Accurate Ionization Potentials and Electron Affinities of Acceptor

Article pubs.acs.org/JCTC

Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods Joseph W. Knight,† Xiaopeng Wang,† Lukas Gallandi,‡ Olga Dolgounitcheva,§ Xinguo Ren,∥ J. Vincent Ortiz,§ Patrick Rinke,⊥ Thomas Körzdörfer,‡ and Noa Marom*,† †

Physics and Engineering Physics, Tulane University, New Orleans, Louisiana 70118, United States Computational Chemistry, University of Potsdam, 14476 Potsdam, Germany § Department of Chemistry and Biochemistry, Auburn University, Auburn, Alabama 36849-5312, United States ∥ Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China ⊥ COMP/Department of Applied Physics, Aalto University School of Science, P.O. Box 11100, FI-00076 Aalto, Finland ‡

S Supporting Information *

ABSTRACT: The performance of different GW methods is assessed for a set of 24 organic acceptors. Errors are evaluated with respect to coupled cluster singles, doubles, and perturbative triples [CCSD(T)] reference data for the vertical ionization potentials (IPs) and electron affinities (EAs), extrapolated to the complete basis set limit. Additional comparisons are made to experimental data, where available. We consider fully self-consistent GW (scGW), partial self-consistency in the Green’s function (scGW0), non-self-consistent G0W0 based on several mean-field starting points, and a “beyond GW” second-order screened exchange (SOSEX) correction to G0W0. We also describe the implementation of the self-consistent Coulomb hole with screened exchange method (COHSEX), which serves as one of the mean-field starting points. The best performers overall are G0W0+SOSEX and G0W0 based on an IP-tuned long-range corrected hybrid functional with the former being more accurate for EAs and the latter for IPs. Both provide a balanced treatment of localized vs delocalized states and valence spectra in good agreement with photoemission spectroscopy (PES) experiments.

■

INTRODUCTION

semiconductors and are increasingly applied to organic semiconductors.17−42 Several studies17,23,26,36−38,43−48 have assessed the accuracy of different GW methods for molecules. Some of these assessed only the accuracy for IPs,23,36,38,45 which is not necessarily indicative of the accuracy for the entire valence spectra or EAs. The latter, in particular, are of interest for electron acceptors. In addition, most studies have focused primarily on small molecules,23,26,36−38,43,45,47,48 which are not necessarily relevant in the context of organic electronics. Finally, most studies have relied on comparisons to photoemission spectroscopy (PES) experiments.17,23,36,37,43,44,46−48 The comparison of GW quasiparticle (QP) excitation energies to experiments may be problematic when aiming for an accuracy of 0.1 eV. GW calculations yield vertical excitation energies, whereas most experiments measure adiabatic values, which include the relaxation of the atomic positions in response to the electronic excitation. The relaxation energy contribution to the adiabatic IP (EA) may be estimated by calculating the total energy difference between the cation (anion) in its relaxed

Computational modeling has the potential to accelerate the discovery and deployment of new organic semiconductors with optimized properties for various applications.1−11 Key parameters for the functionality and efficiency of organic electronic devices, including ionization potentials (IPs), electron affinities (EAs), and the energy level alignment at functional interfaces, are associated with charged excitations, where a particle is added to or removed from the system. The ability to gain meaningful physical insight, propose new materials, and derive design rules for more efficient devices from first-principles simulations hinges on the accurate description of these critical parameters. The desired accuracy of 0.1 eV may be afforded by manybody perturbation theory within the GW approximation, where G represents the one particle Green’s function and W represents the screened Coulomb interaction.12−16 Advances in high performance computing and scalable implementations in several popular electronic structure packages have led to the adoption of GW methods by a growing community of researchers. GW methods have become the de facto standard for the description of charged excitations in inorganic © 2016 American Chemical Society

Received: September 10, 2015 Published: January 5, 2016 615

DOI: 10.1021/acs.jctc.5b00871 J. Chem. Theory Comput. 2016, 12, 615−626

Article

Journal of Chemical Theory and Computation geometry vs the neutral’s geometry.43,44 The measured IPs and EAs are shifted by the zero point energy (ZPE) of the neutral and charged species and broadened due to vibrational replicas. The ZPE and vibrational contributions may also be calculated.49,50 However, it is more difficult to calculate the relaxation, ZPE, and vibrational contributions for the spectrum of deeper lying states beyond the first ionization energy. Additional factors, such as detector resolution, may further contribute to the broadening of measured spectra.51 The combination of these effects may lead to error bars of up to 0.3 eV for IP measurements. The error bars for EA measurements are even larger. Therefore, the accuracy of GW methods is best assessed by comparing to vertical values from coupled cluster singles, doubles, and perturbative triples [CCSD(T)], the gold standard of quantum chemistry.38,45 In the third part of this four-part study, we benchmark the accuracy of several GW methods for a set of 24 acceptors, illustrated in Figure 1, from chemical families commonly used

We consider fully self-consistent GW (scGW), partial selfconsistency in the Green’s function (scGW0),55 non-selfconsistent G0W012 based on various mean-field starting points, and a “beyond GW” second-order screened exchange (SOSEX)56 correction to G0W0.57 We find that the best performers overall are G0W0+SOSEX and G0W0 based on an IP-tuned LC-hybrid functional with the former being more accurate for EAs and the latter for IPs. Both provide a balanced treatment of localized vs delocalized states and valence spectra in good agreement with PES experiments in most cases. However, the behavior of the IP-tuned LC-hybrid functional depends on the character of the highest occupied molecular orbital (HOMO), and its accuracy may deteriorate when the HOMO is highly localized. Partially or non-self-consistent GW methods, based on a judiciously chosen mean-field starting point, are often more accurate than fully self-consistent GW. This is attributed to error cancellation, whereby the underscreening resulting from neglecting the vertex is compensated by overscreening in the mean-field starting point. For the molecules studied here, the mean-field molecular orbitals are similar to the self-consistent Dyson orbitals with few exceptions.

■

COMPUTATIONAL DETAILS Geometry optimizations were performed with the Gaussian 09 code,58 using the Becke 3-parameter exchange with the Lee− Yang−Parr correlation (B3LYP)59 hybrid functional and 6311G** basis sets. The structures were constrained to the highest possible point-group symmetry. The same geometries were used to produce the CCSD(T) reference data52 and the accompanying benchmark studies LC-hybrid functionals53 and electron propagator methods.54 The coordinates are provided in the Supporting Information. All GW calculations were performed using the all-electron numerical atom-centered orbital (NAO) code, FHI-aims.60,61 The NAO basis sets are grouped into a minimal basis, containing only basis functions for the core and valence electrons of the free atom, followed by four hierarchically constructed sets of additional basis functions, denoted by “tier 1−4”, described in detail in ref 60. For example, for first row elements, these hierarchical NAO basis sets are tier 1: [min]+1s1p1d; tier 2: [min]+2s2p2d1f1g; tier 3: [min]+3s3p3d2f1g; tier 4: [min]+4s4p4d3f2g. A detailed account of the all-electron implementation of GW methods in FHI-aims has been given elsewhere.36,47,62 Briefly, a resolution-of-identity (RI) technique is employed, whereby a set of auxiliary basis functions is introduced to represent both the Coulomb potential and the non-interacting response function, allowing for efficient GW calculations with NAO basis functions. The self-energy is first evaluated on the imaginary frequency axis and then analytically continued to the real frequency axis. A two-pole fitting procedure is the default setting of FHI-aims, and the Padé approximation is also available.36,48 The choice of analytical continuation method typically leads to differences well below 0.1 eV in the QP energies (representative examples are provided in the Supporting Information). Non-self-consistent G0W0 calculations were performed based on the following mean-field starting points: (i) the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE),63,64 as a semilocal starting point; (ii) the one-parameter PBE-based hybrid functional (PBE0, also known as PBEh), with 25% of Fock exchange,65 as a hybrid functional starting

Figure 1. Illustrations of the benchmark molecules.

in organic electronics, including acenes, nitro species, nitriles, anhydrides, quinones, and boron-dipyrromethene (BODIPY). The accuracy of vertical IPs and EAs is assessed with respect to CCSD(T) reference data, extrapolated to the complete basis set limit, reported in the first part of this study.52 For completeness, the IPs, EAs, and valence spectra are also compared to experimental data, where available. In the second53 and fourth54 parts of this study, non-empirically tuned long-range corrected (LC) hybrid functionals and electron propagator (EP) methods, respectively, are benchmarked for the same set of acceptor molecules, allowing for a direct comparison of the accuracy of different families of electronic structure methods. 616


Article

Journal of Chemical Theory and Computation

exchange to O(N).36 The G0W0 and scGW implementations in FHI-aims do not yet use the local RI. However, their actual scaling may be better than O(N4) for certain systems, due to efficient matrix algebra. Typically, for a G0W0 calculation based on a hybrid functional, most of the computer time is spent on the self-consistency cycle of the DFT calculation, rather than on the perturbative G0W0 correction, which is performed only once. Therefore, the main difference in computational cost between a hybrid DFT and a G0W0 calculation stems from the larger basis set required to converge the G0W0 results. Presently, systems with over a hundred atoms are within reach for G0W0.80 The formal scaling of SOSEX is O(N5),56 although that too may be reduced in the future by more efficient implementations. Self-consistent GW calculations have the same scaling as G0W0. While the prefactor is larger than in G0W0, the size of the basis set required to converge the results is generally found to be smaller.62 scGW is currently limited by a memory bottleneck, associated with the frequency dependence of the Green’s function.

point; (iii) Hartree−Fock (HF), the typical starting point for quantum chemistry electron propagator methods;66,67 (iv) the long-range corrected (LC) LC-ωPBE functional,68,69 which contains 100% exact exchange with PBE correlation in the longrang and reduces to pure PBE in the short-range, with ω being a range-separation parameter. Here, ω is a system-dependent parameter, tuned to obey the IP-theorem.70−72 The IP tuning procedure is discussed in detail in ref 53, and the values of ω used for all molecules are tabulated in the Supporting Information. In addition, we examine (v) the self-consistent Coulomb hole with screened exchange (COHSEX) starting point for G0W0.12,13,73 Additional details on the COHSEX implementation in FHI-aims, which has not been reported previously, are provided below. Partially self-consistent scGW0 and non-self-consistent G0W0+SOSEX calculations were performed based on a PBE starting point, which has been shown to be optimal for these methods.37,74 Non-self-consistent and partially self-consistent calculations are denoted as [method]@[starting point], for example, G0W0@PBE. G0W0 calculations were conducted with a tier 4 basis set and tight numerical settings.75 For G0W0, this gives QP energies converged to within 0.1 eV.35−37,76 The basis set convergence of the NAO tiers was compared to correlation-consistent NAO77 and Gaussian basis sets.78 The latter were extrapolated to the complete basis set (CBS) limit using a two-parameter inverse cubic fit with respect to the cardinal number79 (the same method was used in ref 54). Presently, CBS extrapolation schemes do not exist for the NAO basis sets of FHI-aims. The more compact tier 4 basis sets were found to provide QP energies of comparable accuracy to quintuple-zeta quality basis sets, which are close to the CBS-extrapolated values. A detailed account of these basis set convergence studies is provided in the Supporting Information. Because of their considerably higher computational cost and memory requirements, scGW0 and scGW calculations were conducted with tier 2 basis sets, which have been shown to be adequately converged.37,62 We emphasize that the results of scGW are completely independent of the mean-field starting point, however starting from HF was found to lead to faster convergence.47 The output of scGW and scGW0 is the (discretized) spectral function: A(ω) = ( −1/π )Im(TrG(ω))

■

COHSEX IMPLEMENTATION IN FHI-AIMS COHSEX,12,13,73 the static limit of GW, has been proposed as an alternative mean-field starting point to DFT and HF for partially or non-self-consistent GW schemes. COHSEX has been shown to be a better starting point than semilocal DFT functionals and comparable to hybrid functionals, in particular for systems involving transition metals with localized d orbitals.43,73,81−84 The COHSEX self-energy has two terms: ΣCOHSEX = ΣCOH + ΣSEX

(2)

The screened exchange term, Σ , has a similar form to the Fock exchange, but with the statically screened Coulomb potential, W(r,r′,ω = 0), instead of the bare Coulomb potential, v(r,r′): SEX

occ.

ΣSEX = − Σ ψm(r)ψm*(r′)W (r,r′,ω = 0) m

(3)

where ψm(r) are mean-field orbitals. The Coulomb hole term, ΣCOH, describes the interaction of a QP with the induced potential due to the polarization response of the surrounding medium: ΣCOH =

(1)

The Green’s function has poles at the QP excitation energies, giving rise to sharp δ-function peaks in the spectral function. The QP excitation energies were extracted by a parabolic fit to the peaks of the spectral function. The spectral function peaks were assigned to mean-field orbitals by analyzing the partial spectral functions corresponding to single diagonal elements of G, represented in the basis of mean-field orbitals. In most cases, these had only one peak, indicating that the contributions of off-diagonal elements were negligible and the diagonal approximation in G0W0 was valid (see further discussion below). The formal scaling of GW is O(N4).48 This is equivalent to the scaling of the non-local Fock exchange in HF and hybrid DFT calculations, considerably lower than the formal O(N7) scaling of the CCSD(T) reference calculations,52 and lower than most of the electron propagator methods benchmarked in ref 54. The actual scaling of GW calculations depends on the specific details of the implementation. The local RI implementation in FHI-aims reduces the scaling of the Fock

1 [W (r,r′,ω = 0) − v(r,r′)]δ(r − r′) 2

(4)

Within a basis set, {ϕi}, the matrix elements of the COHSEX self-energy are given by ΣCOHSEX = ⟨ϕi|ΣCOHSEX (r,r′)|ϕj⟩ ij

(5)

where the COHSEX self-energy is ΣCOHSEX (r,r′) = ΣEX x (r,r′) occ. ⎡1 ⎤ + ⎢ δ(r − r′) − Σ ψm(r)ψ m(r′)⎥Wc(r,r′, ω = 0) ⎣2 ⎦ m

(6) =ΣEX x (r,r′) +

1 Σ[θ(εm − μ) − θ(μ − εm)]ψm(r)ψm(r′)Wc(r,r′, ω = 0) 2m

(7)

Here, ϕi (r) are basis functions; εm and ψm(r) are single-particle eigenvalues and eigenstates, respectively; θ(x) is the Heaviside step function; μ is the chemical potential; ΣxEX = iGv is the exact exchange self-energy; Wc(r, r′, ω = 0) = W(r, r′, ω = 0) − ν(r − r′) is the dynamical 617


Article


Figure 2. Heat map representation of the absolute errors in eV with respect to CCSD(T) of scGW, scGW0@PBE, G0W0@{COHSEX, HF, PBE, PBEh, LC-ωPBE}, and G0W0+SOSEX@PBE for the (a) EAs and (b) IPs of the benchmark molecules.

part of the screened Coulomb interaction at zero frequency. The second term on the right-hand side of eqs 6−7 corresponds to the COHSEX correlation self-energy, ΣCOHSEX . c Equations 6 and 7 are numerically identical only in the complete basis set limit. The COHSEX implementation in the finite basis set of FHI-aims is based on eq 7, where the summation is carried out over all states generated with a given finite basis set. From eqs 5 and 7, the COHSEX correlation selfenergy is given by ΣCOHSEX = c, ij

1 Σ[θ(εm − μ) − θ(μ − εm)]⟨ϕψ |W (ω = 0)|ψmϕj⟩ i m 2m

(8)

36

Using the RI technique, multi-orbital integrals are solved by expanding orbital products {ϕi (r)ψm(r)} in terms of a properly constructed auxiliary basis set {Pμ(r)}, i.e.: ϕi(r)ψm(r) = ΣCiμ,mPμ(r) m

(9)

Eq 8 then becomes ΣCOHSEX = c, ij

1 Σ[θ(εm − μ) − θ(μ − εm)]ΣCiμ,mWμ , υ(ω = 0)Cjυ,m μ 2m

(10)

where Wμ , ν(ω = 0) =

■

∬ drdr′Pμ(r)W (r,r′,ω = 0)Pν(r′)

(11)

RESULTS AND DISCUSSION The errors with respect to CCSD(T) of scGW, scGW0@PBE, G 0 W 0 @{COHSEX, HF, PBE, PBEh, LC-ωPBE}, and G0W0+SOSEX@PBE for the IPs and EAs of the benchmark molecules are presented as heat maps in Figure 2 and as histograms in Figures 3 and 4. Tabulated results are provided in the Supporting Information. We find that scGW has a similar performance to G0W0@PBE, overestimating the EAs and underestimating the IPs by about 0.6 eV on average. The trend of scGW IPs being between those of G0W0@PBE and those of G0W0@HF is

Figure 3. Histograms of the errors with respect to CCSD(T) of scGW, scGW0@PBE, G0W0@ {COHSEX, HF, PBE, PBEh, LC-ωPBE}, and G0W0+SOSEX@PBE for the EAs of the benchmark molecules.

consistent with that reported in refs 47 and 62; however, the present comparison to CCSD(T) vertical values results in somewhat worse agreement than the comparison to experimental values therein. Our findings are also consistent with 618


Article


G0W0@HF for EAs, despite the COHSEX gaps being less overestimated than the HF gaps (mean-field HOMO and LUMO values are provided in the SI). This indicates that the quality of G0W0 results is not simply a function of the meanfield gap. Other contributing factors are discussed below and in the second part of this study.53 The best performers overall are G0W0+SOSEX@PBE and G0W0@LC-ωPBE. G0W0+SOSEX@PBE yields the lowest MAE of 0.07 eV for EAs and a MAE of 0.36 eV for IPs. G0W0@LC-ωPBE yields the lowest MAE of 0.13 eV for IPs and a MAE of 0.19 eV for EAs. Similar accuracy for the IPs and EAs of the present benchmark set is afforded by nonempirically tuned LC-hybrid DFT functionals53 and by EP methods.54 The IPs of two molecules, benzoquinone and maleic anhydride, stand out as outliers, for which the errors of G0W0@LC-ωPBE are particularly large and the errors of scGW are particularly small. The behavior of IP-tuned LC-ωPBE is dictated by the character of the HOMO.85,86 For these molecules, the strong localization of the HOMO leads to particularly large values of ω, close to 0.4 Bohr−1, and to a more HF-like behavior. Typically, scGW yields underestimated ionization potentials. However, strongly localized orbitals may be affected by the self-screening error,87,88 which brings the HOMO peaks of benzoquinone and maleic anhydride to the right place thanks to error cancellation. The treatment of localized vs delocalized orbitals by different GW methods is discussed further below. Additional differences in the performance of GW methods are revealed by examining the valence spectra they produce. In this respect, the comparison to gas-phase PES experiments is informative, despite the aforementioned limitations. We may distinguish between two sources of starting point dependence, illustrated in Figure 5. The first is screening. Overscreening

Figure 4. Histograms of the errors with respect to CCSD(T) of scGW, scGW0@PBE, G0W0@ {COHSEX, HF, PBE, PBEh, LC-ωPBE}, and G0W0+SOSEX@PBE for the IPs of the benchmark molecules.

those of ref 37, underlining that full self-consistency does not necessarily improve the accuracy of IPs and EAs because the vertex is neglected. The scGW results reflect the true performance and limitations of the GW approximation itself. Partially or non-self-consistent GW schemes may benefit from error cancellation that compensates for the effect of neglecting the vertex. The performance of scGW0@PBE is comparable to that of G0W0@PBEh, overestimating the EAs by about 0.45 eV and underestimating the IPs by 0.39 and 0.29 eV, respectively. This is also consistent with the findings of ref 37 that the overscreening in the PBE-based W0 compensates for the under-screening in the self-consistent G. G0W0@COHSEX is slightly worse than G0W0@PBEh for IPs, yielding an underestimation of 0.41 eV on average. For EAs, G0W0@COHSEX has the same mean absolute error as G0W0@PBEh; however, G0W0@COHSEX underestimates EAs whereas G0W0@PBEh overestimates them. G0W0@HF performs well for EAs, overestimating by 0.19 eV on average, but worse for IPs, overestimating by 0.43 eV. The trends in the performance of G0W0@{PBEh, COHSEX, HF} for IPs are consistent with those reported in ref 43 for a set of small transition metal molecules. However, we find different trends for EAs. This is likely because in ref 43 the GW results are compared to vertical values or corrected adiabatic values from experiments, and experimental measurements of EAs are less reliable than those of IPs. Comparisons of the CCSD(T) vertical values to (uncorrected) experimental data for our benchmark set are provided in the Supporting Information. It is perhaps surprising that G0W0@COHSEX performs worse than

Figure 5. Schematic illustration of the effects of screening and SIE on valence spectra. The addition of an increasing fraction of exact exchange reduces the screening, causing a stretch and a shift of the entire spectrum to higher ionization energies. The mitigation of SIE by the exact exchange makes localized orbitals drift to higher ionization energies faster than delocalized orbitals.

generates overstabilization of charges, which reduces ionization energies and increases electron affinities, leading to underestimated gaps and valence spectra that appear too compressed. Underscreening has the opposite effect of destabilizing charges, leading to overestimated gaps and valence spectra that appear too stretched. In hybrid DFT functionals, the screening may be modulated by tuning the fraction of exact (Fock) exchange,89 as illustrated in Figure 5. The second source of differences between valence spectra is due to self-interaction errors (SIE). The single-particle SIE90 is 619


Article

Journal of Chemical Theory and Computation the spurious repulsion of an electron from itself, which arises in DFT because the self-interaction term in the Hartree potential is not exactly canceled out by approximate exchange-correlation functionals. The effect of single-particle SIE is to destabilize highly localized molecular orbitals compared to more delocalized molecular orbitals, which may lead to orbital reordering and distortions of the spectrum.35,37,76,91−95 The inclusion of a fraction of Fock exchange in hybrid functionals mitigates the single-particle SIE, increasing the ionization energies of localized orbitals more than those of delocalized orbitals. The many-particle SIE manifests as the deviation of the total energy from piecewise linearity for fractional particle numbers.96,97 This is rectified by including the long-range part of the Fock exchange in LC-hybrid functionals98 (see ref 53 for a detailed discussion). The effects of both screening and SIE in the mean-field starting point carry over to partially or non-selfconsistent GW schemes and may contribute to favorable error cancellation or lack thereof. Figures 6−9 show representative examples of the chemical families in the benchmark set. All other spectra are provided in

Figure 7. Spectra of nitrobenzene obtained with different GW methods, broadened by a 0.3 eV Gaussian, compared to PES data from ref 103. On the left, unoccupied orbitals are compared to reference data for the EA. Gray stripes represent the range of experimental data from refs 104−115. Dashed blue lines represent CCSD(T) reference values. Illustrations of some frontier orbitals are also shown.

Figure 6. Spectra of phenazine obtained with different GW methods, broadened by a 0.3 eV Gaussian, compared to PES data from ref 99. On the left, unoccupied orbitals are compared to reference data for the EA. Gray stripes represent the range of experimental data from refs 100−102. Dashed blue lines represent CCSD(T) reference values. Illustrations of some frontier orbitals are also shown.

the Supporting Information. The effect of screening leads to systematic trends, observed across the benchmark set. The G0W0@HF spectra are typically shifted to higher ionization energies and appear too stretched in comparison to experiment because of the underscreening in the HF starting point, whereas the G0W0@PBE spectra are typically shifted to lower ionization energies and appear too compressed in comparison to experiment because of the overscreening in the PBE starting point. In most cases, the scGW, scGW0@PBE, G0W0@PBEh, G0W0@COHSEX, G0W0@LC-ωPBE, and G0W0+SOSEX@PBE spectra are of similar quality in terms of screening. The behavior of G0W0@LC-ωPBE varies somewhat, depending on the value of ω, and the resulting spectra sometimes appear too stretched (HF-like).

Figure 8. Spectra of maleic anhydride obtained with different GW methods, broadened by a 0.2 eV Gaussian, compared to PES data from ref 113. On the left, unoccupied orbitals are compared to reference data for the EA. Gray stripes represent the range of experimental data from refs 105 and 116−119. Dashed blue lines represent CCSD(T) reference values. Illustrations of some frontier orbitals are also shown.

Systems with no localizing sites are robust in the sense that different methods produce similar spectra in terms of the relative peak positions and the shape of the frontier region. Systems with localizing sites are sensitive to the effect of SIE for localized orbitals. In the following examples, localized orbitals are shifted to lower ionization energies by G0W0@PBE due to 620


Article


poor agreement with the experiment as the first three peaks are merged into one. Similar SIE effects are observed for all the other anhydrides in the benchmark set and have also been discussed extensively elsewhere.27,91,93−95,131 The quinones, such as benzoquinone, shown in Figure 9, have orbitals localized on the oxygen atoms, colored in red and green. The effect of SIE for these orbitals leads to significant changes across methods in their position with respect to the two π-orbitals, colored in light blue and purple, affecting the distance between the first two spectral peaks. In the G0W0@HF and scGW spectra, these two peaks are practically merged. Similar effects are observed for all the quinones in the benchmark set. In most cases, it is possible to assign the peaks of the scGW0 and scGW spectral functions to mean-field orbitals by analyzing partial spectral functions. This indicates that the mean-field orbitals are similar to the self-consistent Dyson orbitals and that the diagonal approximation in G0W0 is valid. It has also been demonstrated in ref 132 that the off-diagonal contributions are typically small. Cl4-benzoquinone, shown in Figure 10, is an Figure 9. Spectra of benzoquinone obtained with different GW methods, broadened by a 0.2 eV Gaussian, compared to PES data from ref 120. On the left, unoccupied orbitals are compared to reference data for the EA. Gray stripes represent the range of experimental data from refs 121−128. Dashed blue lines represent CCSD(T) reference values. Illustrations of some frontier orbitals are also shown.

the SIE of PBE, which is only partially corrected by G0W0. In contrast, G0W0@HF overbinds localized orbitals, shifting them to higher ionization energies. The scGW spectra are often similar to the G0W0@HF spectra in shape and in the relative positions of localized vs delocalized orbitals, possibly because of self-screening errors. 87,88 scGW 0 @PBE, G 0 W 0 @PBEh, G0W0@COHSEX, G0W0@LC-ωPBE, and G0W0+SOSEX@PBE typically provide a more balanced treatment of localized vs delocalized orbitals. These methods often produce valence spectra of comparable quality in terms of the shape of the frontier region. The G0W0+SOSEX@PBE spectra are often in best agreement with experiment. Phenazine, shown in Figure 6, has an n-orbital localized on the nitrogen lone-pairs, whose position varies significantly between methods. G0W0@PBE shifts this orbital, colored in light blue, to a lower ionization energy with respect to the π‑orbitals, colored in green and magenta, owing to SIE, while G0W0@HF shifts it to a higher ionization energy. Other methods place it somewhere between these two extremes. The shape of the G0W0+SOSEX@PBE spectrum is closest to experiment. Similar trends are observed for the n-orbital of acridine, while other acenes without localizing sites are less sensitive to the effect of SIE. Nitro species, such as nitrobenzene, shown in Figure 7, have orbitals localized on the nitro group, colored in green, light blue, and magenta, whose positions vary across methods due to SIE. This leads, for example, to the significant underestimation of the distance between the first two spectral peaks by G0W0@PBE and a significant overestimation by G0W0@HF. The nitriles are less sensitive to SIE effects. The anhydrides, such as maleic anhydride, shown in Figure 8, have orbitals localized on the anhydride group, colored in purple. The effect of SIE for this orbital and the HOMO leads to significant differences in the positions of the first three spectral peaks of maleic anhydride. Although scGW yields an accurate IP for this molecule, the rest of the frontier region is in

Figure 10. Spectra of Cl4-benzoquinone obtained with different GW methods, broadened by a 0.2 eV Gaussian, compared to PES data from ref 124. On the left, unoccupied orbitals are compared to reference data for the EA. Gray stripes represent the range of experimental data from refs 105 and 128−130. Dashed blue lines represent CCSD(T) reference values. Illustrations of some frontier orbitals are also shown.

exception. In this case, some of the HF molecular orbitals are significantly different than those of other mean-field methods because of a different hybridization between the Cl and O atomic orbitals and the sigma orbitals of the benzene ring. Panels a−f of Figure 11 show the partial spectral functions corresponding to the scGW peaks at −11.47 and −10.77 eV, obtained when starting the calculation from HF, PBE, and PBEh. The Dyson orbitals corresponding to these peaks may be assigned to a mix of two different HF orbitals or to distinct PBE/PBEh orbitals. The peak positions of the total selfconsistent spectral function, shown in panel g of Figure 11, are independent of the mean-field starting point.133 Off-diagonal contributions in scGW are also observed for NDCA and dinitro-benzonitrile. This demonstrates that self-consistency may be necessary in cases where the Dyson orbitals of a system are significantly 621


Article


The best performers overall are G0W0+SOSEX@PBE and G0W0@LC-ωPBE, with the former being more accurate for EAs and the latter for IPs. Both provide a balanced treatment of localized vs delocalized states and valence spectra in good agreement with gas-phase PES experiments. However, the behavior of the IP-tuned LC-hybrid functional depends on the character of the HOMO, and its accuracy may deteriorate when the HOMO is highly localized, as in the cases of maleic anhydride and benzoquinone. In such cases, the G0W0+SOSEX@PBE spectra are in better agreement with experiment than the G0W0@LC-ωPBE spectra, which are too HF-like. Although G0W0+SOSEX@PBE and G0W0@LC-ωPBE provide excellent accuracy for the present benchmark set of isolated molecules, this will not necessarily be the case for more complex systems, including interfaces and nanostructures with dissimilar components. For such systems, it may be difficult to find an optimal mean-field starting point for partially or nonself-consistent schemes. The process of tuning ω in G0W0@LC-ωPBE for complex systems is not straightforward, and G0W0+SOSEX@PBE may fail when the PBE wave function is qualitatively wrong (e.g., when PBE produces spurious charge transfer or a wrong hybridization). For complex systems, selfconsistency may be unavoidable. However, the accuracy of scGW is somewhat limited, and vertex corrections may be computationally unfeasible for such systems. Thus, a general solution for complex systems has yet to be found.

Figure 11. Partial spectral functions of (a,b) HF orbitals associated with peaks at −11.47 and −10.77 eV; PBE orbitals associated with the peaks at (c) −11.47 eV and (d) 10.77 eV; PBEh orbitals associated with the peaks at (e) −11.47 eV and (f) 10.77 eV; panel (g) shows total scGW spectral functions obtained by starting from HF, PBE, and PBEh. The peak positions are independent of the mean-field starting point.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.5b00871. Tabulated IP and EA results for all benchmark molecules compared to CCSD(T) and experimental values; spectra of all molecules not shown in the main text; comparison of the convergence behavior of the NAO tier 1−4 basis sets to that of correlation consistent NAO and Gaussian basis sets; mean-field HOMO and LUMO values; effect of the analytical continuation method; additional references; coordinates of the benchmark molecules. (PDF)

different than the mean-field orbitals. Because the present benchmark is restricted to isolated molecules, only few examples are encountered here; however, for more complex systems, off-diagonal contributions may become more significant. For example, self-consistency has been shown to eliminate spurious charge transfer in donor−acceptor systems.134 We also expect self-consistency to be useful in cases where different mean-field methods produce differently hybridized orbitals for interfaces and nanostructures with dissimilar components.35

■

CONCLUSION In summary, we have assessed the performance of different GW methods for a set of 24 organic acceptors from chemical families commonly used in organic electronics, including acenes, nitro species, nitriles, anhydrides, quinones, and BODIPY. We have considered fully self-consistent GW (scGW); partial self-consistency in the Green’s function (scGW0) based on a PBE starting point; non-self-consistent G0W0 based on HF, PBE, PBEh, COHSEX, and the IP-tuned long-range corrected functional LC-ωPBE; and a “beyond GW” SOSEX correction to G0W0 based on PBE. Errors in the vertical IPs and EAs were evaluated with respect to CCSD(T) reference data extrapolated to the complete basis set limit. The calculated valence spectra were additionally compared to gas-phase photoemission spectra, where available. We find that partially or non-self-consistent GW methods, based on a judiciously chosen mean-field starting point, are often more accurate than fully self-consistent GW. This is attributed to error cancellation, whereby the effect of neglecting the vertex is compensated by the mean-field starting point. The error cancellation is not simply a consequence of starting with a gap that is close to the QP gap, but a result of an interplay of the effects of screening and self-interaction errors.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Oliver Hofmann from TU Graz and Fabio Caruso from the University of Oxford for helpful discussions. Work at Tulane University was supported by the Louisiana Alliance for Simulation-Guided Materials Applications (LA-SiGMA), funded by the National Science Foundation (NSF) award #EPS-1003897. P.R. acknowledges the Academy of Finland through its Centres of Excellence Program (No. 251748 and 284621). Computer time was provided by the National Energy Research Scientific Computing Center (NERSC), which is 622


Article


(19) Tiago, M. L.; Chelikowsky, J. R. First-principles GW-BSE excitations in organic molecules. Solid State Commun. 2005, 136, 333− 337. (20) Neaton, J. B.; Hybertsen, M. S.; Louie, S. G. Renormalization of molecular electronic levels at metal-molecule interfaces. Phys. Rev. Lett. 2006, 97, 216405. (21) Umari, P.; Stenuit, G.; Baroni, S. GW quasiparticle spectra from occupied states only. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 81, 115104. (22) Ma, Y. C.; Rohlfing, M.; Molteni, C. Excited states of biological chromophores studied using many-body perturbation theory: Effects of resonant-antiresonant coupling and dynamical screening. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 241405. (23) Rostgaard, C.; Jacobsen, K. W.; Thygesen, K. S. Fully selfconsistent GW calculations for molecules. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 81, 085103. (24) Umari, P.; Castellarin-Cudia, C.; Feyer, V.; Di Santo, G.; Borghetti, P.; Sangaletti, L.; Stenuit, G.; Goldoni, A. Valence electronic structure of the indene molecule: Experiment vs. GW calculations. Phys. Status Solidi B 2011, 248, 960−963. (25) Qian, X. F.; Umari, P.; Marzari, N. Photoelectron properties of DNA and RNA bases from many-body perturbation theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 075103. (26) Faber, C.; Attaccalite, C.; Olevano, V.; Runge, E.; Blase, X. Firstprinciples GW calculations for DNA and RNA nucleobases. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 115123. (27) Dori, N.; Menon, M.; Kilian, L.; Sokolowski, M.; Kronik, L.; Umbach, E. Valence electronic structure of gas-phase 3,4,9,10-perylene tetracarboxylic acid dianhydride: Experiment and theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 195208. (28) Garcia-Lastra, J. M.; Thygesen, K. S. Renormalization of Optical Excitations in Molecules near a Metal Surface. Phys. Rev. Lett. 2011, 106, 187402. (29) Dias da Silva, L.; Tiago, M. L.; Ulloa, S. E.; Reboredo, F. A.; Dagotto, E. Many-body electronic structure and Kondo properties of cobalt-porphyrin molecules. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 155443. (30) Tiago, M. L.; Kent, P. R. C.; Hood, R. Q.; Reboredo, F. A. Neutral and charged excitations in carbon fullerenes from firstprinciples many-body theories. J. Chem. Phys. 2008, 129, 084311. (31) Palummo, M.; Hogan, C.; Sottile, F.; Bagala, P.; Rubio, A. Ab initio electronic and optical spectra of free-base porphyrins: The role of electronic correlation. J. Chem. Phys. 2009, 131, 084102. (32) Stenuit, G.; Castellarin-Cudia, C.; Plekan, O.; Feyer, V.; Prince, K. C.; Goldoni, A.; Umari, P. Valence electronic properties of porphyrin derivatives. Phys. Chem. Chem. Phys. 2010, 12, 10812− 10817. (33) Blase, X.; Attaccalite, C. Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach. Appl. Phys. Lett. 2011, 99, 171909. (34) Sai, N.; Tiago, M. L.; Chelikowsky, J. R.; Reboredo, F. A. Optical spectra and exchange-correlation effects in molecular crystals. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 161306. (35) Marom, N.; Moussa, J. E.; Ren, X. G.; Tkatchenko, A.; Chelikowsky, J. R. Electronic structure of dye-sensitized TiO(2) clusters from many-body perturbation theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 245115. (36) Ren, X.; Sanfilippo, A.; Rinke, P.; Wieferink, J.; Tkatchenko, A.; Reuter, K.; Blum, V.; Scheffler, M. An accurate resolution of identity approach to Hartree-Fock, hybrid functionals, MP2, RPA, and GW with numeric atom-centered basis functions. New J. Phys. 2012, 14, 053020. (37) Marom, N.; Caruso, F.; Ren, X.; Hofmann, O. T.; Körzdörfer, T.; Chelikowsky, J. R.; Rubio, A.; Scheffler, M.; Rinke, P. Benchmark of GW methods for azabenzenes. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 245127. (38) Bruneval, F.; Marques, M. A. L. Benchmarking the Starting Points of the GW Approximation for Molecules. J. Chem. Theory Comput. 2013, 9, 324−329.

supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-05CH11231.

■

REFERENCES

(1) O’Boyle, N. M.; Campbell, C. M.; Hutchison, G. R. Computational Design and Selection of Optimal Organic Photovoltaic Materials. J. Phys. Chem. C 2011, 115, 16200−16210. (2) Kanal, I. Y.; Owens, S. G.; Bechtel, J. S.; Hutchison, G. R. Efficient Computational Screening of Organic Polymer Photovoltaics. J. Phys. Chem. Lett. 2013, 4, 1613−1623. (3) Halls, M. D.; Djurovich, P. J.; Giesen, D. J.; Goldberg, A.; Sommer, J.; McAnally, E.; Thompson, M. E. Virtual screening of electron acceptor materials for organic photovoltaic applications. New J. Phys. 2013, 15, 105029. (4) Olivares-Amaya, R.; Amador-Bedolla, C.; Hachmann, J.; AtahanEvrenk, S.; Sanchez-Carrera, R. S.; Vogt, L.; Aspuru-Guzik, A. Accelerated computational discovery of high-performance materials for organic photovoltaics by means of cheminformatics. Energy Environ. Sci. 2011, 4, 4849−4861. (5) Hachmann, J.; Olivares-Amaya, R.; Jinich, A.; Appleton, A. L.; Blood-Forsythe, M. A.; Seress, L. R.; Roman-Salgado, C.; Trepte, K.; Atahan-Evrenk, S.; Er, S.; Shrestha, S.; Mondal, R.; Sokolov, A.; Bao, Z.; Aspuru-Guzik, A. Lead candidates for high-performance organic photovoltaics from high-throughput quantum chemistry - the Harvard Clean Energy Project. Energy Environ. Sci. 2014, 7, 698−704. (6) Ornso, K. B.; Garcia-Lastra, J. M.; Thygesen, K. S. Computational screening of functionalized zinc porphyrins for dye sensitized solar cells. Phys. Chem. Chem. Phys. 2013, 15, 19478−19486. (7) Ornso, K. B.; Pedersen, C. S.; Garcia-Lastra, J. M.; Thygesen, K. S. Optimizing porphyrins for dye sensitized solar cells using large-scale ab initio calculations. Phys. Chem. Chem. Phys. 2014, 16, 16246−16254. (8) Ornso, K. B.; Garcia-Lastra, J. M.; De La Torre, G.; Himpsel, F. J.; Rubio, A.; Thygesen, K. S. Design of two-photon molecular tandem architectures for solar cells by ab initio theory. Chem. Sci. 2015, 6, 3018−3025. (9) Cole, J. M.; Low, K. S.; Ozoe, H.; Stathi, P.; Kitamura, C.; Kurata, H.; Rudolf, P.; Kawase, T. Data mining with molecular design rules identifies new class of dyes for dye-sensitised solar cells. Phys. Chem. Chem. Phys. 2014, 16, 26684−26690. (10) Rutledge, L. R.; McAfee, S. M.; Welch, G. C. Design and Computational Characterization of Non-Fullerene Acceptors for Use in Solution-Processable Solar Cells. J. Phys. Chem. A 2014, 118, 7939− 7951. (11) Bérubé, N.; Gosselin, V.; Gaudreau, J.; Côté, M. Designing Polymers for Photovoltaic Applications Using ab Initio Calculations. J. Phys. Chem. C 2013, 117, 7964−7972. (12) Hybertsen, M. S.; Louie, S. G. Electron correlation in semiconductors and insulators - band-gaps and quasi-particle energies. Phys. Rev. B: Condens. Matter Mater. Phys. 1986, 34, 5390−5413. (13) Hedin, L. New method for calculating 1-particle Green’s function with application to electron-gas problem. Phys. Rev. 1965, 139, A796. (14) Onida, G.; Reining, L.; Rubio, A. Electronic excitations: densityfunctional versus many-body Green’s-function approaches. Rev. Mod. Phys. 2002, 74, 601−659. (15) Aryasetiawan, F.; Gunnarsson, O. The GW method. Rep. Prog. Phys. 1998, 61, 237−312. (16) Aulbur, W. G.; Jonsson, L.; Wilkins, J. W. Quasiparticle Calculations in solids. In Solid State Physics: Advances in Research and Applications; Academic Press Inc: San Diego, 2000; Vol. 54, pp 1−218. (17) Blase, X.; Attaccalite, C.; Olevano, V. First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 115103. (18) Grossman, J. C.; Rohlfing, M.; Mitas, L.; Louie, S. G.; Cohen, M. L. High accuracy many-body calculational approaches for excitations in molecules. Phys. Rev. Lett. 2001, 86, 472−475. 623


Article

Journal of Chemical Theory and Computation (39) Droghetti, A.; Cinchetti, M.; Sanvito, S. Electronic structure of metal quinoline molecules from G0W0 calculations. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 245137. (40) Faber, C.; Boulanger, P.; Attaccalite, C.; Duchemin, I.; Blase, X. Excited states properties of organic molecules: from density functional theory to the GW and Bethe−Salpeter Green’s function formalisms. Philos. Trans. R. Soc., A 2014, 372, 20130271. (41) Faber, C.; Boulanger, P.; Duchemin, I.; Attaccalite, C.; Blase, X. Many-body Green’s function GW and Bethe-Salpeter study of the optical excitations in a paradigmatic model dipeptide. J. Chem. Phys. 2013, 139, 194308. (42) Sharifzadeh, S.; Tamblyn, I.; Doak, P.; Darancet, P. T.; Neaton, J. B. Quantitative molecular orbital energies within a G 0 W 0 approximation. Eur. Phys. J. B 2012, 85, 323. (43) Körbel, S.; Boulanger, P.; Duchemin, I.; Blase, X.; Marques, M. A. L.; Botti, S. Benchmark Many-Body GW and Bethe−Salpeter Calculations for Small Transition Metal Molecules. J. Chem. Theory Comput. 2014, 10, 3934−3943. (44) Ke, S.-H. All-electron GW methods implemented in molecular orbital space: Ionization energy and electron affinity of conjugated molecules. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 205415. (45) Krause, K.; Harding, M. E.; Klopper, W. Coupled-cluster reference values for the GW27 and GW100 test sets for the assessment of GW methods. Mol. Phys. 2015, 113, 1952−1960. (46) Pham, T. A.; Nguyen, H.-V.; Rocca, D.; Galli, G. GW calculations using the spectral decomposition of the dielectric matrix: Verification, validation, and comparison of methods. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 155148. (47) Caruso, F.; Rinke, P.; Ren, X.; Scheffler, M.; Rubio, A. Unified description of ground and excited states of finite systems: the selfconsistent GW approach. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 081102. (48) van Setten, M. J.; Caruso, F.; Sharifzadeh, S.; Ren, X.; Scheffler, M.; Liu, F.; Lischner, J.; Lin, L.; Deslippe, J. R.; Louie, S. G.; Yang, C.; Weigend, F.; Neaton, J. B.; Evers, F.; Rinke, P. GW 100: Benchmarking G0W0 for molecular systems. J. Chem. Theory Comput. 2015, 11, 5665. (49) Malagoli, M.; Coropceanu, V.; da Silva Filho, D. A.; Brédas, J. L. A multimode analysis of the gas-phase photoelectron spectra in oligoacenes. J. Chem. Phys. 2004, 120, 7490−7496. (50) Gallandi, L.; Körzdörfer, T. Long-range corrected DFT meets GW: Vibrationally resolved photoelectron spectra from first principles. J. Chem. Theory Comput. 2015, 11, 5391−5400. (51) Lias, S. G.; Bartmess, J. E. Gas-phase ion thermochemistry. http://webbook.nist.gov/chemistry/ion/ (accessed July 1, 2015). (52) Richard, R. M.; Marshall, M. S.; Dolgounitcheva, O.; Ortiz, J. V.; Brédas, J. L.; Marom, N.; Sherrill, C. D. Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules. I. Reference Data at the CCSD(T) Complete Basis Set Limit. J. Chem. Theory Comput. 2016, DOI: 10.1021/acs.jctc.5b00875. (53) Gallandi, L.; Marom, N.; Rinke, P.; Körzdörfer, T. Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules II: Non-empirically Tuned Long-range Corrected Hybrid Functionals. J. Chem. Theory Comput. 2016, DOI: 10.1021/acs.jctc.5b00873. (54) Dolgounitcheva, O.; Díaz-Tinoco, M.; Zakrzewski, V. G.; Richard, R. M.; Marshall, M. S.; Marom, N.; Sherrill, C. D.; Ortiz, J. V. Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules IV: Electron-Propagator Methods. J. Chem. Theory Comput. 2016, DOI: 10.1021/acs.jctc.5b00872. (55) von Barth, U.; Holm, B. Self-consistent GW0 results for the electron gas: Fixed screened potential W_{0} within the randomphase approximation. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 8411−8419. (56) Ren, X.; Marom, N.; Caruso, F.; Scheffler, M.; Rinke, P. Beyond the GW approximation: A second-order screened exchange correction. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 081104. (57) We note that there are additional GW methods that are not benchmarked here, including eigenvalue partial self-consistency and

quasi-particle self-consistent GW. The performance of these methods for the acceptors benchmark set may be examined by others. (58) Frisch, M. J.; G. W. T, Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Ö . Farkas, Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J., Gaussian 09, Revision A.1. Gaussian, Inc.: Wallingford, CT, 2009. (59) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Ab-initio calculation of vibrational absorption and circular-dichroism spectra using density-functional force-fields. J. Phys. Chem. 1994, 98, 11623−11627. (60) Blum, V.; Gehrke, R.; Hanke, F.; Havu, P.; Havu, V.; Ren, X.; Reuter, K.; Scheffler, M. Ab initio molecular simulations with numeric atom-centered orbitals. Comput. Phys. Commun. 2009, 180, 2175− 2196. (61) Havu, V.; Blum, V.; Havu, P.; Scheffler, M. Efficient O(N) integration for all-electron electronic structure calculation using numeric basis functions. J. Comput. Phys. 2009, 228, 8367−8379. (62) Caruso, F.; Rinke, P.; Ren, X.; Rubio, A.; Scheffler, M. Selfconsistent GW: All-electron implementation with localized basis functions. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 075105. (63) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (64) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple (vol 77, pg 3865, 1996). Phys. Rev. Lett. 1997, 78, 1396−1396. (65) Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158−6170. (66) Ortiz, J. V. Electron propagator theory: an approach to prediction and interpretation in quantum chemistry. Wiley Interdisciplinary Reviews: Computational Molecular Science 2013, 3, 123−142. (67) Zakrzewski, V. G.; Dolgounitcheva, O.; Zakjevskii, A. V.; Ortiz, J. V. Ab initio Electron Propagator Calculations on Electron Detachment Energies of Fullerenes, Macrocyclic Molecules, and Nucleotide Fragments. In Adv. Quantum Chem.; John, R. S., Erkki, B., Eds.; Academic Press: 2011; Vol. 62, Chapter 3, pp 105−136. (68) Vydrov, O. A.; Heyd, J.; Krukau, A. V.; Scuseria, G. E. Importance of short-range versus long-range Hartree-Fock exchange for the performance of hybrid density functionals. J. Chem. Phys. 2006, 125, 074106. (69) Vydrov, O. A.; Scuseria, G. E. Assessment of a long-range corrected hybrid functional. J. Chem. Phys. 2006, 125, 234109. (70) Stein, T.; Kronik, L.; Baer, R. Prediction of charge-transfer excitations in coumarin-based dyes using a range-separated functional tuned from first principles. J. Chem. Phys. 2009, 131, 244119. (71) Stein, T.; Kronik, L.; Baer, R. Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory. J. Am. Chem. Soc. 2009, 131, 2818−2820. (72) Baer, R.; Livshits, E.; Salzner, U. Tuned Range-Separated Hybrids in Density Functional Theory. Annu. Rev. Phys. Chem. 2010, 61, 85−109. (73) Bruneval, F.; Vast, N.; Reining, L. Effect of self-consistency on quasiparticles in solids. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 045102. (74) Ren, X.; Marom, N.; Caruso, F.; Scheffler, M.; Rinke, P. Beyond the GW approximation: a second-order screened exchange correction. 624


Article

Journal of Chemical Theory and Computation Phys. Rev. B: Condens. Matter Mater. Phys. 2015, DOI: 10.1103/ PhysRevB.92.081104. (75) We have chosen to use standard settings of FHI-aims (standard NAO basis sets and the default analytical continuation scheme) for the benchmark calculations in order for the results to reflect the performance a typical user can expect out of the code. (76) Marom, N.; Ren, X. G.; Moussa, J. E.; Chelikowsky, J. R.; Kronik, L. Electronic structure of copper phthalocyanine from G0W0 calculations. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 195143. (77) Zhang, I. Y.; Ren, X.; Rinke, P.; Blum, V.; Scheffler, M. Numeric atom-centered-orbital basis sets with valence-correlation consistency from H to Ar. New J. Phys. 2013, 15, 123033. (78) Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007−1023. (79) Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water. J. Chem. Phys. 1997, 106, 9639−9646. (80) Marom, N.; Kö rzdö rfer, T.; Ren, X.; Tkatchenko, A.; Chelikowsky, J. R. Size Effects in the Interface Level Alignment of Dye-Sensitized TiO2 Clusters. J. Phys. Chem. Lett. 2014, 5, 2395− 2401. (81) Jain, M.; Deslippe, J.; Samsonidze, G.; Cohen, M. L.; Chelikowsky, J. R.; Louie, S. G. Improved quasiparticle wave functions and mean field for G0W0 calculations: Initialization with the COHSEX operator. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 115148. (82) Gatti, M.; Bruneval, F.; Olevano, V.; Reining, L. Understanding Correlations in Vanadium Dioxide from First Principles. Phys. Rev. Lett. 2007, 99, 266402. (83) Vidal, J.; Botti, S.; Olsson, P.; Guillemoles, J.-F.; Reining, L. Strong Interplay between Structure and Electronic Properties in CuIn(S,Se)2: A First-Principles Study. Phys. Rev. Lett. 2010, 104, 056401. (84) Boulanger, P.; Jacquemin, D.; Duchemin, I.; Blase, X. Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe−Salpeter Approach. J. Chem. Theory Comput. 2014, 10, 1212− 1218. (85) Körzdörfer, T.; Parrish, R. M.; Marom, N.; Sears, J. S.; Sherrill, C. D.; Brédas, J.-L. Assessment of the performance of tuned rangeseparated hybrid density functionals in predicting accurate quasiparticle spectra. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 205110. (86) Körzdörfer, T.; Brédas, J.-L. Organic Electronic Materials: Recent Advances in the DFT Description of the Ground and Excited States Using Tuned Range-Separated Hybrid Functionals. Acc. Chem. Res. 2014, 47, 3284−3291. (87) Nelson, W.; Bokes, P.; Rinke, P.; Godby, R. W. Self-interaction in Green’s-function theory of the hydrogen atom. Phys. Rev. A: At., Mol., Opt. Phys. 2007, 75, 032505. (88) Aryasetiawan, F.; Sakuma, R.; Karlsson, K. GW approximation with self-screening correction. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 035106. (89) Fonari, A.; Sutton, C.; Brédas, J.-L.; Coropceanu, V. Impact of exact exchange in the description of the electronic structure of organic charge-transfer molecular crystals. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 165205. (90) Perdew, J. P.; Zunger, A. Self-interaction correction to densityfunctional approximations for many-electron systems. Phys. Rev. B: Condens. Matter Mater. Phys. 1981, 23, 5048−5079. (91) Körzdörfer, T.; Marom, N. A Strategy for Finding a Reliable Starting Point for G0W0 Demonstrated for Molecules. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 041110. (92) Salomon, E.; Amsalem, P.; Marom, N.; Vondracek, M.; Kronik, L.; Koch, N.; Angot, T. Electronic structure of CoPc adsorbed on Ag(100): Evidence for molecule-substrate interaction mediated by Co 3d orbitals. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 075407.

(93) Körzdörfer, T.; Kümmel, S.; Marom, N.; Kronik, L. When to trust photoelectron spectra from Kohn-Sham eigenvalues: The case of organic semiconductors. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 201205. (94) Körzdörfer, T.; Kümmel, S.; Marom, N.; Kronik, L. When to trust photoelectron spectra from Kohn-Sham eigenvalues: The case of organic semiconductors (vol 79, 201205, 2009). Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 129903. (95) Körzdörfer, T. On the relation between orbital-localization and self-interaction errors in the density functional theory treatment of organic semiconductors. J. Chem. Phys. 2011, 134, 094111. (96) Ruzsinszky, A.; Perdew, J. P.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E. Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals. J. Chem. Phys. 2006, 125, 194112. (97) Ruzsinszky, A.; Perdew, J. P.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E. Density functionals that are one- and two- are not always many-electron self-interaction-free, as shown for H2+, He2+, LiH+, and Ne2+. J. Chem. Phys. 2007, 126, 104102. (98) Vydrov, O. A.; Scuseria, G. E.; Perdew, J. P. Tests of functionals for systems with fractional electron number. J. Chem. Phys. 2007, 126, 154109. (99) Maier, J. P.; Muller, J. f.; Kubota, T.; Yamakawa, M. Ionisation Energies and the Electronic Structures of the N-oxides of Azanaphthalenes and azaanthracenes. Helv. Chim. Acta 1975, 58, 1641−1648. (100) Hush, N. S.; Cheung, A. S.; Hilton, P. R. Binding energies of πand “lone-pair”-levels in mono- and diaza-phenanthrenes and anthracenes: An He(I) photoelectron spectroscopic study. J. Electron Spectrosc. Relat. Phenom. 1975, 7, 385−400. (101) Amashukeli, X.; Winkler, J. R.; Gray, H. B.; Gruhn, N. E.; Lichtenberger, D. L. Electron-Transfer Reorganization Energies of Isolated Organic Molecules†. J. Phys. Chem. A 2002, 106, 7593−7598. (102) Dillow, G. W.; Kebarle, P. Electron affinities of aza-substituted polycyclic aromatic hydrocarbons. Can. J. Chem. 1989, 67, 1628−1631. (103) Rabalais, J. W. Photoelectron Spectroscopic Investigation of the Electronic Structure of Nitromethane and Nitrobenzene. J. Chem. Phys. 1972, 57, 960. (104) Chowdhury, S.; Heinis, T.; Grimsrud, E. P.; Kebarle, P. Entropy changes and electron affinities from gas-phase electrontransfer equilibria: A- + B = A + B. J. Phys. Chem. 1986, 90, 2747− 2752. (105) Fukuda, E. K.; McIver, R. T. Relative electron affinities of substituted benzophenones, nitrobenzenes, and quinones. J. Am. Chem. Soc. 1985, 107, 2291−2296. (106) Chen, E. C. M.; Chen, E. S.; Milligan, M. S.; Wentworth, W. E.; Wiley, J. R. Experimental determination of the electron affinities of nitrobenzene, nitrotoluenes, pentafluoronitrobenzene, and isotopic nitrobenzenes and azulenes. J. Phys. Chem. 1992, 96, 2385−2390. (107) Chen, E. C. M.; Wiley, J. R.; Batten, C. F.; Wentworth, W. E. Determination of the electron affinities of molecules using negative ion mass spectrometry. J. Phys. Chem. 1994, 98, 88−94. (108) Desfrançois, C.; Périquet, V.; Lyapustina, S. A.; Lippa, T. P.; Robinson, D. W.; Bowen, K. H.; Nonaka, H.; Compton, R. N. Electron binding to valence and multipole states of molecules: Nitrobenzene, para- and meta-dinitrobenzenes. J. Chem. Phys. 1999, 111, 4569−4576. (109) Egdell, R.; Green, J. C.; Rao, C. N. R. Photoelectron spectra of substituted benzenes. Chem. Phys. Lett. 1975, 33, 600−607. (110) Kobayashi, T.; Nagakura, S. Photoelectron Spectra of Substituted Benzenes. Bull. Chem. Soc. Jpn. 1974, 47, 2563−2572. (111) Kobayashi, T. A simple general tendency in photoelectron angular distributions of some monosubstituted benzenes. Phys. Lett. A 1978, 69, 105−108. (112) Palmer, M. H.; Moyes, W.; Spiers, M.; Neil, J.; Ridyard, A. The electronic structure of substituted benzenes ab initio calculations and photoelectron spectra for nitrobenzene, the nitrotoluenes, dinitrobenzenes and fluoronitrobenzenes. J. Mol. Struct. 1979, 55, 243−263. (113) Kimura, K. Handbook of HeI Photoelectron Spectra of Fundamental Organic Molecules: Ionization Energies, Ab Initio Assign625


Article

Journal of Chemical Theory and Computation ments, and Valence Electronic Structure for 200 Molecules; Scientific Societies Press: Tokyo, 1981. (114) Klasinc, L.; Kovac, B.; Gusten, H. Photoelectron spectra of acenes. Electronic structure and substituent effects. Pure Appl. Chem. 1983, 55, 289−298. (115) Katsumata, S.; Shiromaru, H.; Mitani, K.; Iwata, S.; Kimura, K. Photoelectron angular distribution and assignments of photoelectron spectra of nitrogen dioxide, nitromethane and nitrobenzene. Chem. Phys. 1982, 69, 423−431. (116) Galasso, V.; Colonna, F. P.; Distefano, G. Photoelectron spectra of 1,2-indandione, 1,3-indandione and heterocyclic analogues. J. Electron Spectrosc. Relat. Phenom. 1977, 10, 227−237. (117) Almemark, M.; Bäckvall, J. E.; Moberg, C.; Åkermark, B.; Åsbrink, L.; Roos, B. Ab initio calculations and assignment of photoelectron spectra of maleic and succinic anhydride. Tetrahedron 1974, 30, 2503−2508. (118) Paul, G.; Kebarle, P. Electron affinities of cyclic unsaturated dicarbonyls: maleic anhydrides, maleimides, and cyclopentenedione. J. Am. Chem. Soc. 1989, 111, 464−470. (119) Compton, R. N.; Reinhardt, P. W.; Cooper, C. D. Mass spectrometry utilizing collisional ionization of cesium: Maleic anhydride and succinic anhydride. J. Chem. Phys. 1974, 60, 2953. (120) Brundle, C. R.; Robin, M. B.; Kuebler, N. A. Perfluoro effect in photoelectron spectroscopy. II. Aromatic molecules. J. Am. Chem. Soc. 1972, 94, 1466−1475. (121) Dewar, M. J. S.; Worley, S. D. Photoelectron Spectra of Molecules. I. Ionization Potentials of Some Organic Molecules and Their Interpretation. J. Chem. Phys. 1969, 50, 654−667. (122) Kobayashi, T. Photoelectron spectra of p-benzoquinones. J. Electron Spectrosc. Relat. Phenom. 1975, 7, 349−353. (123) Lauer, G.; Schäfer, W.; Schweig, A. Assignment of the four lowest ionized states of p-benzoquinone and the question of “lone pair splitting” in this system. Chem. Phys. Lett. 1975, 33, 312−315. (124) Dougherty, D.; McGlynn, S. P. Photoelectron spectroscopy of carbonyls. 1,4-benzoquinones. J. Am. Chem. Soc. 1977, 99, 3234. (125) Cowan, D. O.; Gleiter, R.; Hashmall, J. A.; Heilbronner, E.; Hornung, V. Interaction between the Orbitals of Lone Pair Electrons in Dicarbonyl Compounds. Angew. Chem., Int. Ed. Engl. 1971, 10, 401−402. (126) Fu, Q.; Yang, J.; Wang, X.-B. On the Electronic Structures and Electron Affinities of the m-Benzoquinone (BQ) Diradical and the o-, p-BQ Molecules: A Synergetic Photoelectron Spectroscopic and Theoretical Study. J. Phys. Chem. A 2011, 115, 3201−3207. (127) Schiedt, J.; Weinkauf, R. Resonant photodetachment via shape and Feshbach resonances: p-benzoquinone anions as a model system. J. Chem. Phys. 1999, 110, 304−314. (128) Heinis, T.; Chowdhury, S.; Scott, S. L.; Kebarle, P. Electron affinities of benzo-, naphtho-, and anthraquinones determined from gas-phase equilibria measurements. J. Am. Chem. Soc. 1988, 110, 400− 407. (129) Sato, N.; Seki, K.; Inokuchi, H. Ultraviolet photoelectron spectra of tetrahalogeno-p-benzoquinones and hexahalogenobenzenes in the solid state. J. Chem. Soc., Faraday Trans. 2 1981, 77, 47−53. (130) Cooper, C. D.; Frey, W. F.; Compton, R. N. Negative ion properties of fluoranil, chloranil, and bromanil: Electron affinities. J. Chem. Phys. 1978, 69, 2367. (131) Dauth, M.; Korzdorfer, T.; Kummel, S.; Ziroff, J.; Wiessner, M.; Scholl, A.; Reinert, F.; Arita, M.; Shimada, K. Orbital Density Reconstruction for Molecules. Phys. Rev. Lett. 2011, 107, 107. (132) Kaplan, F.; Weigend, F.; Evers, F.; van Setten, M. J. Offdiagonal self-energy terms and partially self-consistency in GWcalculations for single molecules: efficient implementation and quantitative effects on ionization potentials. J. Chem. Theory Comput. 2015, 11, 5152. (133) Difference in peak height results from the numerical grid resolution of the discretized spectral function. (134) Caruso, F.; Atalla, V.; Ren, X.; Rubio, A.; Scheffler, M.; Rinke, P. First-principles description of charge transfer in donor-acceptor

compounds from self-consistent many-body perturbation theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 085141.

626


Accurate Ionization Potentials and Electron Affinities of Acceptor

Recommend Documents