \cos{\gamma} = & \frac{\rm xy}{b} \\\end{split}\]

The inverse relationship can be written as follows:

\[\begin{split}{\rm lx} = & a \\ {\rm xy} = & b \cos{\gamma} \\ {\rm xz} = & c \cos{\beta}\\ {\rm ly}^2 = & b^2 - {\rm xy}^2 \\ {\rm yz} = & \frac{b*c \cos{\alpha} - {\rm xy}*{\rm xz}}{\rm ly} \\ {\rm lz}^2 = & c^2 - {\rm xz}^2 - {\rm yz}^2 \\\end{split}\]

The values of a, b, c, \(\alpha\) , \(\beta\), and \(\gamma\) can be printed out or accessed by computes using the thermo_style custom keywords cella, cellb, cellc, cellalpha, cellbeta, cellgamma, respectively.

As discussed on the dump command doc page, when the BOX BOUNDS for a snapshot is written to a dump file for a triclinic box, an orthogonal bounding box which encloses the triclinic simulation box is output, along with the 3 tilt factors (xy, xz, yz) of the triclinic box, formatted as follows:

xlo_bound xhi_bound xy
ylo_bound yhi_bound xz
zlo_bound zhi_bound yz

This bounding box is convenient for many visualization programs and is calculated from the 9 triclinic box parameters (xlo,xhi,ylo,yhi,zlo,zhi,xy,xz,yz) as follows:

xlo_bound = xlo + MIN(0.0,xy,xz,xy+xz)
xhi_bound = xhi + MAX(0.0,xy,xz,xy+xz)
ylo_bound = ylo + MIN(0.0,yz)
yhi_bound = yhi + MAX(0.0,yz)
zlo_bound = zlo
zhi_bound = zhi

These formulas can be inverted if you need to convert the bounding box back into the triclinic box parameters, e.g. xlo = xlo_bound - MIN(0.0,xy,xz,xy+xz).

One use of triclinic simulation boxes is to model solid-state crystals with triclinic symmetry. The lattice command can be used with non-orthogonal basis vectors to define a lattice that will tile a triclinic simulation box via the create_atoms command.

A second use is to run Parrinello-Rahman dynamics via the fix npt command, which will adjust the xy, xz, yz tilt factors to compensate for off-diagonal components of the pressure tensor. The analog for an energy minimization is the fix box/relax command.

A third use is to shear a bulk solid to study the response of the material. The fix deform command can be used for this purpose. It allows dynamic control of the xy, xz, yz tilt factors as a simulation runs. This is discussed in the next section on non-equilibrium MD (NEMD) simulations.