Linear expansion of the eigenvalues of a Hermitian matrix and its application to the analysis of the electronic spectra of 3 d ions in crystals

It is shown that the eigenvalues E_i of a Hermitian matrix H with matrix elements H_ij = Σ_kA^k_ija_k, where A^k_ij are known numbers and a_k a set of parameters, can be exactly expanded as E _i = Σ _k( {∂E _i}/{∂a _k})a _k. This property is applied to the analysis of the optical spectra of transition metal ions in crystals proposed by L. Pueyo, M. Bermejo, and J. W. Richardson ( J. Solid State Chem.31, 217, 1980), and it is shown that this method represents the best fit of the Hamiltonian eigenvalues to the observed (or calculated) spectrum. Further advantages of using this property, in connection with the spectral analysis, are the minimization of the errors associated with the numerical approximations and a reduction in computer time. In the molecular orbital calculation of the optical or uv spectra of these systems, this linear expansion of the eigenvalues give a detailed interpretation of the improvements produced by refined calculations, such as those including configuration interaction. In particular, the changes in one-electron energy and in open-shell repulsion interactions associated with the refinement can be clearly and easily formulated. As examples, the computed spectra of CrF ^4-₆ and CrF ^3-₆ are discussed.