Excitation energies from particle-particle random phase approximation: Davidson algorithm and benchmark studies

Abstract

The particle-particle random phase approximation (pp-RPA) has been used to investigate excitation problems in our recent paper [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. It has been shown to be capable of describing double, Rydberg, and charge transfer excitations, which are challenging for conventional time-dependent density functional theory (TDDFT). However, its performance on larger molecules is unknown as a result of its expensive O(N(6)) scaling. In this article, we derive and implement a Davidson iterative algorithm for the pp-RPA to calculate the lowest few excitations for large systems. The formal scaling is reduced to O(N(4)), which is comparable with the commonly used configuration interaction singles (CIS) and TDDFT methods. With this iterative algorithm, we carried out benchmark tests on molecules that are significantly larger than the molecules in our previous paper with a reasonably large basis set. Despite some self-consistent field convergence problems with ground state calculations of (N - 2)-electron systems, we are able to accurately capture lowest few excitations for systems with converged calculations. Compared to CIS and TDDFT, there is no systematic bias for the pp-RPA with the mean signed error close to zero. The mean absolute error of pp-RPA with B3LYP or PBE references is similar to that of TDDFT, which suggests that the pp-RPA is a comparable method to TDDFT for large molecules. Moreover, excitations with relatively large non-HOMO excitation contributions are also well described in terms of excitation energies, as long as there is also a relatively large HOMO excitation contribution. These findings, in conjunction with the capability of pp-RPA for describing challenging excitations shown earlier, further demonstrate the potential of pp-RPA as a reliable and general method to describe excitations, and to be a good alternative to TDDFT methods.

Noniterative Doubles Corrections to the Random Phase and Higher Random Phase Approximations: Singlet and Triplet Excitation Energies

“…In Refs. 7 and 30 such behavior was observed as well, that is, large underestimation of triplet excitation energies upon inclusion of large amount of exact exchange or by using the RPA method, respectively. In Table also these five states have been excluded from the RPA and RPA(D) results leaving 50 states which seem to behave normally and which results in statistical measures comparable to those of the singlet excited states.…”

“…The same behavior was observed in the TDDFT study from Peach et al upon inclusion of a large amount of exact (Hartree–Fock) exchange in DFT hybrid functionals for the 1 3 B u state of butadiene, the 1 3 B 1 u state of benzoquinone as well as the 1 3 B 2 u state of naphthalene. In a recent study by Yang et al, another seven imaginary triplet RPA excitation energies were reported. In addition to the 10 previously reported unstable RPA states, we observed another six imaginary excitation energies.…”

“…We will show that this problem can be circumvented by combining the apparent stability of the HRPA method with the noniterative doubles correction and we call this method the HRPA(D) method. Other examples of circumventing the issue of triplet instabilities in RPA are to employ the well‐known Tamm–Dancoff approximation (TDA), resulting in the configuration interaction singles (CIS) method or to use the recent particle–particle RPA method …”

Nature of ground and electronic excited states of higher acenes

“…In practice, even though it intrinsically misses non-HOMO excitations, the pp-RPA has been demonstrated to be able to solve such greatly challenging problems as double excitations, Rydberg excitations, charge-transfer excitations, ST gaps for diradical systems, as well as regular HOMO-dominated single excitations (81,83,85,86). According to the benchmark studies (83,85), the Becke-3-Lee-Yang-Parr (B3LYP) functional generally provides an N−2 reference that gives rise to reliable pp-RPA excitation energies and has fewer convergence difficulties. In contrast, the HF functional often yields larger error whereas the Perdew-Burke-Ernzerhof functional frequently encounters N−2 convergence problems.…”

“…It was later further extended to describing excitations (81)(82)(83)(84). For excited states calculation, it starts from a two-electron deficient (N−2) system described with DFT correlation, then recovers a series of neutral states by adding back two explicitly correlated electrons.…”