Fractional coordinates

In crystallography, a fractional coordinate system is a coordinate system in which the edges of the unit cell are used as the basic vectors to describe the positions of atomic nuclei. The unit cell is a parallelepiped defined by the lengths of its edges    and angles between them .

General case

Consider a system of periodic structure in space and use  , , and  as the three independent period vectors, forming a right-handed triad, which are also the edge vectors of a cell of the system. Then any vector  in Cartesian coordinates can be written as a linear combination of the period vectors  

Our task is to calculate the scalar coefficients known as fractional coordinates , , and , assuming , , , and  are known.

For this purpose, let us calculate the following cell surface area vector

then 

and the volume of the cell is

If we do a vector inner (dot) product as follows

then we get 

Similarly, 

we arrive at 

and 

If there are many s to be converted with respect to the same period vectors, to speed up, we can have 

where

In crystallography

In crystallography, the lengths (, , ) of  and angles (, , ) between the edge (period) vectors  (, , ) of the parallelepiped unit cell are known. For simplicity, it is chosen so that edge vector  in the positive -axis direction, edge vector  in the  plane with positive -axis component, edge vector  with positive -axis component in the Cartesian-system, as shown in the figure below.

Then the edge vectors can be written as 

where all , , , ,  are positive. Next, let us express all  components with known variables. This can be done with 

Then 

The last one continues

where

Remembering , , and  being positive, one gets

Since the absolute value of the bottom surface area of the cell is 

the volume of the parallelepiped cell can also be expressed as
.

Once the volume is calculated as above, one has 

Now let us summarize the expression of the edge (period) vectors

Conversion from Cartesian coordinates

Let us calculate the following surface area vector of the cell first

where

Another surface area vector of the cell

where

The last surface area vector of the cell

where

Summarize 

As a result

where , ,  are the components of the arbitrary vector  in Cartesian coordinates.

Conversion to Cartesian coordinates

To return the orthogonal coordinates in ångströms from fractional coordinates, one can employ the first equation on top and the expression of the edge (period) vectors

For the special case of a monoclinic cell (a common case) where  and , this gives:

Supporting file formats
CPMD input
CIF

References

Category:Molecular modelling
Category:Computational chemistry
Category:Crystallography