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

Abstract

The Jacobian method in the refinement of force constants is studied. Theoretical and experimental frequencies and other observables, νs, are matched by minimizing ΣsWs(ν – ν)2, where s = 1, 2, 3,…, proceeds over all normal modes and isotopes, and Ws are weighting factors. Modification of the theoretical frequencies is accomplished with the Jacobian matrix, J, with elements Jsi = ∂νs/∂ki involving each force constant or associated parameter, ki, i = 1, 2, 3,…, by Δν = JΔk. The parameters are adjusted directly with Δk = (JTWJ)−1(JW) Δν, where W is a diagonal matrix which weights the frequencies. The linear dependence problem must be addressed prior to inversion of JTWJ. The approach entails diagonalization of JTWJ, 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 (∂F/∂k)U or J = UT (ΔF/Δk)U, is obtained from the diagonalization of the Hessian, Fmn = ∂2ν/∂pm∂qn, 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 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.