On finite basis set implementation of the exchange-only optimized effective potential method

In this paper, we analyze a structure of the basis set optimized effective potential (OEP) equations from the Fredholm alternative point of view and present one of possible numerical schemes to solve the OEP equation in a stable manner. The solution is constructed as a sum of a unique solution on the subspace of eigenfunctions of the response matrix with non-zero eigenvalues and a non-unique solution on a counterpart subspace with singular eigenvalues. Non-uniqueness of a solution is exploited to obtain a local effective potential that satisfies the condition for the highest occupied molecular orbital (HOMO) without restricting the variational freedom of the optimization procedure. Unlike the existing methods we implement the HOMO condition using the functions of the null-subspace. Numerical results for the total and orbital energies based on the proposed scheme are close to the corresponding literature data.