\(\renewcommand{\AA}{\text{Å}}\)

# compute orientorder/atom command

Accelerator Variants: orientorder/atom/kk

## Syntax

    compute ID group-ID orientorder/atom keyword values ...

- ID, group-ID are documented in compute command

- orientorder/atom = style name of this compute command

- one or more keyword/value pairs may be appended

      keyword = cutoff or nnn or degrees or wl or wl/hat or components or chunksize
        cutoff value = distance cutoff
        nnn value = number of nearest neighbors
        degrees values = nlvalues, l1, l2,...
        wl value = yes or no
        wl/hat value = yes or no
        components value = ldegree
        chunksize value = number of atoms in each pass

## Examples

    compute 1 all orientorder/atom
    compute 1 all orientorder/atom degrees 5 4 6 8 10 12 nnn NULL cutoff 1.5
    compute 1 all orientorder/atom wl/hat yes
    compute 1 all orientorder/atom components 6

## Description

Define a computation that calculates a set of bond-orientational order parameters \(Q_\ell\) for each atom in a group. These order parameters were introduced by Steinhardt et al. as a way to characterize the local orientational order in atomic structures. For each atom, \(Q_\ell\) is a real number defined as follows:

\[\begin{split}\bar{Y}_{\ell m} = & \frac{1}{nnn}\sum_{j = 1}^{nnn} Y_{\ell m}\bigl( \theta( \mathbf{r}_{ij} ), \phi( \mathbf{r}_{ij} ) \bigr) \\ Q_\ell = & \sqrt{\frac{4 \pi}{2 \ell + 1} \sum_{m = -\ell }^{m = \ell } \bar{Y}_{\ell m} \bar{Y}^*_{\ell m}}\end{split}\]

The first equation defines the local order parameters as averages of the spherical harmonics \(Y_{\ell m}\) for each neighbor. These are complex number components of the 3D analog of the 2D order parameter \(q_n\), which is implemented as LAMMPS compute hexorder/atom. The summation is over the nnn nearest neighbors of the central atom. The angles \(\theta\) and \(\phi\) are the standard spherical polar angles defining the direction of the bond vector \(r_{ij}\). The phase and sign of \(Y_{\ell m}\) follow the standard conventions, so that \(\mathrm{sign}(Y_{\ell\ell}(0,0)) = (-1)^\ell\). The second equation defines \(Q_\ell\), which is a rotationally invariant non-negative amplitude obtained by summing over all the components of degree \(\ell\).

The optional keyword cutoff defines the distance cutoff used when searching for neighbors. The default value, also the maximum allowable value, is the cutoff specified by the pair style.

The optional keyword nnn defines the number of nearest neighbors used to calculate \(Q_\ell\). The default value is 12. If the value is NULL, then all neighbors up to the specified distance cutoff are used.

The optional keyword degrees defines the list of order parameters to be computed. The first argument nlvalues is the number of order parameters. This is followed by that number of non-negative integers giving the degree of each order parameter. Because \(Q_2\) and all odd-degree order parameters are zero for atoms in cubic crystals (see Steinhardt), the default order parameters are \(Q_4\), \(Q_6\), \(Q_8\), \(Q_{10}\), and \(Q_{12}\). For the FCC crystal with nnn =12,

\[Q_4 = \sqrt{\frac{7}{192}} \approx 0.19094\]

The numerical values of all order parameters up to \(Q_{12}\) for a range of commonly encountered high-symmetry structures are given in Table I of Mickel et al., and these can be reproduced with this compute.

The optional keyword wl will output the third-order invariants \(W_\ell\) (see Eq. 1.4 in Steinhardt) for the same degrees as for the \(Q_\ell\) parameters. For the FCC crystal with nnn = 12,

\[W_4 = -\sqrt{\frac{14}{143}} \left(\frac{49}{4096}\right) \pi^{-3/2} \approx -0.0006722136\]

The optional keyword wl/hat will output the normalized third-order invariants \(\hat{W}_\ell\) (see Eq. 2.2 in Steinhardt) for the same degrees as for the \(Q_\ell\) parameters. For the FCC crystal with nnn =12,

\[\hat{W}_4 = -\frac{7}{3} \sqrt{\frac{2}{429}} \approx -0.159317\]

The numerical values of \(\hat{W}_\ell\) for a range of commonly encountered high-symmetry structures are given in Table I of Steinhardt, and these can be reproduced with this keyword.

The optional keyword components will output the components of the normalized complex vector \(\hat{Y}_{\ell m} = \bar{Y}_{\ell m}/|\bar{Y}_{\ell m}|\) of degree ldegree, which must be included in the list of order parameters to be computed. This option can be used in conjunction with compute coord_atom to calculate the ten Wolde’s criterion to identify crystal-like particles, as discussed in ten Wolde.

The optional keyword chunksize is only applicable when using the the KOKKOS package and is ignored otherwise. This keyword controls the number of atoms in each pass used to compute the bond-orientational order parameters and is used to avoid running out of memory. For example if there are 32768 atoms in the simulation and the chunksize is set to 16384, the parameter calculation will be broken up into two passes.

The value of \(Q_\ell\) is set to zero for atoms not in the specified compute group, as well as for atoms that have less than nnn neighbors within the distance cutoff, unless nnn is NULL.

The neighbor list needed to compute this quantity is constructed each time the calculation is performed (i.e., each time a snapshot of atoms is dumped). Thus it can be inefficient to compute/dump this quantity too frequently.

Note

If you have a bonded system, then the settings of special_bonds command can remove pairwise interactions between atoms in the same bond, angle, or dihedral. This is the default setting for the special_bonds command, and means those pairwise interactions do not appear in the neighbor list. Because this fix uses the neighbor list, it also means those pairs will not be included in the order parameter. This difficulty can be circumvented by writing a dump file, and using the rerun command to compute the order parameter for snapshots in the dump file. The rerun script can use a special_bonds command that includes all pairs in the neighbor list.

Styles with a gpu, intel, kk, omp, or opt suffix are functionally the same as the corresponding style without the suffix. They have been optimized to run faster, depending on your available hardware, as discussed on the Accelerator packages page. The accelerated styles take the same arguments and should produce the same results, except for round-off and precision issues.

These accelerated styles are part of the GPU, INTEL, KOKKOS, OPENMP, and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package page for more info.

You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.

See the Accelerator packages page for more instructions on how to use the accelerated styles effectively.

## Output info

This compute calculates a per-atom array with nlvalues columns, giving the \(Q_\ell\) values for each atom, which are real numbers in the range \(0 \le Q_\ell \le 1\).

If the keyword wl is set to yes, then the \(W_\ell\) values for each atom will be added to the output array, which are real numbers.

If the keyword wl/hat is set to yes, then the \(\hat{W}_\ell\) values for each atom will be added to the output array, which are real numbers.

If the keyword components is set, then the real and imaginary parts of each component of normalized \(\hat{Y}_{\ell m}\) will be added to the output array in the following order: \(\Re(\hat{Y}_{-m}),\) \(\Im(\hat{Y}_{-m}),\) \(\Re(\hat{Y}_{-m+1}),\) \(\Im(\hat{Y}_{-m+1}), \dotsc,\) \(\Re(\hat{Y}_m),\) \(\Im(\hat{Y}_m).\)

In summary, the per-atom array will contain nlvalues columns, followed by an additional nlvalues columns if wl is set to yes, followed by an additional nlvalues columns if wl/hat is set to yes, followed by an additional 2*(2* ldegree+1) columns if the components keyword is set.

These values can be accessed by any command that uses per-atom values from a compute as input. See the Howto output doc page for an overview of LAMMPS output options.

## Restrictions

none

## Related commands

compute coord/atom, compute centro/atom, compute hexorder/atom

## Default

The option defaults are cutoff = pair style cutoff, nnn = 12, degrees = 5 4 6 8 10 12 (i.e., \(Q_4\), \(Q_6\), \(Q_8\), \(Q_{10}\), and \(Q_{12}\)), wl = no, wl/hat = no, components off, and chunksize = 16384

(Steinhardt) P. Steinhardt, D. Nelson, and M. Ronchetti, Phys. Rev. B 28, 784 (1983).

(Mickel) W. Mickel, S. C. Kapfer, G. E. Schroeder-Turkand, K. Mecke, J. Chem. Phys. 138, 044501 (2013).

(tenWolde) P. R. ten Wolde, M. J. Ruiz-Montero, D. Frenkel, J. Chem. Phys. 104, 9932 (1996).