We present results and analyses for the photoelectron spectra of small copper oxide cluster anions (CuO, CuO2, CuO3, and Cu2O). The spectra are computed using various techniques, including density functional theory (DFT) with semi-local, global hybrid, and optimally tuned range-separated hybrid functionals, as well as many-body perturbation theory within the GW approximation based on various DFT starting points. The results are compared with each other and with the available experimental data. We conclude that as in many metal-organic systems, self-interaction errors are a major issue that is mitigated by hybrid functionals. However, these need to be balanced against a strong role of non-dynamical correlation—especially in smaller, more symmetric systems—where errors are alleviated by semi-local functionals. The relative importance of the two phenomena, including practical ways of balancing the two constraints, is discussed in detail.

Gas-phase photoelectron spectroscopy (PES) and its first-principles simulation using various computational approaches have proven to be a powerful combination for probing the electronic structure of atomic and molecular clusters.1,2 In gas-phase PES, electrons are emitted upon absorption of (typically ultraviolet) light, allowing for the exploration of electron removal energies. Computational methods based on density functional theory (DFT) within the (generalized) Kohn-Sham scheme2–4 or many-body perturbation theory within the GW approximation4–9 are some of the most commonly used approaches to determine the energetics of such charged excitations. The degree of accuracy with which various computational methods can predict these excitations depends not only on the fundamental limitations and practical implementations of the underlying theoretical framework but also on the nature (e.g., chemical composition, size) of the system being studied. From this perspective, transition metal oxide clusters are stringent test cases for state-of-the-art computational methods, owing to enhanced electron correlations inherent in them, as well as their open-shell character,10 which require a balanced and accurate description of dynamical and non-dynamical electron correlation. In addition to the computational challenges in modeling their excited states,5,11,12 bulk and nanostructured transition metal oxides have traditionally been systems of great technological and scientific interest, as they possess a wide range of complex, and often desirable, structural, electronic, and optical properties resulting from the subtle interplay of their orbital, spin, and charge degrees of freedom.13 Motivated by these observations, here we examine the performance of various state-of-the-art DFT and GW methods in modeling the photoelectron spectra of copper oxide anion clusters, by comparing their predictions with each other and with the available experimental data.

We focus on four small clusters, Cu2O, CuO, CuO2, and CuO3, which span a relatively wide range of Cu content and exhibit strong electronic correlations. Photoelectron spectra for these clusters have been available up to photon energies of ∼5-6 eV since the pioneering studies of Polak et al.14 and Wang et al.15,16 Among them, copper monoxide has been extensively studied both experimentally14,16,17 and theoretically.10,18–31 Electronic and optical properties of CuO2 in various charge states have also been of significant interest, especially with regards to its lowest energy structure (bent versus linear) in various experimental16,32–36 and theoretical studies.37–43 There have been fewer investigations on the electronic structures of CuO316,44,45 and Cu2O.10,15,46,47 The first-principles methods explored here include DFT with semilocal and global hybrid exchange-correlation functionals (based on shifted eigenvalue spectra), DFT with optimally tuned range-separated hybrid (OT-RSH) functionals, and many-body perturbation theory techniques within the GW approximation using semilocal or global hybrid functional starting points.

This article is arranged as follows. We begin with an overview of the computational methods and their corresponding parameters, given in Sec. II. This is followed in Sec. III by a brief discussion on the relaxed structures of the copper oxide clusters considered in this study and detailed analyses of the computed photoelectron spectra. Trends in the photoelectron spectra of these clusters are analyzed in terms of various factors, including self-interaction errors (SIEs), spatial extent of the relevant orbitals, amount of exact exchange in the DFT description, importance of non-dynamical correlation, spin-splitting, and starting-point dependence in GW calculations. Finally, we summarize our findings and analyses in Sec. IV.

Computations within DFT were carried out using the NWChem code,48 Version 6.5, using the aug-cc-pVTZ basis set. In these calculations, the geometries of the anions were optimized using both the Perdew-Burke-Ernzerhof (PBE)49 and PBE050 exchange-correlation functionals. The photoelectron spectra were simulated by convoluting the DFT eigenvalue spectra of the anions with a 0.1-eV-wide Gaussian broadening function, without taking photoionization cross sections into account. For the PBE and PBE0 calculations, the eigenvalue spectra were shifted so as to align the first peak with the vertical ionization potential of the anion, computed as the total energy difference between the anionic and the neutral cluster at the fixed anion geometry.51,52 The amounts of the shifts and the vertical ionization potentials for each case are provided in the supplementary material. OT-RSH calculations were based on the combination of full long-range Fock exchange, a fraction of short-range Fock exchange, and PBE-based semi-local short-range and long-range components, along with PBE correlation. The calculations were performed for two different values of the short-range Fock exchange fraction (α), 0.0 and 0.2,53 as in the functionals long-range-corrected (LC)-ωPBE54 and LC-ωPBE0,55 respectively. In difference to the parent functionals, the range-separation parameter, γ, was optimally tuned per system so as to satisfy the ionization potential theorem,56ϵHγopt=IPγopt, where ϵH and IP are the highest occupied molecular orbital (HOMO) energy and (vertical) ionization potential of the anion, respectively.

GW calculations were performed using molgw57 within the perturbative “one-shot” G0W0 methodology. In this approximation, a Green’s function, G0, computed using Kohn-Sham wavefunctions and eigenvalues, and the screened Coulomb interaction, W0, obtained from a dynamical polarizability computed within the random-phase approximation, are used to compute the self-energy Σ = iG0W0. The starting electronic structures were obtained from DFT calculations with PBE and various hybrid functionals performed within the same package. Quasiparticle wavefunctions were represented using atom-centered Gaussian basis sets ranging from aug-cc-pVDZ to aug-cc-pV5Z58 with effective core potentials for Cu.59 The Coulomb interaction terms were evaluated using the resolution-of-the-identity approximation.60,61 The GW energies in the complete basis set limit were obtained by fitting the extrapolated energy E and coefficient c to E(X) = E + cX−2, where X = 2–5 for aug-cc-pVDZ through aug-cc-pV5Z basis sets, as follows: For Cu2O, CuO, and CuO3, the variation in the computed GW quasiparticle energies with a PBE starting point was observed to be non-monotonic as a function of the basis set size when aug-cc-pVQZ and aug-cc-pV5Z basis sets were included. Accordingly, for these three cluster anions, the G0W0@PBE quasiparticle energies were obtained by fitting to X = 2–3 only, while for CuO2 and all other starting points, we extrapolated from X = 3–5.

The optimized structures of copper oxide cluster anions are shown in Fig. 1. With the exception of one of the isomers of CuO3, the computed inter-atomic distances and angles are found to be mostly insensitive to the choice of the exchange-correlation functional. Both CuO and CuO3 of C2v symmetry possess singlet ground states, whereas the linear CuO2 (D∞h) and the CuO3 cluster of Cs symmetry possess triplet ground states. The bent Cu2O (C2v symmetry) is found to have a doublet ground state. Importantly, the slight differences observed in the bond lengths and angles did not have a significant effect on the computed photoelectron spectra. Therefore, differences in the photoelectron spectra computed at various levels of theory, reported next, reflect true electronic structure effects rather than differences inherited from different geometries.

FIG. 1.

Structures and symmetries of the CuxOy clusters considered in this study. O atoms are denoted by smaller red balls and Cu atoms are denoted by larger blue balls. Optimized bond lengths and angles have been computed with the PBE and PBE0 (in parentheses) functionals. For CuO3, the structure with Cs symmetry on the right is lower in energy.

FIG. 1.

Structures and symmetries of the CuxOy clusters considered in this study. O atoms are denoted by smaller red balls and Cu atoms are denoted by larger blue balls. Optimized bond lengths and angles have been computed with the PBE and PBE0 (in parentheses) functionals. For CuO3, the structure with Cs symmetry on the right is lower in energy.

Close modal

We start our analysis of computed photoelectron spectra with Cu2O, which has the largest Cu content of the anions considered in this study. Figure 2 shows the experimental photoelectron spectrum of Cu2O along with spectra computed at various levels of theory. The experimental spectrum consists of a low-energy peak at 1.1 eV (X), followed by a sizeable gap, and sharp peaks at 2.53 (A), 2.66 (B), 2.85 (C), 2.95 (D), and 3.08 eV (E). Wang et al.15 interpreted the large gap between the X and A peaks as an indicator of a large highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap for neutral Cu2O, suggesting that the neutral cluster has a closed-shell electronic structure. The similarity in the spectral features and intensity ratios between the A and C bands, as well as between the B and D bands, led Wang et al. to suggest that these bands arise from the removal of either a spin-down or a spin-up electron from orbitals of the same character.

FIG. 2.

Experimental photoelectron spectrum of Cu2O (adapted from Ref. 15), along with spectra computed with PBE, PBE0, OT-RSH (α = 0, γopt = 0.221 a.u.−1), OT-RSH (α = 0.2, γopt = 0.188 a.u.−1), G0W0@PBE, G0W0@PBE0, G0W0@BHLYP, and equation-of-motion coupled-cluster (EOM-CC) (adapted from Ref. 10). The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster. Contour plots of the selected molecular orbitals (with symmetries), in both majority (↑) and minority (↓) spin channels, are shown on the right-hand side, with matching color codes displayed in the spectra (for G0W0@PBE, the eigenvalues are color-coded). For PBE and G0W0@PBE, the spectra are scaled by half compared with the others to fit them in the panels.

FIG. 2.

Experimental photoelectron spectrum of Cu2O (adapted from Ref. 15), along with spectra computed with PBE, PBE0, OT-RSH (α = 0, γopt = 0.221 a.u.−1), OT-RSH (α = 0.2, γopt = 0.188 a.u.−1), G0W0@PBE, G0W0@PBE0, G0W0@BHLYP, and equation-of-motion coupled-cluster (EOM-CC) (adapted from Ref. 10). The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster. Contour plots of the selected molecular orbitals (with symmetries), in both majority (↑) and minority (↓) spin channels, are shown on the right-hand side, with matching color codes displayed in the spectra (for G0W0@PBE, the eigenvalues are color-coded). For PBE and G0W0@PBE, the spectra are scaled by half compared with the others to fit them in the panels.

Close modal

With both the PBE and PBE0 functionals, the ground state of Cu2O is found to be a doublet and the neutral cluster is found to be a closed-shell singlet in its ground state. An overall comparison of the experimental data with the DFT-computed spectra (Fig. 2) shows that the (shifted) PBE does not provide quantitatively accurate predictions for the photoelectron peak positions, while various hybrid functionals (with or without range separation) perform significantly better. The most obvious discrepancy between experimental data and PBE predictions is the separation of the X band from the AE bands, which is severely underestimated. The XA separation of 0.56 eV computed with PBE is much smaller than the experimental value of 1.43 eV. Introducing a fraction of Fock exchange, either globally or in range-separated form (the latter to correct the asymptotic potential), significantly improves the agreement with the experiment, with the OT-RSH approach additionally removing the need for shifting the spectra.53,56 Specifically, the mean absolute errors (MAEs) averaged over the six measured (XE) peaks for shifted PBE, shifted PBE0, OT-RSH (α = 0), and OT-RSH (α = 0.2) calculations are 0.60, 0.19, 0.12, and 0.10 eV, respectively. These trends are qualitatively the same as those observed previously for a metal-organic Cu-based complex, copper phthalocyanine (CuPc).53,62,63

The predictions from the G0W0 approximation with a PBE starting point (Fig. 2) are strikingly poor when compared with the experimental data. For example, the X and A peaks, as well as their separation, are underestimated severely, by 0.6, 1.44, and 0.59 eV, respectively. In fact, comparison of the spectra displayed in Fig. 2 reveals that the G0W0@PBE performs even worse than the shifted PBE spectrum, leading to a MAE of 1.36 eV. Starting the G0W0 calculation with a DFT calculation that contains some fraction of exact exchange leads to better agreement with experiment. For example, the G0W0@PBE0 predictions are considerably better than the G0W0@PBE results. In spite of this improvement, the G0W0@PBE0 still leads to underbound quasiparticle levels. Similar to the shifted PBE case, the shifted PBE0 predictions are more accurate than the G0W0@PBE0 predictions, for which the MAE over the six experimentally measured peaks is 0.44 eV. To investigate the effect of the amount of exact exchange in the G0W0 starting point further, we performed G0W0 calculations on top of a BHLYP64 starting point (with 50% exact exchange). As shown in Fig. 2, the G0W0@BHLYP results are in nearly perfect agreement with the experimental data, with the largest difference being 0.07 eV (for the X peak) and an overall MAE of less than 0.04 eV. While the importance of using hybrid functional starting points for the ensuing G0W0 calculations has been highlighted in previous studies of sp-bonded molecular systems,7,65–68 of metal-organic complexes,63,69 and of bulk metal oxides,5,70 the results presented here show just how poorly the PBE starting point can be and that basing a GW calculation on it can in fact make things worse rather than better.

To gain further insight into our computational results, Fig. 2 additionally compares them with the quantum chemistry calculations reported in Ref. 10, which were performed within the equation-of-motion coupled-cluster (EOM-CC) approach, at the singles and doubles (CCSD) level, as well as from differences in CCSD with perturbative triples total energies between the doublet anion and the lowest neutral state (to determine the X peak). These calculations predict the photoelectron peaks XE at energies of 1.00, 2.62, 2.68, 3.09, 3.19, and 3.33 eV. Overall, these agree well with the experimental spectrum, with an MAE of 0.16 averaged over the six measured values. Remarkably, both the GW calculations (with an appropriately chosen starting point) and the DFT ones (based on OT-RSH) are overall as accurate as the EOM-CC ones, establishing that quantitative interpretation of the experimental data can be obtained for this cluster anion even without resorting to expensive wavefunction-based approaches.

Figure 2 also shows the DFT-computed molecular orbitals for the highest seven (counting both spin channels) occupied states of Cu2O. Based on the nodal structure along the Cu–O and Cu–Cu directions, these can be viewed as anti-bonding orbitals. The HOMO, possessing a1 symmetry, is a relatively extended orbital with significant Cu content, with appreciable contributions from 3d and 4s atomic orbitals and some from 4p, and smaller (∼15%) O 2p character. In going from PBE to PBE0, the shape of the molecular orbital remains virtually the same, but the Cu 4s content increases at the expense of a decrease in the Cu 3d content. A similar behavior is observed for the lower lying states of b2, b1, and a1 symmetry, for which the Cu 3d content decreases at the expense of an increase in the O 2p content. These observations are in accordance with the expectation that introduction of exact exchange should push the localized Cu 3d orbitals down in energy due to mitigation of self-interaction error (SIE).71 Another difference between PBE and PBE0 predictions is related to the ordering of the various orbitals. In particular, the ordering of the first five occupied orbitals in increasing the binding energy (BE) at the PBE level is a1↑b2↓b2↑b1↓b1↑, while the ordering at the PBE0 level is a1↑b1↓b2↓b2↑b1↑. While in some cases the relevant energy differences are small, such rearrangement of orbitals is generally related to (i) the spatial extent (degree of localization) of the relevant orbitals, which determines the amount that the introduction of exact exchange shifts the orbital eigenvalues in going from PBE to PBE0 to partially correct for SIE,2 and (ii) the tendency of exact exchange to segregate (energetically) orbitals belonging to the same spin channel. In this case, PBE0 leads to a configuration in which HOMO-1/HOMO-3 (and HOMO-2/HOMO-4), associated with A/C (and B/D) bands in the experiment, are in opposite spin channels (unlike PBE). They also have different symmetries (b1 and b2), in agreement with the experimental suggestion that A/C and B/D bands correspond to removal of spin-up/spin-down electrons from orbitals of the same character. When viewed both from the perspective of low MAE and the experimentally suggested ordering of the orbitals, the G0W0@BHLYP prediction with the a1↑b1↓b2↓b1↑b2↑ ordering and the OT-RSH (α = 0, and α = 0.2 being very close) prediction with the a1↑b2↓b1↓b2↑b1↑ ordering have the best overall agreement with the experiment.

In summary, the photoelectron spectra of Cu2O computed with hybrid functionals within DFT, or within the G0W0 approximation with hybrid functional starting points, are in much better agreement with the experimental data and EOM-CC results, as compared with the PBE (or G0W0@PBE) predictions. This is a somewhat expected finding, because as discussed above the introduction of Fock exchange mitigates SIE and binds orbitals of large Cu 3d content more strongly, while PBE typically underbinds such orbitals.

Figure 3 shows the computed and experimental photoelectron spectra of CuO. In the experiments of Polak et al.14 and Wu et al.,16 the first two main peaks (labeled X and Y in Fig. 3) were reported at 1.78 and 2.75 eV, respectively. In previous electronic structure calculations,18–21 the ground state of CuO was found to be a closed-shell singlet (1Σ+) that could roughly be described as 3d102224. Based on these studies and the relative intensity ratios, X and Y states were interpreted as arising from the removal of 2 and 2 electrons, leading to the X2Π ground state and Y2Σ+ excited state of CuO, respectively. The experiments of Wu et al. also revealed a broad and noisy band (labeled Z in Fig. 3) in the 4–6 eV energy range, which was interpreted as being due to the detachment of Cu 3d electrons. In addition, weak features at 1.27 and 3.18 eV were observed, which were attributed to transitions from an electronically excited state of CuO. In the following discussion, we only focus on the X, Y, and Z bands, corresponding to transitions from the ground state of CuO.

FIG. 3.

Experimental photoelectron spectrum of CuO (Ref. 16), along with spectra computed with PBE, PBE0, OT-RSH (α = 0, γopt = 0.225 a.u.−1), OT-RSH (α = 0.2, γopt = 0.138 a.u.−1), G0W0@PBE, G0W0@PBE0, G0W0@BHLYP, and EOM-CC (Ref. 10). The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster. Contour plots of molecular orbitals discussed in the text are shown on the right, with matching color codes displayed in the spectra.

FIG. 3.

Experimental photoelectron spectrum of CuO (Ref. 16), along with spectra computed with PBE, PBE0, OT-RSH (α = 0, γopt = 0.225 a.u.−1), OT-RSH (α = 0.2, γopt = 0.138 a.u.−1), G0W0@PBE, G0W0@PBE0, G0W0@BHLYP, and EOM-CC (Ref. 10). The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster. Contour plots of molecular orbitals discussed in the text are shown on the right, with matching color codes displayed in the spectra.

Close modal

We first discuss DFT and GW results for the X and Y bands. A comparison of the computed spectra with the experimental data surprisingly shows that the PBE provides the best predictions (within 0.05 eV of experiment) for the positions of the first two peaks. The PBE0 and OT-RSH (α = 0.2) predictions are virtually the same, but they significantly underestimate both the IP and the XY separation, by ∼0.5 eV. The OT-RSH predictions with α = 0 for the positions of the first two peaks (both IP and XY separation underestimated by ∼0.2 eV) lie between those of PBE and those of other hybrid functionals. Similar to the case of Cu2O, the G0W0@PBE results are poor, with both X and Y peaks underestimated by ∼1.4 eV. The use of a PBE0 starting point for G0W0 calculations significantly improves the quasiparticle energies, but the X and Y levels are still underestimated by ∼0.4 eV compared with the experiment. Better agreement with the experiment can be obtained at the G0W0@BHLYP level, where the X and Y peaks are predicted to be 0.07 and 0.32 eV below the experimental values.

As in the discussion of Cu2O, we additionally compare our results with those obtained from the EOM-CC calculations.10 The best results obtained from such calculations find transitions at 1.89 and 2.56 eV from the CuO reference. These values are within ∼0.1 and ∼0.2 eV of experimental data. The EOM-CC calculations also provide a convenient point of reference for the Z-band calculations, where the broad and noisy nature of the Z band precludes direct comparisons with the experiment. These higher energy transitions, arising from the ionization of orbitals with large Cu 3d character, were found to be 4.83, 5.11, and 5.57 eV.10 These values fall within the range (4–6 eV) of the Z band observed in the photoemission experiments and therefore indeed provide an adequate benchmark. Comparing this benchmark with the computational approaches employed here, we observe that PBE significantly underestimates the BEs of these higher energy states, while the hybrid functionals do slightly better. The G0W0@PBE predictions are again poor, while hybrid functional starting points, especially G0W0@BHLYP, perform much better. Overall, taking all five peaks (with BEs less than 6 eV) into account, the EOM-CC performs quite well. The G0W0@BHLYP results are also in good agreement with the experiment and quantum chemistry results. The PBE predictions are excellent for the two most loosely bound states, but they do not perform as well for states with higher BEs, while the opposite trend is observed for the case of hybrid functionals.

Figure 3 also provides the orbital characters of the molecular orbitals involved. The doubly degenerate HOMO/HOMO-1 has π character: These antibonding states can be described well as nearly equal mixtures of Cu dxz (dyz) and O 2px (2py) atomic orbitals. The non-degenerate HOMO-2 has anti-bonding σ character with significant Cu dz2 and 4s contribution and O 2pz admixture. The doubly degenerate HOMO-3/HOMO-4 are non-bonding Cu dxy and dx2y2 type orbitals. HOMO-5/HOMO-6 are also doubly degenerate, corresponding to bonding π orbitals, while HOMO-7 is a bonding σ orbital, both with large contributions from Cu d atomic orbitals.

The surprising observation that PBE outperforms PBE0 for the position of the first two (X, Y) peaks can be interpreted in terms of the spatial extent of the relevant orbitals and the compatibility of exact exchange and correlation. Figure 4 shows the unshifted eigenvalue spectra of CuO, computed for various values of exact exchange and semilocal (PBE) correlation. More specifically, using an exchange-correlation energy Exc(α, β) expressed as

Exc(α,β)=αEx,HF+(1α)Ex,PBE+βEc,PBE,
(1)

where Ex,HF is the Hartree-Fock exchange and Ex,PBE and Ec,PBE are semilocal PBE exchange and correlation, respectively, we performed a series of computations by varying α and β from 0 to 1. As discussed above, in general, one excepts stronger effects of adding Fock exchange to PBE calculations for more localized orbitals, as the SIE is more severe in spatially localized orbitals compared with the more extended ones. This is indeed what we observe in CuO too. While all orbitals become more bound (DFT eigenvalues decreasing), the magnitude of eigenvalue decrease correlates directly with the spatial extent of the molecular orbital involved. In particular, the spatial extent of the σ orbital is larger than that of the π orbital. As a result, as more Fock exchange is introduced, the eigenvalue of the π orbital decreases more significantly (by ∼5 eV) than that of the σ orbital (by ∼3.2 eV) in going from α = 0 to 1 (at β = 1). For the more localized lower lying orbitals with large d character, the eigenvalue shifts to lower energies upon increasing Fock exchange are considerably larger than those of σ and π orbitals. Among them, the bonding σ orbital (HOMO-7 at the PBE level) is the least localized one, and this orbital has the smallest eigenvalue shift in going from α = 0 to α = 1, at which point this σ orbital becomes the least bound one among orbitals of large d character, similar to what is observed for the case of σ and π orbitals.

FIG. 4.

Unshifted eigenvalue spectra of CuO computed for various values of exact exchange and PBE correlation, defined by Eq. (1). Orbitals of π, σ, non-bonding d, π, and σ character (same as in Fig. 3) are color-coded to highlight changes in their energies as a function of α and β. See text for more details.

FIG. 4.

Unshifted eigenvalue spectra of CuO computed for various values of exact exchange and PBE correlation, defined by Eq. (1). Orbitals of π, σ, non-bonding d, π, and σ character (same as in Fig. 3) are color-coded to highlight changes in their energies as a function of α and β. See text for more details.

Close modal

Of particular importance in Fig. 4 is the ordering of the orbitals at the HF level (equivalent to α = 1, β = 0), where the HOMO is incorrectly predicted to be of σ character and the doubly degenerate HOMO-1/HOMO-2 has π character. Adding more semilocal correlation to HF decreases the magnitude of the πσ separation slightly, but the HOMO still has σ character even at β = 1. Upon removing some of the exact exchange, however, the πσ ordering gets reversed, and for α ≲ 0.5, HOMO has π character, in agreement with the experimental data. Therefore, the small πσ separation predicted by PBE0 (α = 0.25, β = 1) can be traced to the incorrect description of the ordering at the HF level, with PBE0 still having “too much” exact exchange or “not enough” semilocal exchange. Since semilocal exchange is known to partially represent non-dynamical correlation,72–76 we attribute the apparent success of the PBE predictions (for the first two peaks) to a more accurate accounting of non-dynamical correlation in PBE compared with PBE0.

In order to investigate the effect of the degree of orbital localization on the difference between semilocal and hybrid functional predictions, we have also performed PBE and PBE0 calculations for diatomic molecules CuS and AgO and compared with CuO. Substituting Cu and O with isovalent elements in the next row should result in more extended d− and p−like orbitals, respectively. Figure 5 shows the predicted photoelectron spectra of CuS, AgO, and CuO at the PBE and PBE0 levels. For CuS, the agreement between PBE and PBE0 for the first two peaks (π and σ) is very good, much better than the case for CuO, both in terms of the position of the first peak (doubly degenerate HOMO and HOMO-1) and the πσ separation. Since these three orbitals making up the first two peaks have large S 3p character, they are significantly more spatially extended than the corresponding orbitals in CuO with large O 2p character. Indeed, we have observed no πσ crossing as a function of α for CuS, which has a doubly degenerate HOMO even at the HF level of theory. Lower lying orbitals of large Cu 3d character, on the other hand, are still significantly underbound at the PBE level compared with the PBE0, similar to what is observed in CuO. For AgO, on the other hand, the first two peaks have very similar behavior to that observed in CuO, with significant differences between PBE and PBE0 predictions. Also similar to CuO, the HOMO of AgO at HF level is found to be the singly degenerate orbital of σ character. For the lower lying orbitals of largely d character, on the other hand, the agreement between PBE and PBE0 predictions is quite good, as these molecular orbitals primarily derived from Ag 4d atomic orbitals are significantly more extended than their counterparts in CuO and undergo less severe self-interaction corrections when a fraction of Fock exchange is mixed in at the PBE0 level.

FIG. 5.

Eigenvalue spectra of CuS, AgO, and CuO computed with PBE and PBE0 functionals. The spectra are shifted to align the first peak with the IP of the anion cluster. Orbitals of π, σ, non-bonding d, π, and σ character are same as in Figs. 3 and 4.

FIG. 5.

Eigenvalue spectra of CuS, AgO, and CuO computed with PBE and PBE0 functionals. The spectra are shifted to align the first peak with the IP of the anion cluster. Orbitals of π, σ, non-bonding d, π, and σ character are same as in Figs. 3 and 4.

Close modal

In summary, as in the case of Cu2O, the photoelectron spectrum of CuO is well captured by the G0W0@BHLYP and the hybrid functionals do well in predicting the excitation energies of lower lying states of predominantly Cu 3d character. However, for the first two peaks, the semi-local PBE functional provides a much better prediction. This somewhat surprising finding is interpreted in terms of a stronger SIE effect for the lower lying states, as compared with a larger role of non-dynamical correlation for the higher lying states.

For the linear CuO2 structure shown in Fig. 1, DFT calculations at both the PBE and the PBE0 level show that the molecule does not possess a closed-shell singlet configuration as originally assumed,16,34 but rather it has a triplet ground state. At the PBE level, the (Σg3) triplet state of CuO2 is energetically favorable than the Σg+1 singlet configuration by 0.71 eV, in good agreement with the value of 0.68 eV obtained previously by Deng et al.39 At the PBE0 level, the triplet state is energetically favorable by a larger energy difference of 1.22 eV.

Removing an electron from the triplet CuO2 to find its IP requires more careful considerations, as the electron removal can lead to either a doublet or a quartet configuration of the neutral cluster. With PBE, the quartet state (4Πu) is lower in energy than the doublet state (2Πg), by a relatively small 0.09 eV. However, we find that the quartet configuration possesses unstable vibrational modes, again in agreement with the results of Deng et al.39 Furthermore, the doubly degenerate HOMO of CuO2 is in the majority spin channel, and the charge density difference between the triplet state of CuO2 and the doublet state of neutral CuO2 indeed corresponds to the charge density distribution associated with the HOMO of CuO2. PBE0 also predicts the quartet state of the neutral to be lower in energy than the doublet state but with a much larger energy difference of 1.36 eV. Furthermore, PBE0 calculations do not reveal any unstable vibrational modes, and because the doubly degenerate HOMO of CuO2 is in the minority spin channel, the charge density difference between the triplet state of CuO2 and the quartet state of neutral CuO2 does correspond to the charge density distribution associated with the HOMO of CuO2.

Figure 6 shows the computed and experimental photoelectron spectra of CuO2. In the experiments of Wu et al.,16 sharp photoelectron peaks were observed at 3.47 (X), 3.79 (A), 4.10 (B), 4.28 (C), 4.67 (D), and 5.16 eV (E). In light of the above discussion, the PBE and PBE0 spectra have been shifted according to the total energy difference between the triplet Σg3 state of CuO2 and the doublet or quartet state of CuO2, respectively. As explained above, for the OT-RSH results, no shift of eigenvalues is needed. Nevertheless, the neutral energy still comes into play as part of the optimal tuning process. As in PBE0, the OT-RSH calculations also found the quartet to be lower in energy and with a charge density difference compatible with the HOMO of the anion.

FIG. 6.

Experimental photoelectron spectrum of CuO2 (Ref. 16) along with spectra computed with PBE, PBE0, OT-RSH (α = 0, γopt = 0.314 a.u.−1), OT-RSH (α = 0.2, γopt = 0.230 a.u.−1), G0W0@PBE, G0W0@PBE0, and G0W0@BHLYP. The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster. Also shown is the PBE spectrum shifted by the IP of the quartet configuration. See the text for details. Contour plots of the molecular orbitals discussed in the text, with matching color codes displayed in the spectra, are shown on the right.

FIG. 6.

Experimental photoelectron spectrum of CuO2 (Ref. 16) along with spectra computed with PBE, PBE0, OT-RSH (α = 0, γopt = 0.314 a.u.−1), OT-RSH (α = 0.2, γopt = 0.230 a.u.−1), G0W0@PBE, G0W0@PBE0, and G0W0@BHLYP. The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster. Also shown is the PBE spectrum shifted by the IP of the quartet configuration. See the text for details. Contour plots of the molecular orbitals discussed in the text, with matching color codes displayed in the spectra, are shown on the right.

Close modal

Comparison of the computed peaks with the measured photoelectron spectrum reveals interesting trends. With PBE, the onset energy of 4.01 eV is much too high in comparison with the measured value at 3.47 eV. However, the overall line shape of the spectrum appears to correspond to the experiment fairly well. Indeed, Fig. 6 also reveals that if the PBE data are shifted by the IP (3.92 eV) of the “incompatible” quartet configuration anyway, improvement with the experiment improves drastically. We note that in this scenario the PBE spectrum is shifted so as to align the doubly degenerate HOMO-1/HOMO-2 in the minority spin channel with the IP of the quartet configuration, so that the removal of one of the electrons in HOMO-1/HOMO-2 of CuO2 would lead to the 4Πu configuration of CuO2. However, this comes at the cost of predictive power, as strict validity of the theoretical procedure demands that the doublet be used. With PBE0 or OT-RSH, while the onset energy is close to the measured value [especially with OT-RSH (α = 0) for which the onset is 3.50 eV], separations between the first two computed peaks, which are 0.85, 1.03, and 1.27 eV for PBE0, OT-RSH (α = 0), and OT-RSH (α = 0.2), respectively, are much too large compared with the experimental XA separation of 0.32 eV and, in fact, are in much worse agreement with the experiment than the prediction of PBE. Turning to the GW results, as before the G0W0@PBE quasiparticle energies with an onset energy of 2.81 eV are significantly underestimated with respect to the experimental data. However, unlike the case for Cu2O and CuO, G0W0@PBE0 provides very good agreement with the experimental data: The onset energy is only 0.09 eV higher than the X peak, and the MAE over the six measured peaks is 0.17 eV. Interestingly, the G0W0@BHLYP, which was shown earlier to lead to very good agreement with the experiment for Cu2O (Fig. 2) and CuO (Fig. 3), does not perform well for CuO2: The separation of 1.53 eV between the first two peaks computed with G0W0@BHLYP is much too large compared with the experimental XA separation, resulting in a spectrum where all (AE) but the lowest energy (X) transitions are predicted above 4.75 eV, approximately 1 eV higher than the experimental spectrum.

As the differences between the PBE and PBE0 predictions are striking, it is instructive to examine them in more detail. At the PBE level, the HOMO of CuO2 occurs in the majority spin channel and is a doubly degenerate orbital of eg symmetry that is a nearly equal mixture of O 2p and Cu 3d states. The LUMO in this case corresponds to the same (spin-split) eg orbital in the minority spin channel. At the PBE0 level, on the other hand, while LUMO is still the eg orbital, HOMO occurs in the minority spin channel and has eu symmetry, which is almost purely composed of O 2p states (in a slightly π bonding configuration). In Fig. 7, we plot the (unshifted) eigenvalue spectra of CuO2 at the PBE, PBE0, and HF levels. As observed in the figure, this switch in the orbital character of HOMO occurs as the eg orbital gets pushed down by as much as 1.98 eV, while the eu orbital is pushed down by only 0.6 eV in going from PBE to PBE0. At first sight, it is tempting to interpret this large difference in the downward shift of the eigenvalues in terms of the character of the relevant orbitals, namely, since the eg orbital has approximately 50% Cu 3d character, while the eu orbital has practically no contribution from the Cu 3d states, the SIE is expected to affect the eg orbital more than the eu orbital, leading to the observed change in the HOMO orbital character. However, SIE cannot be the sole explanation, as the orbital of eu symmetry in the majority spin channel (eu), which has the same nearly pure O 2p composition as eu, is observed to shift down by the large amount of 1.87 eV, unlike its eu counterpart, in going from PBE to PBE0. Note that this difference in the amount by which spin-split orbitals get pushed down in energy becomes most obvious in the extreme case of HF level of theory; e.g., the eu and eu orbitals shift down by the significantly different amounts of 7.88 and 1.93 eV, respectively, in going from PBE to HF.

FIG. 7.

(Unshifted) Eigenvalue spectra of CuO2 (along with symmetries of the associated orbitals) in the majority (↑) and minority (↓) spin channels at the PBE, PBE0, and HF level of theory. Contour plots of the molecular orbitals discussed in the text, with color codes matching the levels shown in the spectra, are shown on the left.

FIG. 7.

(Unshifted) Eigenvalue spectra of CuO2 (along with symmetries of the associated orbitals) in the majority (↑) and minority (↓) spin channels at the PBE, PBE0, and HF level of theory. Contour plots of the molecular orbitals discussed in the text, with color codes matching the levels shown in the spectra, are shown on the left.

Close modal

In addition to SIE, another factor that contributes to the different ordering of orbitals at the PBE and PBE0 level is the amount of spin-splitting. In general, introduction of exact exchange is expected to increase spin-splitting. This is indeed what we observe in CuO2, especially for doubly degenerate orbitals for which the increase in spin-splitting as a function of the amount of exact exchange is particularly large. For example, the 1.22 eV spin-splitting of the eu orbitals (of predominantly O 2p character) at the PBE level increases to 2.50 and 7.20 at the PBE0 and HF levels, respectively. Similarly, the 1.38 eV spin-splitting of the eg orbital mentioned earlier (HOMO and LUMO at PBE) increases to 4.53 and 15.35 eV at PBE0 and HF, respectively. Accordingly, as more exact exchange is introduced, the eigenvalue spectrum becomes more segregated with respect to the spin channel; namely, the spectrum consists of clusters of several states in the majority/minority spin channels before switching to the other spin channel. For example, as shown in Fig. 7, the six highest energy-occupied orbitals at the HF level all occur in the minority spin channel, followed by six orbitals in the majority spin channel, and so on. This is significantly different from the much more homogeneous ordering of orbitals (with respect to spin channel) at the PBE level. This observation can be qualitatively understood as follows: Removing an electron from the minority (as opposed to majority) spin channel of a given orbital in the anion cluster would result in a neutral cluster configuration with a larger spin imbalance and hence be energetically more favorable due to the increased magnitude of the exchange energy. Because the HF eigenvalues are related to (unrelaxed) electron removal energies, it makes sense that in all the occupied spin-split orbitals, those in the minority channel appear at higher energies (less negative eigenvalues) than those in the majority spin channel, as observed in Fig. 7. However, one should notice that this exchange interaction has a profound effect, at the HF level, not just on spin-split orbitals but also on orbitals of significantly different shape and atomic orbital content. For example, Fig. 7 shows that at the HF level the six highest eigenvalues all occur in the minority spin channel. In particular, the doubly degenerate HOMO-2/HOMO-3 of eg symmetry in the minority spin channel, where the Cu atom is in a pdπ bonding configuration with the O atoms, is located ∼3.30 eV higher than the eu orbitals in the majority spin channel, in spite of the fact that eg orbitals have ∼85% Cu 3d character, while the eu orbitals have almost purely O 2p content.

Because the amount by which spin-split levels are pushed down in energy with more exact exchange is spin-channel-dependent, we focus on how the centers of spin-split states change as a function of exact exchange (α), as shown in Fig. 8. This averaging over spin-splitting provides a clearer effect of SIE. Indeed, the trends observed in Fig. 8 are consistent with the orbital characters of the states. The states with symmetries of a1g (σ bonding), b1g, b2g (non-bonding Cu dxy and dx2y2 orbitals), and eg (π bonding), which have pure or very large Cu 3d content, fall into one category, which are affected the most by exact exchange. Above them are three states with symmetries of a2u, a1g (σ anti-bonding), and eu (π bonding of O 2p orbitals), which have very little or (practically) no Cu 3d content. Finally, although the center of the eg state (nearly equal mixture of O 2p and Cu 3d orbitals in an anti-bonding π configuration) does not show a significant variation as a function of α, the reason for this is that the unoccupied eg orbital (LUMO at PBE and PBE0 levels and LUMO+5 at HF) is pushed up in energy to maintain the symmetry of spin-splitting. As shown in Fig. 7, the occupied eg orbital does experience significant SIE, as the level moves down by ∼8.3 eV.

FIG. 8.

DFT eigenvalue centers (average of majority and minority spin channel eigenvalues) of CuO2 orbitals mentioned in the text computed with differing amounts of exact exchange [α in Eq. (1) varying from 0 to 1, with β = 1]. The orbitals have the same color code as in Figs. 6 and 7. The eigenvalue centers for HF calculations (α = 1, β = 0) are also shown.

FIG. 8.

DFT eigenvalue centers (average of majority and minority spin channel eigenvalues) of CuO2 orbitals mentioned in the text computed with differing amounts of exact exchange [α in Eq. (1) varying from 0 to 1, with β = 1]. The orbitals have the same color code as in Figs. 6 and 7. The eigenvalue centers for HF calculations (α = 1, β = 0) are also shown.

Close modal

The above analysis shows that on the one hand, there is a significant role played by SIE, indicating a preference for hybrid functionals. But on the other hand, the disagreement between the eigenvalue spectra produced by the hybrid functionals and experiment, along with the good agreement (up to a shift) between the PBE data and the experiment, suggests a strong role of non-dynamical correlation. This suggests that a useful compromise between these two constraints may be sought by using a Fock exchange fraction α that is lower than the default 25% in PBE0. This idea has precedence in the context of energy differences between high-spin and low-spin energy splittings in spin-crossover molecules, such as Fe(ii)–S complexes, where an exact exchange admixture of 10%-15% was recommended.77–79 Figure 9 shows the shifted spectra computed with α values ranging from 0 (PBE) to 0.25 (PBE0). We observe that it is indeed possible to get very good agreement with the experimental data if one uses a Fock exchange fraction of 15%. In this case, the largest discrepancy of 0.25 eV is in the predicted position of the A peak and the MAE averaged over the six experimental peaks is 0.09 eV.

FIG. 9.

The experimental photoelectron spectrum of CuO2 along with DFT eigenvalue spectra computed with Fock exchange fractions α varying from 0 (PBE) to 0.25 (PBE0). In each case, the spectrum is shifted to align the first peak with the IP of the anion cluster.

FIG. 9.

The experimental photoelectron spectrum of CuO2 along with DFT eigenvalue spectra computed with Fock exchange fractions α varying from 0 (PBE) to 0.25 (PBE0). In each case, the spectrum is shifted to align the first peak with the IP of the anion cluster.

Close modal

In summary, unlike in the previous cases, the photoelectron spectrum of CuO2 is not well captured by PBE, or PBE0/OT-RSH, or even by GW based on a BHLYP starting point. This is likely owing to a large role of both non-dynamical correlation and SIE considerations, with the former benefiting from lack of Fock exchange and the latter suffering from it. However, using a smaller fraction of exact exchange (15% for PBE0 and 25% as the starting point for GW) results in a quantitatively useful compromise between the two conflicting requirements.

For CuO3, the two lowest energy isomers we have found, shown in Fig. 1, have Cs (Iso1) and C2v (Iso2) symmetries. Iso1 possesses a triplet ground state, while the ground state of Iso2 is a singlet. Within both PBE and PBE0, Iso1 is lower in energy than Iso2, by 0.05 and 0.43 eV, respectively. Both Iso1 and Iso2 feature an O2 unit bonded to a CuO unit. This bonding scenario is consistent with the work of Wu et al.,16 who suggested it based on photodissociation events observed at a 355 nm photon energy for both CuO3 and Cu(O2) complex.

Figures 10 and 11 show the experimental photoelectron spectrum for the CuO3 cluster, as compared with the theoretical spectra computed for Iso1 (Fig. 10) and Iso2 (Fig. 11). In the experiments of Wu et al.,16 sharp photoelectron peaks were recorded at 3.39 (A) and 4.34 eV (D), as well as weaker ones at 3.19 (X), 3.51 (B), 4.02 (C), and ∼5.8 eV (E), with the possibility of feature B (separated from A by 0.12 eV) being a vibrational replica of A. The spectrum of CuO3 is basically divided into two main groups, with features X, A, and B forming the first group and features C and D forming the second. These two groups are followed by the broader and weaker feature E. The observed separation between the two main groups of peaks is roughly the same as the XY separation in CuO. Accordingly, Wu et al. suggested that these two groups could be interpreted as derived from the XY states of CuO perturbed by an O2 molecule and assigned features X through D to O 2p orbitals and the E feature at high BE due to Cu 3d orbitals.

FIG. 10.

Experimental photoelectron spectrum of CuO3 (Ref. 16), along with spectra computed using PBE, PBE0, OT-RSH (α = 0, γopt = 0.349 a.u.−1), OT-RSH (α = 0.2, γopt = 0.284 a.u.−1), G0W0@PBE, and G0W0@PBE0 for the lower energy (Iso1) Cs structure of CuO3. The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster.

FIG. 10.

Experimental photoelectron spectrum of CuO3 (Ref. 16), along with spectra computed using PBE, PBE0, OT-RSH (α = 0, γopt = 0.349 a.u.−1), OT-RSH (α = 0.2, γopt = 0.284 a.u.−1), G0W0@PBE, and G0W0@PBE0 for the lower energy (Iso1) Cs structure of CuO3. The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster.

Close modal
FIG. 11.

Experimental photoelectron spectrum of CuO3 (Ref. 16), along with spectra computed with PBE, PBE0, OT-RSH (α = 0, γopt = 0.246 a.u.−1), OT-RSH (α = 0.2, γopt = 0.165 a.u.−1), G0W0@PBE, G0W0@PBE0, and G0W0@BHLYP for the Y-shaped Iso2 structure of CuO3 (with C2v symmetry). The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster. Contour plots of the molecular orbitals (with associated symmetries) discussed in the text are shown on the right, with matching color codes displayed in the computed spectra.

FIG. 11.

Experimental photoelectron spectrum of CuO3 (Ref. 16), along with spectra computed with PBE, PBE0, OT-RSH (α = 0, γopt = 0.246 a.u.−1), OT-RSH (α = 0.2, γopt = 0.165 a.u.−1), G0W0@PBE, G0W0@PBE0, and G0W0@BHLYP for the Y-shaped Iso2 structure of CuO3 (with C2v symmetry). The PBE and PBE0 spectra are shifted to align the first peak with the IP of the anion cluster. Contour plots of the molecular orbitals (with associated symmetries) discussed in the text are shown on the right, with matching color codes displayed in the computed spectra.

Close modal

Comparison of the spectra computed with PBE, PBE0, OT-RSH, G0W0@PBE, and G0W0@PBE0 for Iso1 (Fig. 10) with the experimental data shows that none of the DFT or GW methods results in satisfactory agreement with the experiment. For example, the onset energy with PBE (2.67 eV) is much smaller than the position of the X peak. While it may be possible to assign some of the peak positions predicted with hybrid functionals and G0W0@PBE0 to photoelectron data (e.g., the first two peaks of PBE0 and G0W0@PBE0 are within ∼0.15 and ∼0.04 eV of the experimental X and A bands, respectively), the overall shapes of the predicted spectra do not resemble the experimental spectrum that has two broad peaks centered at 3.39 and 4.34 eV and no appreciable signals between 4.34 and 5.80 eV.

Agreement between theory obtained for Iso2 and experiment, though not perfect, is much more satisfactory. Assuming that the experimental feature B is indeed a vibrational level of feature A, we can compare the first four computed peaks at each level of theory with features X, A, C, and D in the experiment. The MAEs for PBE, PBE0, OT-RSH (α = 0), and OT-RSH (α = 0.2) are then found to be 0.17, 0.15, 0.24, and 0.34 eV, respectively. We therefore focus on Iso2, although we comment further on the isomer issue below. We also note that using a Fock exchange fraction α between PBE and PBE0 values leads to only slightly better agreement with the experiment. The MAEs for α = 0.05, 0.10, 0.15 are found to be 0.085, 0.12, and 0.10 eV, respectively. The computed binding energies are reported in the supplementary material. Therefore, for the analysis and discussion to follow, we focus on PBE, PBE0, OT-RSH, and G0W0 levels of theory.

For Iso2, the first peak is predicted quite well (within ±0.1 eV) with both PBE and hybrid functionals. While PBE and PBE0 predictions for the first two peaks are very close to each other and agree well with the experiment, the positions of the next two peaks are overestimated and underestimated by PBE and PBE0, respectively, by 0.2-0.3 eV. Surprisingly, the experimental peak at ∼5.8 eV is best predicted with PBE, while the predictions from PBE0 and range-separated hybrid functionals are not as good, even though the relevant orbitals in this energy range indeed have large (>90%) Cu 3d character. Similar to the other clusters considered, the G0W0@PBE predictions are poor, with the first two peaks underestimated by 0.8-1.0 eV and an averaged MAE of 0.73 eV. The G0W0@PBE0, on the other hand, provides excellent agreement with the experimental data for the first four peaks with an MAE of 0.07 eV. The fifth peak, however, lies too low in the G0W0@PBE0 calculation. Similar to the case of CuO2, the G0W0@BHLYP results, while better than the G0W0@PBE, are not good; the predicted quasiparticle levels are found to be much more bound compared with the experiment, and the averaged MAE is 0.51 eV.

Analysis of the different orbital characters for Iso2, also shown in Fig. 11 at various levels of theory, additionally lends support to the assumption of a CuO unit perturbed by an O2 unit: In particular, at the PBE level, HOMO-1 of b1 symmetry and HOMO-2 of a1 symmetry have very similar shapes and orbital content as those of X (π) and Y (σ) states of CuO (see Fig. 3), respectively, as they contain very little (less than 3%) contribution from the O2 unit. We note, however, that these b1 and a1 states of CuO3 do have appreciable (∼40 and ∼70%, respectively) Cu 3d content, similar to π and σ states of CuO. Orbitals of large O 2p character at the PBE level are HOMO (a2 symmetry) and HOMO-3 (b2 symmetry) with very little (less than 10%) Cu 3d contribution.

One of the significant changes in the ordering of the orbitals in going from PBE to PBE0 for Iso2 of CuO3 is the switching of a1 and b2 orbitals. This behavior can again be understood in terms of the degree of spatial localization of the relevant orbitals. In going from PBE to PBE0, orbitals of a2, b1, and a1 symmetry are pushed down in energy by the similar amounts of 1.67, 1.60, and 1.45 eV, respectively, while the b2 orbital becomes more bound by only 0.76 eV (keep in mind that the spectra plotted in Fig. 11 are shifted eigenvalue spectra). From the shapes and spatial extents of the molecular orbitals plotted in Fig. 11, one can easily observe that the b2 is the least localized orbital, which accordingly gets affected the least by SIE, consistent with the difference in the shifts of the corresponding eigenvalues.

Another important observation about the PBE and PBE0 spectra of CuO3 is related to the separation of the b1 and a1 orbitals, which correspond to the π and σ orbitals of CuO, as mentioned above. With PBE, the b1a1 separation in CuO3 has exactly the same value of 0.88 eV as the πσ separation of CuO. However, with PBE0, while the πσ separation of CuO drops significantly to 0.46 eV, the b1a1 separation in CuO3 is only reduced to 0.73 eV and remains close to the PBE value. Earlier, we attributed the large change in the πσ separation of CuO upon addition of exact exchange to a more accurate accounting of non-dynamical correlation in PBE compared with PBE0. Our finding of similar b1a1 separation in CuO3 with PBE and PBE0 is then in accordance with the expectation that non-dynamical correlation should play a less significant role in larger and less symmetric molecular systems. Accordingly, both the PBE and the PBE0 do a reasonably good job of describing the first two main (broad) experimental peaks, while PBE is clearly much better for the CuO molecule.

The results additionally show that while starting points that involve some fraction of exact exchange are definitely needed for satisfactory agreement of one-shot G0W0 predictions with experiment, the amount of that fraction is highly dependent on the cluster composition: Clusters with larger Cu content (Cu2O and CuO) appear to need DFT starting points with a larger exact exchange fraction, such as BHLYP, than clusters with a smaller Cu content (CuO2 and CuO3) for which the PBE0 starting point works quite well.

Finally, we note that because Iso1 is slightly lower in energy than Iso2, one could have expected it to be the experimentally pertinent isomer. Here, we note that generally photoemission measurements of clusters are not necessarily taken at thermal equilibrium and that higher energy clusters may dominate the experimental spectrum owing to kinetic considerations.51 Still, the remaining differences between theory for Iso2 and experiment may indicate some presence of Iso1 under the experimental setup. Furthermore, the Iso1 structure is inherently floppy and its orbital arrangement may depend on the exact geometry, and therefore change with temperature. The overall observed spectrum may then reflect contributions from both the isomers.80 Exploring this scenario, however, would require extensive molecular dynamics simulations that are outside the scope of this article.

In summary, we provided a systematic comparison of DFT- and GW-computed eigenvalue spectra to experimental photoelectron spectra of four small copper oxide cluster anions–Cu2O, CuO, CuO2, and CuO3—with a focus on comparison between semi-local and hybrid functional (within DFT and as a starting point for a G0W0 calculation). A main theme found in the comparison between theory and experiment is that the theoretical success hinges on a number of constraints that are difficult to fulfill simultaneously. On the one hand, hybrid functionals have a clear advantage in mitigating self-interaction errors. On the other hand, removal of some semi-local exchange, which is necessary for creating a hybrid functional, results in a less successful treatment of non-dynamical correlation. Generally, we found non-dynamical correlation to increase for smaller, more symmetric clusters but to bear some importance also for the larger ones. As a consequence, for some clusters (notably Cu2O), conventional hybrid functionals were very successful, for others (notably CuO2), a semi-local hybrid (with some rigid shift) was quite accurate. For others yet, a hybrid functional with a reduced amount of Fock exchange turned out to be a successful compromise. These trends reflected themselves in the GW calculations as well: Whereas Cu-rich clusters, where SIE is a major issue, required more (50%) exact exchange in the hybrid functional starting point, Cu-poor clusters required less exact exchange (25%). These results obtained with the one-shot G0W0 method suggest that it would be interesting future work to examine the effects of various levels of GW self-consistency as well as the effects of beyond-GW corrections. Generally, the importance of non-dynamical correlation is expected to decrease with the system size, explaining the well-documented success of DFT for larger clusters. However, larger clusters can and very often do encounter a significant problem of multiple isomers, which has also been reflected here in the case of CuO3. Taken together, these findings call for careful assessment of the opposing roles of non-dynamical correlation and self-interaction errors in functional choice for small transition metal oxide and metal-organic clusters. They also call for further development of DFT and GW approaches that can balance the two issues in a natural way.

See supplementary material for tabulated PBE and PBE0 atomic coordinates, and experimental, DFT (PBE, PBE0, OT-RSH), and G0W0 (with PBE, PBE0, and BHLYP starting points) binding energies for all copper oxide cluster anions considered.

B.S. and S.Ö. would like to acknowledge support from the U.S. Department of Energy Grant No. DE-SC0017824, as well as the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, for computational resources. F.B. acknowledges High Performance Computing resources from GENCI-CCRT-TGCC (Grant No. 2017-096018). Work in Rehovoth was supported by the European Research Council. We also would like to thank Natalie Orms and Anna I. Krylov for useful discussions.

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