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

Thus far, the Casimir-Eckart condition (CEC) has been considered to be the general requirement in a rigorous transformation of differential tensor qua ntities like dipole moment derivatives or polarizability derivatives from C artesian coordinates to internal coordinates. It is well known that this co nventional transformation matrix (A-matrix) based on the Wilson method, whi ch always guarantees the CEC, depends on the atomic masses and provides ato mic mass-dependent differential quantities. Since this conventional method may give abnormal isotope effects in dipole derivatives or polarizability d erivatives even in mass-free internal coordinates, we have re-examined the validity of the CEC and propose a new transformation method that does not i mpose this condition. We show that the A-matrix obtained by the method used in internal coordinate molecular dynamics (ICMD) formalisms is mass-free f or mass-free generalized (internal and external) coordinates and does not c ause such an abnormal isotope effect. Since the CEC itself depends on the c hoice of a moving coordinate frame, the transformed internal coordinate dip ole derivatives by the Wilson method also depend on the choice of the movin g coordinate frame, while those by the ICMD method are independent of the c hoice of the moving coordinate frame. As Eckart comments, the Casimir condi tion is not valid in general but its validity depends on the type of intern al coordinates used. All the additional degrees of freedom in defining a mo ving coordinate frame can be satisfactorily removed by relations from the o rthogonality 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. (C) 2001 Elsevier Scie nce BN. All rights reserved.