Linear Scaling Calculations of Excitation Energies with Active-Space Particle–Particle Random-Phase Approximation

Abstract

We developed an efficient active-space particle− particle random-phase approximation (ppRPA) approach to calculate accurate charge-neutral excitation energies of molecular systems. The active-space ppRPA approach constrains both indexes in particle and hole pairs in the ppRPA matrix, which only selects frontier orbitals with dominant contributions to low-lying excitation energies. It employs the truncation in both orbital indexes in the particle− particle and the hole−hole spaces. The resulting matrix, whose eigenvalues are excitation energies, has a dimension that is independent of the size of the systems. The computational effort for the excitation energy calculation, therefore, scales linearly with system size and is negligible compared with the ground-state calculation of the (N − 2)-electron system, where N is the electron number of the molecule. With the active space consisting of 30 occupied and 30 virtual orbitals, the active-space ppRPA approach predicts the excitation energies of valence, charge-transfer, Rydberg, double, and diradical excitations with the mean absolute errors (MAEs) smaller than 0.03 eV compared with the full-space ppRPA results. As a side product, we also applied the active-space ppRPA approach in the renormalized singles (RS) T-matrix approach. Combining the non-interacting pair approximation that approximates the contribution to the self-energy outside the active space, the active-space G RS T RS @PBE approach predicts accurate absolute and relative core-level binding energies with the MAEs around 1.58 and 0.3 eV, respectively. The developed linear scaling calculation of excitation energies is promising for applications to large and complex systems.

Accurate Excitation Energies of Point Defects from Fast Particle–Particle Random Phase Approximation Calculations

“…It has been shown that using an active space of only 30 occupied and 30 virtual orbitals, active-space ppRPA achieves fast convergence to within 0.05 eV compared to full ppRPA for molecular excitations of different characters, including charge transfer, Rydberg, double, and valence excitations as well as diradicals. 54 As a result, the ppRPA calculations for molecular excitations become linear scaling and are more efficient than the ground-state SCF calculations of the same molecules.…”

“…Thus, we choose to use the active space composed of 200 occupied and 200 virtual orbitals in all ppRPA calculations in this work. Compared with the previous work of using the active-space ppRPA approach for molecular systems, 54 the number of orbitals needed in the Figure 1. Vertical excitation energies of NV − in diamond obtained from ppRPA@PBE with respect to the number of orbitals in the active space.…”

“…Therefore, the excitation energy of the N -electron system can be obtained in two different manners: (a) the particle–particle channel with the difference between two-electron addition energies of the ( N – 2)-electron system and (b) the hole–hole channel with the difference between two-electron addition energies of the ( N + 2)-electron system, ,, where N is the number of electrons. It has been shown that ppRPA predicts accurate excited-state properties of molecular systems, including valence, double, Rydberg, and charge transfer excitation energies, ,− analytic gradients, conical intersections, oscillator strengths, and spin-state energetics . Recently, ppRPA with the Tamm–Dancoff approximation (TDA) has been applied in the multireference DFT approach to describe dissociation breakings and to predict excitation energies. , ppRPA is also used in Green’s function formalism, where the self-energy in the T matrix approximation is formulated with the ppRPA eigenvalues and eigenvectors to calculate quasiparticle energies. ,− …”

“…As introduced in ref , the ppRPA matrix can be constructed in an active space that constrains indices of both particle and hole pairs. This means that for the singlet ppRPA matrix in eqs –, the indices are constrained as a < b ≤ N vir , act .25em and .25em c < d ≤ N vir , act i < j ≤ N occ , act .25em and .25em k < l ≤ N occ , act where N occ,act and N vir,act are the numbers of occupied and virtual orbitals in the active space, respectively.…”

“…For the triplet ppRPA matrix in eqs and , the indices are constrained in the same way. As shown in ref , the active space described above includes only the particle and hole pairs with large contributions to low-lying excitation energies, which greatly reduces the computational cost of the ppRPA calculations. The scaling of active-space ppRPA is scriptO ( N act 4 ) using the Davidson algorithm, , where N act is a small number of active-space orbitals.…”