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

The Jacobian method in the refinement of force constants is studied. Theoretical and experimental frequencies and other observables, ν_s, are matched by minimizing Σ_sW_s(ν – ν)², where s = 1, 2, 3,…, proceeds over all normal modes and isotopes, and W_s are weighting factors. Modification of the theoretical frequencies is accomplished with the Jacobian matrix, J , with elements J_si = ?ν_s/?k_i involving each force constant or associated parameter, k_i, i = 1, 2, 3,…, by Δν = J Δ k . The parameters are adjusted directly with Δ k = ( J ^T WJ )^?1( JW ) Δν, where W is a diagonal matrix which weights the frequencies. 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, 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 = U ^T (? F /? k ) U or J = U ^T (Δ F /Δ k ) U , is obtained from the diagonalization of the Hessian, F_mn = ?²ν/?p_m?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 the ability to evaluate the Jacobian matrix by these methods are discussed. Applications to CH₄, H₂CO, C₂H₄, and C₂H₆ are presented. Linearly dependent and ill-conditioned parameters are identified and removed. The procedure is general for any observable quantity. © 1994 by John Wiley & Sons, Inc.