The determination of Wilson–Decius F matrix elements from Cartesian force constants

Abstract

This work presents a new and consistent derivation of a completely general algorithm for the calculation of the F matrix of the Wilson–Decius FG method of vibrational analysis from a Cartesian force constant matrix, FCART. The latter is routinely determined using a computer program such as Gaussian03 once a molecular geometry optimization has successfully converged. For a molecule containing NOAT atoms, the total number of degrees of freedom, NA, is equal to 3*NOAT and the number of normal modes of vibration, NVIB, is equal to (NA − 5) or (NA − 6) for, respectively, a linear or a non-linear molecule. If one utilizes NOB internal coordinates to describe the normal vibrations then NOB must be greater than or equal to NVIB. In the former case, NRED (=NOB − NVIB) redundancies, having frequency values of zero, will be included in the solution of the problem. The algorithm utilizes two newly defined matrices, BIN and GIN, which are determined by the following two relationships: \( \begin{aligned} {\mathbf{BIN}} = {\mathbf{M}}^{ - 1} {\mathbf{B}}^{{\mathbf{t}}} {\mathbf{GIN}}, {\mathbf{GIN}} = \left( {{\mathbf{D}}_{{{\mathbf{NVIB}}}} } \right)\left( {\Upgamma_{{{\mathbf{NVIB}}}} } \right)^{ - 1} \left( {{\mathbf{D}}_{{{\mathbf{NVIB}}}} } \right)^{{\mathbf{t}}} \\ \end{aligned} \) where M−1 is a (NA × NA) diagonal matrix containing the inverses of the atomic masses three times each (to account for motions in the x, y and z directions), B is the well known (NOB × NA) rectangular matrix of the Wilson–Decius method which defines the transformation from Cartesian to internal coordinates and the superscript “t” indicates that the transpose of the matrix is to be taken. The DNVIB and (ΓNVIB)−1 matrices are determined from the diagonalization of the (NOB × NOB) Wilson–Decius G matrix: \( {\mathbf{GD}}_{{{\mathbf{NOB}}}} = {\mathbf{D}}_{{{\mathbf{NOB}}}} \Upgamma_{{{\mathbf{NOB}}}} \) where DNOB and ΓNOB are the (NOB × NOB) eigenvector and eigenvalue matrices, respectively, of G in which the eigenvalues (and their corresponding eigenvectors) have been reorganized from highest to lowest (i.e., zero) magnitude. By this process the eigenvector matrix, DNOB, is then partitioned into two sections representing the symmetry coordinates (the first NVIB columns) and the redundant coordinates (the last NRED columns). The DNVIB matrix is then defined as the (NOB × NVIB) rectangular portion of DNOB which corresponds to the symmetry coordinates (that is, the first NVIB columns of DNOB) and (ΓNVIB)−1 is a (NVIB × NVIB) diagonal matrix composed of the inverses of the non-zero eigenvalues of the G matrix arranged from lowest to highest magnitude. With these matrices at hand, it is then possible to calculate the Wilson–Decius F matrix with the following transformation: \( {\mathbf{F}} = {\mathbf{BIN}}^{{\mathbf{t}}} {\mathbf{F}}_{{{\mathbf{CART}}}} {\mathbf{BIN}} \). With properly determined F and G matrices, it is then possible to perform a complete normal coordinate analysis of a molecule whose optimized geometry and Cartesian force constant matrix were originally determined through ab initio or density-functional calculations. The method is completely general and allows for the choice between a set of non-symmetrized internal coordinates, a set of symmetry adapted linear combinations of internal coordinates or a set of symmetry coordinates. The procedure is illustrated, via the Supplementary Material 1, with a number of practical examples. A discussion of the criteria necessary for a proper choice of a set of internal coordinates which may be used for a vibrational analysis is also included in this paper.