## Translation Vector for defining Polymers, Layer systems, and Solids

The translation vector is the distance through which an atom must be moved (translated) in order to be in the next unit cell. The symbol for a translation vector is Tv. The Tv should be specified at the end of the geometry. No real or dummy atoms should come after a Tv, although symmetry data, etc, are allowed. The order of the data is thus: (real and dummy atoms) (translation vector(s)) (blank line) (symmetry data or MECI data etc, if needed).

Translation vectors can be defined in internal coordinates or Cartesian coordinates.  Both definitions give rise to the same result, but the methods of definition are very different.  Before proceeding, remember that atom positions are defined as being in internal coordinates if there is a connectivity.  If there is no connectivity, the atom position is defined in Cartesian coordinates. The utility MAKPOL is provided to simplify construction of MOPAC data sets for solids.

The number of translation vectors determines the type of infinite solid.  A polymer has one Tv, a layer system has two Tv, and a solid has three Tv.

### Internal coordinate definition of translation vector

If a Tv has a connectivity, then its position is defined using internal coordinates.  The translation vector is then given by the difference in position of the atom the Tv  is connected to and the position defined by Tv.   For example, consider polyethylene:

```        Polyethylene, -[C12H24]-

C         0.00000000 +0    0.0000000 +0    0.0000000 +0                        -0.2638
C         1.53502951 +1    0.0000000 +0    0.0000000 +0     1     0     0      -0.2639
C         1.53434080 +1  111.4738515 +1    0.0000000 +0     2     1     0      -0.2638
C         1.53486713 +1  111.4515730 +1 -179.5035734 +1     3     2     1      -0.2638
C         1.53436383 +1  111.4544128 +1 -179.5609499 +1     4     3     2      -0.2639
C         1.53487336 +1  111.4657224 +1 -179.9694611 +1     5     4     3      -0.2638
C         1.53438258 +1  111.4286979 +1 -179.7424610 +1     6     5     4      -0.2638
C         1.53483201 +1  111.4669973 +1  179.6187779 +1     7     6     5      -0.2638
C         1.53442061 +1  111.4159036 +1  179.6200571 +1     8     7     6      -0.2639
C         1.53492245 +1  111.4448809 +1  179.5385773 +1     9     8     7      -0.2638
C         1.53443267 +1  111.3905812 +1  179.4842656 +1    10     9     8      -0.2639
C         1.53474400 +1  111.3843156 +1 -179.9610941 +1    11    10     9      -0.2638
H         1.10720431 +1  109.8104572 +1  -57.5906521 +1     1     2     3       0.1319
H         1.10698586 +1  109.8681806 +1  116.0168453 +1     1     2    13       0.1319
H         1.10709315 +1  109.8700918 +1  122.0438631 +1     2     1     3       0.1319
H         1.10713575 +1  109.8236237 +1  115.9314480 +1     2     1    15       0.1319
H         1.10712021 +1  109.8652470 +1 -122.0100491 +1     3     2     4       0.1319
H         1.10709847 +1  109.8813887 +1 -116.0033729 +1     3     2    17       0.1319
H         1.10711978 +1  109.8682857 +1  122.0383223 +1     4     3     5       0.1319
H         1.10711511 +1  109.8397525 +1  115.9441858 +1     4     3    19       0.1319
H         1.10713850 +1  109.8677152 +1 -122.0280078 +1     5     4     6       0.1319
H         1.10712347 +1  109.8680744 +1 -115.9603262 +1     5     4    21       0.1319
H         1.10712096 +1  109.8603331 +1  122.0225998 +1     6     5     7       0.1319
H         1.10706606 +1  109.8553502 +1  115.9629566 +1     6     5    23       0.1319
H         1.10712955 +1  109.8765480 +1 -121.9985907 +1     7     6     8       0.1319
H         1.10711605 +1  109.8542406 +1 -115.9994488 +1     7     6    25       0.1319
H         1.10707238 +1  109.8604415 +1  122.0413124 +1     8     7     9       0.1319
H         1.10709630 +1  109.8633232 +1  115.9604117 +1     8     7    27       0.1319
H         1.10708893 +1  109.8924995 +1 -122.0076508 +1     9     8    10       0.1319
H         1.10711581 +1  109.8477424 +1 -116.0065534 +1     9     8    29       0.1319
H         1.10704898 +1  109.8675671 +1  122.0217258 +1    10     9    11       0.1319
H         1.10708168 +1  109.8627808 +1  115.9892025 +1    10     9    31       0.1319
H         1.10704508 +1  109.9042865 +1 -122.0607012 +1    11    10    12       0.1319
H         1.10715046 +1  109.8566042 +1 -115.9742091 +1    11    10    33       0.1319
H         1.10698039 +1  109.8868070 +1  -58.2346895 +1    12    11    10       0.1319
H         1.10741358 +1  109.7138737 +1  116.0025351 +1    12    11    35       0.1320
Tv        15.21694969 +1   34.2865646 +1   -0.0375057 +1     1     2     3     ```
The 'bond length', 15.22 Ångstroms, is the distance of Tv from atom 1. The direction is given by the angle (here 34 degrees) and dihedral (-0.04 degrees). So atom 1, on translation, would be moved to the position of Tv. All other atoms would then be moved the same way.

But this way of defining Tv has a severe drawback - a small change in the angle or dihedral of Tv would produce a large change in position of the Tv.  An easy way around this is to put a dummy atom at the position of Tv, and then define Tv in terms of atom 1 and the dummy atom.  To illustrate this, consider polyethylene again.  By adding a dummy atom, the end of the data set now looks like this:

```  H         1.10698039 +1  109.8868070 +1  -58.2346895 +1    12    11    10       0.1320
H         1.10741358 +1  109.7138737 +1  116.0025351 +1    12    11    35       0.1320
XX         1.53482200 +1  111.5010360 +1  122.0188610 +1    12    11    36
Tv        15.21694969 +1    0.0000000 +0    0.0000000 +0     1    37    35 ```
Now the angle and dihedral of Tv can be defined using the dummy atom.  The angle Tv - atom 1 - dummy atom can be fixed at zero degrees, and marked "not to be optimized."  The dihedral is not important, but needs to be given, so select any other atom for the connectivity and lock the dihedral at zero degrees.  This gives a perfectly general and robust method of Tv in terms of internal coordinates.

### Cartesian coordinate definition of translation vector

If the connectivity is missing Tv is defined as Cartesian coordinates.  The absolute position of Tv defines the motion of all atoms.  Consider polyethylene again.  In Cartesian coordinates, the data set would be:

```Polyethylene, -[C12H24]-

C         0.00000000 +1    0.0000000 +1    0.0000000 +1                         0.0000
C         1.53502951 +1    0.0000000 +1    0.0000000 +1                         0.0000
C         2.09671572 +1    1.4278341 +1    0.0000000 +1                         0.0000
C         3.63153284 +1    1.4272569 +1   -0.0123771 +1                         0.0000
C         4.19338646 +1    2.8550080 +1   -0.0014355 +1                         0.0000
C         5.72820359 +1    2.8547421 +1   -0.0145707 +1                         0.0000
C         6.28919628 +1    4.2827881 +1    0.0028023 +1                         0.0000
C         7.82402390 +1    4.2834120 +1   -0.0008140 +1                         0.0000
C         8.38373432 +1    5.7120857 +1    0.0070968 +1                         0.0000
C         9.91863588 +1    5.7134285 +1    0.0149906 +1                         0.0000
C        10.47706263 +1    7.1426306 +1    0.0100363 +1                         0.0000
C        12.01179057 +1    7.1438002 +1    0.0169597 +1                         0.0000
H        -0.37524221 +1    0.5583029 +1    0.8794274 +1                         0.0000
H        -0.37621723 +1    0.5451135 +1   -0.8869777 +1                         0.0000
H         1.91131794 +1   -0.5524188 +1    0.8825507 +1                         0.0000
H         1.91048784 +1   -0.5523063 +1   -0.8830279 +1                         0.0000
H         1.72795936 +1    1.9771690 +1    0.8876740 +1                         0.0000
H         1.71422443 +1    1.9828794 +1   -0.8782324 +1                         0.0000
H         4.01464282 +1    0.8652321 +1    0.8611611 +1                         0.0000
H         3.99989284 +1    0.8845857 +1   -0.9042980 +1                         0.0000
H         3.82488823 +1    3.3975418 +1    0.8905409 +1                         0.0000
H         3.81002536 +1    3.4169220 +1   -0.8749396 +1                         0.0000
H         6.11169319 +1    2.2888993 +1    0.8563337 +1                         0.0000
H         6.09647982 +1    2.3164462 +1   -0.9091127 +1                         0.0000
H         5.91538732 +1    4.8232426 +1    0.8938195 +1                         0.0000
H         5.91095296 +1    4.8460271 +1   -0.8720690 +1                         0.0000
H         8.20242684 +1    3.7254711 +1    0.8773217 +1                         0.0000
H         8.19832236 +1    3.7375404 +1   -0.8882750 +1                         0.0000
H         8.00284989 +1    6.2609926 +1    0.8898619 +1                         0.0000
H         8.01178944 +1    6.2664053 +1   -0.8761314 +1                         0.0000
H        10.29078179 +1    5.1665370 +1    0.9026699 +1                         0.0000
H        10.29979525 +1    5.1573344 +1   -0.8631362 +1                         0.0000
H        10.09560328 +1    7.6993624 +1    0.8875824 +1                         0.0000
H        10.10444119 +1    7.6888972 +1   -0.8779548 +1                         0.0000
H        12.38474913 +1    6.6007161 +1    0.9065479 +1                         0.0000
H        12.38973517 +1    6.5834579 +1   -0.8602742 +1                         0.0000
Tv        12.57289793 +1    8.5719164 +0    0.0067071 +0  ```

Now Tv is defined as X=12.57, Y=8.57, and Z=0.01.  On translation, every atom would be displaced by this amount.  Although it is the default, atom 1 does not need to be at the origin.

The definition of Tv, Cartesian or internal, does not depend on the definitions of any of the other atoms, that is, some atoms can be in Cartesian coordinates and some in internal coordinates.  The definitions gives here refer to the Tv only.

For polymers, internal coordinates are easier than Cartesian coordinates.  For solids, Cartesian coordinates are recommended, unless the system has high symmetry, in which case internal coordinates are preferred, as that allows extensive use of symmetry functions to reduce the number of geometric parameters to be optimized.  In most high symmetry cases, only one or two geometric parameters need to be optimized.