Thermochemistry and Geometries for Transition-Metal Chemistry from

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Thermochemistry and geometries for transition metal chemistry from the random phase approximation Craig Waitt, Nashali M. Ferrara, and Henk Eshuis J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00756 • Publication Date (Web): 17 Oct 2016 Downloaded from http://pubs.acs.org on October 20, 2016

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Journal of Chemical Theory and Computation

Thermochemistry and geometries for transition metal chemistry from the random phase approximation Craig Waitt, Nashali M. Ferrara, and Henk Eshuis∗ Department of Chemistry and Biochemistry, Montclair State University, Montclair, NJ 07043, USA E-mail: [email protected]

Abstract The performance of the random phase approximation (RPA) is tested for thermochemistry and geometries of transition metal chemistry using various benchmarks obtained either computationally or experimentally. Comparison is made to popular (semi-)local, meta- and hybrid density functionals as well as to second-order MøllerPlesset perturbation theory (MP2) and its spin-component-scaled derivatives. The benchmarks sets include reaction energies, barrier heights and dissociation energies of prototype bond-activation reactions, dissociation energies for a set of large transition metal complexes, bond lengths and dissociation energies of metal hydride ions, and bond lengths and angles of a set of closed-shell first-row transition metal complexes. The emphasis is on first-row transition metal chemistry, though for energies secondrow elements are included. Attention is paid to the basis set convergence of RPA. For thermochemistry RPA performs on par or better than the DFT functionals presented and is significantly more accurate than MP2. The largest errors are observed ∗

To whom correspondence should be addressed

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in dissociation energies where the electronic environment is altered substantially. For structural parameters very good results are obtained and RPA meets the high quality of structures from DFT. In most cases well-converged structures are obtained with basis sets of triple zeta quality. MP2 optimized values can often not be obtained and are on average of inferior quality. Though chemical accuracy is not reached, the RPA method is a step forward towards a systematic, parameter-free all-round method to describe transition metal chemistry.

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Introduction The random phase approximation (RPA) method for the calculation of ground-state electron correlation energies has seen a recent revival. 1–3 RPA has distinct advantages over other popular computational approaches which make it a promising method. Compared to semilocal Density Functional Theory (DFT) 4 it is parameter-free and of includes dispersion interactions seamlessly. In comparison to wave function based Møller-Plesset perturbation theory, it is non-perturbative and therefore applicable to systems with small HOMO-LUMO gaps. Using density fitting methods RPA achieves a scaling of N 4 log N with system size N which allows computation of systems with over 100 atoms. 5 This is in sharp contrast to most coupled-cluster methods 6 and multi reference methods 7 which are limited to smaller systems. The performance of RPA for mid- and long-range correlation is an order of magnitude better than semi-local DFT and on par with the best-performing dispersion corrected functionals. 8 For reaction energies and barrier heights RPA performs similarly to the widely used B3-LYP 9,10 functional. 8 RPA works for a range of systems, 11 including molecules, 1,12–17 solids, 18–23 and surfaces. 24–26 The applicability of RPA was greatly enhanced by the recent availability of analytical gradients. 27,28 Benchmark results for small molecules on structures, vibrational frequencies and dipole moments showed that the good performance of RPA for 2


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ground-state energies carries over to these properties. The implementation also has the favorable N 4 log N scaling, 28 though be it with a larger pre-factor. The aforementioned qualities of RPA make it a promising candidate for the study of transition metal systems: small-gap situations can be handled, dispersion is included and good accuracy is achieved for energies and structures. However, for molecular systems, the majority of systems involved in the benchmark calculations were composed of main-group elements and only a limited number of results were published for transition metal systems. The predissociation energy of a Grubbs-II catalyst was reproduced well by RPA 8 as well as the structural parameters for a set of small transition metal complexes. 28 RPA produced very good agreement with experiment in a study of C-H bond activation reactions of bimetallic rare-earth hydride complexes. 16 Overall, very little is known of the performance of RPA for molecular systems and reactions involving transition metal elements. This study aims to provide an assessment of the performance of RPA for transition metal chemistry. Transition metals are a challenge for computational chemistry. 29,30 The quality of performance of a method on main-group elements is not a clear predictor for its performance for systems involving transition metals. 31–33 The inclusion of electrons in d- (and f -) orbitals makes these elements particularly difficult to describe computationally as one often has to deal with near-degeneracies. 30 The performance of DFT is also much more erratic across a row of the periodic table than for main-group chemistry and results for one group are no predictor for the performance of a different group. It is thus important to test a method carefully for well studied transition metal systems before applying it broadly to new chemical processes. Benchmark studies on systems with transition metals are less common than studies on main-group chemistry. First, because of the larger number of electrons involved it is much harder and often impossible to obtain high level theoretical benchmarks, such as CoupledCluster Singles, Doubles with Perturbative Triples 6 (CCSD(T)) results. Second, the amount of gas-phase experimental data that is available for comparison is limited and often uncer-

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tainties in these experimental values are too large for the data to be useful. However, in recent years several benchmark sets were designed. 31–40 The study of transition metal chemistry was greatly facilitated by the rise of hybrid and semi-local DFT due to its efficiency and surprising quality and DFT has become a general tool to understand many aspects of chemical reactivity and catalytic activity. 41 Structures are predicted particularly well and reaction energies are generally quite accurate. For an overview of the performance of DFT a broad of range of properties, see the recent review by Tsipis. 41 However, several issues remain, notably the lack of dispersion interactions that can play a large role in transition metal complexes. Also, the description of near degeneracies can be problematic. As mentioned before, the performance of DFT across a row of the periodic table is typically much more erratic than for main-group elements. The only feasible alternative from wave-function based methods is some variant of Møller-Plesset second order perturbation theory 42 (MP2); coupled cluster methods are often prohibitively expensive for the relatively sizable transition metal complexes. MP2 is not able to describe systems with a small HOMO-LUMO gap. In spite of the successes of the mentioned DFT methods, it is still of great interest to find a method for transition metal chemistry that is efficient with a consistently accurate performance for a wide-range of transition metals. Many attempts have been made to design such a method. Without being complete, some of the most notable are mentioned here. The fundamental flaw of semi-local DFT not to include dispersion can be largely remedied by applying a posteriori a Grimme-type correction, 43 which have become popular on account of their efficiency and accuracy. The Minnesota functionals 44–46 are based on fitting a set of parameters to several sets of fitting data. Some versions perform well for transition metal chemistry. 44,47,48 Other examples are the double-hybrid functionals which combine a certain amount of post-HF correlation with a DFT hybrid functional 49–52 and range-separated hybrids. 53 There is an extensive literature on the performance of these methods for transition metal chemistry. 41,53–60 On the side of the wavefunction based methods it is worth mentioning the spin-component-scaled

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MP2 methods, 61,62 which often outperform MP2. Coupled-cluster methods are generally too computationally expensive to be practical. In this work we perform a benchmark study to investigate the potential of RPA as a viable alternative to DFT or MP2 for transition metal chemistry. For practical reasons we will focus mainly on first row transition metals. First, energy gradients are not available for second and third row transition metals and relativistic effects are small for first row transition metals and will not affect energies and structures significantly. Second, first row transition metals present a greater challenge to computation since near degeneracies play a larger role. It thus seems reasonable to conclude that the quality of a method can be well judged by its performance on first row transition metal chemistry. Energetics results will also be presented for systems that involve second- and third-row transition metals, as they are commonly used in catalytic systems. Effort has been make to compare RPA results to either very high level theory or to experimental results. For all cases comparison is made to DFT with popular functionals and, if possible, to MP2 theory. The focus of this study is on structural parameters and on energetics. Other interesting properties, such as spin-spin splittings, (hyper)polarizabilities and vibrational modes, have been postponed to future work to limit the length of this study. For transition metal catalysis, the correct prediction of the structure and energy of reactants, products, intermediates and transition states is of crucial importance. A useful method ought to be able to describe these properties to a satisfactory degree. We will look at the performance of RPA for thermochemistry, including reaction energies, barrier heights, and dissociation energies. First, results will be presented for a benchmark set of reactions containing small model systems for which high level theoretical results are available. Second, RPA is applied to a set of dissociation energies for large, realistic catalytic systems. In the second part of the study we study the performance of RPA for structural parameters for metal hydride ions and a set of transition metal complexes. Finally, we draw a conclusion and present an outlook to the future.

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Computational Methods For a complete overview of the available RPA methods we refer the reader to recent reviews. 1,2 Here it suffices to say that the RPA total energy is given as

E RPA = E HXX + E CRPA ,

(1)

where E HXX is the expectation value of the Hamiltonian operator of the reference determinant and E CRPA is the RPA correlation energy. Both pieces are evaluated in a post-Kohn-Sham fashion using a self-consistent Kohn-Sham reference determinant. The RPA energy is thus evaluated non-self-consistently. In this work the RPA energy is evaluated in the so-called direct manner which ignores all exchange contributions to the kernel. 2 The resolution-ofthe-identity (RI) method is used to approximate four-center integrals, which results in a N 4 log N scaling for RPA. 5 RPA analytical gradients are available for closed-shell systems and are evaluated non-self-consistently in a post-Kohn-Sham fashion. 28 The implementation also has the N 4 log N scaling, but with a larger pre-factor. The current implementation of RPA does not allow the use of ECPs in conjunction with analytical gradients. This study therefore does not present structural parameters for systems with elements beyond the first transition metal row, for which ECPs are crucial to capture relativistic effects. All calculations were performed using the TURBOMOLE program package. 63 Self-consistent orbitals were computed using DFT with the Perdew-Burke-Ernzerhof 64 (PBE) functional, unless indicated otherwise. Tight convergence criteria were used for both density and energy. From experience it is known that the choice of (meta-)GGA functional has relatively little effect on the RPA results. 8 We therefore use the PBE functional by default and do not always present results from other functionals. Subsequent RPA calculations were performed with the rirpa module in TURBOMOLE. 63 Density fitting for Coulomb integrals was used for both DFT and RPA calculations. It is known that for both methods the error due to the RI-approximation is small and well controlled. 5 Input orbitals were computed using the same 6


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basis set as was used for the RPA calculation. For energy calculations the frozen core option was used for RPA and MP2. The nearest noble gas core was used for all elements. This option is not yet available for structure optimizations and therefore all electrons were included in the RPA structure optimizations. The grid size for the RPA frequency integration was chosen such that the correlation energies were accurate to 1×10−5 hartree. MP2 results were obtained from self-consistent Hartree-Fock orbitals. The ricc2 65,66 module was used with density fitting for the Coulomb integrals for MP2, SCS-MP2 and SOS-MP2. The scaling parameters were cOS = 1.2 and cSS = 0.3333 for SCS-MP2 and cOS = 1.3 for SOS-MP2. Karlsruhe basis sets 67,68 were used (def2-TZVP, def2-QZVP and def2-QZVPP) as indicated. They strike a good balance between accuracy and efficiency. Corresponding auxiliary basis set were used for RPA and MP2. 69–72 For the Coulomb fitting in DFT the def2-universal basis set was used. 73 For comparison and to study basis set convergence behavior correlationconsistent Dunning basis sets 74–78 were used ((aug)-cc-pVXZ, where X=D,T,Q, or 5). For transition metals beyond the first row relativistic effective core potentials (ECPs) were used for the core electrons. ecp-28 79 was used in conjunction with the def2- basis sets and defppecp 80 for the Dunning basis sets ((aug)-cc-pVXZ-PP). 81

Results Thermochemistry Model catalytic reactions There is a relative dearth in benchmark results for insertion reactions involving transition metals. To get a more complete picture of a method’s performance, Steinmetz and Grimme designed a set of 13 prototype reactions involving the catalysts Pd, PdCl− , PdCl2 and Ni. 38 The reactions include various types of bond activations, namely C-H, C-C, B-H, NH, and C-Cl. The activated molecules (H2 O, NH3 , C3 H6 , C2 H2 , C2 H4 , BH3 , H2 , C6 H6 ,

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CH4 , CH3 Cl and C2 H6 ) are all small so that high level theory benchmark energies could be obtained (CCSD(T) extrapolated to the complete basis set limit (CBS)). For each reaction the reactants form a complex which proceeds through a transition state to a metal-bond inserted product – See Figure 1. The relevant data for each reaction are the dissociation energy of the reaction complex, the forward and reverse barrier heights, and the reaction energy. Not all 13 reactions were included in the test set for every catalyst since for certain combinations it proved impossible to find a converged CCSD(T) reference. The resulting test set was used to benchmark a wide range of density functionals as well as MP2 methods. The best performing functional was the hybrid PBE0 functional with Grimme’s dispersion correction (-D3). 43 Grimme and Steinmetz used this benchmark in a recent study on a double hybrid density functional based on RPA and for comparison included some results for RPA. 82 Our results are in line with their findings and provide a more extensive picture for this benchmark. Figure 1: Representative example of the prototype reactions by Steinmetz and Grimme. 38 M represents the catalyst (Pd, PdCl− , PdCl2 , Ni).

M H

M + C2 H 4

H

M H

M, C2H4 H

Reactants

H

Complex

H

Transition State

H

H

Product

Figure 2 and Table 1 show the RPA results for these prototype reactions. The semi-local DFT reference for the Ni atom violates the aufbau principle and RPA results could not be obtained, therefore dissociation energies for the Ni-complexes are not included in these results. The table also presents results for the PBE functional and for the MP2 method including spin-component-scaled (SCS)-MP2 61 and scaled-opposite-spin (SOS)-MP2. 62 RPA performs slightly better than PBE for these reactions. The smallest average deviation is

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are well converged with respect to basis set size: going from an augmented quadruple zeta basis set (aug-cc-pVQZ) to an augmented quintuple basis set (aug-cc-pV5Z) changes the mean absolute deviation from 4.0 kcal/mol to 3.7 kcal/mol. The change in MAD of 0.3 kcal/mol is much smaller than the MAD itself. The changes in error distribution for RPA with respect to basis set are small as shown in Figure 3 for the barrier heights subset. Even though some basis set incompleteness error persists at the quadruple zeta level, it is sufficiently small to justify the use of these basis sets for RPA calculations. When compared with MP2, the performance of RPA is outstanding. MP2 shows larger deviations for reaction energies and barrier heights than RPA and has a MAD of 13.3 kcal/mol. The use of a spin-component scaled MP2 methods somewhat improves the results, but RPA is still more accurate. MP2 fails to incorporate the multi-reference character of these complexes, whereas RPA is able to do so to some extent and thus yield improved results. The effect of excluding some core electrons from the RPA correlation treatment is small compared to the inherent error of the method. Including all electrons (but still using ECPs for Pd) in the calculation of the RPA energy yields a MAD of 3.4 kcal/mol with the augcc-pVQZ(-PP) basis set, compared to 4.0 kcal/mol when core electrons are excluded for the same basis set. To better capture the contribution of the core electrons we also calculated all-electron RPA results using the aug-cc-pwCVQZ(-PP) basis set, which was designed to describe core-valence contributions accurately. 81,85 The MAD was also 3.4 kcal/mol. A more extensive study is needed to investigate the effect of freezing core electron and its dependence on the size of the core, but we leave that for a future study. Dissociation energies of large complexes Many benchmark sets have the drawback that they consists of small complexes and thus contrast strongly in size with realistic catalytic systems which can contain more than 100 atoms. Since it is impossible to obtain theoretical benchmark results for such large, realistic systems and experimental results are hard to obtain, there are far fewer benchmark sets

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available that contain large systems. It is crucial, though, to evaluate a method’s performance for large systems, since the effect of size can have a considerable impact on a method’s performance. For example, the contribution of non-covalent interactions in a system with large ligands is larger than in a small model system. This was clearly demonstrated in a study on the performance of DFT for the predissociation of a Grubbs II catalyst and derived model systems. 86 The B3-LYP functional performed well for the small model catalysts, but much worse for the real catalyst. In recent work Weymuth and Reiher 39 compiled a ligand dissociation energy database of large cationic transition metal complexes called the WCCR10 test set and analyzed the performance for the test set of a variety of density functionals for both structures and dissociation energies. The test set was compiled as a benchmark for large transition metal complexes which mimic complexes used in catalysis better than the smaller complexes that are part of other benchmark sets. The database consists of 10 dissociation reactions with a variety of transition metals (Au, Pt, Pd, Ru, Cu, and Ag) and ligand types. The complexes range in size from 42 to 174 atoms with ligands ranging in size from 3 to 52 atoms. Experimental dissociation energies from mass spectroscopy are available for these reactions; the values range from 29 to 52 kcal/mol. In all reactions the metal-ligand bond is broken. Such a dissociation is often a crucial part of the catalytic cycle and since the coordination of the metal center frequently changes in the process, this step presents a challenge for computational chemistry. In addition, the presence of large ligands leads to a much larger contribution of non-covalent interactions to the dissociation energy. These reactions thus present a strict test for a computational model. All molecules in the database are singlet closed-shell systems. The best performing functionals were found to be PBE0 and TPSSh, 39 but even these functionals yield a mean absolute deviation of about 7 kcal/mol. For structures it was found that the orientation of the ligand can vary considerably with functional and that empirical dispersion corrections frequently result in worse structures than obtained by the functional

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without dispersion corrections. Here, we will use WCCR10 to benchmark RPA. We limit ourselves to energies as gradients are not available beyond first row transition metals. Table 2: Mean deviation (MD), mean absolute deviation (MAD) and maximum absolute deviation (MAX) in kcal/mol in dissociation energies for the WCCR10 testset for various methods. The experimental dissociation energies for the reactions are provided as well (Ref.). The deviation is calculated as Emethod − Eref . The calculated energies were corrected for zero-point vibrational energies. Def2-QZVPP basis sets were used and the RPA results were obtained using self-consistent PBE orbitals. PBE structures taken from Weymuth et al. 39 were used. B3-LYP and PBE0 results were taken from Ref. 39 Reaction 1 2 3 4 5 6 7 8 9 10 MD MAD MAX

Ref. 28.7 48.9 49.5 32.7 46.0 50.6 52.2 46.5 38.9 24.5

RPA -1.1 12.2 11.8 19.6 3.6 15.3 7.3 3.4 -1.1 -1.1 7.0 7.7 19.6

TPSS -6.3 -5.7 -5.9 -4.9 -9.5 14.7 10.5 3.9 -4.4 -5.9 -1.3 7.2 14.7

B3-LYP -10.8 -7.9 -8.3 -11.7 -20.0 5.2 0.5 -4.0 -12.8 -12.5 -8.2 9.4 20.0

PBE0 -10.3 -5.8 -6.2 -1.1 -13.1 9.8 2.7 -1.5 -5.6 -8.2 -3.9 6.4 13.1

MP2 -3.3 15.9 15.5 23.9 13.8 26.7 16.3 11.6 13.2 4.9 13.9 14.5 26.7

SCS-MP2 -2.6 11.4 10.9 16.6 5.8 21.2 11.2 6.7 4.4 0.4 8.6 9.1 21.2

SOS-MP2 -2.3 9.1 8.6 13.0 1.8 18.5 8.6 4.3 0.1 -1.9 6.0 6.8 18.5

Table 2 shows results for the WCCR10 test set. The results are based on structures optimized at the PBE level of theory. For all methods zero-point vibrational energy corrections were made using the BP86 values computed by Weymuth et al. 39 The ZPVEs vary little with the method used and it is thus a reasonable assumption to use the BP86 values to avoid the computationally demanding calculation of second derivatives for these large systems. Comparison to the experimental reference values shows that for most systems RPA overestimates the binding energy (positive error equals overbinding), sometimes to a large extent. For reaction 4 the relative error is 60%. Compared to PBE, RPA does not show an improved performance, though the error distribution is very different for the two methods. For reactions 2-5 PBE underestimates, whilst RPA overestimates the binding energy. For reaction 6-8 both methods overestimate. RPA has a MD of 7.0 kcal/mol and a MAD of 7.7 13



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kcal/mol, whereas PBE has MD of -1.3 kcal/mol and a MAD of 7.2 kcal/mol. The error for PBE is thus more random than for RPA which mostly overbinds. MP2 performs on the whole significantly worse than PBE and RPA with a MAD of 14.5 kcal/mol. Like RPA it overestimates the binding energy for most reactions. Significant improvement is obtained by using a spin-component-scaled variant of MP2. MADs of 9.1 and 6.8 kcal/mol are obtained for SCS-MP2 and SOS-MP2, respectively. Even though the performance of RPA is by no means great, it clearly outperforms MP2 for these large systems. For weakly bound systems RPA is known to underestimate the binding energy in the basis set limit. 87 The origin of the overbinding of RPA for these systems is not clear. Further analysis of the different contributions to the RPA energy is needed to shed light on this issue. Reactions 6-8 are all the same, except for the metal involved (Au, Ag, Cu, respectively). All methods show the same trend, giving the smallest error for the first-row transition metal Cu and the largest for the third-row transition metal Au. Since ECPs were used, it is likely that relativistic effects are not correctly accounted for. The def2-QZVPP basis sets used in our calculations are very large and the effect of basis set superposition error is thus expected to be small. To get a sense of the basis set convergence of the results the calculations were repeated using Dunning’s correlation consistent basis sets (cc-pVXZ). Full results are reported in the supporting information. RPA yields a MAD of 7.2 kcal/mol for the cc-pVQZ basis set and MAD of 6.9 kcal/mol for the cc-pV5Z basis set. For all reactions the increase in basis set size leads to a smaller dissociation energy, but the effect is small compared to the error in the method. The trend is in line with the previously observed basis set convergence of RPA for non-covalently bound dimers. 87 The minimum energy structures vary sometimes considerably with the functional. 39 The RPA dissociation energies for the structures of all the functionals are given the in the supporting information. The variation in RPA dissociation energy with functional used to optimize the structures, is much smaller than the error in RPA energies. We therefore decided not to emphasize this aspect.

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Structures First-row transition metal hydrides Table 3: Deviations in bond lengths (in ˚ A) and in dissociation energies D (in kcal/mol) of + metal hydride ions (MH ) compared to experimental values for the methods B3-LYP, RPA, and MP2. Deviations are computed as rcalc − rexpt and Dcalc − Dexpt , respectively. For each method the mean deviation (MD), mean absolute deviations (MAD) and absolute maximum deviation (MAX) is presented. All calculations were performed with the def2-QZVPP basis sets. RPA results were calculated from self-consistent PBE orbitals. Hydride ScH+ VH+ CrH+ MnH+ FeH+ CoH+ MD MAD MAX

State 2∆ 4∆ 5 Σ+ 6 Σ+ 5∆ 4Φ

Atom 3D 5D 6S 7S 6D 3F

re ( ˚ A) 1.829 1.661 1.604 1.652 1.603 1.547

B3-LYP -0.069 -0.031 -0.004 -0.053 -0.040 -0.027 -0.037 0.037 0.053

RPA -0.039 -0.014 0.006 -0.046 -0.052 -0.001 -0.024 0.026 0.052

MP2 -0.062 -0.040 -0.046 -0.047 -0.063 -0.034 -0.049 0.049 0.062

D (kcal/mol) 56.3 47.3 31.6 47.5 48.9 45.7

B3-LYP 3.7 6.8 4.9 1.1 8.5 3.0 4.4 4.4 8.5

RPA 2.2 0.1 8.2 -1.1 0.4 3.3 2.2 2.6 8.2

MP2 6.0 -5.1 -12.8 -7.8 -3.3 3.6 -3.2 6.4 12.8

Some of the simplest possible test systems for transition metal complexes are the metal hydrides MH+ , where M represents the metal. The first-row transition metal series of hydrides has been studied experimentally 88 and computationally. 29,89,90 Experimental data are available for equilibrium bond lengths and dissociation energies (to M+ and H). 88 Blomberg et al. included this set in their benchmark study on hybrid functionals. 29 The dissociation of these systems, though small, presents a challenge for RPA since the only bond present is broken and the spin state of the metal cation is different from the hydride. These systems also allow for the simplest possible investigation of the performance of RPA for metal-hydrogen bond lengths. Since only a single bond is involved, it is straightforward to find the minimum energy structure numerically. This set therefore also allows the study of the quality of RPA structural parameters for open-shell systems. In Table 3 deviations in bond lengths and in dissociation energies are shown for a set of first-row transition metal hydrides for the methods B3-LYP, RPA and MP2. The metals Ti and Ni are omitted since the DFT semi-local reference for RPA violated the aufbau principle. 15



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The spin states of the hydride ion and the corresponding metal cation are shown as well; these states are the lowest lying states. The amount of spin contamination of each PBE reference is less than 2% of the eigenvalue of the Sˆ2 operator for the corresponding spin state. The experimental dissociation energies have an average uncertainty of 2.1 kcal/mol. 88 There is little difference in the performance of B3-LYP and RPA for bond lengths and both methods are in excellent agreement with experiment. MP2 yields slightly larger errors but is not significantly worse. All methods underestimate the bond lengths (except RPA for CrH+ ). For the dissociation energies RPA surprisingly performs somewhat better than B3-LYP with a MAD of 2.6 kcal/mol. Given that the only bond present is broken and different spin states are involved, the RPA results are outstanding even though there are still substantial errors. The largest deviation is 8.2 kcal/mol for the chromium hydride ion. Chromium has multiple near-degenerate states and is notoriously difficult to describe with a single-reference method. 91 With a MAD of 6.4 kcal/mol MP2 does not describe these dissociation accurately. First-row transition metal complexes B¨ uhl and Kabrede 40 published a benchmark study for geometries of first-row transitionmetal complexes from DFT. They used a set of 32 molecules for which accurate gas-phase experimental structural parameters were available (either from gas-phase electron diffraction or microwave spectroscopy). Three of the complexes have a doublet spin state, all others are singlets. The emphasis is on bond-lengths, but about 15 angles are included as well. The set covers a wide range of bonding situations and thus forms a good test set for a computational method. The authors assessed the performance of various popular functionals and found that the BP86 functional was the best performing GGA and TPSS a slightly better metaGGA. Since the implementation of RPA in TURBOMOLE used for this work does not allow optimization of open-shell molecules, we will remove those from the test set and use the 29 remaining closed-shell structures. It should be noted that the experimental bond

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Table 4: Deviations in bond lengths (in pm) compared to experimental values for various density functionals, RPA, and MP2. Deviations are computed as rcalc − rexpt . The mean deviation (MD), mean absolute deviations (MAD), standard deviation (STD) and absolute maximum deviation (MAX) for these data are presented in Figure 4. All calculations were performed with the def2-QZVP basis sets. RPA results were calculated from self-consistent PBE orbitals. Compound Sc(acac)3 TiCl4 TiMeCl3

Distance Ref. PBE BP86 B3-LYP TPSS RPA Sc-O 207.6 3.5 3.3 3.1 3.1 0.5 Ti-Cl 216.9 1.6 1.6 1.3 1.7 1.5 Ti-C 204.7 0.2 0.3 -0.3 1.1 1.3 Ti-Cl 218.5 1.0 1.1 1.2 1.2 0.5 TiMe2 Cl2 Ti-C 205.8 0.0 0.1 -0.5 0.9 0.3 Ti-Cl 219.8 0.9 0.9 1.4 1.2 0.2 VOF3 V=O 157.0 0.5 0.6 -1.4 0.6 3.0 V-F 172.9 0.8 0.7 -0.2 0.5 0.4 VF5 V-F ax 173.4 3.0 2.9 1.5 2.5 2.7 V-F eq 170.8 1.9 1.8 0.5 1.6 1.5 VOCl3 V=O 157.3 0.0 0.0 -1.9 0.0 2.2 V-Cl 213.8 1.3 1.4 1.0 1.3 1.7 VCp(CO)4 V-CCO 196.3 -3.5 -3.0 -0.9 -1.8 -1.6 CrO2 F2 Cr=O 157.4 -0.3 -0.3 -2.3 -0.4 2.4 Cr-F 171.9 0.4 0.3 -0.5 0.1 0.3 CrO2 Cl2 Cr=O 157.7 -0.5 -0.4 -2.4 -0.5 2.4 Cr-Cl 212.2 0.4 0.4 0.2 0.3 0.4 CrO2 (NO3 )2 Cr=O 158.4 -1.0 -0.9 -3.2 -1.0 2.2 Cr-O 195.4 -2.0 -2.2 -3.1 -2.3 -2.5 Cr(C6 H6 )2 Cr-C 215.0 -0.6 0.3 2.2 -0.7 -1.1 Cr(C6 H6 )(CO)3 Cr-C Ar 220.8 0.7 1.6 3.9 0.0 -1.5 Cr-C co 186.3 -2.1 -1.9 -0.3 -0.8 0.0 Cr(NO)4 Cr-N 175.0 -0.8 -0.8 -1.7 -0.8 1.7 MnO3 F Mn=O 158.6 0.2 -0.8 -2.9 -1.0 2.3 Mn-F 172.4 -0.4 -0.5 -1.0 -0.8 -0.8 MnCp(CO)3 Mn-C Cp 214.7 1.1 1.7 3.4 0.2 -0.5 Mn-C Co 180.6 -1.9 -1.7 0.0 -0.8 -0.1 Fe(CO)5 Fe-C mean 182.9 -2.5 -2.2 -0.7 -1.7 -1.7 Fe(CO)3 (tmm) Fe-C Co 181.0 -2.8 -2.5 1.2 -1.8 -0.6 Fe-C cent 193.8 1.1 1.5 1.0 0.8 -0.4 Fe-C CH2 212.3 1.3 2.0 1.4 0.2 -0.8 Fe(CO)2 (NO)2 Fe-C mean 187.2 -5.5 -5.2 -3.7 -4.7 -2.9 Fe-N 167.4 -0.3 -0.2 -1.5 -0.4 3.1 FeCp2 Fe-C 206.4 -1.8 -1.3 1.5 -2.2 -1.1 Fe(C2 H4 )(CO)4 Fe-C et 211.7 1.7 2.5 3.0 1.1 -0.9 Fe-C ax 181.5 -1.4 -1.1 0.9 -0.5 -0.2 Fe-C eq 180.6 -1.6 -1.3 -0.1 -0.7 0.7 Fe(C5 Me5 )(P5 ) Fe-P 237.7 -0.3 0.0 3.2 -1.5 1.3 CoH(CO)4 Co-C eq 179.8 -0.7 -0.4 1.0 -0.1 1.0 Co(CO)3 (NO) Co-N 165.8 0.4 0.4 -1.1 0.1 5.3 Co-C 183.0 -2.3 -2.1 -0.3 -1.7 -0.2 Ni(CO)4 Ni-C 182.5 -0.1 0.1 1.9 0.3 1.7 Ni(acac)2 Ni-O 187.6 -2.0 -2.1 -1.6 -2.6 -0.3 Ni(PF3 )4 Ni-P 209.9 1.2 1.7 3.2 0.8 0.5 CuCH3 Cu-C 188.4 1.0 1.0 3.3 -1.8 1.7 CuCN Cu-C 183.2 -1.1 -1.3 1.9 -0.9 1.4 a MP2 results omitted due to failed HF or CC convergence. acac=acetylacetonate, ax=axial, 19 cent=central,Cp =cyclopentadienyl ACS Paragon Plus Environment eq=equatorial. et=ethylene, tmm=trimethylenemethane

MP2a 1.8 2.2 2.8 -0.3 1.0 -0.8

17.9 5.6 5.5 1.0 -4.3 -0.2 0.8 0.2 -0.3

-3.1 -1.9 -0.4 13.4 1.1

3.3

-8.2 11.2 8.1 1.8 7.9

-9.1 -5.4 -2.2 -0.5 -0.9


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Table 5: Deviations in bond angles (in degrees) compared to experimental values for B3LYP, RPA, and MP2. Deviations are computed as anglecalc − angleexpt . For each method the mean deviation (MD), mean absolute deviations (MAD), standard deviation (STD) and absolute maximum deviation (MAX) is presented. All calculations were performed with the def2-QZVP basis sets. RPA results were calculated from self-consistent PBE orbitals. Compound TiMeCl3 TiMe2 Cl2

Angle Cl-Ti-C C-Ti-C Cl-Ti-Cl O-V-F Cl-V-Cl O-Cr-O F-Cr-F O-Cr-O Cl-Cr-Cl CCO -Cr-CCO O=Mn-F Fe-Ccent -CCH2 Ceq -Fe-Ceq H-Co-Ceq

Ref. 105.6 102.8 117.3 107.7 111.4 107.8 111.9 108.5 113.3 87.4 108.5 76.4 111.7 80.3

B3-LYP RPA MP2 -0.6 -1.3 -0.6 0.9 0.8 1.4 1.1 2.1 1.2 VOF3 0.8 0.3 -6.0 VOCl3 -0.8 -0.8 1.3 CrO2 F2 0.5 -0.5 -0.3 -2.2 0.2 2.8 CrO2 Cl2 0.6 0.5 0.3 -2.5 -1.6 0.9 Cr(C6 H6 )(CO)3 1.6 0.8 -6.9 MnO3 F -0.1 1.7 -4.9 Fe(CO)3 (tmm) 0.5 0.0 Fe(C2 H4 )(CO)4 1.8 0.7 -9.1 CoH(CO)4 0.8 1.2 -0.9 MD 0.2 0.3 -1.6a MAD 1.1 0.9 2.7a STD 1.3 1.1 3.8a MAX 2.5 2.1 9.1a a Values calculated with missing MP2 data cent=central, eq=equatorial, tmm=trimethylenemethane

Table 6: Timings for RPA Energy and Gradient calculations (hrs) for 2 systems from Table 4 for 2 basis sets. VOF3 has 58 electrons, CrO2 (NO3 )2 has 102 electrons. Timings are wall times for calculations on a single 2.3 GHz CPU with 12Gb of memory.

Energy (hr) Gradient (hr) Total (hr)

VOF3 def2-QZVP aug-cc-pVQZ 3.4 × 10−3 0.4 −2 2.4 × 10 2.2 2.7 × 10−2 2.6

20

CrO2 (NO3 )2 def2-QZVP aug-cc-pVQZ 0.2 2.1 1.6 12.7 1.8 14.8


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indicates an overall underestimation of RPA of bond lengths, whereas for the Karlsruhe basis sets used for Table 4 we obtain a positive MD and thus an overall overestimation. A more extended study is necessary to investigate the basis set behavior of RPA for structures. Finally, in Table 6 representative timings for RPA energy and gradient calculations are presented to give the reader an impression of the effort associated with these optimizations.

Conclusion The performance of RPA for transition metal chemistry was studied using various test sets. For reaction energies and barrier heights of small prototype bond-activation reactions RPA performs very well, particularly in comparison with MP2. Even though the RPA results are satisfactory chemical accuracy of less than 1 kcal/mol is not reached. The dissociation energies for this set and the metal hydrides are more of a challenge which is a reflection of the large changes in electronic environment. A similar conclusion can be drawn for the dissociations of the large systems in the WCCR10 set. RPA performs similarly to the TPSS functional and again improves upon MP2, but the relatively large MAD of 7.2 kcal/mol demonstrates the need for additional corrections to RPA. The results for the metal hydride ions are surprising in view of RPA’s poor performance for atomization energies. 83 RPA performs somewhat better than B3-LYP even though spin state changes are involved and the only available bond is broken. In a catalytic cycle the addition or dissociation steps are typically the most challenging for computational chemistry. Hence, the results shown in this work are expected to represent the largest deviations for a catalytic process. One expects the deviations for all other steps in the cycle to be equal or smaller. This will be the subject of a future study. Bond distances and angles were optimized at the RPA level for small metal hydrides and a set of first-row transition metal complexes. RPA shows a marked improvement over MP2 and performs on par with the density functionals tested. Even though RPA only includes

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ring diagrams in its description of the correlation and no exchange it is able to describe structures with good accuracy. In all, we conclude that RPA is a step forward towards a systematic description of transition metal chemistry. RPA offers a strong alternative to MP2 with better performance for both energies and structures at lower computational cost. In contrast to MP2, stable reference wavefunctions could be obtained for all systems except for single atoms such as Ni and Ti and small-gap systems are not a problem. Compared to commonly used functionals, RPA performs similar or better for all systems studied and thus offers a viable alternative that from the offset includes dispersion and is parameter-free. A possible use of RPA is in the study of reaction mechanisms involving transition metals, for example in transition metal catalysis. The systems involved often are too large for high-level theoretical methods and RPA provides a good alternative. Though not without error, it behaves more systematically than most density functionals. Alongside commonly used functionals, RPA can be used to calculate the relative energies of reaction pathways. In addition, it can be used to check the quality of structures, especially when non-covalent interactions are involved. However, improvement of RPA is certainly needed to reduce errors in energies and achieve good quantitative accuracy; this is particularly clear from the dissociation energies of the large systems studied. The performance of RPA is not sufficiently uniform and balanced to be relied on as a benchmark method. The clear theoretical framework of RPA allows for systematic improvements. Several such methods have already been proposed 17,82,84,92–94 and are likely to lead to an improved description of transition metal chemistry. This work is a first overview of the performance of RPA for transition metal chemistry. Several areas were excluded from study. A more comprehensive overview of second- and third-row transition metal chemistry including relativistic effects as well as the study of other properties such as frequencies and (hyper)polarizabilities is needed to get a more complete picture. It is also desirable to further investigate the contributions to the fairly large deviations in dissociation energies of large transition metal complexes to understand

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the origin of the overbinding of RPA. The basis set convergence of RPA for thermochemistry of transition metal systems is similar to that of main-group elements. 87 Some basis set incompleteness error persists at the quadruple zeta level, but it is typically much smaller than the deviation of RPA with the benchmark. When very accurate results are required and when system size permits, one can use basis set extrapolation to estimate the basis set limit. However, in most cases the use of quadruple zeta quality basis sets suffices. For energies it is not recommended to use basis sets of lower quality. Structural parameters converge more quickly with basis set size. Some improvement is observed when using quadruple zeta basis sets vs. triple zeta basis sets, but the effect is small. Good quality structures can therefore be obtained relatively efficiently at the RPA level using triple zeta basis sets.

Acknowledgement This work was supported by the National Science Foundation (Grant CHE-1464960). H.E. thanks Jefferson E. Bates for useful comments.

Supporting Information Available Full results for the WCCR10 set and for bond lengths and angles of the optimized complexes are available in the Supporting Information.

This material is available free of charge via

the Internet at http://pubs.acs.org/.

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