The Fukui matrix: a simple approach to the analysis of the Fukui function and its positive character

The Fukui matrix is introduced as the derivative of the one-electron reduced density matrix with respect to a change in the number of electrons under constant external potential. The Fukui matrix extends the Fukui function concept: the diagonal of the Fukui matrix is the Fukui function. Diagonalizing the Fukui matrix gives a set of eigenvectors, the Fukui orbitals, and accompanying eigenvalues. At the level of theory used, there is always one dominant eigenvector, with an eigenvalue equal to 1. The remaining eigenvalues are either zero or come in pairs with eigenvalues of the same magnitude but opposite sign. Analysis of the frontier molecular orbital coefficient in the eigenvector with eigenvalue 1 gives information on the quality of the frontier molecular orbital picture. The occurrence of negative Fukui functions can be easily interpreted in terms of the nodal character of the dominant eigenvector versus the characteristics of the remaining eigenvectors and eigenvalues.