Eckart Conditions - Relation To The Harmonic Approximation

Relation To The Harmonic Approximation

In the harmonic approximation to the nuclear vibrational problem, expressed in displacement coordinates, one must solve the generalized eigenvalue problem

where H is a 3N × 3N symmetric matrix of second derivatives of the potential . H is the Hessian matrix of V in the equilibrium . The diagonal matrix M contains the masses on the diagonal. The diagonal matrix contains the eigenvalues, while the columns of C contain the eigenvectors.

It can be shown that the invariance of V under simultaneous translation over t of all nuclei implies that vectors T = (t, ..., t) are in the kernel of H. From the invariance of V under an infinitesimal rotation of all nuclei around s, it can be shown that also the vectors S = (s x R₁0, ..., s x R_N0) are in the kernel of H :

Thus, six columns of C corresponding to eigenvalue zero are determined algebraically. (If the generalized eigenvalue problem is solved numerically, one will find in general six linearly independent linear combinations of S and T). The eigenspace corresponding to eigenvalue zero is at least of dimension 6 (often it is exactly of dimension 6, since the other eigenvalues, which are force constants, are never zero for molecules in their ground state). Thus, T and S correspond to the overall (external) motions: translation and rotation, respectively. They are zero-energy modes because space is homogeneous (force-free) and isotropic (torque-free).

By the definition in this article, the non-zero frequency modes are internal modes, since they are within the orthogonal complement of R_ext. The generalized orthogonalities: applied to the "internal" (non-zero eigenvalue) and "external" (zero-eigenvalue) columns of C are equivalent to the Eckart conditions.