Low scaling EOM-CCSD and EOM-MBPT(2) method with natural transition orbitals

A low-scaling method is presented for the equation-of-motion coupled-cluster theory with single and double (EOM-CCSD) excitations and its second-order many-body perturbation theory [EOM-MBPT(2)] approximations. For a simple description of an excited state, the particular orbitals, ϕĨ and ϕÃ, are selected from the natural transition orbitals (NTOs, ϕ), where Ĩ and à refer to NTO occupied and virtual orbital indices. They are chosen based on the largest eigenvalues of the transition density matrix. We expect the ϕĨ and ϕà pair to be dominant in representing excited states in EOM calculations. Therefore, the double excitation vector, R2 which scale as ∼O2V2, can be modified to keep only a few dominant excitations. Our work indicates that the most important contributions of the R2 vector define smaller subspaces that scale as ∼OV, ∼O2V, and ∼OV2, where O and V refer to the occupied and virtual orbitals in the NTO basis. Thus, the scaling for the EOM part becomes ∼M5. The energy changes due to R2 truncation are small (the mean average deviation from untruncated EOM-CCSD is ∼0.03 eV). We show that this approach works relatively well with various types of NTOs, ranging from configuration singles to time-dependent density functional theory making ∼M5 scaling calculations possible with the use of MBPT(2) as the reference state.