Symmetry of the dynamical matrix

The crth component of the displacement u nn) of the /v.th atom in the th unit cell due to the normal mode of vibration (q / is a function of a and time t, 

The wave vector q in the transformed displacement (E i)ua(nn) appears in the scalar product with a in the eigenvalue of the function operator ( a/), and is unaffected by a pure translation, as we should expect. Under the pure rotation (/ 0), the rotated displacement 

Equation (4) follows from eq. (3) because the scalar product (SP) in eq. (3) is invariant under the rigid rotation R. 

Equation (6) verifies that under the rotation (7 0) the wave vector q is rotated into /x q, just as we might have anticipated in the active representation. The transformed function 

Since 91/ is a symmetry operator, (nn) and (NK) are equivalent sites occupied by the same kind of atom, so that Mk=Mk. It follows from eq. (7) that the Fourier-transformed dynamical matrix D(q) is transformed by 91/ into D(7 q). The AK, BK element of D(7 q) is 

---

The potential is expanded up to second order in terms of the Aj( ). It is then possible to express all force constants (translational, librational, and interaction) in terms of the translational force constants only. However, the components of the vectors R(W), R(k l ) also appear in-these expressions. If, for example, we consider the problem of one molecule per primitive unit cell, the dynamical equation is of dimension six and there are, in general, 36 force constants. In the extended point-mass approximation considered here, on the other hand, there are only nine force constants, as would appear in the dynamical equation containing translational motions only. In addition, there are the components of R(0, R(/ ) (for the interacting molecules I, / ) which represent six further parameters. If we regard the force constants as parameters, the total number of these is 15 in this model, as compared to 36 in the general case. Actually, the symmetry of the dynamical matrix further reduces this number in both cases. 

---