Derivatives of the energy levels of a molecular system with respect to parameters of the Hamiltonian, and their application to magnetic susceptibility calculations and least-squares fitting

Abstract

Formulae are given for the first, second, and third partial derivatives, with respect to the parameters a_i, of the eigenvalues of a matrix ℋ=Σa_iℋ_i(where ℋ_i are Hermitian or real symmetric matrices), in terms of the eigenvalues and eigenvectors of ℋ. The matrix ℋ may also include bilinear terms of the type a_ia_jℋ_ij. If ℋ is the Hamiltonian matrix for a molecular system in a magnetic field, so that one of the a_i is the applied field strength and the others represent features of the model such as crystal-field, spin–orbit coupling, and exchange parameters, these results can be used to obtain expressions for the paramagnetic susceptibility which are exact even at very low temperatures, within the limitations of the model and the accuracy with which the eigenvalues and eigenvectors of ℋ are known. The formulae can also be used to set up least-squares normal equations determining the values of the parameters that give the best fit to experimental susceptibility data (or other observed properties of the system), and to solve those equations iteratively. The formulae are likely to be especially useful whenever it is impossible, because of the complexity of the secular equation, to write explicit expressions for the energy levels and to calculate their derivatives by analytic differentiation, but the generality of the method may make it useful even when analytic differentiation is possible. Examples of its use in the least-squares fitting of magnetic susceptibility data are given.