A Natural Orbital Diagnostic for Multiconfigurational Character in Correlated Wave Functions

It is well known that not all chemical species can be well represented by a single configuration wave function. Equivalently, an adequate qualitative description of such species requires more than one simple Lewis structure. Since the use of a wave function that is based on a single configuration description can lead to a poor description of the properties of the system one is studying, it is of interest to have one or more diagnostics that are general enough to apply to any level of theory as predictive tools to flag potentially suspect wave functions. Of particular import in this regard are two landmark contributions by Lowdin. The first of these was the demonstration that for a given atomic basis, the exact wave function may be represented by a complete configuration interaction ~CI! within that basis. The second important contribution was the definition of the natural orbitals $f i% as the eigenvectors of the first order density matrix r, and the demonstration that these natural orbitals provide a unique description in the limit of the exact wave function. The eigenvalues of the first-order density matrix are called the natural orbital occupation numbers l i ~NOON!. Because of the Pauli exclusion principal, no orbital may have an occupation number greater than 2 ~one electron for each spin!. Since the first-order density matrix is positive definite, there are no negative NOONs. In a simple closed shell, single configuration wave function, the occupied orbitals all have l i52.0, while the unoccupied ~virtual! orbitals have 0.0 occupation numbers. Several diagnostics have been proposed to aid in determining when a multiconfigurational description of the wave function might be important. Pulay has made the very important observation that in a multiconfigurational wave function, if a putative virtual orbital has a NOON of 0.1 or greater, one should not rely on a single configuration description of the species of interest. Of course, this determination requires construction of a multiconfigurational wave function. For neutral molecule restricted Hartree–Fock ~RHF! wave functions, a negative virtual orbital energy is frequently taken to be an indication that the single configuration wave function does not provide an adequate descrip-