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Effect of Complex-Valued Optimal Orbitals on Atomization Energies with the Perdew−Zunger Self-Interaction Correction to Density Functional Theory Susi Lehtola,*,†,‡ Elvar Ö . Jónsson,† and Hannes Jónsson†,§ †

COMP Centre of Excellence and Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, Finland ‡ Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States § Faculty of Physical Sciences, University of Iceland, 107 Reykjavík, Iceland S Supporting Information *

ABSTRACT: The spurious interaction of an electron with itselfself-interaction error is one of the biggest problems in modern density-functional theory. Some of its most glaring effects, such as qualitatively incorrect charge separation upon dissociation, can be removed by an approximate self-interaction correction due to Perdew and Zunger (PZ) (Perdew, J.; Zunger, A. Phys. Rev. B 1981, 23, 5048). However, the correction introduces an explicit dependence on the occupied orbital densities, which makes proper minimization of the functional difficult. Previous work (Vydrov et al., J. Chem. Phys. 2006, 124, 094108) has suggested that the application of the PZ correction results in worse atomization energies than those obtained with the uncorrected parent functional. But, it has only recently been found that the correct minimization of the PZ energy functional requires complex-valued orbitals, which have not been used in previous work on atomization energies. Here, we study the effect of the proper use of complex-valued orbitals on the atomization energies of molecules in the W4-11 data set (Karton, A.; Daon, S.; Martin, J. M. Chem. Phys. Lett. 2001, 510, 165). We find that the correction has a tendency to weaken the binding of molecules. The correction using complex-valued orbitals is invariably found to yield better atomization energies than the correction with realvalued orbitals. The correction applied to the PBEsol exchange-correlation functional results in a functional that is more accurate than the (uncorrected) PBE functional.

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INTRODUCTION Kohn−Sham density-functional theory1,2 (KS-DFT) is arguably one of the most powerful tools in computational chemistry and materials science, as it is able to capture chemically important dynamical correlation effects with only modest computational cost. However, there are classes of systems where the theory does not work well, one class being systems exhibiting strong correlation (for which the use of KS-DFT is ill-founded3) and another being molecular systems with charge transfer4−9 as well as defect states in solids.10−12 The problems with the latter class of systems can be attributed to self-interaction error,6,8,13−15 which arises from the imperfect cancellation of the Coulomb and the exchange-correlation interactions, the first of which is known exactly but the latter only approximatively. The importance of self-interaction error can be reduced with range-separated hybrid functionals.16 However, the error cannot be completely removed by this approach and may still result in incorrect results.9 Furthermore, the evaluation of exact exchange is computationally challenging for solid-state systems, which motivates the search for alternative approaches. Instead of trying to develop new approximate exchange-correlation functionals with a lesser degree of self-interaction error for KSDFT, it is also possible to approximately remove the self© XXXX American Chemical Society

interaction error from the KS-DFT functional as suggested by Perdew and Zunger17 (PZ) as EPZ ‐ SIC[nα ,nβ ] = EKS ‐ DFT[nα ,nβ ] −

∑ (J[niσ ] + K[niσ ]) iσ

(1)

where nα and nβ are the spin-up and spin-down densities, respectively, and niσ are occupied orbital densities of spin σ that span the total spin densities as nσ = ∑iniσ. The minimization of the energy functional for the PZ self-interaction correction (PZ-SIC) is challenging due to the introduced dependence on the set of occupied orbital densities.18−28 Importantly, it has only recently been shown that the proper implementation of PZ-SIC requires the use of complex-valued orbitals even in the molecular case28 (although this had been suspected based on earlier calculations24−27,29,30), requiring re-examination of the validity of the results obtained in the literature using real-valued orbitals (denoted as PZ-RSIC in the rest of the manuscript). PZ-SIC with complex-valued orbitals has been shown to reproduce accurate charge transfer,9,25,26 Rydberg states of Received: June 20, 2016

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DOI: 10.1021/acs.jctc.6b00622 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

Table 1. Mean Errors (ME), Mean Absolute Errors (MAE), and Root Mean Square (RMS) Errors of Atomization Energies in eV for Molecules in Classes (A−C)a KS-DFT ME LDA PW91 PBE PBEsol BLYP TPSS PBE0 B3LYP ωB97X TPSSh LRC-ωPBE LRC-ωPBEh a

2.537 0.583 0.536 1.248 0.158 0.106 −0.054 −0.056 −0.097 −0.087 0.185 −0.048

MAE 2.537 0.631 0.607 1.259 0.300 0.195 0.150 0.157 0.115 0.195 0.301 0.155

PZ-RSIC RMS 2.875 0.772 0.748 1.476 0.418 0.254 0.215 0.234 0.161 0.273 0.377 0.214

ME

MAE

1.245 −0.669 −0.882 −0.268 −1.634 −1.423 −1.113 −1.410 −1.452 −1.435 −0.974 −1.090

1.321 0.767 0.917 0.645 1.655 1.477 1.118 1.443 1.478 1.487 0.983 1.093

PZ-SIC RMS 1.734 1.058 1.241 0.854 2.069 1.811 1.399 1.812 1.797 1.818 1.285 1.376

ME 1.443 −0.378 −0.563 0.029 −1.218 −1.091b −0.882 −1.108 −1.166 −1.153b −0.713 −0.860

MAE 1.452 0.470 0.588 0.393 1.220 1.119b 0.882 1.122 1.178 1.184b 0.714 0.861

RMS 1.814 0.655 0.790 0.551 1.470 1.322b 1.074 1.365 1.394 1.391b 0.894 1.048

The smallest number in each column is shown in bold. bExcluding AlCl3 due to convergence issues.

molecules,31−33 and the ground state of a dipole bound anion,34 as well as defect states in solids.12 As far as we know, the basic chemical performance of PZSIC, as proxied by atomization energies, has not been thoroughly investigated in the literature, especially not with complex-valued orbitals. Vydrov and co-workers studied the sixmolecule AE6 test set35 with fixed geometries and real-valued orbitals and found that PZ-SIC overcorrects and proposed an orbital dependent method for scaling down PZ-SIC.36,37 Klüpfel and co-workers have studied 17 small (mostly diatomic) molecules with complex-valued orbitals at optimized geometries and found that PZ-SIC overcorrects and that scaling down the correction by one-half yields better results while still undershooting the correct values.29 Borghi and co-workers have examined the G2 test set38 at optimized geometries with complex-valued orbitals using plane waves and pseudopotentials and found that PZ-SIC yields larger deviation from experiment than the parent functional,27 in agreement with the results of Klüpfel et al.29 A thorough study of atomization energies using all-electron PZ-SIC with variational, complex-valued optimal orbitals has not yet been reported. In the present manuscript, we report the application of all-electron PZ-SIC using real-valued and complex-valued optimal orbitals on the atomization energies of the W4-11 test set,39 which contains 140 small molecules. The organization of the manuscript is as follows. In the next section, Approach, we outline the way the present calculations have been performed. Then, in the Results section, we present the findings of the work. The study concludes with a Summary and Discussion section.

The Cholesky decomposition technique28,46 is used to speed up the calculation of Coulomb and exact exchange matrices, thus avoiding the need for an auxiliary basis set; a decomposition threshold of Δ = 10−5 is used. As is wellknown, PZ-SIC requires the use of larger grids than those used in KS-DFT.22,25,47 Here, an unpruned (100,590) grid is used for the initial calculations in the cc-pVDZ basis set, the first number in the parentheses denoting the number of radial shells and the second denoting the number of points on the angular grid, while an unpruned (225,770) grid is used in the cc-pVQZ basis set. We have found the two grids to yield energies converged to microhartree accuracy for first-row molecules and second-row molecules, respectively. Exchange-correlation functionals are evaluated using the LIBXC library.48,49 Orbital rotations are performed consecutively in the occupied−occupied block until the gradient norm goo vanishes (goo ≤ 10−4) and in the occupied−virtual (and virtual−occupied) block until the gradient norm gov vanishes (gov ≤ 10−5). Alternatively, convergence is deemed to have been achieved if a line search results in a change of the energy |ΔE| ≤ − 10−10.

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RESULTS We study the atomization energy defined as ⎛ ⎞ Eat. = ⎜⎜ ∑ Ei⎟⎟ − Emolecule > 0 ⎝ atoms i ⎠

(2)

for the 140 molecules in the W4-11 database39 at fixed geometries. The molecules in W4-11 are divided into four classes by the importance of static correlation:39 (A) no static correlation (51 molecules), (B) mild static correlation (49 molecules), (C) moderate static correlation (26 molecules), and (D) strong static correlation (14 molecules). The analysis is split into two parts, corresponding to, first, the 126 molecules in the first three sections with no to moderate static correlation, and the 14 multireference molecules with strong static correlation effects, respectively. Twelve different approximate exchange-correlation functionals are studied: the pure LDA50−52 functional; the pure generalized gradient approximation53 (GGA) functionals PW91,54,55 PBE,56,57 PBEsol,58 and BLYP;59−61 the hybrid GGA functionals PBE062,63 and B3LYP;64 and the range-separated GGA hybrids LRC-ωPBE,65

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APPROACH The procedure used to minimize the PZ-SIC functional has been described elsewhere.25,26,28 The ERKALE program40,41 is used for all calculations, which are performed in two steps. In the first step, a KS-DFT calculation is performed for the molecule with the cc-pVDZ basis set.42,43 Next, the occupied orbitals are localized44 using the Foster−Boys criterion,45 after which the (real-valued) orbitals are optimized to minimize the PZ-SIC energy. In the second step, the cc-pVDZ orbitals are read in, and the optimization of the real-valued orbitals is redone in the cc-pVQZ basis set.42,43 Complex-valued orbitals are formed from the converged real-valued orbitals by the use of stability analysis.28 B


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Table 2. Mean Errors (ME), Mean Absolute Errors (MAE), and Root Mean Square (RMS) Errors of Atomization Energies in eV for Molecules in Class (D)a KS-DFT ME LDA PW91 PBE PBEsol BLYP TPSS PBE0 B3LYP ωB97X TPSSh LRC-ωPBE LRC-ωPBEh a

2.682 1.001 0.996 1.642 0.462 0.272 −0.195 −0.285 −0.460 −0.172 0.090 −0.262

MAE 2.682 1.024 1.013 1.642 0.617 0.440 0.240 0.339 0.472 0.302 0.573 0.309

PZ-RSIC RMS 3.029 1.205 1.193 1.894 0.785 0.531 0.387 0.474 0.664 0.412 0.647 0.474

ME

MAE

−0.154 −1.267 −1.528 −1.173 −2.485 −2.135 −1.951 −2.487 −2.254 −2.275 −1.712 −1.927

0.542 1.268 1.528 1.186 2.485 2.135 1.951 2.487 2.254 2.275 1.712 1.927

PZ-SIC RMS

ME

MAE

0.756 1.542 1.806 1.479 2.782 2.424 2.241 2.781 2.554 2.572 1.976 2.209

−0.135 −1.350 −1.534 −1.156 −2.445 −2.073b −1.992 −2.484 −2.358 −2.227 −1.764 −1.985

0.416 1.377 1.559 1.196 2.445 2.091b 2.003 2.484 2.358 2.234 1.777 1.994

RMS 0.537 1.717 1.902 1.509 2.823 2.530b 2.366 2.868 2.783 2.649 2.099 2.352

The smallest number in each column is shown in bold. bExcluding S4 due to convergence issues.

LRC-ωPBEh,65 and ωB97X;66 as well as the pure meta-GGA67 (mGGA) functional TPSS68,69 and its hybrid TPSSh.70 The resulting errors in the atomization energy, which we define as ref. ΔEat. = Eat. − Eat.

It is interesting that range-separation does not appear to benefit any functional within the PZ-SIC scheme. While ωB97X as an empirical functional is not expected to work well in combination with PZ-SIC,28 the LRC-ωPBE and LRCωPBEh functionals are almost ab initio functionals and thus should be compatible with PZ-SIC. Still, both LRC-ωPBE-SIC and LRC-ωPBEh-SIC are overperformed by PBE-SIC, the lattermost showing significantly smaller errors. The surprisingly large difference between LRC-ωPBE-SIC and LRC-ωPBEhSIC results is also noteworthy. However, the two functionals differ not only in the amount of short-range exact exchange (0% in LRC-ωPBE and 20% in LRC-ωPBEh) but also in the range-separation parameter (0.3 bohr−1 in LRC-ωPBE vs 0.2 bohr−1 in LRC-ωPBEh). Results for the LDA and PBE functionals with real-valued and complex-valued orbitals have also been reported by Borghi and co-workers on the G2 test set.27 However, the results of Borghi et al. are not all electron and used optimized geometries, for which reason a comparison with our results is not sensible. For molecules dominated by static correlation (Table 2), the results are fundamentally different. While the same analysis as above still holds at the KS-DFT level of theory, when the PZ correction is used the LDA becomes the best functional. In contrast to the dynamical correlation dominated molecules for which the error of all functionals was reduced in PZ-SIC compared to PZ-RSIC, here only the LDA result improves. Because KS-DFT and, as a result, PZ-SIC, is not expected to work for static correlation,3 this finding is not really surprising, and we do not consider these results to be consequential. The rationale for the results for the molecules dominated by dynamical correlation can be seen in the error histograms, which are shown in Figures 1 and 2 for the pure and hybrid functionals, respectively. From the figures, it is clear that the PZ correction has a tendency to weaken the binding of molecules. Thus, functionals that have a tendency to overbind the molecules, such as the LDA, PW91, and PBEsol, benefit from the PZ correction, while functionals that are as likely to overbind or underbind, such as BLYP or TPSS, do not.

(3)

where Eref. at. is the high level ab initio reference value for the atomization energy given in W4-11, are summarized in Tables 1 and 2 for the dynamical and static correlation dominated molecules, respectively, for the 10 different functionals under study. Full scatter plots of the results are available as part of the Supporting Information. Examining the results in Table 1, the known pattern at the KS-DFT level of theory is seen: LDA gives the worst results, with GGA functionals reducing the error by a factor of 3−7. Hybrid functionals reduce the error by a further factor 2−3; a similar reduction can be achieved by introducing dependence on the kinetic energy density as in the meta-GGA functionals. The most accurate results can be achieved by range separation, after which density-functional theory is used for small length scales where it fares well, but far-away interactions are modeled using Hartree−Fock theory. This behavior is exemplified by the ωB97X functional that shows the smallest root mean square (RMS) error by a large margin, as it is an empirical functional that has been fit to reproduce atomization energies among others. Next, for the Perdew−Zunger self-interaction correction with real-valued optimal orbitals, it is seen that while significant improvements are seen for LDA and PBEsol, the correction deteriorates the performance of other functionals. This result is in line with earlier AE6 results on LDA and PBE in the literature.36,37 However, when complex-valued optimal orbitals are used, the situation changes: the RMS errors for all functionals except the LDA go down by roughly half an eV. PW91-SIC is more accurate than PW91, and while PBEsol is the worst functional of all, PBEsol-SIC turns out to be the best PZ functional, in line with earlier speculations.25,71 While the PBEsol-SIC value is only 0.13 eV worse than that of the best pure KS-DFT functional, BLYP, the RMS error for PBEsol-SIC is still over three times larger than of the best KS-DFT functional, ωB97X. Still, PBEsol-SIC is more accurate than the PBE functional, which is one of the main results of this article.

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SUMMARY AND DISCUSSION We have reported atomization energy calculations on the W411 data set using all-electron PZ-SIC with variational, complexvalued optimal orbitals. We have found that the PZ correction tends to weaken the binding of molecules and that the best C


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Figure 1. Histograms for the LDA, PW91, PBE, PBEsol, TPSS, and BLYP functionals for molecules in groups (A)−(C).

results are obtained when the correction is applied to a functional that tends to overbind. The results with complexvalued orbitals were seen to differ greatly from those obtained with real-valued orbitals: The errors in the atomization energies with complex-valued optimal orbitals were invariably smaller than the ones given by calculations with real-valued optimal orbitals. Good results are obtained when the PZ correction is applied to the PBEsol functional; the resulting PBEsol-SIC method gives better results than the PBE functional. The improved performance of PBEsol-SIC can be ascribed to two factors. First, because the errors of PBEsol-SIC are smaller than of PBEsol-RSIC, the complex degrees of freedom are helpful for the accuracy of PZ-SIC. Second, as PBEsol-RSIC is much better than PBE-RSIC, and PBEsol-SIC is better than PBE-SIC,

the smaller gradient enhancement factor in PBEsol compared to PBE is clearly beneficial for PZ-SIC. While the results with PBEsol-SIC are encouraging, the results overall suggest that the PZ-SIC approach is far from perfect for applications in thermochemistry. Fitting a functional especially parametrized for PZ-SIC might help by making PZSIC calculations more accurate for this purpose.71 In particular, a combinatorial approach that has recently been shown to yield excellent results for KS-DFT72,73 could be used also for PZSIC. However, due to the large computational requirements of PZ-SIC, self-consistent optimization of an exchange-correlation functional is a large task. Furthermore, as some of us have recently shown,28 even if a more accurate exchange-correlation functional were developed, PZ-SIC may still lead to broken molecular symmetries. D


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Figure 2. Histograms for the PBE0, B3LYP, ωB97X, and TPSSh functionals for molecules in groups (A)−(C).

Finally, we note that an alternative to PZ-SIC based on Fermi orbitals74 (FO-SIC) has been recently proposed.75−79 The FO-SIC formulation involves fewer number of degrees of freedom that need to be optimized, which may lead to a smaller prefactor for self-interaction corrected calculations. As a fully variational formulation of FO-SIC has not yet been presented to our knowledge, it is not clear whether the approach will break molecular symmetries as PZ-SIC does.28 However, due to the mathematical connection of Fermi orbitals with real orthogonal transformations, we expect the thermochemical performance of FO-SIC to be similar to that of PZ-SIC with real-valued orbitals.

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Scatter plots for the atomization energy errors for the 10 functionals studied in the present work. (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].fi. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.L. thanks Narbe Mardirossian for discussions and comments on the manuscript. The computer time spent on the calculations were provided by CSC − IT Center for Science, Ltd. (Espoo, Finland) and the University of Iceland Computer Services (RH), which are gratefully acknowledged. This work was funded by the Academy of Finland through its Centres of

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.6b00622. E


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