## Relation between eigenvector and atomic displacements

The eigenvalue problem is described as,

 (3.19)

where and are an eigenvalue and an eigenvector of this problem, respectively. The coordinate system is Cartesian. The is called polarization vector which corresponds to the normalized atomic displacements weighted by the square root of the atomic mass as written by,

 (3.20)

where is the amplitude vector of an atomic displacement. The corresponds to vibrational frequency. The is complex number and the phase factor at given instant is expressed by . The normalization is expressed as,

 (3.21)

The normalized atomic displacement is obtained by,

 (3.22)

The origin where the radial vector is measured can be set at an arbitrary point in the real space. An eigenvector calculated at an exact k-point can be represented by the atomic modulations in the calculated supercell at an arbitrary instance . However the amplitude of the displacements is not described by Eq. (3.24), but is calculated by the thermodynamic properties.

togo 2009-02-12