Please use `MAKPOL` for
building data-sets for solids, it is easier to use than constructing the Tv "by hand"
If you do want to construct the translation vectors, the following information
will be useful:

Translation vectors define the distance vectors that all atoms in the cluster are translated through to form another cluster in the solid. The Tv can be in Cartesian or internal coordinates. If a Tv is defined in Cartesian coordinates, then it is used directly. If it is defined in internal coordinates, then it is converted to Cartesian coordinates at run time, and the Cartesian coordinates of atom 1 (real or dummy) is subtracted from it. If atom 1 is at the origin, i.e., at (0.0, 0.0, 0.0) then the coordinates defined by Tv are the translation vectors, this simplifies visualizing the translation vectors using a GUI, in that a line drawn from atom 1 to the Tv would represent the translation vector.

Most GUI's cannot display the Tv because Tv is not a chemical symbol. To allow a GUI to display the Tv, edit the geometry to replace the Tv by an element that is not already in the system. A good choice would be He or Rn. If the translation vectors are to be displayed, make sure that atom 1 is real, not a dummy atom, and is at the origin.

In solids, the three translation vectors should define a cluster unit cell
that is big enough to hold a sphere of diameter about 10 - 15 Å.
This is to allow every atom in the cluster to interact with atoms in adjacent
clusters as if they were different atoms. If a cluster unit cell was not
large enough, an atom might interact incorrectly with its equivalent atom in an
adjacent unit cell. If the cluster unit cell is too small, the error
message "**Volume of unit cell unreasonably small**" will be printed and the
job stopped.

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. `

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 3The '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 define Tv in terms of atoms near to its desired position. To illustrate this, consider polyethylene again. By using the carbon atom at the end of the unit cell and two other atoms for angle and dihedral, the end of the data set would now look 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 Tv 1.53482200 +1 111.5010360 +1 122.0188610 +1 12 11 36This is a perfectly general and robust method of defining Tv in terms of internal coordinates.

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, however if it is not at the origin, visualization of the translation vectors can be misleading.

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.