The Casimir–Eckart condition and the transformation of dipole moment derivatives revisited

Abstract

Thus far, the Casimir–Eckart condition (CEC) has been considered to be the general requirement in a rigorous transformation of differential tensor quantities like dipole moment derivatives or polarizability derivatives from Cartesian coordinates to internal coordinates. It is well known that this conventional transformation matrix ( A-matrix) based on the Wilson method, which always guarantees the CEC, depends on the atomic masses and provides atomic mass-dependent differential quantities. Since this conventional method may give abnormal isotope effects in dipole derivatives or polarizability derivatives even in mass-free internal coordinates, we have re-examined the validity of the CEC and propose a new transformation method that does not impose this condition. We show that the A-matrix obtained by the method used in internal coordinate molecular dynamics (ICMD) formalisms is mass-free for mass-free generalized (internal and external) coordinates and does not cause such an abnormal isotope effect. Since the CEC itself depends on the choice of a moving coordinate frame, the transformed internal coordinate dipole derivatives by the Wilson method also depend on the choice of the moving coordinate frame, while those by the ICMD method are independent of the choice of the moving coordinate frame. As Eckart comments, the Casimir condition is not valid in general but its validity depends on the type of internal coordinates used. All the additional degrees of freedom in defining a moving coordinate frame can be satisfactorily removed by relations from the orthogonality of internal coordinates to external translations and rotations. Moreover, an analogy with normal coordinates for internal modes provides the very equivalent to Eckart's original derivation.