LINEAR DEPENDENCY IN THE REFINEMENT OF FORCE-CONSTANTS WITH THE JACOBIAN METHOD

The Jacobian method in the refinement of force constants is studied. T heoretical and experimental frequencies and other observables, v(s),, are matched by minimizing Sigma(s)w(s)(v(s)(exp) - v(s)(th))(2), where s = 1, 2, 3,..., proceeds over all normal modes and isotopes, and w(s ) are weighting factors. Modification of the theoretical frequencies i s accomplished with the Jacobian matrix, J, with elements J(si) = part ial derivative v(s)/partial derivative k(i) involving each force const ant or associated parameter, k(i), i = 1, 2, 3,..., by Delta v = J Del ta k. The parameters are adjusted directly with Delta k = (J(T) WJ)(-1 )(JW)Delta v, where W is a diagonal matrix which weights the frequenci es. The linear dependence problem must be addressed prior to inversion of J(T) WJ. The approach entails diagonalization of J(T) WJ, analysis of the components of the eigenvectors associated with zero and small eigenvalues, identification of the linearly dependent parameters, succ essive elimination of selective parameters, and a repeat of this proce dure until linear dependency is removed. The Jacobian matrices are obt ained by differencing the frequencies when the parameters are varied a nd by numerical and analytical evaluation of the derivative of the pot ential. The unitary transformation, U, used to calculate J = U-T(parti al derivative F/partial derivative k)U or J = U-T(Delta F/Delta k)U, i s obtained from the diagonalization of the Hessian, F-mn = partial der ivative(2)V/partial derivative p(m) partial derivative q(n), where p, q = x, y, z are the Cartesian coordinates for atoms m, n = 1, 2, 3,... , at the initial value of k(i), i = 1, 2, 3,.... The accuracy of and t he ability to evaluate the Jacobian matrix by these methods are discus sed. Applications to CH4, H2CO, C2H(4), and C2H6, are presented. Linea rly dependent and ill-conditioned parameters are identified and remove d. The procedure is general for any observable quantity. (C) 1994 by J ohn Wiley & Sons, Inc.