We calculate
by the small-displacement
method. In harmonic approximation the Cartesian component of
the force exerted on the atom at position
is

(2) |

by displacing once at a time all the atoms of the lattice along the three Cartesian components by , and calculating the forces induced on the atoms in . Eqn.( 3) computes the force constant matrix using forward differences; for numerical reasons, it can be advantageous in some cases to use central differences, in which case the force constant matrix can be calculated as:

Since the crystal is invariant under traslations of any lattice vector, it is only necessary to displace the atoms in one primitive cell and calculate the forces induced on all the other atoms of the crystal. In what follows we will assume this as understood and put simply .

It is important to appreciate that the in the formula for is the force-constant matrix in the infinite lattice, with no restriction on the wavevector , whereas the calculations of can only be done in supercell geometry. Without a further assumption, it is strictly impossible to extract the infinite-lattice from supercell calculations, since the latter deliver information only at wavevectors that are reciprocal lattice vectors of the superlattice. The further assumption needed is that the infinite-lattice vanishes when the separation is such that the positions and lie in different Wigner-Seitz (WS) cells of the chosen superlattice. More precisely, if we take the WS cell centred on , then the infinite-lattice value of vanishes if is in a different WS cell; it is equal to the supercell value if is wholly within the same WS cell; and it is equal to the supercell value divided by an integer if lies on the boundary of the same WS cell, where is the number of WS cells having on their boundary. With this assumption, the elements will converge to the correct infinite-lattice values as the dimensions of the supercell are systematically increased.

It is not always necessary to displace all the atoms in the primitive
cell, since the use of symmetries can reduce the amount of work
needed. This is done as follows. We displace one atom in the primitive
cell, let's call it 'one', and we calculate the forces induced by the
displacement on all the other atoms of the supercell. Then we pick up
one other atom of the primitive cell, atom 'two'. If there is a
symmetry operation (not necessarily a point group symmetry
operation) such that, when is applyed to the crystal atom two is
sent into atom one and the whole crystal is invariant under such
transformation, then it is not necessary to displace atom two, and the
part of the force constant matrix associated with its displacement can
be calculated using

(5) |

In principle each atom has to be displaced along the three Cartesian
directions. It is sometimes convenient to displace the atoms along
some special directions so as to maximize the number of symmetry
operations still present in the 'excited' supercell, in this way the
calculations of the forces are less expensive. This can always be
done, as long as one displaces the atoms along three linearly
independent directions. The forces induced by the displacements along
the three Cartesian directions is easily reconstructed by the linear
combination

(6) |

Using symmetries it is possible to reduce the number of displacements
even further: if applying a point group symmetry operation to the
displacement vector one obtains a vector which
is linearly independent from , then the force field that
would be induced by the displacement can be calculated by

(7) |

The force constant matrix is invariant under the point group symmetry
operations of the crystal. This is not automatically garanteed by the
procedure just described, because in general the crystal is not
harmonic, and therefore eqns.( 3, 4) are
only an approximation. So, the force constant matrix must be
symmetrized with respect to the point group operations of the crystal:

(8) |

As an example of the procedure just described let's consider the h.c.p. crystal. There are two atoms in the primitive cell, so in principles we would need six independent calculations. We will see that the number of calculations needed is equal to two. In first place one can easily recognize that only one atom needs to be displaced: if we traslate the crystal from one atom to the other and we perform a spatial inversion the crystal remains unchanged. Secondly, by applying a clockwise rotation of degrees, for example, to a displacement in the direction, one obtains an independent displacement. So only one additional displacement along the direction is needed.