Reference Determinant Dependence of the Random Phase

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Reference determinant dependence of the random phase approximation in 3d transition metal chemistry Jefferson E Bates, Pál Dániel Mezei, Gabor I. Csonka, Jianwei Sun, and Adrienn Ruzsinszky J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00900 • Publication Date (Web): 06 Dec 2016 Downloaded from http://pubs.acs.org on December 9, 2016

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Journal of Chemical Theory and Computation

Reference determinant dependence of the random phase approximation in 3d transition metal chemistry J. E. Bates,∗,† P. D. Mezei,‡ G. I. Csonka,‡ J. Sun,¶ and A. Ruzsinszky† Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA, Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, H-1521 Budapest, Hungary, and Department of Physics, University of Texas El Paso, El Paso, Texas 79968, USA E-mail: [email protected]

Abstract Without extensive fitting, accurate prediction of transition metal chemistry is a challenge for semilocal and hybrid density funcitonals. The Random Phase Approximation (RPA) has been shown to yield superior results to semilocal functionals for main group thermochemistry, but much less is known about its performance for transition metals. We have therefore analyzed the behavior of reaction energies, barrier heights, and ligand dissociation energies obtained with RPA and compare our results to several semilocal and hybrid functionals. Particular attention is paid to the reference determinant dependence of RPA. We find that typically the results do not vary much between semilocal or hybrid functionals as a reference, as long as the fraction of exact exchange (EXX) mixing in the hybrid functional is small. For large fractions of EXX mixing, ∗

To whom correspondence should be addressed Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA ‡ Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, H-1521 Budapest, Hungary ¶ Department of Physics, University of Texas El Paso, El Paso, Texas 79968, USA †

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however, the Hartree-Fock like nature of the determinant can severely degrade the performance. Overall, RPA systematically reduces the errors of semilocal functionals and delivers excellent performance from a single reference-determinant for inherently multireference reactions. The behavior of dual-hybrids that combine RPA correlation with a hybrid exchange energy was also explored, but ultimately did not lead to a systematic improvement compared to traditional RPA for these systems. We rationalize this conclusion by decomposing the contributions to the reaction energies, and briefly discuss the possible implications for double-hybrid functionals based on RPA. The correlation between EXX mixing and spin-symmetry breaking is also discussed.

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Introduction

The accuracy of a density functional approximation used in Kohn-Sham (KS) density functional theory 1,2 (DFT) typically increases with the information used to construct it. Starting with local functionals that depend only on the density, 3 generalized gradient approximations 4–6 (GGAs) include density gradients to improve their accuracy, and kinetic-energy densities 7–9 (meta-GGAs) can also be included to push the accuracy limits even further. The strongly constrained and appropriately normed (SCAN) meta-GGA 10 represents the most sophisticated, non-empirical semilocal functional developed to date, and sits on the third rung of Jacob’s ladder. 11 The highest degree of accuracy for modern functionals typically requires nonlocal Kohn-Sham orbital dependence such as in hybrid functionals 5,12,13 that utilize Hartree-Fock exchange or approaches such as the Random Phase Approximation 14–16 (RPA), fourth and fifth rung functionals. For main-group thermochemistry, RPA has been shown to lead to significant improvements compared to semilocal functionals, albeit at a slightly higher computational cost. 17,18 In the solid state, RPA is one of the few non-perturbative methods that can be applied to metallic systems and has been used to study weak interactions, adsorption and catalysis, 19–23 among other physical properties. 24–27 Though RPA is a promising method, it can require correcting for non-isogyric processes. 28,29 2


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For systems with stronger multireference character, such as those found in transition metal complexes, fully non-local functionals such as RPA are needed to provide accurate predictions because hybrid functionals do not always systematically improve upon semilocal functionals, 30 and high-level wavefunction methods are too expensive. RPA is non-empirical, one-electron self-interaction free in the exchange energy, and accounts for some static correlation 17,18,31 all from a single reference determinant, so it is a natural method for quality-control of DFT when wavefunction methods such as MP2 or coupled-cluster approaches break down due to deficiencies in the Hartree-Fock (HF) reference. Due to its non-perturbative nature, RPA is applicable to small-gap systems where perturbative methods fail. This also implies that a double-hybrid functional 32,33 which freely combines exchange and correlation energy contributions from (dispersion-corrected) semilocal functionals and RPA will be more robust than previous double-hybrids based on second-order perturbation theory. 34 Self-consistent implementations of RPA are not in widespread use; pilot applications for atoms and diatoms have been reported in the literature, 35–38 though an efficient scheme for general systems is presently unavailable. Consequently, the freedom to choose any reference determinant to evaluate the RPA energy introduces an implicit dependence on the orbitals and orbital eigenspectrum of the reference method. Previous works have shown the difference in performance for RPA evaluated with LDA and (meta-)GGA references is typically small for atoms, molecules, and simple solids indicating that semilocal functionals deliver very similar orbitals and orbital energies. 17,39–42 Since hybrid functionals mix non-local (Fock) exchange with semilocal exchange, the differences in the orbital eigenvalues change more dramatically than from one semilocal functional to another because full Hartree-Fock single-particle excitations are related to ionization processes, not N -electron excited states. Hybrid functionals are preferred for band gap calculations in solids for precisely this reason, 43,44 and also for prediction of ionization potentials in aromatic molecules. 45,46 This behavior is directly tied to the reduction of many-electron self-interaction error and linearity of the energy as a function of the electron

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number. 47–49 Hybrid functionals are therefore attractive since they can reduce both one- and many-electron self-interaction errors in the reference state used to evaluate RPA. The use of a Hartree-Fock (HF) reference to compute RPA-optimized geometries was previously shown to yield inferior results compared to RPA based on a semilocal DFT reference for several molecules, 50,51 and results for RPA energy differences with a HF reference also appeared to be inaccurate for a small set of organic molecules. 52 The impact hybrid references have on RPA total energies has since remained unexplored, but is important to understand if one is trying to construct double or dual 53 hybrid functionals using RPA correlation. To investigate the implicit dependence on the reference, we studied the performance of RPA for three literature test sets 30,54,55 using input-orbitals from (meta-)GGA and hybrid functionals. The impact of changing the hybrid mixing for dual-hybrids of RPA were also investigated. Throughout we use the notation Method@DFT to indicate the DFT reference used to evaluate the method energy. Spin and spatial symmetry breaking are also important points to consider when studying transition metal chemistry or systems away from equilibrium such as in stretched bonds. 17,56–60 We have accounted for the influence of spatial symmetry breaking by using C1 symmetry throughout, however spin-symmetry breaking can arise for the reference determinant of closed-shell systems when a hybrid functional is used. Semilocal functionals and hybrids with small amounts of Fock exchange tend to be stable with resepect to spin-symmetry breaking, 61,62 but large fractions of exchange can lead to well-known triplet instabilities from HF theory. 17,63–66 Triplet instabilities indicate that a spin-symmetry broken solution to the Kohn-Sham or Hartree-Fock equations is lower in energy than the spin-restricted reference state. We have explored this point by performing stability analyses for closed-shell molecules in two test sets as a function of exchange mixing.

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Methods

In this study we used the generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof 6 (PBE) and its one-parameter hybrid PBE0, 12 as well as the newly developed SCAN meta-GGA 10 to prepare self-consistent input orbitals and orbital eigenvalues for the RPA calculations. We could expect the accuracy of the functionals themselves to be RPA > PBE0 ≈ SCAN > PBE, and in fact SCAN has been recently shown to be a systematic improvement compared to PBE and on the level of PBE0 for a wide range of main-group chemical reactions. 10,67,68 RPA has also been shown to improve upon PBE and deliver MAEs for main-group thermochemistry on the order of a few kcal/mol 17,18,69 and accurately reproduced equilibrium bond lengths, vibrational frequencies, and dissociation energies for a set of eight 3d transition metal monoxides. 29 For bulk solids and surfaces, RPA has also been applied with great success, considering it is one of few correlation methods that can be directly applied to metallic systems with a zero Kohn-Sham gap. 21,25–27,41 While very accurate for isogyric processes, non-isogyric processes may require a correction beyond RPA that takes into account exchange interactions in the correlation energy. 29,31,42,60,70–75 We do not pursue such corrections here, however, as efficient implementations of these methods are still in their early stages. While semilocal DFT is well established, RPA is still a relatively new method for chemical applications. Herein RPA, or direct RPA (dRPA), refers to the variant of the method that neglects exchange or kernel contributions to the correlation energy. 17 Compared to semilocal functionals RPA is particularly attractive because it can safely be applied to small-gap systems, naturally incorporates dispersion interactions, and eliminates the one-electron selfinteraction errors of semilocal exchange by using 100% exact exchange (EXX) in the total energy, i.e.

E RPA = E EXX + ECRPA ,

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where E EXX is the HF energy evaluated with orbitals from a (hybrid) semilocal DFT calculation. The correlation energy can be computed in the formalism of the adiabatic connection 15,16,76,77 or from a ring-only coupled-cluster doubles (rCCD) approach. 78 Both approaches yield equivalent correlation energies, 78 though the implementation of each is quite different. Efficient methods from the adiabatic-connection utilize numerical frequency integration, 79–81 while iterative procedures must be used in the rCCD approach. 31,78,82 In both cases the O(N 6 ) scaling of brute force RPA is reduced to O(N 4 ), where N is a measure of the basis set size, using auxiliary basis set expansions or Cholesky decomposition. This is in contrast to MP2 which scales as O(N 5 ) and tends to fail miserably for transition metal reactions. Algorithms that further reduce the formal scaling of RPA with respect to the basis set size have also been reported in the literature. 83–86 For a more complete discussion of many of these topics we refer the reader to the review articles Refs. 17 and 18 and references therein. Building on RPA, the “dual-hybrid”, dRPA75, 53 seeks to hybridize the RPA energy on two levels: (1) self-consistent orbitals are generated using PBE75, PBE with 75% EXX mixing, and then (2) the exchange-correlation (xc) energy is evaluated taking 75% EXX and 25% PBE exchange energies with full RPA correlation as

dRPA75 PBE75 EXC = EX + ECRPA ,

(2)

PBE EXX PBE75 + (1 − ax )EX for ax = 0.75. All components of the dRPA75 = ax EX where EX

energy are evaluated with PBE75 orbitals. This is in contrast to a “double-hybrid” that hybridizes a post-KS correlation method and exact-exchange only in the energy expression and utilizes input orbitals from a traditional self-consistent calculation from a given (semilocal) functional. 32,33 The results for organic and main group thermochemistry reported for dRPA75 are an improvement over the standard RPA for a variety of processes, though some challenges remain due to the self-interaction error introduced by using PBE exchange and

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the self-correlation error inherent in the RPA correlation energy. 53 Here we explore whether this good behavior is also inherited for transition metal chemistry. Since we evaluate RPA non self-consistently, the dependence of ECRPA on the reference determinant principally enters through the orbital energy differences because the KS excitation energies appear in the denominantor of the correlation energy. 29 It follows then that functionals that yield larger orbital eigenvalue differences will result in less negative (smaller in magnitude) RPA correlation energies. Using a hybrid functional or pure HF as a reference will therefore lead to less negative RPA correlation energies in comparison to those obtained using a semilocal functional. In the end, however, it is the sum of the EXX and correlation energies that determines the resulting performance for energy differences. The EXX energy for a given system should become more negative as the exchange mixing parameter increases since the orbitals are approaching Hartree-Fock orbitals, indicating that the total RPA energy may be less sensitive to these changes than the individual components. Dual hybrid functionals such as dRPA75 will also exhibit variable behavior with changes in the fraction of exchange mixing, though it is not clear how the combination of PBE exchange and EXX will counterbalance RPA correlation. In Section 3.4 we explore the variations of RPA and the dual-hybrid with changes in the exchange mixing for CuF, CrO3 , and a methane insertion reaction.

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Results

Three test sets from the literature were used to assess the performance of the methods described above. We will refer to them as Furche-Perdew 18 (FP18), Johnson-Becke 32 (JB32), and Steinmetz-Grimme 13 (SG13). 30,54,55 Each set contains a different collection of tests that are described briefly followed by the results. These sets represent different paradigms of 3d transition metal chemistry extending from diatomics to carbonyl complexes to models of catalytic bond breaking in light atoms via transition metal catalysts, and

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therefore should provide a broad view of the performance of each method.

3.1

FP18

The 18 reaction energies of FP18 mostly focus on small diatomic binding energies as well as some polyatomic reactions such as Fe(CO)5 −→ Fe(CO)4 + CO. They have been back corrected for zero-point vibrational energies and first-order relativistic effects and are intended to serve as an initial assessment of a new method since a variety of bond types are included. 30 Examples for each metal from Sc to Cu are included providing a good sampling of the different occupations that arise from a partially filled d-shell. The full list is given in the supporting information. We used the optimized TPSS geometries for polyatomic species from Ref. 30. These reactions are challenging for DFT due to the need to compute atomic energies, though FP do note that for the reactions where atoms are not required, semilocal functionals tend to perform better than hybrids. We would like to clarify two points related to this test set. The first is related to the geometry of Fe2 Cl4 and its impact on the reaction energy, while the second is related to the ground-state of CoCl3 . Jiménez-Hoyos et al. reported 87 results for a different structure of Fe2 Cl4 than that used in the original FP work, and additionally found that a brokensymmetry, 56 anti-ferromagnetic coupling between the iron atoms is lower in energy than the ferromagnetically coupled state. We have used this updated spin state and geometry, 88 reported in the supporting information, to compute our results. With respect to CoCl3 , Jiménez-Hoyos et al. report reaction energies that are ∼10 kcal/mol larger than those reported by Furche and Perdew and this is due to a different ground-state occupation. Our semilocal results agree with Ref. 87, so we are confident that we have also found the lowest energy ground-states. To facilitate the analysis, a modified box and whisker plot is given in Fig. 1. Results for individual systems and a summary table are reported in the supporting information. Comparing PBE and PBE0 for this set, the mixture of non-local and semilocal exchange 8


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results in frequent underbinding for PBE0 and increases the distribution of errors in comparison to PBE. SCAN performs well overall, delivering a mean absolute error (MAE) below 10 kcal/mol which is difficult for semilocal or hybrid functionals to achieve on this set. Evaluating RPA from PBE, PBE0, and SCAN yields an analogous error trend to that of the functionals themselves with RPA@PBE0 yielding the largest MAE. The performance of dRPA75 is similar to that of RPA@PBE0, though the reaction energies are slightly larger on average, which reduces the overall absolute error. From the position of the box in Fig. 1, the reaction energies are generally larger in magnitude for RPA@SCAN compared to RPA evaluated with the other references. One example of large differences between references is for V2 dissociation where RPA@PBE or RPA@PBE0 underbind by ∼10-15 kcal/mol compared to RPA@SCAN, which itself is about 10 kcal/mol underbound. The reasons for these relative behaviors trace back to the different contributions of exchange and correlation, and in this particular system dRPA75 benefits from a strong cancellation of errors in the PBE hybrid exchange and RPA correlation contributions. Still, the MAEs for all of the RPA methods are consistent with the errors reported previously for RPA@TPSS applied to transition metal monoxides in Ref 29. A qualitative trend with respect to the reference determinant is difficult to establish since the differences in reaction energies depend both on the differences in EXX@DFT and in the correlation contributions for each system. Still, the magnitudes of the correlation energies (|ECRP A |) tend to behave as dRPA75 < RPA@PBE0 < RPA@SCAN < RPA@PBE. Thus the larger the HOMO-LUMO gap for a given reference determinant, the more positive the RPA correlation energy. The magnitudes of the EXX energy roughly follow the opposite trend. Evidently for RPA approaches as well as semilocal functionals, the mixing of EXX in the reference does not necessarily reduce the absolute errors in this test set, 30 though it can increase the RPA reaction energy magnitudes. Overall RPA does yield a systematic improvement compared to the reference determinant method and dRPA75 performs equally well on average.

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superior accuracy compared to semilocal functionals. Figure 2 shows the modified boxplot for the JB32 test set while individual results and a summary table are provided in the supporting information. SCAN yields roughly the same improvement over PBE with both functionals delivering MAEs of 9.5 and 12.7 kcal/mol, respectively. We checked that addition of empirical dispersion 68 to SCAN did not significantly impact the results and therefore ignore this correction for the other semilocal (hybrid) methods as well. PBE0 is a huge improvement over PBE, lowering the MAE by 8 kcal/mol, and yielding some of the smallest errors for this set. Hybrids do provide some systematic improvement for these LDEs as JB also reported MAEs of 4.3 and 4.5 for B3LYP and TPSSh, respectively. RPA evaluated on top of PBE, PBE0, and SCAN delivers an MAE on the order of 34 kcal/mol, and, in contrast to FP18, there is less than a 1 kcal/mol difference between them. The tighter error distribution for different determinants is not unreasonable considering that all of the complexes in JB32 have closed shell ground states, while those in FP18 are generally spin-polarized with near degeneracies. Since the same atomic species (and errors) largely appear in both test sets, we infer that the differences in performance for RPA reaction energies with different references for FP18 and JB32 must trace back to the differences in errors for the closed and open shell complexes. PBE75 shows large errors here due to the high fraction of EXX, and RPA@PBE75 inherits the underbinding of the input orbitals leading to a large MAE. Considering the entire set, dRPA75 and RPA based methods with a small ax in the reference yield results similar to hybrid functionals. Spin symmetry breaking is a concern for the complexes when large fractions of EXX are mixed with semilocal exchange. In the supporting information we report the lowest eigenvalue of the triplet stability matrix 90 for PBE, PBE0, and PBE75. We generally find that the triplet stability eigenvalues decrease with an increase in ax . Both PBE and PBE0 yield stable, closed-shell singlet reference states for all of the complexes, except for Fe(CO)4 where PBE0 yields a small, negative eigenvalue. Unfortunately PBE75 yields triplet instabilities

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single-point energies and energy differences; 27 for Ni, 39 for Pd, 30 for PdCl− , and 27 for PdCl2 . While the dissociation energies for the Pd catalysts are likely correct, the Ni dissociation energies are highly questionable and one of our main motivations for leaving them out. The reference Ni atom state used by Steinmetz and Grimme corresponds to a 3d10 (1 S) singlet occupation, which is actually an excited state, and thus the use of ground-state semilocal functionals is strictly an approximation. Experimentally the ground state is 4s2 3d8 due to relativistic stabilization of the s-orbitals, 91 but both correlated wavefunction methods 92–94 and semilocal functionals 30 predict the 4s1 3d9 (3 D) occupation to be the non-relativistic ground-state. The HF method fails in this regard, however, and predicts the 4s2 3d8 (3 F) occupation to be lowest in energy. 95 Electron correlation is therefore explicitly required to obtain the proper state ordering in the atom, though multireference approaches are not necessarily required. 91 As originally noted by Steinmetz and Grimme, PBE0 performs remarkably well producing the lowest MAE (1.1 kcal/mol) over the entire set. Non-empirical semilocal functionals and double-hybrids tend to yield MAEs closer to 4 kcal/mol. The authors also studied MP2 and found that it is unable to account for the complicated electronic structure encountered during these reactions yielding MAEs much closer to 10 kcal/mol depending on the variant of MP2 used. SG also noted that because of the failures of pure Hartree-Fock and MP2, double-hybrids with a large admixture of EXX or perturbative correlation perform poorly. Grimme and Steinmetz have recently developed a new double-hybrid (PWRB95) based upon RPA instead of MP2 using a similar truncation of SG13, and showed it also delivers errors of similar quality to PBE0 and improves upon RPA@PBE as well. 34,96 Given the similarity between RPA@PBE from Refs. 34 and 96 and our RPA@SCAN results, we compare only RPA@SCAN to RPA@PBE0 and dRPA75 below. We averaged the results for each catalyst separately and plotted them in Fig 3. Though SCAN is a more sophisticated functional than PBE, for this particular test set the errors

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mational energy differences, though these methods are noticably different for barrier heights and atomization energies, 34 as well as for the SG13 set. RPA evaluated with PBE, SCAN, or PBE0 does yield accurate results, but dRPA75 is much less accurate overall. The dual-hybrid performs reasonably well for Pd and PdCl− reactions, reducing the errors of RPA@SCAN by approximately 1 kcal/mol, but extremely poorly for the Ni and PdCl2 reaction sets. This drastic difference in performance compared to the other RPA methods is directly linked to the 75% admixture of EXX in the reference determinant (PBE75) and used to evaluate the energy expression. SG also point out there is to a certain extent a breakdown of perturbative correlation for the Ni reactions due to the perturbative nature of MP2. 55 This problem is entirely avoided by using RPA correlation in dRPA75 and PWRB95, but the improvement is not enough for dRPA75 to overcome the exceedingly large errors of EXX. Though PWRB95 also uses 50% EXX, the energy is evaluated from semilocal orbitals and so the deficiencies of HF-like orbitals are avoided and the method can still yield accurate results. For the Ni reactions in particular, some caution is required due to the borderline multireference cases included in the test set where CCSD(T) might exhibit errors larger than 1 kcal/mol, and perhaps the errors are not as large as they appear here. We also investigated the stability of each closed-shell complex with PBE, PBE0, and PBE75, and report the lowest triplet stability eigenvalue in the supporting information. In general, the reaction complexes and transition states are more prone to exhibit instabilities than the product complexes. The difficulty of these reactions is reinforced by the fact that even PBE predicts some of these complexes to be triplet unstable. In total, PBE predicts 6 instabilities; 3 for the Ni set, 2 for Pd, 0 for PdCl− , and 1 for PdCl2 . PBE0 exhibits slightly more instabilities with a total of 10, 8 of which come from the Ni subset, but PBE75 exhibits 40 instabilities. Not only does PBE75 exhibit instabilities more frequently, but the entire Ni subset is triplet unstable. This reinforces our earlier discussion of the multireference character encountered for these systems, and is clear evidence that CC results on the triplet

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surfaces should be compared to the singlet surfaces in order to determine the true reaction paths. Similarly large thermochemical errors were also reported for meta-GGA hybrids and to a lesser degree for standard double-hybrids in Ref 55. In particular the Minnesota family of functionals shows larger errors with larger fractions of exchange, while the double-hybrids exhibit the smallest errors when using between 20% and 25% EXX. To gain insight as to why dRPA75 performs so poorly for two out of four catalysts in SG13, yet yields decent results for FP18 and JB32, we investigated the contributions to a few reaction energies as a function of EXX mixing.

3.4

EXX mixing analysis

To analyze the impact of EXX mixing on the performance of RPA-based methods we selected three systems, CuF (FP18/JB32), CrO3 (JB32), and reaction 9 (methane insertion) from SG13, and systematically varried the mixing parameter ax to generate self-consistent orbitals. We then computed the PBE hybrid (PBEh), RPA, and dual-hybrid total energies finding a smooth behavior in each case, though the variation is not necessarily linear. Breaking the total energies into exchange and correlation contributions further facilitates the comparison. The different contributions for each reaction are plotted in Figures 4 and 5, and the behaviors of the dRPA75 and RPA@PBE for each catalyst in Figure 6. All one-electron terms from the kinetic energy and the external and Hartree potential energies are included in the exchange contribution plots below, i.e. they are not pure exchange energies (EX ), but rather the total energy differences of each system at the exchange only level (EX + VH + Vext + T ). CuF is largely ionic, the dominant contribution from the metal arising from the d10 configuration of Cu+ . 30 Semilocal functionals such as PBE and SCAN are accurate and underbind by less than 5 kcal/mol, yet RPA based methods underbind by 10-15 kcal/mol. From Fig. 4 it is clear this is due to the extremly low binding energy for CuF of EXX@PBE, ∼40 kcal/mol, about half that of the PBE exchange (PBEx) contribution. RPA@PBE 16


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both Ni and Pd, the EXX and RPAc contributions counterbalance each other leading to accurate results, but the dual-hybrid does not inherit this behavior as a different value of ax is needed for each catalyst to maximize the cancellation of errors between the PBEh exchange energy and RPAc@PBEh. Comparing RPA@PBEh to the dual-hybrid method for all catalysts reiterates that RPA@PBEh varies less dramatically with ax . Even for the Ni reaction energy, RPA@PBEh changes by less than 15 kcal/mol, whereas the dual-hybrid varies by nearly 20 kcal/mol. For the Pd based catalysts the variation can be even larger, between 30 and 40 kcal/mol for Pd and PdCl2 . The Pd based catalysts benefit from a large fraction of EXX in the dual-hybrid, but choosing larger values of ax results in large errors for the Ni system, further supporting the idea that no single dual-hybrid can be accurate for the breadth of transition metal chemistry. Fitting ax to minimize the error for some training set is certainly a possibility, but given the accuracy of traditional RPA evaluated with semilocal functionals for the systems studied here there seems to be little systematic improvement to be made by hybridizing the reference determinant or energy and performing such a fit.

4

Discussion and Conclusions

Taking the above reactions as a whole, the most systematic RPA method for transition metal chemistry tends to be the original formulation that uses input orbitals from a semilocal functional. While we have seen that in some cases, such as the JB32 set or the Pd and PdCl− reactions, using a hybrid functional to generate the reference determinant can lead to improved performance of RPA, the general conclusion from the other reactions is that no systematic improvement can be attained because of the variability in performance of HF for transition metal chemistry. This is likely due to the fact that for transition metal complexes the xc hole tends to be short-ranged, but non-local. 30 Mixing EXX with semilocal functionals introduces some long-ranged behavior not present in the real xc hole and so global hybrids are

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not as systematic an improvement as they typically are for main-group, organic chemistry. 97 Mixing EXX into the reference determinant or PBE exchange in the ACFD energy tends to reduce the interplay of EXX and RPA correlation contributions to the total energy and can lead to larger errors compared to the traditional RPA. Compared to hybrid functionals, RPA contains no fitted parameters, naturally incorporates dispersion, and only requires one evaluation of Fock exchange to compute the total energy while delivering equally accurate or superior results for transition metal reactions. From our studies of the reaction energy contributions it is clear that the behavior of semilocal and Fock exchange, as well as semilocal and RPA correlation, in transition metal systems is very different. Often EXX severely underbinds or does not bind the complexes at all because it over stabilizes the fragments. RPA correlation makes up for this deficiency, meaning it gives large contributions to the energy differences. For PBE, the exchange part generally yields the dominant contribution of the energy difference and PBEc simply introduces a relatively smaller shift in the result than RPAc. Consequently, any hybridized method of RPA with semilocal functionals, such as dual- or double-hybrids, essentially tries to find the combination of energy contributions that yields the smallest errors by maximizing the cancellation of errors any one piece makes with the others, e.g. dRPA75 for CrO3 . Though a fit for different mixing parameters can yield an overall decent double-hybrid (e.g. PWRB95), any physical intuition as to why the method is yielding accurate results for transition metal chemistry is likely completely lost in the fitting procedure. Furthermore, dual-hybrids that utilize large fractions of EXX are prone to triplet instabilities in the reference state much like Hartree-Fock references, which is clearly an additional drawback of these methods compared to RPA evaluated with semilocal reference states. In order to improve the RPA results presented here without hybridization, beyond-RPA correlation methods are needed that incorporate the correct short-ranged correlation, yet remain cost-effective to study large systems. In conclusion, we have studied the performance of the Random Phase Approximation for

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several transition metal test sets paying close attention to the dependence on the reference determinant. Exact exchange and RPA correlation tend to counterbalance each other as the reference determinant changes, resulting in relatively insensitive total energy differences. While dual-hybrid variants such as dRPA75 can improve results for main-group chemistry, for transition metal reactions the traditional approach based on a self-consistent semilocal reference is preferred. Compared to semilocal functionals themselves, RPA yields a systematic improvement for predicting general transition metal thermochemistry, and can serve as a useful benchmark when high-level wavefunction results cannot be obtained.

Supporting Information Available Individual results and average errors for each test set can be found in Tables S1 and S3S6. Table S2 contains the optimized coordinates for the Fe2 Cl4 species used in this work. Plots of the exchange mixing dependence for the methane insertion reaction for Ni, Pd, PdCl− , and PdCl2 are also provided in Figures S1 and S2. Triplet instability analyses can be found in Tables S8-S12. This information is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgments We would like to thank F. Furche and C. A. Jiménez-Hoyos for insightful discussions. Funding for J.E.B., P.D.M., and A.R. was provided by the US Department of Energy under grant #DE–SC0010499. J.S. acknowledges support from the Center for the Computational Design of Functional Layered Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award # DE–SC0012575. Figures were made using matplotlib. 98

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Computational Details A developer’s version of turbomole 99 V7.0 was used for all SCAN calculations.

Karlsruhe

quadruple-zeta basis sets with additional polarization functions (def2-QZVPP) 100,101 were used for all atoms and quasi-relativistic effective core potentials were used for Pd. 102 Energies and densities were converged to at least 10−7 a.u. and fine quadrature grids 103 (size 4 or greater) were used to compute the density functional contributions. Resolution of the Identity (RI) methods 104,105 and the corresponding auxiliary basis sets 106 were used to accelerate the computation of the energy. To compute the RI-RPA 79 correlation energy in turbomole we typically used 50-80 frequency points, but grids larger than 100 points were sometimes needed for small gap systems to achieve sensitivity measures smaller than 10−4 a.u., which is a rough quantification of the error of the numerical integration. The 3s/3p and 4s/4p orbitals were included in the valence for the 3d and 4d transition metals, respectively, and the remaining core electrons were kept frozen in the RPA calculations. The calculations were performed in C1 symmetry. Basis set incompleteness errors for RPA were checked by studying the FP18 and SG13 test sets with extrapolated correlation consistent triple- and quadruple-zeta basis set results, and we found that the QZVPP results are reasonably close to the CBS limit so we did not do any further extrapolations. SCAN can be sensitive to the integration grid in turbomole, 68 and we checked each set accordingly. Only for FP18 was a larger than standard radial grid size needed; we used a radial size of 40 to ensure the results were fully converged. To study the dependence on the exchange mixing parameter in turbomole, the XCfun interface was used. 107 PBE, PBE0, RPA@PBE, RPA@PBE0, and dRPA75 were computed using mrcc 108 with the same basis sets described above. Radial and angular integration grids are constructed using literature methods. 109–111 The energy and density-matrix convergence thresholds used in turbomole were also used for mrcc. For RPA@PBE(0) and dRPA75, the RPA correlation energy was computed as in Ref. 82, in combination with density-fitting, 112 and the plasmon formula was used for any small-gap systems to avoid numerical instabilities. The fitting of integral lists was performed before the dRPA iterations. The same frozen core definitions described above were also adopted in mrcc.

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