Pseudopotential Approaches to Localized Orbitals for Polyatomic Systems

We present a method for constructing optimized pseudopotentials for general polyatomic systems which yield the “smoothest” orbitals localized in particular regions of space. (“Smoothest” orbitals are those for which the expectation value of the kinetic energy operator is a minimum.) We specifically consider the general case in which one can have several occupied orbitals localized in a particular region, rather than a single orbital. Recently, Anderson presented a pseudopotential formalism for constructing localized orbitals for a single isolated band in a polyatomic system. We have extended his formalism to the general multiorbital case, and show that for a specific choice of pseudopotential the Anderson equation yields optimal orbitals. We also show that the Hückel form of the Anderson pseudopotential equation yields identical eigenvalues (neglecting terms of order $S^2$ ) to the optimal form, and orbitals differing from optimal orbitals by terms linear in $S$ . The correction terms to the Hückel orbitals are obtained by first‐order perturbation theory. Finally we examine the Adams–Gilbert equation, which is valid only for systems representable by a single Slater determinant, and indicate how one may solve this equation for the localized orbitals using perturbation theory with Anderson orbitals as the basis set.