Linear dependency in the refinement of force constants with the Jacobian method

Abstract

The Jacobian method in the refinement of force constants is studied. The linear dependence problem must be addressed prior to inversion of J T WJ.The approach entails diagonalization of JT WJ, analysis of the components of the eigenvectors associated with zero and small eigenvalues, identification of the linearly dependent parameters, successive elimination of selective parameters, and a repeat of this procedure until linear dependency is removed. The Jacobian matrices are obtained by differencing the frequencies when the parameters are varied and by numerical and analytical evaluation of the derivative of the potential. The unitary transformation, U, used to calculate J = UT(d F/dk)U or J = UT(AF/Ak)U, is obtained from the diagonalization of the Hessian, F, , = d2V/dp,dq,, where p , q = x, y, z are the Cartesian coordinates for atoms m, n = 1,2,3,. . . , at the initial value of ki, i = 1,2,3,. . . . The accuracy of and the ability to evaluate the Jacobian matrix by these methods are discussed. Applications to CH,, H,CO, C2H4, and C2H, are presented. Linearly dependent and ill-conditioned parameters are identified and removed. The procedure is general for any observable quantity.