Combining Renormalized Singles GW Methods with the Bethe–Salpeter Equation for Accurate Neutral Excitation Energies

Abstract

We apply the renormalized singles (RS) Green’s function in the Bethe–Salpeter equation (BSE)/GW approach to predict accurate neutral excitation energies of molecular systems. The BSE calculations are performed on top of the G_RSW_RS method, which uses the RS Green’s function also for the computation of the screened Coulomb interaction W. We show that the BSE/G_RSW_RS approach significantly outperforms BSE/G₀W₀ for predicting excitation energies of valence, Rydberg, and charge-transfer (CT) excitations by benchmarking the Truhlar–Gagliardi set, Stein CT set, and an atomic Rydberg test set. For the Truhlar–Gagliardi test set, BSE/G_RSW_RS provides comparable accuracy to time-dependent density functional theory (TDDFT) and is slightly better than BSE starting from eigenvalue self-consistent GW (evGW). For the Stein CT test set, BSE/G_RSW_RS significantly outperforms BSE/G₀W₀ and TDDFT with the accuracy comparable to BSE/evGW. We also show that BSE/G_RSW_RS predicts Rydberg excitation energies of atomic systems well. Besides the excellent accuracy, BSE/G_RSW_RS largely eliminates the dependence on the choice of the density functional approximation. This work demonstrates that the BSE/G_RSW_RS approach is accurate and efficient for predicting excitation energies for a broad range of systems, which expands the applicability of the BSE/GW approach.

Graphical Abstract

1. INTRODUCTION

The accurate prediction of neutral excitation energies from first-principle calculations plays a critical role in guiding the understanding and new developments in chemistry, biochemistry, and material science, such as drug delivery^1,2 and design of an organic sensitizer.^3,4 In the past decades, many theoretical approaches have been developed to compute accurate excitation energies. Time-dependent density functional theory^5–7 (TDDFT) ranks among the most popular approaches. Because of the good accuracy and the affordable computational cost, TDDFT has been widely implemented in modern quantum chemistry packages to calculate energies, structures, and other properties of excited states for molecular and periodic systems.^8–10 However, TDDFT has several well-known shortcomings. For example, TDDFT with commonly used density functional approximations (DFAs) poorly predicts excitation energies of Rydberg and charge-transfer (CT) excitations.^11,12 The failure of TDDFT for these excitations must be attributed to the incorrect description of the long-range behavior using conventional DFAs.^11,12 Attempts to address this issue include the usage of range-separated or Coulomb-attenuated functionals^13–16 and tuning the fraction of the Hartree–Fock (HF)^17,18 exchange in DFAs.¹⁹ In addition, the accuracy of TDDFT strongly depends on the exchange–correlation (XC) kernel, which is the second derivative of the XC energy from the given DFA with respect to the electron density.^10,20 The difference in the calculated excitation energies obtained from TDDFT with different DFAs can exceed 1.0 eV for valence excitations and even exceed 2.0 eV for Rydberg excitations.^10,21 Recent developments to improve the accuracy of TDDFT include multireference DFT,^22,23 orbital-optimized DFT,²⁴ and mixed-reference spin-flip TDDFT.²⁵

In the last years, the Bethe–Salpeter equation^26–28 (BSE) formalism in the Green’s function many-body perturbation theory^29,30 has become increasingly popular for computing the optical excitations of molecules³¹ and has been most recently also applied to K-edge transitions.³² BSE is commonly performed on top of a GW calculation^30,33,34 using the GW quasiparticle (QP) energies as the input. We denote this approach as BSE/GW. Recently, BSE in combination with the T-matrix approximation for predicting neutral excitation energies has also been reported.³⁵ In the BSE/GW approach, the electron–hole interaction is described by the screened Coulomb interaction W instead of the bare Coulomb interaction. The GW QP energies substantially improve upon the Kohn–Sham (KS) orbital energies for predicting fundamental HOMO–LUMO gaps and ionization potentials in general, which are the critical quantities in BSE to calculate excitation energies. BSE/GW has been successfully applied to calculate excitation energies for systems of different sizes including molecules, solids, and low-dimensional materials.^32,36–49

However, the BSE/GW approach still suffers from several problems. First, although BSE has a computationally favorable scaling of O(N⁴) with respect to the system size N, which is the same as in TDDFT, the preceding GW calculation is computationally demanding. In the fully analytical treatment of the GW self-energy, the diagonalization of the random phase approximation (RPA) equation scales as O(N⁶) and the self-energy evaluation scales as O(N⁵).^34,50 Thus, the computationally expensive GW step dominates the computational cost in a BSE/GW calculation. Many techniques have been developed to reduce the computational cost of GW calculations, for example, the contour deformation technique⁵¹ or analytic continuation of the self-energy,^52,53 which scales O(N⁴) for valence states. The computational cost is further reduced in cubic scaling GW implementations, which recently emerged for localized basis set codes^54–57 enabling GW calculations for large systems of more than thousand atoms.

Second, the accuracy of the commonly used BSE/G₀W₀ approach has an undesirable dependence on the choice of the DFA,^58,59 which is due the perturbative nature of the G₀W₀ scheme. In BSE/G₀W₀, the dependence on the DFA starting point is in the range of 0.5 eV for predicting valence excitation energies and can even exceed 1.0 eV for predicting CT excitation energies.⁵⁹ It has been shown that BSE/G₀W₀ based on range-separated functionals and tuned hybrid functionals provides more accurate excitation energies^32,60 than BSE/G₀W₀ based on GGA functionals. This dependence can be largely reduced by introducing self-consistency into the GW calculations, such as the eigenvalue-self-consistent GW (evGW) method, where the eigenvalues are iterated in G and W, the quasiparticle-self-consistent GW (qsGW) scheme,c^61,62 or the fully self-consistent GW (scGW) approach.^63,64 It has been shown that the BSE/evGW approach provides an accuracy comparable to TDDFT for predicting valence excitations and significantly outperforms BSE/G₀W₀ and TDDFT for predicting CT and Rydberg excitations.^38,59,65 The DFA dependence in BSE/evGW is largely washed out.^38,59,65 In practice, evGW calculations can converge within a few steps by using the linear mixing method.⁶² However, evGW might have convergence difficulties for systems with multiple solutions.^66,67 In addition, the extra computational cost in evGW is expensive for large systems.

To reduce the computational cost, efforts were recently made to approximate the GW QP energies by improved orbital energies from DFT and to use the latter as the BSE input. For example, we recently employed the localized orbital scaling correction⁶⁸ (LOSC), which minimizes the delocalization error in DFAs to predict orbital energies. We showed that the BSE/LOSC approach provides good accuracy for predicting excitation energies of different systems⁵⁹ and yields significantly better results than BSE/G₀W₀. The BSE/LOSC approach scales as O(N⁴) complexity. Another computationally cheaper BSE approach is based on Koopmans-compliant (KC) functionals, where the orbital energies are derived from KC functionals and the screened interaction is obtained via direct minimization on top of a maximally localized Wannier function basis.⁶⁹ The KC-based BSE yields similar accuracy as the BSE/G₀W₀ method.⁶⁹

In this work, we applied the recently developed renormalized singles (RS) Green’s function^70,71 in BSE/GW to compute accurate excitation energies with affordable computational cost. The idea of the RS Green’s function approach is to compute the HF self-energy with the KS orbitals from DFT^72–74 instead of the HF orbitals. Because of Brillouin’s theorem,¹⁸ there is no singles contribution to the HF self-energy in the perturbation. However, the HF Green’s function is not a good starting point for G₀W₀ calculations.^70,75 Therefore, the HF Hamiltonian is constructed with KS orbitals and diagonalized separately in the occupied and virtual subspaces. This renormalization procedure absorbs all singles contributions into the self-energy to reduce the dependence on the choice of the DFA. The resulting RS Green’s function is constructed with RS orbital energies from the renormalization process, which has the same form as the KS Green’s function. From the viewpoint of the RPA, higher order contributions are also included in the infinite summation by this renormalization process. Compared with the self-consistent GW methods that solve Hedin’s equations iteratively to eliminate the starting point dependence, the one-shot RS process captures all contributions of the single excitations while hardly increasing the computational compared to G₀W₀. The RS Green’s function has been used in the GW methods and T-matrix methods to predict accurate QP energies for valence and core states.^59,70,76 The concept of RS has also been used in the multireference DFT approach²³ to describe the static correlation in strongly correlated systems. The RS Green’s function shares similar thinking as the RS excitation in the RPA calculation for correlation energies.^77–79

The RS Green’s function has been applied in the GW approximation in two flavors. The first one is the G_RSW₀⁷⁰ method, which uses the RS Green’s function as a new starting point and calculates the screened interaction with the KS Green’s function. We found that the G_RSW₀ method provides a considerable improvement over G₀W₀ for predicting valence ionization potentials and electron affinities with a reduced starting point dependence⁷⁰ but fails to restore the correct physics for deep core excitations.⁷¹ The second one is the G_RSW_RS method,⁷¹ which uses the RS Green’s function as a new starting point and calculates the screened interaction with the RS Green’s function. We found that G_RSW_RS also yields an improvement over G₀W₀,⁸⁰ that is, it opens the fundamental gaps compared to G₀W₀ similar to evGW; see also Section 1 in the Supporting Information. Because of the nonlinear nature of the QP equation, multiple solutions can be found due to the unphysical discontinuities and intruder states,^66,67,81 especially for core states. G_RSW_RS properly separates the main core QP state from the satellites and provides a solution to the QP equation corresponding to the desired core state,⁷¹ which is not the case for G₀W₀ with GGA or hybrid functionals with a low amount of exact exchange. Even though the absolute core-level energies are overestimated by several electronvolts, the relative shifts are predicted with a reasonable accuracy.⁷¹ For valence as well as core states, the starting point dependence is significantly reduced compared to G₀W₀.

In the present work, we benchmark both RS flavors, BSE/G_RSW₀ and BSE/G_RSW_RS, for the prediction of neutral excitations of molecular systems, including valence, CT, and Rydberg excitations.

2. THEORY

2.1. RS Green’s Function.

The goal of the RS Green’s function approach is to improve the accuracy of perturbative GW methods by including exactly the exchange self-energy, which is a one-electron contribution, and reduce the starting point dependence of the orbital energies on the DFA. The general idea is to construct the HF self-energy from KS orbitals, followed by a separate diagonalization in the occupied orbital subspace and the virtual orbital subspace.⁷⁰ Effectively, the RS approach treats the one-body exchange self-energy exactly, or nonperturbatively, by diagonalization, unlike the commonly used G₀W₀ approach, in which the one-body exchange is included perturbatively along with the many-body correlation self-energy. The RS Green’s function G_RS is defined as the solution of the two projected equations in the occupied orbital subspace and the virtual orbital subspace⁷⁰

$$P\left(G_{\text{RS}}^{-1}\right)P=P\left(G_0^{-1}\right)P+P\left({\mathrm{\Sigma }}_{\text{Hx}}\left[G_0\right]-v_{\text{Hxc}}\right)P$$ (1)

and

$$Q\left(G_{\text{RS}}^{-1}\right)Q=Q\left(G_0^{-1}\right)Q+Q\left({\mathrm{\Sigma }}_{\text{Hx}}\left[G_0\right]-v_{\text{Hxc}}\right)Q$$ (2)

where $P={\sum }_i^{occ}|{\psi }_i\rangle \langle {\psi }_i|$ is the projection into the occupied orbital space and Q = I − P is the projection into the virtual orbital space, ψ is the KS orbital, Σ_Hx is the Hartree-exchange self-energy, and v_Hxc is the Hartree-exchange–correlation potential. Σ_Hx[G₀] means that the HF self-energy (Hartree and exchange contribution) is constructed from the KS density matrix. Equivalently, the RS Green’s function is obtained by using the DFA density matrix in the HF Hamiltonian, namely, H_HF[G₀], and solving two projected HF equations in the occupied and virtual subspaces⁷⁰

$$P\left(H_{\text{HF}}\left[G_0\right]\right)P|{\psi }_i^{\text{RS}}\rangle ={ϵ}_i^{\text{RS}}P|{\psi }_i^{\text{RS}}\rangle$$ (3)

and

$$Q\left(H_{\text{HF}}\left[G_0\right]\right)Q|{\psi }_a^{\text{RS}}\rangle ={ϵ}_a^{\text{RS}}Q|{\psi }_a^{\text{RS}}\rangle$$ (4)

where ϵ^RS is the RS orbital energy and ψ^RS is the RS orbital. Here, we use i, j, k, l for occupied orbitals, a, b, c, d for virtual orbitals, p, q, r, s for general orbitals. The subspace diagonalization of the HF Hamiltonian is performed only once. The resulting RS Green’s function is diagonal in the occupied and virtual subspaces⁷⁰

$${\left[G_{\text{RS}}(\omega )\right]}_{pq}={\delta }_{pq}\frac{1}{\omega -{ϵ}_p^{\text{RS}}+\text{i}\eta \mathrm{sgn}\left({ϵ}_p^{\text{RS}}-\mu \right)}$$ (5)

Here, μ is the chemical potential and η is the broadening parameter. As shown in eq 5, the RS Green’s function has a similar form as the KS Green’s function except that the KS orbital energies in the denominator are replaced by the RS orbital energies. Therefore, the RS Green’s function can be directly implemented in existing GW codes.

2.2. G_RSW₀ and G_RSW_RS.

The RS Green’s function is used in two GW flavors: G_RSW₀⁷⁰ and G_RSW_RS.⁷¹ In both methods, KS orbitals instead of RS orbitals are used for simplicity^70,71 because using the RS orbitals does not change the results significantly. Therefore, the exchange part of the self-energy in G_RSW₀ and G_RSW_RS is the same as G₀W₀. In G_RSW₀,⁷⁰ the RS Green’s function is used as the starting point and the screened interaction is calculated with the KS Green’s function, which means the KS Green’s function is used in the RPA calculation. The correlation part of the self-energy in G_RSW₀ is⁷⁰

$$\begin{gathered} {[{\mathrm{\Sigma }}_{\text{c}}^{G_{\text{RS}}{W}_0}(\omega )]}_{pp} \\ =\sum _m\sum _q\frac{{|\left(pq\mid {\rho }_m^{\text{KS}}\right)|}^2}{\omega -{ϵ}_q^{\text{RS}}-({\mathrm{\Omega }}_m^{\text{KS}}-\text{i}\tilde{\eta })\mathrm{sgn}\left({ϵ}_q^{\text{RS}}-\mu \right)} \end{gathered}$$ (6)

where ${\mathrm{\rho }}_m^{\text{KS}}$ and ${\mathrm{\Omega }}_m^{\text{KS}}$ are the transition density and the excitation energy from RPA calculated with the KS Green’s function, m is the index for the RPA excitation, and $\tilde{\eta }=3\eta$.⁵⁰

With the self-energy in eq 6, the QP equation for G_RSW₀ is⁷⁰

$${ϵ}_p^{\text{QP}}={ϵ}_p^0+\langle p|{\sum }_{\text{xc}}^{{\text{G}}_{\text{RS}}{W}_0}({ϵ}_p^{\text{QP}})-v_{\text{xc}}|p\rangle$$ (7)

where ${ϵ}_p^0$ is the KS orbital energy. In eq 7, the QP energy ${ϵ}_p^{\text{QP}}$ appears in both sides, which means eq 7 needs to be solved iteratively. To reduce the computational cost, eq 7 can be linearized^30,70 as

$${ϵ}_p^{\text{QP}}={ϵ}_p^0+Z_p^{G_{\text{RS}}{W}_0}\langle p|{\mathrm{\Sigma }}_{\text{xc}}^{G_{\text{RS}}{W}_0}({ϵ}_p^{\text{RS}})-v_{\text{xc}}|p\rangle$$ (8)

with the factor $Z_p^{G_{\text{RS}}{W}_0}={\left(1-{\frac{\partial {[{\mathrm{\Sigma }}_{\text{c}}^{{\text{G}}_{\text{RS}}{\mathrm{W}}_0}(\omega )]}_{pp}}{\partial \omega }|}_{\omega ={ϵ}_p^{\text{RS}}}\right)}^{-1}$.

The G_RSW_RS method⁷¹ uses the RS Green’s function as a new starting point and calculates the screened interaction with the RS Green’s function, which means that the RS Green’s function is used in the RPA calculation. The correlation part of the self-energy in G_RSW_RS⁷¹

$$\begin{gathered} {[{\mathrm{\Sigma }}_{\text{c}}^{G_{\text{RS}}{W}_{\text{RS}}}(\omega )]}_{\text{pp}} \\ =\sum _m\sum _q\frac{{|(pq\mid {\rho }_m^{\text{RS}})|}^2}{\omega -{ϵ}_q^{\text{RS}}-({\mathrm{\Omega }}_{\text{m}}^{\text{RS}}-\text{i}\tilde{\eta })\mathrm{sgn}({ϵ}_q^{\text{RS}}-\mu )} \end{gathered}$$ (9)

where ${\rho }_m^{\text{RS}}$ and ${\mathrm{\Omega }}_m^{\text{RS}}$ are the transition density and the excitation energy from RPA calculated with the RS Green’s function. Therefore, the QP equation for G_RSW_RS is⁷¹

$${ϵ}_p^{\text{QP}}={ϵ}_p^0+\langle p|{\mathrm{\Sigma }}_{\text{xc}}^{G_{\text{RS}}{W}_{\text{RS}}}({ϵ}_p^{\text{QP}})-v_{\text{xc}}|p\rangle$$ (10)

Equation 10 can also be linearized as

$${ϵ}_p^{\text{QP}}={ϵ}_p^0+Z_p^{G_{\text{RS}}{W}_{\text{RS}}}\langle p|{\mathrm{\Sigma }}_{\text{xc}}^{G_{\text{RS}}}{W}_{\text{RS}}({ϵ}_p^{\text{RS}})-v_{\text{xc}}|p\rangle$$ (11)

with the factor $Z_p^{G_{\text{RS}}{W}_{\text{RS}}}={\left(1-{\frac{\partial {[{\mathrm{\Sigma }}_{\text{c}}^{{\text{G}}_{\text{RS}}{W}_{\text{RS}}}(\omega )]}_{pp}}{\partial \omega }|}_{\omega ={ϵ}_p^{\text{RS}}}\right)}^{-1}$.

As shown in ref 34, the linearized QP equation gives small errors for valence QP energy calculations, which are important in BSE. In Section 2 in the Supporting Information, we show that using linearized QP equations defined in eqs 8 and 11 gives small differences around 0.01 eV compared to eqs 7 and 10 for the type of excitations studied here. Therefore, the linearized QP equations are solved to reduce the computational cost in the present work.

2.3. BSE/G_RSW₀ and BSE/G_RSW_RS Approaches.

The QP energies obtained from G_RSW_RS are used in BSE to calculate the excitation energies. With the static approximation for the screened interaction that treats the frequency as zero,^31,82,83 the working equation of BSE is a generalized eigenvalue equation,^31,82,83 which is similar to the Casida equation in TDDFT^6,7

$$\left[\begin{array}{ll}\mathbf{A}\hfill & \mathbf{B}\hfill \\ {\mathbf{B}}^{*}\hfill & {\mathbf{A}}^{*}\hfill \end{array}\right]\left[\begin{array}{l}\mathbf{X}\hfill \\ \mathbf{Y}\hfill \end{array}\right]=\mathrm{\Omega }\left[\begin{array}{ll}\mathbf{I}\hfill & \mathbf{0}\hfill \\ \mathbf{0}\hfill & -\mathbf{I}\hfill \end{array}\right]\left[\begin{array}{l}\mathbf{X}\hfill \\ \mathbf{Y}\hfill \end{array}\right]$$ (12)

where Ω is the excitation energy. In eq 12 the A, B matrices are defined as

$$A_{ia,jb}={\delta }_{ij}{\delta }_{ab}({ϵ}_a^{\text{QP}}-{ϵ}_i^{\text{QP}})+v_{ia,jb}-W_{ij,ab}(\omega =0)$$ (13)

$$B_{ia,jb}=v_{ia,bj}-W_{ib,aj}(\omega =0)$$ (14)

where v is the Coulomb interaction and W(ω = 0) is the static screened interaction. v is the Coulomb interaction defined as

$$v_{pq,rs}=\int \text{d}{x}_1\text{d}{x}_2\frac{{\psi }_p^{*}\left(x_1\right){\psi }_r^{*}\left(x_2\right){\psi }_q\left(x_1\right){\psi }_s\left(x_2\right)}{\left|{\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_2\right|}$$ (15)

where {ψ_p} is the set of input orbitals and x is the combined space-spin variable for (r, σ). W is the screened interaction defined as

$$W_{pq,rs}=\sum _{tu}{\left(D^{-1}\right)}_{pq,tu}{v}_{tu,rs}$$ (16)

where the dielectric function D is calculated by the static response function χ^82,83

$$D_{pq,rs}={\delta }_{pr}{\delta }_{qs}-v_{pq,rs}{\chi }_{rs,rs}$$ (17)

$${\chi }_{ia,ia}={\chi }_{ai,ai}={({ϵ}_i^{\text{QP}}-{ϵ}_a^{\text{QP}})}^{-1}$$ (18)

The BSE working equation in eq 12 is analogous to the Casida equation^6,7 in TDDFT. The only difference is that the BSE kernel replaces the XC kernel. Thus, the scaling of solving eq 12 is O(N⁴) by using the canonical Davidson algorithm.^84,85

3. COMPUTATIONAL DETAILS

We implemented the BSE/G_RSW₀ and BSE/G_RSW_RS approaches in the QM4D quantum chemistry package⁸⁶ and applied them to calculate the excitation energies of different systems. As discussed in ref 59, the application of the Tamm–Dancoff approximation (TDA) improves the accuracy of BSE/G₀W₀ for predicting both singlet and triplet excitation energies because BSE/G₀W₀ largely underestimates the excitation energies of molecular systems. However, for BSE/G_RSW_RS and BSE/evGW, which predict larger excitation energies than BSE/G₀W₀, using TDA leads to similar or worse triplet results and worsens singlet results as shown in Section 4 in the Supporting Information and ref 38. In addition, as shown in recent studies, using TDA in BSE/GW can lead to blueshifts in nanosized systems^87–89 and worse accuracy for singlet-triplet energy gaps in organic molecules.³⁶ Therefore, TDA is not used in the present work. We tested three different sets: the comprehensive Truhlar–Gagliardi test set⁹⁰ that contains singlet, triplet, valence, CT, and Rydberg excitations, the Stein CT test set⁹¹ that contains 12 intramolecular CT excitations between an aromatic donor and the tetracyanoethylene acceptor, and a test set for Rydberg excitations that contains three atomic systems. For the Truhlar–Gagliardi test set,⁹⁰ the aug-cc-pVTZ basis set^92,93 was used for all molecules, except for naphthalene, pNA, and DMABN, for which the augcc-pVDZ basis set^92,93 was employed. It has been shown that the aug-cc-pVTZ basis sets yield converged neutral optical excitations,⁴⁶ and even aug-cc-pVDZ results were found to deviate by not more than 0.2–0.3 eV from the basis set limit.⁴⁶ B-TCNE was excluded due to the high computational cost. Reference values for pNA and DMABN were taken from refs 94 and 95 for the remaining molecules in the Truhlar–Gagliardi test set. The reference values are the theoretical best estimates, for example, using FCI or CCSDTQ.^94,95 Geometries were all taken from ref 90. Note that geometries used in ref 95 are slightly different from those in the Truhlar–Gagliardi set. As shown in refs 95 and 96, the difference between theoretical best estimates obtained with two slightly different geometries is around 0.01 eV. Thus, we do not expect the small differences to change the conclusion. For the Stein CT test set,⁹¹ the cc-pVDZ⁹² basis set was used. Because theory best estimates for the Stein CT test set are not available, the experimental values in the gas phase⁹¹ were taken as the reference, which can be a source of errors. For the test of Rydberg excitation energies of B⁺, Be, and Mg, the aug-cc-pVQZ basis set^92,93 was employed. Experimental reference values were taken from ref 97. TDDFT calculations were performed with the GAUSSIAN16 A.03 software.⁹⁸ BSE/G₀W₀, BSE/G_RSW₀, BSE/G_RSW_RS, and evGW calculations were performed with QM4D. QM4D uses Cartesian basis sets and uses the resolution of identity technique^52,99,100 to compute two-electron integrals. All basis sets and corresponding fitting basis sets were taken from the basis set exchange.^101–103

4. RESULTS

4.1. Truhlar–Gagliardi Test Set.

We first examine the performance of the BSE/G_RSW_RS approach for predicting the excitation energies of molecules from the Truhlar–Gagliardi test set. Excluding B-TCNE, this test set contains 18 valence excitations, two Rydberg excitations, and two CT excitations. The valence excitations in this set refer to n → π* and π → π* excitations. The mean absolute errors (MAEs) and mean signed errors (MSEs) of excitation energies obtained from TDDFT, BSE/G₀W₀, BSE/G_RSW₀, BSE/G_RSW_RS, and BSE/evGW with HF, BLYP, PBE, B3LYP, PBE0, and PBEh(0.75) are shown in Figure 1 and Table 1. The PBE-based hybrid functional PBEh(0.75) that has 75% HF exchange is shown as the optimal starting point for G₀W₀ to predict IPs in the GW100 set.¹⁰⁴ The signed error is defined as the difference between the calculated value and the reference value, that is, E^calc − E^reference. Because this test set mainly contains valence excitations, TDDFT with conventional DFAs provides good accuracy. However, TDDFT based on the DFA with a large percentage of the HF exchange has large errors and can suffer from triplet instability. As evident from Figure 1, BSE/G₀W₀ greatly underestimates the excitation energies and provides large MAEs, which were also reported previously.^31,59 This error can be attributed to the overscreening problem in G₀W₀, that is, the screened interaction W is calculated with an underestimated KS gap.³¹ The underestimated fundamental gap in G₀W₀ leads to the underestimated optical gap in BSE/G₀W₀. In addition, BSE/G₀W₀ has a strong starting point dependence. The difference between MAEs of BSE/G₀W₀ with GGA and hybrid functionals is larger than 0.7 eV. BSE/G₀W₀, based on PBEh(0.75), that is the optimal starting point for valence QP energy provides a small MAE of 0.27 eV. However, the percentage of the HF exchange needs to be reoptimized for excitations of different characters and different species.³² Recent work has shown that the optimally tuned range-separated hybrid DFAs can be a good starting point for BSE/G₀W₀.¹⁰⁵ The BSE/G_RSW₀ approach improves upon BSE/G₀W₀. Using G_RSW₀ instead of G₀W₀, the MAEs are reduced by around 0.6 eV with GGA functionals and by around 0.3 eV with hybrid functionals. However, there is still an undesired starting point dependence in BSE/G_RSW₀ because the screened interaction in G_RSW₀ is calculated at the chosen DFA level. The BSE/G_RSW_RS approach significantly outperforms BSE/G₀W₀ and BSE/G_RSW₀. The MAEs from BSE/G_RSW_RS with conventional DFAs are around 0.4 eV. They are similar to the MAEs from TDDFT with hybrid functionals and are slightly better than the ones from BSE/evGW. BSE/G_RSW_RS with the optimal starting point PBEh(0.75) provides the smallest MAE of 0.29 eV. As shown in Section 3 in the Supporting Information, fundamental gaps obtained from RS orbital energies are always larger than those obtained from KS orbital energies. By inserting the RS Green’s function into the RPA equation to formulate the screened interaction, BSE/G_RSW_RS greatly reduces the overscreening error and provides excellent accuracy. The starting point dependence in BSE/G_RSW_RS is largely reduced, which is similar to BSE/evGW. The different DFAs induce only small changes of less than 0.1 eV in the BSE/G_RSW_RS MAEs. We find that BSE/G_RSW_RS, BSE/G₀W₀, and BSE/evGW yield triplet excitation energies which are significantly too low, which is in agreement with previous work.³¹

Figure 1.

MAEs and MSEs of excitation energies in the Truhlar–Gagliardi test set obtained from TDDFT, BSE/G₀W₀, BSE/G_RSW₀, BSE/G_RSW_RS, and BSE/evGW based on HF, BLYP, PBE, B3LYP, PBE0, and PBEh(0.75). Reference values for pNA and DMABN were taken from ref 94 and for the remaining molecules from ref 95. The reference values are the theoretical best estimates. The aug-cc-pVDZ basis set was used for naphthalene, pNA, and DMABN, and the aug-cc-pVTZ basis set was used for the remaining systems. B-TCNE was excluded due to the high computational cost. Total MAEs and total MSEs were calculated by averaging all systems with equal weights. The error for system i is defined as ${\text{Error}}_i=E_i^{\text{calc}}-E_i^{\text{reference}}$.

Table 1.

Total MAEs of Excitation Energies in the Truhlar–Gagliardi Test Set Obtained from TDDFT, BSE/G₀W₀, BSE/G_RSW₀, BSE/G_RSW_RS, and BSE/evGW Based on HF, BLYP, PBE, B3LYP, PBE0, and PBEh(0.75)^a

	HF	BLYP	PBE	B3LYP	PBE0	PBEh
BSE/G0W0	0.76	1.43	1.40	0.99	0.88	0.27
BSE/GRSW0	0.76	0.83	0.82	0.67	0.61	0.28
BSE/GRSWRS	0.76	0.37	0.36	0.35	0.34	0.29
BSE/evGW	0.74	0.48	0.47	0.52	0.53	0.26
TDDFT	1.54	0.52	0.51	0.33	0.31	0.85

^aReference values for pNA and DMABN were taken from ref 94 and for the remaining molecules from ref 95. The reference values are the theoretical best estimates. The aug-cc-pVDZ basis set was used for naphthalene, pNA, and DMABN, and the aug-cc-pVTZ basis set was used for the remaining systems. B-TCNE was excluded due to the high computational cost. Total MAEs were calculated by averaging all systems with equal weights. The error for system i is defined as ${\text{Error}}_i=E_i^{\text{calc}}-E_i^{\text{reference}}$.

4.2. Stein CT Test Set.

We further study the performance of BSE/G_RSW_RS in predicting CT excitation energies by testing the Stein CT test set. This test set contains 12 intramolecular CT systems. The MAEs and MSEs of excitation energies obtained from TDDFT, BSE/G₀W₀, BSE/G_RSW₀, and BSE/G_RSW_RS with HF, BLYP, PBE B3LYP, PBE0, and PBEh(0.75) are listed in Table 2. It shows that TDDFT with conventional DFAs fails to predict CT excitation energies due to the incorrect description of the long-range behavior. TDDFT with both GGA and hybrid functionals greatly underestimates the CT excitation energies and gives MAEs larger than 1.0 eV. TDDFT with PBEh(0.75) that has a large percentage of the HF exchange provides a small MAE of 0.19 eV. In addition, TDDFT has a strong starting point dependence. BSE/G₀W₀ provides improved results over TDDFT because of the correct long-range behavior from the screened interaction. However, BSE/G₀W₀ still suffers from a strong dependence on the choice of the DFA and yields relatively large errors. As shown in Section 6 in the Supporting Information, BSE/G₀W₀ always underestimates the excitation energies of CT systems. Slightly larger excitation energies can be obtained when using the TDA.⁵⁹ The BSE/G_RSW₀ approach improves again upon BSE/G₀W₀. Compared with BSE/G₀W₀, the MAEs of BSE/G_RSW₀ are 1.0 and 0.4 eV smaller with GGA and hybrid functionals, respectively. The dependence on the DFA is reduced to only 0.09 eV in the BSE/G_RSW₀ scheme. The BSE/G_RSW_RS approach provides the most accurate results with the smallest starting point dependence. The MAEs of BSE/G_RSW_RS with all tested DFAs are only around 0.15 eV, which are comparable to the accuracy of BSE/evGW as reported in ref 41. In addition, the dependence on the choice of the DFA in BSE/G_RSW_RS is largely eliminated. The difference originating from using different DFAs is only around 0.04 eV.

Table 2.

MAEs and MSEs of CT Excitation Energies in the Stein CT Test Set Obtained from TDDFT, BSE/G₀W₀, BSE/G_RSW₀, and BSE/G_RSW_RS with HF, BLYP, PBE, B3LYP, PBE0, and PBEh(0.75)^a

	HF		BLYP		PBE		B3LYP		PBE0		PBEh
	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE
TDDFT	0.78	0.78	1.44	−1.44	1.45	−1.45	1.16	−1.16	1.08	−1.08	0.19	0.02
BSE/G0W0	0.10	−0.06	1.28	−1.28	1.31	−1.31	0.74	−0.74	0.65	−0.65	0.14	0.09
BSE/GRSW0	0.10	−0.06	0.35	−0.35	0.35	−0.37	0.29	−0.28	0.29	−0.29	0.16	−0.15
BSE/GRSWRS	0.10	−0.06	0.17	0.14	0.18	0.11	0.14	0.09	0.17	0.14	0.11	−0.07

^aAll values in eV. Geometries were taken from ref 91. Experiment values in the gas phase were taken as the reference values.⁹¹ Gas-phase experimental references were used. The error for system i is defined as ${\text{Error}}_i=E_i^{\text{calc}}-E_i^{\text{experiment}}$ The cc-pVDZ basis set was used.

4.3. Rydberg Excitations.

We further investigate the performance of the BSE/G_RSW₀ and the BSE/G_RSW_RS approaches for predicting Rydberg excitation energies of atomic systems by testing B⁺, Be, and Mg. The signed errors of Rydberg excitation energies obtained from TDDFT, BSE/G₀W₀, BSE/G_RSW₀, BSE/G_RSW_RS, and BSE/evGW with HF, BLYP, PBE, B3LYP, and PBE0 are listed in Figure 2. Numerical results are shown in Section 7 in the Supporting Information. Similar to the CT excitation energies, it is well-known that TDDFT with common DFAs has relatively large errors for predicting Rydberg excitation energies.³¹ The latter largely underestimates with GGA as well as hybrid functionals in TDDFT. Compared to TDDFT, the BSE/G₀W₀ approach yields only slightly better predictions, reducing the MAE by around 0.1 eV. TDDFT and BSE/G₀W₀ both show strong starting point dependence. BSE/G_RSW₀ slightly improves upon BSE/G₀W₀, reducing the MAE by 0.1 eV with respect to BSE/G₀W₀. The BSE/G_RSW_RS approach provides further improvements over BSE/G_RSW₀. As shown in Section 7 in the Supporting Information and Figure 2, BSE/G_RSW_RS shows significant improvements over TDDFT for predicting singlet Rydberg excitation energies with reduced errors around 0.4 eV. However, the improvements on triplet Rydberg excitation energies are small. The MAEs of BSE/G_RSW_RS with all different DFAs are around 0.6 eV, which is close to the BSE/evGW level. The DFA starting point dependence is reduced to round 0.2 and 0.1 eV with BSE/G_RSW₀ and BSE/G_RSW_RS, respectively.

Figure 2.

Signed errors of B⁺, Be and Mg obtained from TDDFT, BSE/G₀W₀, BSE/G_RSW₀, BSE/G_RSW_RS, and BSE/evGW with HF, BLYP, PBE, B3LYP, and PBE0. All values in eV. Experimental values were taken as the reference values.⁹⁷ The aug-cc-pVQZ basis set was used. The signed error for system i is defined as ${\text{Error}}_i=E_i^{\text{calc}}-E_i^{\text{experiment}}$.

5. CONCLUSIONS

In this work, we applied the BSE formalism on top of the G_RSW_RS method to calculate valence, Rydberg, and CT excitation energies of molecular systems. The G_RSW_RS method provides improved fundamental gaps compared to G₀W₀ and largely reduces the dependence on the choice of the density functional approximation. In the BSE/G_RSW_RS approach, the QP energies from G_RSW_RS are used in BSE. For the Truhlar–Gagliardi test set, we found that BSE/G_RSW_RS provides excellent accuracy for excitations of different characters (valence, CT, and Rydberg excitations) with MAEs around 0.4 eV. The accuracy of BSE/G_RSW_RS is similar to TDDFT and slightly better than BSE/evGW. Using the Stein CT test set, we further showed that BSE/G_RSW_RS is significantly more accurate for predicting CT excitation energies than BSE/G₀W₀ and TDDFT. We also found that the predictions compare well to BSE/evGW results reported in the literature.⁴¹ We also showed that BSE/G_RSW_RS predict accurate Rydberg excitation energies for atomic systems. We found for all three test sets that the dependence on the choice of the DFA is also largely eliminated. The computational cost of BSE/G_RSW_RS is similar to BSE/G₀W₀, which has a much lower computational cost than BSE/evGW.

This work demonstrates that the BSE/G_RSW_RS approach is accurate and efficient for predicting all three types of excitation energies of a broad range of systems. Therefore, the BSE/G_RSW_RS approach is expected to expand the applicability of the BSE/GW approach.