# [lammps-users] confusion about scaled coordinates for triclinic cells

Hi! I am trying to calculate interparticle distances from a configuration-dump file with header

ITEM: TIMESTEP

14000000

ITEM: NUMBER OF ATOMS

100000

ITEM: BOX BOUNDS xy xz yz pp pp pp

-2.7226138904763260e+01 2.7158265908439908e+01 -4.3446974625327561e-01

-2.6695721711161067e+01 2.6681029457996679e+01 3.6659674992992375e-01

-2.6524840642222021e+01 2.6524840642222021e+01 -1.4692253164388701e-02

Using the definitions on the doc page (https://docs.lammps.org/Howto_triclinic.html), I get

xlo_bound = -27.2261
xhi_bound = 27.1583
xy = -0.43447
ylo_bound = -26.6957
yhi_bound = 26.681
xz = 0.366597
zlo_bound = -26.5248
zhi_bound = 26.5248
yz = 0.0146923

Converting these to the cell shape parameters gives

xlo = xlo_bound - min(0, xy, xz, xy + xz) = -26.7917
xhi = xhi_bound - max(0, xy, xz, xy + xz) = 26.7917
ylo = ylo_bound - min(0,yz) = -26.681
yhi = yhi_bound - max(0,yz) = 26.681

zlo = zlo_bound = -26.5248
zhi = zhi_bound = 26.5248

Using these definitions, the cell-edge (a, b, and c) vectors are

a = {xhi - xlo, 0, 0} = {53.5833, 0, 0}
b = {xy, yhi - ylo, 0} = {-0.43447, 53.3621, 0}
c = {xz, yz, zhi - zlo} = {0.366597, -0.0146923, 53.0497}

Then the h-matrix (the shape matrix whose column vectors are a, b, c) is

53.5833 -0.43447 0.366597
0 53.3621 -0.0146923
0 0 53.0497

and its inverse h_inv is

0.0186625 0.000151949 -0.000128924
0. 0.0187399 5.19007*10^-6
0 0 0.0188503

Here’s where I ran into trouble. I would expect the scaled coordinates of the cell corners {xlo, ylo, zlo} and {xhi, yhi, zhi} to be respectively

hinv {xlo, ylo, zlo} = {-.5, -.5, -.5}

and

hinv {xhi, yhi, zhi} = {.5, .5, .5}.

However, for the above hinv, they are

hinv {xlo, ylo, zlo} = {-.500634, -.500138, -.5}

and

hinv {xhi, yhi, zhi} = {.500634, .500138, .5}.

Thus the scaled cell isn’t a unit cube, so I can’t calculate interparticle distances using the usual textbook method of multiplying differences in unit-cube-scaled coordinates by h

Any idea what’s going on here? I’ve checked the math several times so I assume I’ve misunderstood something, but can’t figure out what that might be…

Rob

You are working with numbers that have about 5 digits precision. because you take differences of values of different magnitude, for smaller magnitude values that precision drops further.
So the precision on your results is disappointingly low, but I would suspect that if you output the data with full 15 digits precision that is available to floating point numbers, that error should drop.

Hi Axel. Thanks for the quick reply. And good suggestion, but that doesn’t seem to be the issue here. The C++ code I used to perform this calculation (attached in case you’re interested) reads in the data in the LAMMPS header to double precision. When I edited the code to output the h-matrix and scaled box bounds to high precision I got

rshoy@hoyoffice ONanalyze-triclinic_cell % ./ONanalyze-triclinic 100000 1 10 10 10 jammedconfig.fastcompression.purelyrepulsive

Double-precision h matrix:

53.58333831702 -0.434469746253276 0.366596749929924

0 53.3620589159934 -0.0146922531643887

0 0 53.049681284444

Scaled box bounds: xlo -0.50063445673268 xhi 0.50063445673268 ylo -0.500137665726013 yhi 0.500137665726013 zlo -0.5 zhi 0.5

Execution took 2 seconds.

So, ???

Thanks,
Rob

ONanalyze-triclinic.cpp (13.9 KB)

I don’t have time to look into the code, but could it be that you are missing a transpose somewhere?