The three primitive translation vectors are , , . Their representation in Cartesian coordinates (atomic units) is:
The atomic positions are specified by the coordinates for where is the number of atoms natom.
Representation in reduced coordinates:
The volume of the primitive unit cell (called ucvol in the code) is
The scalar products in the reduced representation are valuated thanks to
where is the metric tensor in real space stored in
The three primitive translation vectors in reciprocal space are , ,
This definition is such that
For historical reasons, the internal implementation uses the convention . This means that a factor must be taken into account in the Fortran code. We don’t use this convention in the theory notes to keep the equations as simple as possible.
Reduced representation of vectors (K) in reciprocal space
e.g. the reduced representation of is (1, 0, 0).
The reduced representation of the vectors of the reciprocal space lattice is made of triplets of integers.
The scalar products in the reduced representation are evaluated thanks to
where is the metric tensor in reciprocal space called
gmet inside the code.
Taking into account the internal conventions used by the code, we have the correspondence:
Fourier series for periodic lattice quantities¶
Any function with the periodicity of the lattice i.e any function fullfilling the property:
can be represented with the discrete Fourier series:
where the Fourier coefficient, , is given by: