How To Arrive at Accurate Benchmark Values for Transition Metal

Article Cite This: J. Chem. Theory Comput. 2017, 13, 5291-5316

pubs.acs.org/JCTC

How To Arrive at Accurate Benchmark Values for Transition Metal Compounds: Computation or Experiment? Yuri A. Aoto,† Ana Paula de Lima Batista,‡ Andreas Köhn,*,† and Antonio G. S. de Oliveira-Filho*,§ †

Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany Departamento de Química Fundamental, Instituto de Química, Universidade de São Paulo, 05508-000 São Paulo, SP, Brazil § Departamento de Química, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, 14040-901 Ribeirão Preto, SP, Brazil ‡

S Supporting Information *

ABSTRACT: With the objective of analyzing which kind of reference data is appropriate for benchmarking quantum chemical approaches for transition metal compounds, we present the following, (a) a collection of 60 transition metal diatomic molecules for which experimentally derived dissociation energies, equilibrium distances, and harmonic vibrational frequencies are known and (b) a composite computational approach based on coupled-cluster theory with basis set extrapolation, inclusion of core−valence correlation, and corrections for relativistic and multireference effects. The latter correction was obtained from internally contracted multireference coupled-cluster (icMRCC) theory. This composite approach has been used to obtain the dissociation energies and spectroscopic constants for the 60 molecules in our data set. In accordance with previous studies on a subset of molecules, we find that multireference corrections are rather small in many cases and CCSD(T) can provide accurate reference values, if the complete basis set limit is explored. In addition, the multireference correction improves the results in cases where CCSD(T) is not a good approximation. For a few cases, however, strong deviations from experiment persist, which cannot be explained by the remaining error in the computational approach. We suggest that these experimentally derived values require careful revision. This also shows that reliable reference values for benchmarking approximate computational methods are not always easily accessible via experiment and accurate computations may provide an alternative way to access them. In order to assess how the choice of reference data affects benchmark studies, we tested 10 DFT functionals for the molecules in the present data set against experimental and calculated reference values. Despite the differences between these two sets of reference values, we found that the ranking of the relative performance of the DFT functionals is nearly independent of the chosen reference.

1. INTRODUCTION

are energetically near-degenerate and contribute strongly to the actual wave function. Kohn−Sham density functional theory (DFT) is a popular and cost-efficient approach to compute the solutions of the electronic Schrödinger equation for compounds from all over the periodic table.13−16 The typical density functionalsused to approximate the exchange-correlation potentialmodel a rather local exchange-correlation hole, and the method thus works very well for cases with a small electronic gap, as often found for transition metal compounds. In contrast to this, wave function methods based on a Hartree−Fock reference break down in these cases because of artifacts from the strongly non-local exchange potential of Hartree−Fock. On the other hand, traditional Kohn−Sham DFT still relies on a single-reference determinant, which becomes questionable whenever strong configuration mixing or non-high-spin spin-coupling of the open

The open d-shell of transition metal atoms is key for the rich chemistry and the special physical properties of their derived compounds. These are of particular interest in catalysis and biocatalysis and also in materials science, e.g., for the exploitation of magneto-electrical or magneto-optical effects.1−8 For instance, in order to understand the details of catalytic mechanisms or the origin of specific physical properties, the theoretical modeling of such systems has found increased interest.9−12 The parameters for such models, however, are often hard to obtain from experiment alone, and ab initio computation has, in recent decades, become an important tool to provide such missing information. The problem of predicting the properties of transition metal compounds is a difficult one because of their complex electronic structure. Often, the spin-coupling of the open d-shell has to be taken into account, and in many cases, it is not possible to understand the electronic structure on the basis of a single electronic configuration. Rather, several configurations © 2017 American Chemical Society

Received: July 4, 2017 Published: September 27, 2017 5291

DOI: 10.1021/acs.jctc.7b00688 J. Chem. Theory Comput. 2017, 13, 5291−5316

Article

Journal of Chemical Theory and Computation

The authors reported rather disappointing results for coupledcluster methods and concluded that coupled-cluster theory is not appropriate for benchmarking purposes when transition metal compounds are involved. More recently, this view was challenged by the works of Cheng et al.89 and Fang et al.90 Both studies demonstrated that reliable results can be derived when using the CCSD(T) approach and sufficiently large basis sets. In addition, they showed that the large deviations observed between theoretical and experimental data for some molecules present in the set of ref 85 were due to inaccurate experimental values. In this work, we set out to further explore the limits of applicability of single-reference CCSD(T) for transition metal compounds. In addition, we will provide results using a multireference analogue of this method, namely, the internally contracted multireference CCSD(T) [icMRCCSD(T)] method,91−94 which is being actively developed by two of the present authors. This approach to MRCC theory is orbital-invariant, sizeextensive, and size-consistent and treats all of the reference determinants on equal footing. For comparison, a number of density functionals is studied as well. We collected an extended set of benchmark data that not only consists of thermochemical data but also takes into account the prediction of equilibrium structures and vibrational frequencies, which provide additional critical tests of the method’s validity.

shell plays a role. For recent attempts to extend DFT to multiconfigurational systems, see refs 17−22. Generally, DFT suffers from the problem of not providing a systematic hierarchy of approximations, and functionals might fail dramatically for unexpected cases.23−28 Hence, although DFT is often still the only way to treat large-scale systems, dedicated benchmarking of the functionals is inevitable whenever authoritative predictions are attempted. For simple closed-shell systems, there is a well-established hierarchy of methods, in particular involving coupled-cluster (CC) theory, that allows one to systematically improve the accuracy of the theoretical description. Although this, of course, comes with a strong increase in the required computational resources, these methods still offer the possibility to benchmark less accurate methods (in particular DFT) in a reliable manner, without the need to resort to experimental data. On the other hand, a number of embedding schemes have been reported29−35 that enable the application of accurate quantum-chemical methods in the core region of interest, while the environment (ligands or protein environment) can be treated at lower levels of theory. Furthermore, the recent progress in the development of linear-scaling coupled-cluster methods has pushed the limits of application to quite large molecules.36−42 To summarize, the situation for closed-shell molecules is such that the well-known CCSD(T) method (CC theory with singles, doubles, and perturbative triples clusters) used with sufficiently large basis sets is believed to provide accurate thermochemical data within 1 kcal mol−1 accuracy. More advanced methods that, in particular, take care of correlation effects beyond CCSD(T) by additive schemes can even reach the 1 kJ mol−1 regime and may even be used to challenge the validity of experimentally derived results.43−55 Examples are the high accuracy extrapolated ab initio thermochemistry approach (HEAT53), the Weizmann-4 protocol (W456,57), focal point analysis (FPA58−60), the Feller− Peterson−Dixon method (FPD43,61,62), and the correlation consistent composite approach (ccCA49,50) and its multireference version (MR-ccCA63). For open-shell systems with pronounced multireference character, the situation in terms of accurate computational schemes is less clear. The single-reference CC hierarchy breaks down whenever a single determinant becomes an improper zeroth-order description of the system, whereas multireference methods lack a systematic hierarchy comparable to CC theory. The standard schemes, beyond second-order perturbation theory, still rely on multireference configuration interaction (MRCI) with approximate size-extensivity corrections.64−68 The extension of coupled-cluster theory to multireference cases, on the other hand, is not straightforward, and only recently have a number of implementations been reported that are capable of treating realistic systems.69−73 In this context, the multireference coupled-cluster (MRCC) theory71,73 should provide a very reliable way to treat systems with such complex electronic structure. The choice of a suitable level of theory for a specific task is central to routine quantum chemical calculations. This choice is non-trivial, and to establish the quality of an electronic structure method, its performance with respect to experimental or a high-level theoretical reference values must be measured. These benchmark studies are consistently reported in the literature.61,74−88 Recently, Xu et al.85 studied a number of small transition metal complexes, using both coupled-cluster theory (CCSD(T) and also some methods that go beyond this level) and Kohn−Sham DFT with a selection of density functionals.

2. EXPERIMENTAL REFERENCE VALUES The comparison to experimentally determined values is of central importance in the present study, and a new database of reference values was set up, as shown in Table 1. It consists of 60 diatomic molecules containing d-block metals and main group elements for which experimentally determined values of internuclear equilibrium distances, harmonic frequencies, and dissociation energies are known. Only the most common isotope of each element was considered. The database contains only data derived from experiment as compiled in standard collections of spectroscopic data.139,208−210 We carefully revised the values listed in these collections, and updated values are used whenever newer and reliable experimental works were available. For instance, we used the revised values for dissociation energies of some hydrides proposed by Cheng et al.89 Note that our database overlaps significantly with the 3dMLBE20 collection by Truhlar and co-workers85 and with the collection of heats of formation of Wilson and co-workers.211 However, we have also found a number of new experimental references for molecules present in these databases. A comparison of reference values collected in the 3dMLBE20 database and in our work is given in Table S2 of the Supporting Information. In particular, great care was taken to ensure that all reference values in our database have experimental origin. For instance, the dissociation energy of ZnS shown in Table 1 differs by about 15 kcal mol−1 from the value in the 3dMLBE20 database.85 As has been observed by Fang et al.,90 the dissociation energy used in the 3dMLBE20 database does not have experimental origin: its value for the dissociation energy of ZnS is from a thermochemical analysis by von Szentpály,212 which actually uses calculated (at the MRCI+Q/CBS level of theory) values for the dissociation energy.213 A complete version of the database, including the reduced masses used in all our calculations, as well as the bond dissociation energies at 0 K and first vibrational anharmonic constants for all 60 molecules, is given in Table S1 of the Supporting Information. 5292


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Journal of Chemical Theory and Computation Table 1. Database of Spectroscopic Constants for 60 Diatomic Molecules Containing d-Block Metalsa molecule ScH ScO ScF ScS TiH TiN TiO TiF TiS TiCl VN VO VCl CrH CrO CrF CrCl MnH MnO MnF MnS MnCl FeH FeO FeS FeCl CoH CoO CoS CoCl

electronic state

Re (Å)

ωe (cm−1)

De (kcal mol−1)

ref

molecule

Σ 2 + Σ 1 + Σ 2 + Σ 4 Φ 2 + Σ 3 Δ 4 Φ 3 Δ 4 Φ 3 Δ 4 − Σ 5 Δ 6 + Σ 5 Π 6 + Σ 6 + Σ 7 + Σ 6 + Σ 7 + Σ 6 + Σ 7 + Σ 4 Δ 5 Δ 5 Δ 6 Δ 3 Φ 4 Δ 4 Δ 3 Φ

1.7754 1.6661 1.787 2.1353 1.777 1.5802 1.6203 1.8311 2.0827 2.2697 1.5703c 1.5893 2.2145 1.6554 1.615 1.7839 2.194 1.7309 1.6446c 1.836 2.0663 2.2352 1.606 1.6164 2.0140c 2.1742 1.5327 1.6286 1.9786 2.0656

1596.00 972.2 735.33 565.2 1567.3 1049.73 1009.18 658b 562.40 406.98 1033 1011.3 417.4 1656.05 898.4 664.11 400.3b 1546.85 839.55 624.2 491.05 386.30 1831.80 880.41 518 408.01 1924.53 862.4 515 433.79

50 ± 4 161.0 ± 0.2 143 ± 3 114.2 ± 2.5 47.4 ± 1.4 115 ± 8 159.9 ± 1.6 136 ± 8 99.8 ± 0.7 101.0 ± 2.0 116.70 ± 0.05 150.9 ± 2.0 102.5 ± 2.0 51.1 ± 1.6 104.8 ± 1.4 110.4 ± 2.4 90.6 ± 1.3 41.1 ± 1.4 89.3 ± 1.8 107.3 ± 1.8 72 ± 4 80.4 ± 1.3 41.6 ± 1.7 97.6 ± 0.2 78 ± 3 79.9 ± 1.3 51 ± 3 95.3 ± 2.1 78.9 ± 2.3 81.9 ± 1.3

95, 96 99, 100 104, 105 108−110 98, 113 115−117 120, 121 125−127 131−133 135−137 140−142 139, 146 149, 150 89, 153 98, 139 160, 161 112, 160 98, 164 167−169 171−173 176, 177 112, 180 89, 183 184, 185 187−189 112, 192 89, 194 197−199 139, 202, 203 112, 205

NiH NiO NiF NiCl CuH CuO CuF CuS CuCl ZnH ZnO ZnS ZnCl RuC RuO RhC AgH AgO AgF AgCl IrC IrO PtH PtC PtO AuH AuO AuF AuS AuCl

1 +

electronic state

Re (Å)

ωe (cm−1)

De (kcal mol−1)

ref

Δ 3 − Σ 2 Π 2 Π 1 + Σ 2 Π 1 + Σ 2 Π 1 + Σ 2 + Σ 1 + Σ 1 + Σ 2 + Σ 1 + Σ 5 Δ 2 + Σ 1 + Σ 2 Π 1 + Σ 1 + Σ 2 Δ 2 Δ 2 Δ 1 + Σ 3 − Σ 1 + Σ 2 Π 1 + Σ 2 Π 1 + Σ

1.4538 1.6271 1.7387 2.0615c 1.4626 1.7246 1.7449 2.0499 2.0512 1.5935 1.7047 2.0464 2.1300 1.6055 1.714 1.6133 1.6179 2.0030 1.9830 2.2808 1.6830 1.7231 1.5285 1.6767 1.7272 1.5237 1.8488 1.9184 2.156 2.1990

2001.31 839.69 655b 425.63 1940.27 640.14 620.87 414.83 417.64 1603.18 730.5 459.25 392.12 1100.0 863.5 1049.87 1759.96 489.8 513.45 343.60 1060.0 909.35 2387b 1051.13 851.07 2305.50 624.59 563.61 410.19 383.61

59.2 ± 1.9 90.3 ± 0.7 105.3 ± 1.4 88.7 ± 1.3 62.7 ± 1.4 71.0 ± 0.7 96.8 ± 0.7 65 ± 3 90.2 ± 1.0 21.9 ± 0.2 38.2 ± 0.9 49 ± 3 55.3 ± 1.9 147.1 ± 2.5 124 ± 10 140.0 ± 1.5 57.5 ± 0.7 46.3 ± 0.2 82.2 ± 1.0 74.9 ± 1.0 151.4 ± 1.2 85 ± 5 82.7 ± 0.2 138.7 ± 1.6 100.4 ± 2.8 77 ± 3 55 ± 4 77 ± 8 60 ± 4 67 ± 3

97, 98 101−103 106, 107 111, 112 98, 114 118, 119 122−124 128−130 107, 134 138, 139 143−145 144, 147, 148 151, 152 154−156 157−159 139, 162 163 139, 165, 166 107, 170 107, 174, 175 178, 179 181, 182 139 139, 186 186, 190, 191 95, 193 195, 196 200, 201 196, 204 206, 207

2

a Ground electronic states, equilibrium bond lengths (Re, in Å), harmonic vibrational frequencies (ωe, in cm−1), and equilibrium dissociation energies (De, in kcal mol−1). bObtained from the reported ΔG1/2 and from estimated ωexe; see the Supporting Information. cObtained from the reported B0 and the estimated αe; see the Supporting Information.

3. COMPUTATIONAL DETAILS 3.1. Spectroscopic Constants. All spectroscopic constants (dissociation energy, equilibrium bond length, and harmonic vibrational frequency) were determined from a set of seven single-point energy calculations at displaced geometries around the experimental equilibrium bond length. The displacement values are −0.3, −0.2, −0.1, 0, +0.1, +0.3, and +0.5a0. This limited potential energy curve was adjusted to a polynomial of fourth degree that allows one to readily obtain the spectroscopic constants.214 The dissociation energy was computed from the energy at the minimum of the potential energy curve and the electronic energy of the separated atoms. For a selection of molecules, the accuracy of this approach was assessed by a comparison to a direct determination of the bond lengths and harmonic frequencies using analytic gradients and Hessians at the Hartree−Fock level, as implemented in the MOLPRO package.215,216 In the geometry optimization, the root mean square of the gradient was required to be less than 10−6 Eh/a0. The polynomial expansion of fourth degree was found to be sufficiently accurate, with a mean absolute deviation (MAD) of 0.0003 Å for equilibrium bond lengths and 3 cm−1 for harmonic vibrational frequencies of seven transition metal diatomic molecules (ScH, ScF, CuH, CuF, CuCl, ZnO, and ZnS) using the RHF/aug-cc-pVDZ level of theory. The polynomial

expansion was chosen because it is much more practical for diatomic molecules, as it requires only seven single-point energy calculations for each combination of molecule and level of theory, contrasting with the usual approach that may require several steps with additional gradient evaluations, depending on the initial geometry guess. Also, in the case of wave function theory calculations, the construction of potential energy curves at predetermined internuclear distances allows one to use additivity schemes without any further complications. To facilitate direct comparison between theory and experiment, the calculated dissociation energies were approximately corrected for spin−orbit effects. For the dissociated atoms, the spin−orbit correction was computed as a weighted sum over the J-multiplets of the ground state term atomic ΔESO =−

∑J (2J + 1)E(J ) ∑J (2J + 1)

(1)

Atomic term energies from the NIST Handbook of Basic Atomic Spectroscopic Data217 and eq 1 were used to calculate the atomic spin−orbit corrections given in Table S3 of the Supporting Information. For the molecular spin−orbit correction, a CASSCF/aug-cc-pVQZ(-PP) calculation with a valence active space ((n − 1)d ns orbitals for transition metal atoms plus 1s orbital for hydrogen or ns np orbitals for other non-metals) 5293


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obtain an estimate of the error associated with the frozen-core approximation, and they allowed us to make further additive corrections to the energy in cases where including the corecorrelation is too expensive (see below). A correction for scalar relativistic effects was obtained by the Douglas−Kroll−Hess approximation of second order (DK),229−231 obtained in a valence only CCSD(T) calculation with the aug-cc-pwCVTZ-DK basis set.223−225,232 For the 4d and 5d elements, the effective core potentials already take relativistic effects into account. In these cases, the DK calculations can be seen as a further correction to the approximation made by using effective core potentials. For each single-point energy calculation, we defined a scalar relativistic correction as the difference between the DK-CCSD(T)/aug-cc-pwCVTZ-DK and CCSD(T)(FC)/aug-cc-pwCVTZ(-PP) electronic energies

was made at the experimental equilibrium bond length, which is given in Table 1. The two components of the electronic state, in C2v symmetry, were mixed within the state interacting approach using the full Breit−Pauli operator.218 The molecular spin−orbit correction, ΔEmolecular (AB), was defined as the difference between SO the energy of the spin−orbit ground state and the non-spin−orbit energy molecular ΔESO (AB) = ESO(AB) − Eelectronic(AB)

(2)

The calculated molecular spin−orbit corrections are given in Table S4 of the Supporting Information. Throughout this work, all calculated dissociation energies, De, are corrected for the spin−orbit effect according to atomic atomic De = Deelectronic + ΔESO (A) + ΔESO (B ) molecular − ΔESO (AB)

ΔDK(R ) = EDK ‐ CCSD(T)/awCVTZ ‐ DK (R )

(3)

where Delectronic is the dissociation energy of the AB diatomic e molecule calculated directly, without the spin−orbit correction. The mixing of calculated and experimental spin−orbit splitting is a procedure adopted in several previous benchmark studies85,89,90 and usually does not pose a problem. Our test calculations on the atomic spin−orbit splitting at the CASSCF level show differences within 0.6 kcal mol−1 to the experimental values, with the exception of Ir with a difference of 1.5 kcal mol−1. For the Ir atom, the strong interaction of configurations makes both the experimental assignment of spectral lines and the calculation of spin−orbit coupling rather difficult.219 3.2. Coupled-Cluster Calculations. The single-reference coupled-cluster calculations were based on a restricted open-shell Hartree−Fock reference, and the spin-unrestricted formulation of coupled-cluster theory with singles and doubles clusters and a perturbative estimate of triples clusters, CCSD(T), was used.220 The correlation consistent basis sets aug-cc-pwCVnZ (n = T, Q, and 5) for main-group and 3d elements221,222 and aug-ccpwCVnZ-PP (n = T, Q, and 5) for 4d and 5d metals223−225 were employed, with the corresponding pseudopotential for the inner core electrons. We note that the aug-cc-pwCVnZ basis set already has tight d functions, similar to the ones used in the cc-pV(n+d)Z family;226 thus, no special basis is required for third-row elements. The complete basis set (CBS) limit for the electronic energies was estimated by extrapolation schemes. A systematic study of several extrapolation formulas showed that there is no single expression that is superior in all cases and that all schemes improve the final results.227 Therefore, as we did not intend to exhaustively test extrapolation schemes, total electronic energies were extrapolated at each nuclear configuration with the formula

En = ECBS +

A n3

− E CCSD(T)(FC)/awCVTZ(‐PP)(R )

(5)

Potential energy curves from a composite approach including core−valence correlation, an estimate of the CBS limit, and scalar relativistic effects were obtained as E CCSD(T)(CV)/CBS +ΔDK (R ) = E CCSD(T)(CV)/CBS(R ) + ΔDK(R )

(6)

It has been pointed out that the correlation of the 4f orbitals can be important for the correct description of the 5d transition metals, especially for the early transition metals.225,233 Therefore, we have performed test calculations at the triple-ζ level for the platinum and iridium compounds from our data set. These show, however, that the effect of 4f correlation is within 1 kcal mol−1, 0.004 Å, and 10 cm−1 for De, Re, and ωe, respectively. This is of the same order as the error associated with the additivity approximation; thus, the correlation of the 4f orbitals of the 5d transition metals is not included in the present work. In addition, we investigated correlation effects beyond the CCSD(T) level using a multireference coupled-cluster method. These computations were performed with the internally contracted multireference coupled-cluster theory with singles, doubles, and perturbative treatment of triples, icMRCCSD(T),93,234 and were carried out with the aug-cc-pwCVTZ(-PP) basis set within the frozen-core approximation. These results were added as a further correction on top of our additive scheme. Analogously to the ΔDK correction, for each single-point calculation we defined a multireference correction ΔMR(R ) = EicMRCCSD(T)/awCVTZ(‐PP)(R ) − E CCSD(T)(FC)/awCVTZ(‐PP)(R )

(4)

(7)

The active orbitals for the icMRCC calculations were optimized by the CASSCF method using Hartree−Fock core orbitals. The use of the same core orbitals as in the CCSD(T)(FC) calculations is important for the validity of the additivity assumption of the core−valence correlation effects in our final composite scheme. For the hydrides, we also carried out calculations with the aug-cc-pwCVQZ(-PP) basis sets, and the ΔMR dependence with basis set was found to be small; see the discussion in Section 4.2. Whenever possible, state-specific CASSCF calculations with the full valence active space were carried out. For some cases, the icMRCC calculations on top of a full valence active space became too expensive, and a reduced active space was used. Moreover, because of the existence of

where n is the cardinal number of the basis sets.228 The two largest basis sets, namely, aug-cc-pwCVQZ(-PP) and aug-ccpwCV5Z(-PP), were used to estimate the basis set limit from eq 4, denoted as CCSD(T)/CBS in the following. Core−valence correlation effects were considered in the single-reference coupled-cluster calculations. For these calculations, indicated by CCSD(T)(CV), we included the valence electrons (i.e., (n − 1)d ns for d-block metals, 1s for hydrogen, and ns np for the other elements) and the outer core subshells, (n − 1)s (n − 1)p, in the correlation treatment. The calculations in which only the valence electrons were correlated are denoted as CCSD(T)(FC). With these calculations, we were able to 5294


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Journal of Chemical Theory and Computation several low-lying excited states and strong coupling among them, some cases required the use of orbitals optimized with stateaveraged CASSCF. A detailed description of the active spaces and the states averaged in the CASSCF calculations is given in Table S5 of the Supporting Information. For some molecules (namely, MnH, MnF, MnCl, ZnH, and ZnCl), the CASSCF reference wave function is exactly the restricted open-shell Hartree−Fock wave function. This happens because the reference space contains only CSFs related to each other by single excitations; thus, the reference wave function can always be described by a single CSF for an appropriate choice of active orbitals. Although these cases are single-reference cases, the corresponding potential energy curves have a small difference at the CCSD(T) and icMRCCSD(T) levels of theory, giving a small but non-zero ΔMR correction. The reason for this is that the icMRCCSD(T) theory as employed here is spin-adapted, whereas the spin-unrestricted CCSD(T) wave function has a small spin contamination (the cases mentioned above are all open-shell molecules). Thus, ΔMR is in these cases a correction for spin-contamination effects. Finally, our best estimate to the potential energy curves was obtained as

This correction was added to the non-relativistic potential energy curves that were refitted to a fourth-order polynomial, yielding a new set of spectroscopic constants, as described above. All DFT calculations were performed with the MOLPRO program package.215,216

4. RESULTS AND DISCUSSION This section is structured as follows. First, we discuss the general performance of the composite coupled-cluster approach, as described in Section 3.2. In particular, we systematically investigate the basis set convergence, relativistic contributions, and the importance of core−valence and higher-order correlation effects. For selected systems, deviations with respect to experimental reference values are further explored. In addition, we investigate multireference diagnostics in light of the observed values for the multireference correction, ΔMR. Finally, we discuss the assessment of different DFT functionals, either in comparison to theoretical best estimates (from our composite coupled-cluster approach) or in comparison to experimental data. We discuss how far certain trends are stable with respect to uncertainties in the reference data set. 4.1. Single-Reference Coupled-Cluster Calculations. An overview of the single-reference coupled-cluster results is given in Figure 1. The computed dissociation energies show a remarkably uniform basis-set convergence (see Figure 1A); they increase for larger basis sets (the only exception is the PtH, whose De slightly decreases). The smallest basis set used in this study, awCVTZ, leads to differences of up to 7 kcal mol−1 with respect to the estimated CBS values. This illustrates once more the importance of exploring the basis set limit in correlated wave function theories,89,90 whereas results from “practical” coupledcluster calculations with triple-ζ quality basis sets should only be used with care.85 At the awCV5Z basis set level, on the other hand, the maximum difference from the CBS result is just 1.6 kcal mol−1. The relativistic correction ΔDK for the dissociation energy does not have the same sign for all molecules. It is negative for most cases; the largest correction is found for cobalt compounds, but it is positive for chromium and nickel molecules and a few other cases. We recall that for 4d and 5d molecules, the ΔDK correction only includes effects beyond the ECP approximation and is therefore rather small. The equilibrium distances are overestimated at smaller basis sets (Figure 1B), but the truncation error is always smaller than 0.015 Å. The contribution of the ΔDK correction to the equilibrium distance is, when appreciable, always negative. It is larger for nickel and copper compounds, reaching −0.056 Å for CuO. As expected, the ΔDK contribution is small for the diatomics treated with the effective core potential, but it is also very small for the scandium, titanium, and vanadium compounds. For the vibrational frequencies, the basis set truncation error generally leads to an underestimation of the value as compared with the CBS limit (Figure 1C). The deviation can reach up to 22 cm−1 in the worst case (for VN), but it is usually small compared to the ΔDK correction or errors associated with the frozen-core approximation (see below). The ΔDK correction has the effect of increasing ωe, with few exceptions where this correction is small. The effect of the frozen-core approximation can be seen in Figure 2. The largest contributions for the dissociation energy are for some of the titanium, scandium, and vanadium compounds, reaching 5 kcal mol−1 in the CBS limit. For these cases and for the iron and cobalt containing molecules, this core-correlation

E CCSD(T)(CV)/CBS +ΔDK +ΔMR (R ) = E CCSD(T)(CV)/CBS(R ) + ΔDK(R ) + ΔMR(R )

(8)

All single-reference coupled-cluster calculations were done with the MOLPRO program package,215,216 and the icMRCC calculations were performed by the General Contraction Code (GeCCo), used to implement the icMRCC theory in one of our groups as described in ref 93. 3.3. KS-DFT Calculations. We selected a number of density functionals widely used in electronic structure calculations. These cover the group of generalized gradient approximation (GGA) functionals (BP86,235,236 BLYP,236,237 PBE238,239), metaGGA functionals (TPSS240), hybrid functionals (B3LYP,241,242 PBE0243), and more strongly empirically optimized functionals of either purely meta-GGA type (M06L244), hybrid type (B97245), or hybrid meta-GGA type (M06,246 M06-2X246). Empirical damped dispersion corrections to Kohn−Sham calculations were considered in test calculations but were removed from the final data set because the inclusion of the D3 dispersion correction developed by Grimme247 did not yield significant changes in the calculated spectroscopic constants. This is, in fact, expected as the dispersion correction is damped for short and medium range internuclear distances. We note, however, that the inclusion of dispersion corrections may be important for larger systems. We also considered a number of different basis sets, namely, aug-cc-pVnZ (n = D, T, and Q)222,248−250 for main-group elements and 3d metals and aug-cc-pVnZ-PP (n = D, T, and Q)223−225 for 4d and 5d elements, along with the appropriate pseudopotential for the inner electrons. The def2-TZVP, def2TZVPP, and def2-QZVP basis sets,251 developed by Ahlrichs and co-workers, were also used. As for the coupled-cluster calculations, scalar relativistic effects were calculated using the aug-cc-pwCVTZ-DK basis set and the Douglas−Kroll−Hess approach to second order. We defined a scalar relativistic correction as the difference between the DK-KS-DFT/aug-cc-pwCVTZ-DK and DFT/aug-cc-pwCVTZ(-PP) electronic energies ΔDK(R ) = EDK ‐ KS ‐ DFT/awCVTZ ‐ DK (R ) − EKS ‐ DFT/awCVTZ(‐PP)(R ) 5295


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Figure 1. Deviation, with respect to the experimental values, for the dissociation energies, equilibrium distances, and vibrational frequencies of the 60 molecules in the data set with several basis sets and core−valence correlation effects. Molecules are ordered by increasing deviation of CCSD(T)(CV)/CBS+ΔDK. Experimental error bars (for dissociation energies) are indicated in gray.

molecules, the core-correlation contribution to the dissociation energy is not very sensitive to the basis set. For equilibrium distances, the contribution from the core−valence correlation is larger (but also negative) for the molecules containing

contribution is basis-set-dependent and increases by about 1.5 kcal mol−1 when going from the smallest to the largest employed basis set. The platinum compounds show a decrease of the core-correlation contribution, and for the remaining 5296


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Figure 2. Contribution from the correlation of the outer-core electrons to the dissociation energy, equilibrium distance, and harmonic frequency.

4.2. Multireference Coupled-Cluster Calculations. The possibility of systematically increasing the accuracy is one of the strengths of wave function based theories and coupled-cluster theory in particular. Up to this point, calculations at the highest level of theory include core−valence correlation, extrapolation to the CBS limit, and scalar relativistic effects at the CCSD(T) level.

scandium, titanium, and vanadium, but it is small for nickel, copper, and zinc. The reader is referred to the Supporting Information for detailed graphs showing the contributions to each spectroscopic constant and for detailed tables with the present results for the individual molecules. 5297


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Journal of Chemical Theory and Computation Table 2. Multireference Correction, ΔMR, for the Spectroscopic Constants with the awCVTZ Basis Seta ΔMR De

molecule ScH ScO ScF ScS TiH TiN TiO TiF TiS TiCl VN VO VCl CrH CrO CrF CrCl MnH MnO MnF MnS MnCl FeH FeO FeS FeCl CoH CoO CoS CoCl a

1.9 2.5 1.6 3.5 3.5 3.6 1.4 3.4 3.1 2.9 1.5 −0.6 −1.4 0.6 −0.2 0.4 1.9 0.1 −0.1 0.3 0.3 0.2 −0.7 2.1 3.0 1.8 2.8 7.7 16.4 3.7

ωe

Re [2.0]

[3.7]

[0.5]

[0.0]

[−0.7]

[2.8]

0.2 0.0 0.3 0.3 −0.4 0.4 0.0 −0.7 2.0 −0.6 1.3 0.1 0.0 0.8 0.3 0.3 3.6 0.1 0.4 0.0 0.4 0.0 0.6 0.4 0.8 0.0 −0.4 3.2 −0.1 −1.2

[0.1]

[−0.5]

[0.7]

[0.1]

[0.5]

[−0.5]

−14 4 −27 −3 4 −2 1 23 −25 4 −53 −22 −4 −23 −4 −7 −69 3 −13 0 −8 0 −2 −26 −17 0 −8 −50 −11 2

De

molecule [−15]

[9]

[−15]

[3]

[−1]

[−3]

NiH NiO NiF NiCl CuH CuO CuF CuS CuCl ZnH ZnO ZnS ZnCl RuC RuO RhC AgH AgO AgF AgCl IrC IrO PtH PtC PtO AuH AuO AuF AuS AuCl

3.0 0.9 3.4 3.7 1.4 0.8 −0.5 0.3 1.0 −0.4 1.0 0.7 −0.1 5.0 0.6 2.2 −0.3 0.4 −0.1 −0.2 2.0 15.3 −0.6 1.6 5.0 −0.5 1.1 −0.1 0.5 −0.1

ωe

Re [3.2]

[1.5]

[−0.3]

[−0.4]

[−0.5]

[−0.4]

0.8 −0.3 0.0 0.4 0.8 −0.2 0.4 0.7 −0.2 0.1 0.2 0.0 0.0 1.4 0.3 0.7 0.6 −0.8 0.1 0.1 0.0 −1.8 0.0 0.1 0.7 0.2 0.3 0.1 0.1 0.2

[0.8]

[0.8]

[0.1]

[0.5]

[0.0]

[0.2]

−22 −30 −15 −11 −23 −28 −4 1 −1 4 3 −1 −1 −72 −9 −68 −21 7 −2 −1 −1 26 −11 −4 21 −16 −6 −3 14 −3

[−22]

[−25]

[4]

[−21]

[−12]

[−16]

For the hydrides systems, corrections employing the awCVQZ basis set are shown in square brackets. Values are given in kcal mol−1, pm, and cm−1.

introduced in the present work has a similar effect as the higherorder excitation corrections and can be used to further improve the single-reference coupled-cluster results. As the diatomic molecules near the equilibrium region have larger multireference character than the separated atoms, the calculated dissociation energy is, in most of the cases, increased after including the ΔMR correction. This improves the agreement with the experimental value in several cases where the best single-reference coupled-cluster results underestimate the experiment. However, it also slightly increases the disagreement when the experimental results are overestimated at the single-reference level. For a number of cases, the equilibrium distances are also improved, especially those for which the single-reference coupledcluster calculation underestimates the experimental value. For instance, results very close to the experiments are obtained for CoO and TiS after adding the multireference correction. For the subset of cases treated in the works of Cheng et al.89 and Fang et al.,90 the multireference corrections to the equilibrium distances and harmonic frequencies are rather small, and our values are in good agreement with theirs. A clear exception is the CrCl molecule, for which the multireference character is more pronounced. The final values of the Re and ωe after the inclusion of the ΔMR correction are much closer to the experiment than the results from single-reference calculations. 4.3. Error Distribution of Coupled-Cluster Calculations. We will now turn the attention to a statistical analysis of the

We expect that the leading error in the calculations comes from the neglect of higher-order excitations and/or the singlereference nature of the CCSD(T) method. Therefore, we will in the following investigate the effect of going beyond the single-reference approach by including static correlation via icMRCC calculations. The differences between the icMRCCSD(T) and CCSD(T)(FC) dissociation energies and spectroscopic constants, ΔMR, are shown in Table 2. They are interpreted as manifestations of multireference character. We first note that the ΔMR corrections calculated at awCVTZ and awCVQZ for the hydrides are essentially the same, and we thus expect that the ΔMR correction computed at the awCVTZ level is sufficiently reliable. Figure 3 shows the deviations from the experimental value for the individual molecules of our composite approaches. We also include the results of Cheng et al.89 and Fang et al.90 for the 3dMLBE20 data set for comparison. For instance, Cheng et al. have shown that the contribution of higher-order excitations to the dissociation energy is small for the transition metal diatomics of this data set and that the full triple and quadruple excitations very often cancel each other.89 For most of the cases, their results are in very good agreement with our best single-reference coupled-cluster estimate. The exceptions are the CoH and CoCl molecules, which have large contributions from higher excitations (of 1.8 and 1.4 kcal mol−1, respectively). After the inclusion of the ΔMR contribution, our final results match the value calculated by Cheng et al. This suggests that the ΔMR correction 5298


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Figure 3. Deviation with respect to the experimental values for the dissociation energies of the 60 molecules in the data set, ordered by increasing deviation of CCSD(T)(CV)/CBS+ΔDK. Dissociation energies that are indicated as SFX2C-1e-CCSD(T)(CV)/CBS+ΔT+ΔQ are from the work of Cheng et al.89 Data indicated as FDP are from the work of Fang et al.90

for instance that of different DFT functionals. The ordinary mean absolute deviation (MAD) and mean signed deviation (MSD) are usually employed for this purpose, and this procedure is used in the present discussion. However, a few drawbacks must be kept in

present data. Because this is a relatively large data set, it is valuable to use a representative descriptor of the error to evaluate the performance of the several coupled-cluster approximations and, in a later step, the performance of more approximate methods, 5299


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Journal of Chemical Theory and Computation mind to avoid a misleading interpretation of the results. The MADs should not be taken as uncertainties associated with the methods, as the conventional 95% confidence interval is much larger than the MAD; for this reason, its use to benchmark electronic structure methods has been criticized.252 (When the MSD is not too large, the MAD has approximately the same value as the standard deviation σ, whereas the 95% confidence interval is approximately 2σ.) On the other hand, the experimental reference values are also associated with uncertainties; this is in particular true for the dissociation energies, for which the error bars are shown in Figure 1A. In quite a number of cases, the deviation of the computed values lies within these error bars, and it might be argued that alternative measures like weighted averages could be more appropriate and meaningful. However, we are not aware of any established alternative procedure, and developing and testing such procedures would clearly go beyond the scope of this work. When working with standard measures of error analysis, the most important question is whether the data set to be analyzed (here, the errors with respect to the reference data) is close to a normal distribution. In order to test this, we will use the cumulative distribution function (CDF), which is obtained as the number of molecules with deviation to the experimental value equal to or lower than x plotted against x. In Figure 4, the CDFs for the dissociation energies, equilibrium bond distances, and harmonic vibrational frequencies are shown for our most accurate results, with and without the multireference correction ΔMR. The CDF can be directly compared to the error function associated with the MSD and the standard deviation σ of the results. Large differences between the CDF and the error function indicate that the error distribution is far from a normal distribution. There are other means to evaluate how close an error distribution is to a normal distribution, like the use of histograms or the ratio of the mean deviation to the standard deviation.253 Compared to histograms, the CDF has the advantage of being independent of a chosen binning. Compared to the ratio of the mean deviation to the standard deviation, the plots of the cumulative distribution function allows one to easily identify individual outliers. As can be seen from Figure 4A, the error distributions for the dissociation energy closely follow a normal distribution. The distribution is rather broad (the MAD is around 3.5 kcal mol−1 in both cases), and quite a number of molecules have errors much larger than the MAD. It should be kept in mind, however, that the experimental error bars for the dissociation energies are rather large in many cases, on the order of 2.5 kcal mol−1 on average. Furthermore, a closer analysis of a number of cases also shows that some experimental values are probably inaccurate (see Section 4.3.1). We also briefly analyzed how the inclusion of experimental error bars and the presence of strong outliers affect the error analysis by considering a subset of our database excluding 10 molecules for which our results suggest a lack of reliability of the experimental results. We found that the error distribution, for this subset, also closely follows a normal distribution. The reader is referred to the Supporting Information for details. In remaining part of this work, the calculated average errors are calculated for the full data set of 60 molecules. For the equilibrium distances and the harmonic frequencies, the CDFs are much narrower. However, a few outliers lead to significantly larger values for the standard deviation, such that the obtained error function fits the CDF less well. These outliers also call for individual analysis.

Figure 4. Cumulative distribution function (CDF), with respect to experiment, for dissociation energies (A), equilibrium bond distances (B), and harmonic vibrational frequencies (C) calculated with coupledcluster theory. Solid lines are the error functions associated with a normal distribution with the same average and same standard deviation of the results. Top panels show the error distributions, including results with different basis sets.

Nonetheless, we can conclude that the deviations from experimental values are approximately normally distributed; the standard measures MSD and MAD might be used as a measure to discuss overall trends. The resulting MSDs and MADs are summarized in Table 3. Increasing the basis set and extrapolating to the basis set limit reduces the MAD for both frozen-core and core−valence correlated calculations. The scalar relativistic correction ΔDK slightly increases the MAD for dissociation energies if it is used in conjunction with the frozen-core calculations, but if core−valence correlation is included as well, it consistently reduces the MADs for all three properties. The results from icMRCCSD(T)/awCVTZ 5300


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In Table 4, the dissociation energies at several levels of theory are collected. Similar tables can be found for all molecules in the Supporting Information. [Note that the additivity scheme is carried out pointwise for the energy curves (see Section 3.1), such that the differences (ΔCV, ΔDK, ΔMR) reported in Table 4 and comparable tables do not necessarily add up to give the final best estimates.] At the CCSD(T)(CV)/CBS+ΔDK level of theory, our result for the VN dissociation energy is 114.4 kcal mol−1 and thus deviates by −2.3 kcal mol−1 from the experimental value. However, after adding the multireference correction, this deviation decreases to only −1.0 kcal mol−1. As the experimental value is indeed very accurate, this deviation is solely attributed to shortcomings in our approach. Inspecting the contributions in Table 4, we assume that the main deficiency of our protocol is the additivity assumption for the ΔDK and ΔMR corrections, where in particular the latter may be improved by conducting icMRCC computations with larger basis sets and including core−valence correlation effects directly. Another source of error could be spin−orbit effects (a correction of −0.44 kcal mol−1 is included in all theoretical values for VN in Table 4). Nevertheless, the deviation of −1.0 kcal mol−1 is rather satisfactory at this stage, still being at the edge of the usual range for “chemical accuracy”. A somewhat more loose bound of ±3.0 kcal mol−1 for “transition metal chemical accuracy” has previously been suggested by DeYonker et al.78 This accuracy is indeed reached for a number of cases in our test set, in particular when taking into account the experimental error bars. In fact, as mentioned in the previous section, there are a number of cases where the experimental error bars are rather large, such as ±4 kcal mol−1 (for MnS) or even ±10 kcal mol−1 (for RuO). For these cases (specifically TiN, RuO, AuF, and MnS), there is a large discrepancy between our computed dissociation energies and the experimental ones, but the computed values still lie within the experimental error. In the statistical analysis, however, these cases contribute to the large MAD. More precise measurements of these molecules would certainly be welcome. A particular class of molecules with large deviations between computation and experiment are the hydrides. Specifically, these are the molecules (NiH, FeH, ScH, CoH, CuH, and TiH) whose De values are consistently overestimated in our computations by around 3−12 kcal mol−1. However, ΔMR corrections for these molecules were computed using full valence active spaces in the icMRCC calculations, and the basis set dependence of ΔMR is small (see Table 2) and cannot account for the large deviations. Furthermore, for a number of cases previous computational studies exist, which are in line with our findings. For instance, for FeH, DeYonker and Allen254 carried out focal point analysis using single-reference coupled-cluster methods with up to hextuple excitations and considering corrections to the Born−Oppenheimer approximation. The contributions to the dissociation energy due to higher excitations, i.e., beyond CCSD(T), and due to the diagonal Born−Oppenheimer correction (DBOC) amount to only −0.14 and +0.30 kcal mol−1, respectively. Their final value, De = 45.6 kcal mol−1, is very close to our best estimate of 44.5 kcal mol−1 (including a small correction of −0.5 kcal mol−1 for multireference effects; see Table 4), in particular, if we consider that we can use their DBOC result to correct our value to 44.8 kcal mol−1. Cheng et al.89 and Fang et al.90 obtained results of 45.7 and 44.6 kcal mol−1, respectively, for the dissociation energy of FeH, which is similar to ours and that of DeYonker and Allen. Also, for the other hydrides present in the 3dMLBE20 database

Table 3. Mean Absolute Deviations (MAD) and Mean Signed Deviations (MSD) for the Dissociation Energies (De, in kcal mol−1), Equilibrium Bond Lengths (Re, in Å), and Harmonic Vibrational Constants (ωe, in cm−1) of 60 Diatomic Molecules Containing d-Block Metals Calculated with Different Approaches Based on Coupled-Cluster Theory De basis set

MAD

ωe

Re MSD

MAD

MSD

MAD

Valence Correlation (CCSD(T)(FC)) awCVTZ 5.8 −5.2 0.019 0.016 21 DK/awCVTZ-DK 6.7 −6.3 0.015 0.009 18 awCVQZ 4.6 −3.3 0.017 0.014 19 awCV5Z 4.3 −2.6 0.017 0.013 19 CBS 4.0 −1.9 0.016 0.012 18 CBS+ΔDK 4.3 −3.0 0.013 0.005 17 Outer-Core and Valence Correlation (CCSD(T)(CV)) awCVTZ 5.3 −4.3 0.013 0.007 19 awCVQZ 4.2 −2.2 0.012 0.004 20 awCV5Z 3.9 −1.5 0.011 0.003 20 CBS 3.7 −0.8 0.011 0.001 21 CBS+ΔDK 3.6 −1.8 0.008 −0.006 19 Results with Multireference Calculations icMRCCSD(T)/ 4.7 −3.3 0.021 0.019 27 awCVTZ CCSD(T)(CV)/ 3.8 1.2 0.011 0.004 21 CBS+ΔMR CCSD(T)(CV)/ 3.3 0.1 0.007 −0.004 14 CBS+ΔDK+ΔMR

MSD −15 −2 −11 −10 −8 4 −3 0 1 2 14 −25 −8 5

calculations have a smaller MAD for dissociation energies when compared to the single-reference counterpart, CCSD(T)(FC)/awCVTZ, but slightly larger MADs for Re and ωe. However, when the ΔMR correction is employed on top of the CCSD(T)(CV)/CBS+ΔDK result, we observe a general improvement of the MADs for De, Re, and ωe. In summary, by approaching the CBS limit and correcting for core−valence correlation and relativistic and multireference effects, we consistently approach the experimental values. Nonetheless, our best estimates from the composite approach CCSD(T)(CV)/CBS+ΔDK+ΔMR still have a sizable MAD relative to the experimental reference values, e.g., 3.3 kcal mol−1, for dissociation energies. This is a relatively large deviation, considering that a high level of theory and large basis sets have been employed. Probably more disturbing is the presence of strong outliers with deviations as large as 15 kcal mol−1. The mean deviations for equilibrium distances (0.007 Å) and harmonic frequencies (14 cm−1) are somewhat more acceptable (albeit not fully satisfactory), but here also there are some extreme outliers in our data set. Obviously, two main sources of errors have to be considered: Either the computations still neglect one or more significant contributions or the experimental results are not sufficiently accurate. Although it is outside the scope of the present work to make a molecule-by-molecule study, a deeper analysis of some cases is useful to understand the results and guide further experimental and theoretical investigations. 4.3.1. Discussion of Particular Cases. We start our discussion with dissociation energies De. Before turning to the problematic cases, we would like to focus on a case where a presumably very precise measurement is available, namely, VN. Its dissociation energy was recently obtained by measuring the predissociation threshold.141 This is a very precise direct measurement, with an estimated error of only 0.05 kcal mol−1. How well does our composite approach work for this case? 5301


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Table 4. Dissociation Energies (in kcal mol−1) for Selected Transition Metal Containing Diatomic Molecules Calculated with Different Approaches Based on Coupled-Cluster Theory VN awCVTZ CCSD(T)(FC) CCSD(T)(CV) ΔCV ΔDK icMRCCSD(T) ΔMR best estimate exptl.

FeH awCVQZ

105.2 107.9 2.7 −0.4 106.7 1.5 114.4 116.70 ± 0.05

108.5 111.7 3.3

awCV5Z

CBS

109.6 113.2 3.6

110.8 114.8 4.0

(+MR = 115.7)

awCVTZ CCSD(T)(FC) CCSD(T)(CV) ΔCV ΔDK icMRCCSD(T) ΔMR best estimate exptl.

42.8 44.6 1.8 −2.7 42.1 −0.7 45.0 41.6 ± 1.7

ZnO awCVTZ CCSD(T)(FC) CCSD(T)(CV) ΔCV ΔDK icMRCCSD(T) ΔMR best estimate exptl.

35.0 34.5 −0.4 −2.1 36.0 1.0 35.8 38.2 ± 0.9

awCVQZ

awCV5Z

CBS

37.6 37.2 −0.4

38.2 37.9 −0.4

(+MR = 36.8)


31.1 30.8 −0.2 −1.9 31.8 0.7 33.8 49 ± 3

TiO CCSD(T)(FC) CCSD(T)(CV) ΔCV ΔDK icMRCCSD(T) ΔMR best estimate exptl.

151.7 154.9 3.2 −0.4 153.1 1.4 159.6 159.9 ± 1.6 awCVTZ 80.8 83.2 2.4 −4.5 88.5 7.7 84.4 95.3 ± 2.1

CBS

44.4 46.9 2.5

44.9 47.7 2.8

awCV5Z

CBS

34.8 34.6 −0.2

35.9 35.6 −0.2

43.3 −0.7 (+MR = 44.5)

awCVQZ 33.9 33.6 −0.2

(+MR = 34.4) TiS

awCVQZ 153.9 157.7 3.9

awCV5Z

CBS

154.6 158.8 4.2

155.4 160.0 4.6

(+MR = 160.9)


98.8 102.3 3.5 −0.5 101.9 3.1 107.5 99.8 ± 0.7

CoO CCSD(T)(FC) CCSD(T)(CV) ΔCV ΔDK icMRCCSD(T) ΔMR best estimate exptl.

awCV5Z

ZnS

36.9 36.5 −0.4

awCVTZ

awCVQZ 44.0 46.2 2.2

awCVQZ 101.6 105.5 3.9

awCV5Z

CBS

102.6 106.7 4.1

103.6 108.0 4.4

awCV5Z

CBS

64.5 67.4 2.9

65.7 68.9 3.2

(+MR = 110.3) CoS

awCVQZ 83.7 86.4 2.7

awCV5Z

CBS

84.7 87.6 2.9

85.7 88.9 3.2


(+MR = 91.7)

59.7 61.9 2.2 −5.7 76.1 16.4 63.3 78.9 ± 2.3

awCVQZ 63.4 66.0 2.6

(+MR = 79.7)

cobalt hydrides. In case of FeH and CoH, deviations of 2.9 and 5 kcal mol−1 still persist after the revision. Given the agreement with previous studies, which partly use complementary approaches to estimate relativistic effects and correlation effects beyond singlereference CCSD(T) theory, a major failure of the theoretical approach would be rather surprising. We thus agree with the statement of previous authors89,90 that a careful revision of the experimental determination would be useful to shed further light on the source of this discrepancy. Comparable deviations between computed and experimental De values are also observed for the dissociation energies of some hydrides that are not present in the 3dMLBE20 database, like ScH, TiH, and NiH. For the reasons given above, our approach is expected to be accurate for these cases too, and the remaining uncertainties in our computations cannot account for the deviations to the experimental reference. For NiH, in particular, our calculations differ from the experimental value by about 12 kcal mol−1. This is a rather large disagreement, and we

(CrH, CoH, CuH, and ZnH), our results agree well with theirs. Both author teams have reanalyzed the experimental bond dissociation energies of the hydrides based on more recent and reliable values for the heterolytic bond dissociation energies of the alkanes and acetaldehydes used in the measurements. These alkanes and acetaldehydes, abbreviated as RH in the following, are used as reactants in the hydrogen transfer reaction M+ + RH → MH + R+, and the hydride dissociation energy is then obtained via a thermodynamic cycle that requires knowledge of the metal ionization potential (which is known accurately) and the said heterolytic RH dissociation energies. For instance, Cheng et al. provided corrected values for these energies, using both a statistical analysis of existing thermochemical data and high-level computations, and suggested an updated experimental value of De = 41.6 ± 2 kcal mol−1 for FeH. This is considerably closer to the calculated dissociation energies in our study and previous ones. In fact, our database uses the revisited values suggested by Cheng et al.89 for the chromium, iron, and 5302


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on the reliability of the measurements is clearly outside the focus of this work, the deviations between our results and the newest measurements call for a critical revision of these and previous results. The equilibrium distances and harmonic frequencies of all molecules in this study have (to our knowledge) exclusively been obtained from fitting observed spectroscopic transitions to model Hamiltonians, typically via the Dunham expansion. Although we are inclined to consider the thus derived experimental values very accurate, some comments are warranted. Concerning equilibrium distances, again a significant fraction of the outliers are hydrides (FeH, CoH, CrH, TiH, and ScH, with deviations greater than 1 pm). The case with largest deviation is the FeH molecule, for which our composite approach underestimates the experimental value by 5 pm. For this case, the ΔMR contribution is small and we agree well with previous singlereference coupled-cluster calculations. In particular, DeYonker and Allen254 have obtained an estimate of the Re for FeH with the inclusion of effects up to the CCSDTQP level of theory. However, higher excitations affect the equilibrium distance by less than 1 pm, and they still underestimate the experimental value by about 4 pm. The reason for this large discrepancy in unclear, and DeYonker and Allen state that “a serious disparity is now apparent between the theoretical Born−Oppenheimer Re value for FeH (X4Δ) and the experimental Re and R0 distances derived by spectroscopic analyses based on effective Hamiltonians. Further research on this issue is clearly warranted”.254 For the harmonic frequencies, the results for the hydrides seem to be in better agreement with experimentally determined ωe values, but we found strong disagreement for CrH, where our best estimate deviates by +100 cm−1. Interestingly, MRCI calculations carried out by Bauschlicher et al.153 give values much closer to experiment for ωe. These MRCI calculations, however, do not explore the basis set limit but use ANO basis sets corresponding approximately to our awCVTZ calculations. The main difference between their study and ours is a much smaller core−valence correlation contribution (of only about 24 cm−1 in the MRCI study), whereas the core−valence correlation contribution at the CCSD(T) level amounts to around 80 cm−1 in the present work. We suppose that the core−valence separation and the corresponding non-additivity of ΔCV and ΔMR contributions could be the source of the large deviation from experiment. We note that in our study all determinations of Re and ωe neglect spin−orbit effects. For the molecules containing 4d or 5d transition metals, this might be another source of error. The experimental values refer to the lowest-lying component whenever spin−orbit splitting occurs. There is no proper way of averaging different levels to obtain a spin−orbit free Re or ωe, as shown computationally in the case of AuO.255 On the other hand, the proper treatment of spin−orbit effects by ab initio calculations requires the inclusion of several interacting electronic states, which is outside the scope of the present work. For a number of cases, the less satisfactory performance of our approach for Re and ωe may be attributed to the use of reduced active spaces in order to keep the size of the problem tractable. This particularly concerns the cases of CuO, CuS, NiO, IrO, and AgO molecules. Although a restriction of the active space often works very well for energies within the icMRCCSD(T) approach, harmonic frequencies tend to be more sensitive.256 In fact, for the above-mentioned cases, the agreement of the equilibrium distance with the experimental value is decreased after including the multireference correction. Although the

suggest that a revision of the experimental dissociation energies of these hydrides is also needed. We note that our De for NiH is in good agreement with the upper bound estimate (from the predissociation of an excited state of NiH) given by Huber and Herzberg,139 amounting to D0 ≤ 70.8 kcal mol−1. Apart from the hydrides, there are further cases with large deviations between computed and experimentally derived De values. These are ZnS (−14.6 kcal mol−1), PtO (−9.9 kcal mol−1), CuO (−8.4 kcal mol−1), TiS (10.5 kcal mol−1), CrF (5.0 kcal mol−1), PtC (5.2 kcal mol−1), ZnCl (−5.2 kcal mol−1), VCl (−5.0 kcal mol−1), FeCl (3.7 kcal mol−1), CoO (−3.6 kcal mol−1), and NiCl (+3.6 kcal mol−1), where the deviations are given in parentheses. The origin of these discrepancies is difficult to assert in general, but there are some indications that our computational approach should be more accurate than implied by these deviations. First, for the molecules ZnS, ZnCl, VCl, and FeCl, also present in the 3dMLBE20 database, our results agree very well with the values from Cheng et al. The latter results also include an estimate of high-level correlation effects (up to quadruple excitations), which nearly always matches our ΔMR correction. The only exception is the NiCl molecule, for which our result is higher by 2 kcal mol−1 as compared to the value given by Cheng et al. The main difference comes from the estimate of higher-level correlation effects in this case, where our ΔMR correction predicts a significantly larger correction (+3.7 kcal mol−1) than their estimate based on CCSDT and CCSDTQ (+0.3 kcal mol−1). Further differences come from different approaches to estimate the spin−orbit correction. Second, there is, for each case mentioned above, an isovalent molecule for which our composite approach gives very good agreement with experiment. These are ZnO (as isovalent counterpart for ZnS), CuS and AgO (for CuO), TiO (for TiS), CrCl (for CrF), CoS (for CoO), and NiF (for NiCl). Three such pairs are shown in Table 4: ZnO and ZnS, TiO and TiS, and CoO and CoS. We see that in all cases the main difference in De between the oxide and the sulfide is already described at the CCSD(T)(FC)/ CBS level. Core valence contributions are very similar (and actually rather small in case of ZnO/ZnS), and the ΔDK corrections also differ only slightly. The differences in the spin−orbit correction (which is implicitly included in all computational results) mainly come from the different splitting of the 3P term of oxygen (Δatomic = SO = −0.56 kcal mol−1). These −0.22 kcal mol−1) and sulfur (Δatomic SO values are actually taken from experiment and are hence not a source of error. For ZnO and ZnS, the ΔMR correction is small as well, and it is hence difficult to rationalize why ZnO is in rather good agreement with experiment (−1.4 kcal mol−1) while ZnS deviates by −15 kcal mol−1. This clearly calls for a revision of the experimentally derived value. For TiO and TiS, ΔMR is slightly larger (particularly for TiS), but again there is no clear indication why the computational result for TiS should be wrong by +10 kcal mol−1 while that for TiO lies within the experimental error bar of ±1.6 kcal mol−1. For CoO and CoS, a more substantial multireference correction is found (see Table 4). In both cases, the correction drastically improves the single-reference result. Although for CoS (with the larger correction) our best estimate lies within the error bar, the result for CoO is still a bit too low. Of course, for corrections of this size, the present approach with the ΔMR computed at the triple-ζ level only might not be fully sufficient, and the perfect match for CoS is fortuitous to some degree. Finally, there also exist older measurements of the dissociation energy for the molecules CuO, PtC, and PtO that are in closer agreement with our calculations.139,162 Although any statement 5303


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Journal of Chemical Theory and Computation selection of the active orbitals was done with care to include all configurations with largest contribution into the active space, there is still the possibility that this approach is slightly biased. We also emphasize that, for some cases, state averaging is crucial to obtain proper active orbitals. Without this, the orbitals drastically change along the range of distances used in the calculations due to low-lying excited states, leading to poor potential energy curves. We observed the same effects in MRCI calculations, where the dependence on the orbitals was found to be much more drastic than for icMRCC. In summary, we conclude that our present approach should be very accurate for dissociation energies and the cases where strong deviations to experimental reference values occur (with absolute deviations of, say, 6 kcal mol−1 and larger) call for revision. For other spectroscopic properties, icMRCC more strongly inherits the active-space dependence of multireference methods and requires careful analysis and unfortunately veers away from being a black box method. More detailed studies of some of the above cases with more extended icMRCC computations directly including core−valence correlation and basis set extrapolation in direct comparison to high-order correlated coupled-cluster treatments will be helpful to further develop a reliable benchmark tool for transition metal compounds. 4.3.2. (A Short Note on) Multireference Diagnostics. In the present work, we used a composite approach that includes multireference effects through the ΔMR correction, rigorously obtained from MRCC calculations. However, MRCC calculations are still computationally expensive and not straightforward to conduct, in particular for larger molecules. Therefore, a reliable and cheap estimate of the magnitude of multireference corrections would be extremely valuable as an indicator whether the single-reference treatment is valid and sufficient. In fact, many multireference diagnostics have been described in the literature. Recently, a number of DFT-based diagnostics have been proposed,257−261 but here we will only consider measures that more or less naturally occur in correlated methods, like the T1,262 D1,263,264 and M diagnostics.265 These measures will be compared to the magnitude of the ΔMR corrections for dissociation energies (ΔMRDe), bond distances (ΔMRRe), and harmonic frequencies (ΔMRωe) obtained in the present work from icMRCC calculations. All diagnostics were applied to the molecules in the data set in frozen-core calculations using the awCVTZ basis set and the experimental equilibrium geometry. The T1 and D1 diagnostics262−264 are obtained from the CCSD single excitation amplitudes t1, and they are probably the most frequently employed multireference diagnostics in computational chemistry. While T1 is the Frobenius norm of the singles amplitudes, D1 is the matrix 2-norm.263 The rationale behind both criteria is that an inappropriate reference will lead to strong changes in the orbitals of the correlated wave function, leading to large t1 amplitudes.262 Historically, the single-reference wave function is considered reliable if T1 ≤ 0.02 and D1 ≤ 0.05, but recently new values have been proposed for TM-containing systems:266,267 T1 ≤ 0.05 and D1 ≤ 0.15 for the 3d TM-containing species and T1 ≤ 0.045 and D1 ≤ 0.12 for 4d TM-containing species. We will assume that the last criterion is the most appropriate for the 5d TM-containing molecules. The more recent M diagnostic is defined by265 1 M = (2 − N< 2 + N> 0 + 2

It is based on the occupation numbers obtained from the one-particle reduced density of the CASSCF wave function, where N0 is the largest occupation number, referring to an orbital that is unoccupied in the leading configuration. N1(j) are the occupation numbers of singly occupied orbitals (nSOMO is the number of these orbitals). Thus, the M diagnostic measures the deviation from integer occupation numbers as they would occur for a single configuration reference. Along with its definition, the threshold of M < 0.05 was suggested to consider single-reference wave functions as reliable. In the present work, the M diagnostics have been obtained from CASSCF wave functions optimized with the full valence active space, state-specific orbitals, and Hartree−Fock core orbitals. Note that, for a number of cases, this CASSCF wave function differs from the one used for the icMRCC calculations. It was chosen this way, however, to obtain a more or less black box procedure. The T1, D1 and M diagnostics of all molecules are listed in Table S6 of the Supporting Information. We also list the bond dissociation error with respect to experiment, ΔBDE, for the CCSD(T)(CV)/CBS+ΔDK approach. If we assume that the experimental data contains reliable values and that the major flaw of CCSD(T)(CV)/CBS+ΔDK is a deficient treatment of multireference systems, ΔBDE should correlate with the multireference diagnostics and with the multireference corrections ΔMR. In Table 5, we show the Pearson linear correlation Table 5. Pearson Linear Correlation Coefficients among the Different Diagnostics and the Absolute Values of ΔBDE and ΔMRDe T1 D1 M ΔBDE

j=1

M

|ΔBDE|

|ΔMRDe|

0.36 0.28

0.10 0.09 0.34

0.22 0.11 0.45 0.53

coefficients between the different diagnostics, |ΔBDE|, and the absolute value of the multireference correction for the dissociation energy, ΔMRDe. There is no good correlation between ΔMRDe and the diagnostics or the ΔBDE; that is, one cannot say that their magnitude is directly related to the magnitude of the multireference character. Although such a direct correlation is absent, the diagnostics, along with appropriate thresholds, could still be useful as indicators for the reliability of the singlereference treatment. Strong multireference corrections most often lead to a strong change of the computed dissociation energy, but we also found examples in our test set where this energy did not change significantly, as energy contributions from higher excitations often tend to cancel each other, as discussed by Cheng et al.89 However, the equilibrium distance and harmonic frequency are potentially more sensitive, and in fact we found a number of cases where these quantities change strongly due to the multireference correction, although the dissociation energy does not. In order to condense this into a single number, we define the dimensionless measure ⎧ |ΔMR De| |ΔMR R e| |ΔMR ωe| ⎫ ⎬ , ΔMR max = max⎨ , −1 1 pm 15 cm−1 ⎭ ⎩ 1 kcal mol

nSOMO

∑

D1 0.92

|N1(j) − 1|) (9)

(10) 5304


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Figure 5. Absolute value of the multireference correction, ΔMR, versus the multireference diagnostics: (A) T1, (B) D1, and (C) M. The vertical lines separating the colored regions are placed at the recommended threshold for a given multireference diagnostic (0.05 for T1, 0.15 for D1, and 0.04 for M). The horizontal lines are placed at ΔMRmax = 1. The molecules placed at green and red regions, respectively, are classified in accordance or not with the established threshold of a particular diagnostic. For the diatomic molecules containing 4d of 5d transition metals, their T1 and D1 diagnostics have been multiplied by 0.9 and 0.8, respectively, to have a common threshold with the molecules containing 3d transition metals. See the text and ref 267.

low multireference character but the actual ΔMRmax is large (false negatives). A large number of points in this region indicates a lack of reliability of the diagnostic. On the other hand, points on the bottom right red region are associated with systems that are well described by a single-reference approach but which have a large value for the diagnostic (false positives). Points in this region show that the diagnostic is not efficient because it would suggest the need for more accurate calculations when they are not really needed. The T1 and D1 diagnostics, with the most recently suggest thresholds for transition metal compounds, are in fact not very reliable. Almost half of molecules fall into the upper left region (false negatives). These have at least one property not well described by the single-reference approach, but this is not indicated by the diagnostics. The number of these cases is surprisingly high for diagnostics that are used so often. It is interesting to note that the traditional criteria of T1 ≤ 0.02 and D1 ≤ 0.05 make these diagnostics very reliable, but much less

The measure is larger than unity if the multireference correction for one of the parameters is larger than one of the following limits: For the dissociation energy, 1 kcal mol−1; for the equilibrium distance, 1 pm; and for the harmonic frequency, 15 cm−1. Figure 5 shows plots of ΔMRmax versus the three diagnostics, T1, D1, and M. Each plot is divided into four regions that can be used to evaluate the diagnostics. The vertical lines are placed at the recommended threshold for a given multireference diagnostic; therefore, molecules that are to the right of this line are supposed to have multireference character according to the diagnostic criterion. The horizontal lines are placed at ΔMRmax = 1, and points above this line correspond to molecules that, according to the above definition, indeed have considerable multireference character. An ideal multireference diagnostic would have all molecules in the green areas (large diagnostic together with large ΔMRmax or small diagnostic together with small ΔMRmax) and none in the red areas. Points on the left upper red region are associated with cases for which the diagnostic criterion indicates 5305


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Dunham expansion) that are not real observables at all. The accuracy of these derived experimental quantities is not always obvious without close inspection of the original literature. Of course, it is possible to extend the theoretical treatment and aim for “real” observables. For instance, the computation of rovibrational levels and thus the actual transition energies is rather straightforward for diatomic molecules. It becomes, however, significantly more difficult for larger molecules, and even for diatomics, the computational effort is much higher than for the approach taken in the present study as a significantly larger part of the potential energy curve must be computed. Whenever sufficiently accurate experimental reference values are not available, results from highly accurate computations are an alternative. These have the advantage of directly providing one-to-one comparable reference values. However, good reference values come at a rather high computational cost, as demonstrated in this and previous works.89,90 They require the minimization of all sources of error, in particular basis set extrapolation, accounting for higher-order correlation effects, and including relativistic effects. Each of the effects comes with some residual error margin that is not always simple to estimate. Finally, and this is probably the main drawback, the size of systems, for which such accurate computations are possible, is seriously restricted. In order to overcome this restriction, one may settle with more approximate methods as reference, arguing that a fully converged theoretical reference value is not always needed. In fact, if similar approximations are made (like omitting relativistic effects), benchmarks can still be meaningful. This requires, however, that the approximations have a very similar and controlled effect on both the test and reference models. For instance, it makes perfect sense to benchmark, say, Møller−Plesset second-order (MP2) energies obtained with a cc-pVTZ basis and the frozen-core approximation against CCSD(T)(FC)/cc-pVTZ reference values. For DFT computations, however, such a reference is (close to) meaningless. This is because DFT and coupled-cluster theory implements a completely different approach to treating the correlation problem. The basis set dependence strongly differs from that of CCSD(T), and the frozen-core approximation is meaningless in DFT. Furthermore, the decrease of accuracy for strongly correlated or low energy gap systems is very different for CCSD(T) and DFT. Thus, coupled-cluster computations as benchmarks for DFT results are suitable only when the expected intrinsic error of the coupled-cluster computations is much lower than the targeted accuracy in the study, for which the DFT results are calibrated. This particularly requires CCSD(T) results at the basis set limit, either by extrapolation or by employing explicitly correlated versions of coupled-cluster theory.270 In the following, we want to discuss the results of the DFT computations using four different sets of reference values. We will use (i) the experimental values, (ii) comparably inexpensive (“practical”) CCSD(T)(FC)/awCVTZ calculations, and (iii) the best estimates either from the pure single-reference approach CCSD(T)(CV)/CBS+ΔDK or (iv) including multireference corrections CCSD(T)(CV)/CBS+ΔDK+ΔMR. Before doing so, we will have a short look at the basis set dependence and the impact of relativistic corrections at the DFT level. For this discussion, we will stick to experimental reference values. Table 6 collects the results obtained for the B97 functional, which later will be shown to be among the best performing functionals in our study. Detailed results for all functionals can be found in the Supporting Information. First of all, we point out that the mean deviations have a small dependence on the basis

efficient. Indeed, there are very few cases below these thresholds. The M diagnostic is more reliable, but still more than a third of the molecules fall into the red regions. For the present data set, we have collected a few other quantities259,266 that have been suggested in the literature to measure the quality of the single-reference coupled-cluster calculations, and plots similar to the Figure 5 are given in the Supporting Information. The first is 1 − C20, where C20 is the dominant coefficient in the CASSCF wave function. However, this measure can be very unreliable for larger systems. For instance, for a system of noninteracting water molecules (a clear single-reference system), C20 approaches zero in the limit of an infinite number of monomers.268 Thus, one cannot define a threshold limit for classifying multireference cases solely based on C0. The second is the percentage of the dissociation energy accounted by the triples correction, %De[(T)]. Too large of a contribution of triple excitations could be interpreted as inadequacy of the single-reference approach.266 However, as shown by Cheng et al.,89 many transition metal dimers feature large triples contributions to De but the higher-order correlation corrections still turn out to be very small. Finally, one can base the diagnostic on the largest doubles amplitude, max{|t2|} (or the norm of the doubles amplitudes ∥t2∥, where the former is preferred as it does not scale with the system size). The motivation is that exceptionally large amplitudes indicate a large contribution of configurations that could have been included in the reference wave function. An orbital-invariant alternative of max{|t2|} is the D2 diagnostic.269 However, we did not find any correlation between these quantities and the multireference correction. The analysis carried out in this section shows the poor performance of common diagnostics to predict the applicability of single-reference methods. Not only is there no correlation between the multireference contributions and the diagnostics but there are also several exceptions to the rules defined by the thresholds. The traditional T1 and D1 diagnostics fail for almost 50% of the cases and are seldom reliable. Cheng et al.89 reached similar conclusions, showing that large t1 amplitudes in the CCSD wave function for the diatomic molecules in the 3dMLBE20 database do not reflect a failure of single-reference methods. Although the M diagnostic has a larger number of correct predictions, it still fails for about 30% of the molecules. These “exceptions” to the diagnostic rules are so numerous that one should take extreme care when drawing conclusions from them. 4.4. How To Assess DFT Functionals? In this section, we want to address the following question: Is there a best practice for assessing the performance of approximate quantum chemical methods, in particular DFT computations with different functionals? Or, if there are several equivalent procedures, how much does the ranking of the functionals really depend on the chosen reference? Of course, the most obvious choice for reference values is experimental data. This is the actual target of the computations: We want to model, understand, and predict experimentally observable processes rather than just doing theory as art for art’s sake. There is only one thing to be kept in mind (which we already discussed above): We are often talking about quantitieshere, dissociation energies, equilibrium distances, harmonic frequenciesthat are not observed directly in experiments. Rather, these are the result of a more or less complex analysis, including thermodynamic cycles (with sometimes hidden dependence on the quality of other experiments) or fits to model Hamiltonians. The latter leads to model parameters (like equilibrium distances and harmonic frequencies from a 5306


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Table 6. Mean Absolute Deviation (MAD) and Mean Signed Deviation (MSD, in Parentheses) of Dissociation Energies, Equilibrium Distances, and Harmonic Vibrational Frequencies, with Respect to Experiment, Calculated with the B97 Functional, Different Basis Sets, with and without Including Scalar Relativistic Corrections (ΔDK) Calculated with the Douglas−Kroll−Hess Hamiltonian and the aug-cc-pVTZ-DK Basis Set basis set De in kcal mol−1 +ΔDK Re in Å +ΔDK ωe in cm−1 +ΔDK

TZVP

TZVPP

QZVP

aVDZ

aVTZ

aVQZ

5.1 (−3.2) 5.4 (−3.8) 0.020 (0.008) 0.017 (0.002) 42 (−4) 37 (5)

4.6 (−1.9) 4.6 (−2.6) 0.020 (0.007) 0.017 (0.001) 40 (−3) 35 (5)

4.2 (−1.3) 4.3 (−2.0) 0.019 (0.006) 0.016 (0.000) 38 (−3) 33 (5)

4.7 (−2.3) 4.9 (−3.0) 0.023 (0.015) 0.021 (0.009) 40 (−7) 34 (2)

4.1 (−1.2) 4.1 (−1.9) 0.020 (0.008) 0.016 (0.002) 38 (−3) 33 (6)

4.1 (−1.0) 4.0 (−1.7) 0.019 (0.006) 0.016 (0.000) 38 (−2) 33 (6)

set, in particular for Re and ωe. Only for dissociation energies are somewhat increased MADs (by up to 1.4 kcal mol−1) found for the smaller basis sets (aVDZ, TZVP, and TZVPP). The better performance of aVTZ in comparison to TZVPP is most likely due to the inclusion of diffuse basis functions for the former. Relativistic corrections (always computed using triple-ζ type basis sets) have only little influence on the overall performance of the functional. This has also been pointed out by Truhlar and coauthors.85 The MAD for De decreases by only 0.3 kcal mol−1, and for Re and ωe, it decreases by 0.003 Å and 5 cm−1, respectively. In fact, all functionals improve by typically the same margin (relative to the experimental reference values; see the Supporting Information), and we will not further discuss this in detail. Instead, all further DFT results discussed in this work are based on the aVQZ basis set and include the relativistic correction. In Table 7, we summarize the MAD and MSD of the different functionals with different choices of reference values. In short, the main conclusion from this table is that the order of performance among all f unctionals is almost independent of the reference. The only exception is the CCSD(T)(FC)/awCVTZ reference, which is strongly limited by the remaining basis set error of the triple-ζ type basis set. The results in Table 7 also clearly indicate a different performance of the functionals for the dissociation energies on the one hand and equilibrium distances and harmonic frequencies on the other hand. For De, the hybrid functionals clearly have the best performance, in particular B97, B3LYP, and PBE0. This conclusion in fact holds for all references, but in comparison to experimental values or highly accurate computations, B97 is markedly better than the other two. The M06 and M06-L functionals have slightly larger error measures for De, where M06-L is still the best local DFT functional. The remaining local functionals, in particular BP86 and PBE, have significantly larger MADs. Their MSDs are always positive, indicating that the binding energies are overestimated. The results for the double exchange functional M06-2X are rather poor as well; in this case, the MSD is negative, and the binding energies are underestimated on average. In fact, M06-2X was optimized for main-group thermochemistry,246 and the present results confirm that it is not a suitable density functional for transition metal compounds. M06 and M06-L, on the other hand, have been recommended for organometallic and inorganometallic thermochemistry.246 However, the results for

these are not exceptionally good, performing worse than B3LYP for the molecules selected for the present work. For the equilibrium distances, Table 7 reveals a rather similar performance for all density functionals, with MAD errors with respect to the experimental reference lying between 0.014 Å (for TPSS) and 0.019 Å (for M06). The only exception is M06-2X, which has significantly larger errors. The same picture is found with accurate computations as reference; here, the MAD lies between 0.011 Å (TPSS) and 0.017 Å (M06). The CCSD(T)(FC)/awCVTZ calculations are not a suitable reference, as the basis set (and frozen-core) error is larger than the typical errors of DFT functionals, as the comparison to CCSD(T)(CV)/ CBS+ΔDK reveals. The good performance of the local GGA functionals like BP86 and PBE justifies their frequent use for structure optimizations in applications to large systems. The errors for harmonic frequencies are slightly more broadly distributed than the errors for Re. The sequence of functionals, however, is very similar and again not very dependent on the reference data set. As before, GGA and meta-GGA functionals perform quite well, whereas hybrid functionals lead to slightly larger errors. The M06 functionals show the largest errors, particularly the M06-2X functional.

5. CONCLUSIONS In the present work, we investigated how reliable reference data for benchmark test sets can be obtained, either from experiment or from highly accurate computations. For this purpose, we compiled a new set of experimental data for 60 diatomic transition metal compounds. The new data set extends the 3dMLBE20 set85 used in previous studies85,89,90 by additional compounds, also including selected 4d and 5d elements, and it also contains, in addition to dissociation energies, equilibrium distances and harmonic vibrational frequencies for all 60 compounds. Furthermore, we proposed a composite approach based on coupled-cluster theory with basis set extrapolation to the complete basis set limit, including core−valence correlation effects and scalar relativistic and multireference corrections. The main difference from previous coupled-cluster based schemes (like the FPD,43,61,62 HEAT,53 FPA,58−60 and W456,57 protocols) is the use of a multireference coupled-cluster theory, the internally contracted multireference coupled-cluster (icMRCC) theory,91−94 to compute corrections beyond the CCSD(T) level. Although it 5307


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Table 7. Ranking Levels of Theory (KS-DFT/aVQZ+ΔDK, CCSD(T)(FC)/awCVTZ, and CC+DKb) According to the Mean Absolute Error (MAD) with Respect to the Experimental Data or with Respect to Coupled-Cluster Theory Based Approaches for Transition Metal Containing Diatomic Molecules dissociation energy, in kcal mol−1 experiment

CCSD(T)(FC)/awCVTZ

method

MAD

MSD

method

MAD

CC+DKb B97 B3LYP CC/TZa PBE0 M06 M06-L BLYP TPSS BP86 M06-2X PBE

3.6 4.0 5.6 5.8 5.9 7.9 7.9 10.0 10.1 12.4 12.1 13.6

−1.8 −1.7 −3.5 −5.2 −2.5 −6.2 3.0 7.7 9.4 11.4 −11.0 12.8

CC/TZa CC+DKb B3LYP B97 M06 PBE0 M06-2X M06-L BLYP TPSS BP86 PBE

0.0 3.5 4.0 4.7 5.7 5.2 7.6 9.0 13.7 14.6 16.7 18.0

MSD

method

experiment method

MAD

B3LYP B97 BLYP BP86 CC+DKb CC/TZa M06 M06-2X M06-L PBE PBE0 TPSS

0.015 0.016 0.016 0.015 0.008 0.019 0.019 0.038 0.016 0.015 0.017 0.014

b

CC+DK CC/TZa BLYP BP86 PBE TPSS B3LYP B97 PBE0 M06-L M06 M06-2X a

MAD 19 21 27 27 27 28 32 33 37 40 46 59

method

0.0 CC+DKb 3.4 B97 1.7 B3LYP 3.5 CC/TZa −1.0 PBE0 2.7 M06 −5.8 M06-L 8.2 BLYP 12.9 M06-2X 14.6 TPSS 16.6 BP86 18.0 PBE equilibrium distance, in Å

CCSD(T)(FC)/awCVTZ a

0.001 0.000 0.006 −0.011 −0.006 0.016 −0.005 0.019 −0.006 −0.010 −0.012 −0.008

CC/TZ B3LYP B97 BLYP BP86 CC+DKb M06 M06-2X M06-L PBE PBE0 TPSS

MSD

method

experiment method

CCSD(T)/CBS+ΔDK

MSD

MAD 0.0 0.020 0.020 0.020 0.028 0.022 0.029 0.030 0.026 0.028 0.028 0.026

14 −15 −9 15 14 20 9 6 28 14 16 1

a

CC/TZ CC+DKb BLYP B97 B3LYP PBE BP86 TPSS M06-L PBE0 M06 M06-2X

0.0 29 30 33 34 36 37 38 42 44 50 52

MSD

method

MAD

MSD

0.0 2.3 3.4 3.5 4.3 5.7 6.6 10.7 10.3 11.2 13.4 14.6

0.0 0.1 −1.7 −3.4 −0.7 −4.4 4.8 9.5 −9.2 11.2 13.2 14.6

CC+DKb B97 B3LYP CC/TZa PBE0 M06-L M06 BLYP TPSS BP86 M06-2X PBE

2.1 3.2 4.7 5.4 5.3 6.7 7.1 9.1 9.6 11.7 12.2 12.8

−1.9 −1.8 −3.6 −5.3 −2.6 2.9 −6.3 7.6 9.3 11.3 −11.1 12.7

MAD

MSD

method

MAD

MSD

0.0 0.013 0.014 0.015 0.012 0.022 0.017 0.038 0.014 0.012 0.012 0.011

0.0 0.006 0.006 0.012 −0.005 0.022 0.001 0.025 0.000 −0.004 −0.006 −0.002

B3LYP B97 BLYP BP86 CC+DKb CC/TZa M06 M06-2X M06-L PBE PBE0 TPSS

0.014 0.014 0.014 0.012 0.005 0.021 0.017 0.038 0.014 0.012 0.013 0.011

0.004 0.004 0.010 −0.007 −0.002 0.020 −0.001 0.023 −0.002 −0.006 −0.008 −0.004

CCSD(T)/CBS+ΔDK

MSD

method b

0.0 CC+DK −0.016 B3LYP −0.016 B97 −0.010 BLYP −0.027 BP86 −0.022 CC/TZa −0.021 M06 0.003 M06-2X −0.022 M06-L −0.027 PBE −0.028 PBE0 −0.025 TPSS harmonic frequency, in cm−1

CCSD(T)(FC)/awCVTZ MAD

CCSD(T)/CBS+ΔDK+ΔMR

MAD

CCSD(T)/CBS+ΔDK+ΔMR

CCSD(T)/CBS+ΔDK

MSD

method b

0.0 29 6 21 24 29 30 35 29 43 30 15

CC+DK TPSS PBE BP86 B3LYP B97 CC/TZa PBE0 BLYP M06-L M06 M06-2X

MAD 0.0 20 21 22 27 29 29 29 30 34 39 57

CCSD(T)/CBS+ΔDK+ΔMR MSD 0.0 5 −1 1 −6 −8 −29 13 −23 −0 1 −14

method b

CC+DK CC/TZa TPSS BP86 PBE B3LYP BLYP B97 PBE0 M06-L M06 M06-2X

MAD

MSD

13 25 26 27 27 30 31 32 34 39 44 59

9 −20 15 10 9 4 −14 1 23 9 10 −5

CCSD(T)(FC)/awCVTZ. bCCSD(T)(CV)/CBS+ΔDK.

mandatory, however, to reach this high accuracy. In these cases, the multireference correction serves the same purpose as the higher-order cluster corrections (from CCSDT and CCSDTQ computations) used by Cheng et al.89 We believe, however, that the multireference correction used in this work is more robust and efficient for more strong multireference cases. Our test set also includes cases with strong multireference character, such as CoS. For this molecule, a multireference correction of 15 kcal mol−1 to the dissociation energy was computed, leading again to a good agreement with the experimentally determined value.

has been shown that other composite approaches are able to handle multireference systems,62,63 we are not aware of any composite approach that employs multireference coupled-cluster calculations. For quite a number of cases in the test set, the multireference correction (ΔMR) turns out to be small, and in agreement with recent studies by Cheng et al.89 and Fang et al.,90 we conclude that CCSD(T) can indeed give accurate results for transition metal compounds. Basis set extrapolation, the inclusion of core−valence correlation, and scalar relativistic corrections are 5308


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also prove useful in further studies of transition metal thermochemistry. A number of revisions and improvements of this approach are possible, of course, such as a more explicit treatment of spin−orbit effects, which is important for compounds with 4d and 5d transition metals. In particular, it might be criticized that the single-reference CCSD(T) calculations are not a good reference point for true multireference systems, e.g., when low-spin coupled open-shell states are involved. A more efficient implementation of icMRCC theory is under development in one of the involved groups. This will open the perspective of including core−valence correlation and basis set extrapolation directly at the icMRCC level.

In such a case, of course, CCSD(T) alone is not a good approximation. We also investigated the effectiveness of diagnostics to predict the multireference character and thus the size of multireference corrections before conducting computationally demanding icMRCC calculations. The usual measures, based on t1 and t2 amplitudes of CCSD, did not show a good correlation with the size of ΔMR, as already pointed out by Cheng et al.89 The latter authors also showed that the size of the triples correction in CCSD(T) is not a good measure for that. A further measure, the M diagnostic proposed by Tishchenko et al.,265 based on the CASSCF wave function, is slightly better than the CCSD-based diagnostics, but it still does not have satisfactory predictive power. Comparing the results from our composite approach, including ΔMR corrections, to the experimental values, we find a good agreement for the majority of cases, with deviations within ±3 kcal mol−1 for dissociation energies (often with deviations lower than ±1 kcal mol−1 from the bounds of the experimental error bar), ±0.01 Å for equilibrium distances, and ±15 cm−1 for harmonic frequencies. We note, however, that for a number of cases, particularly for dissociation energies, a surprisingly high deviation persists. This also includes a number of cases with very small multireference correction, for which also previous computational studies arrived at values close to ours. Typically, in these cases the experimental values have been obtained indirectly and required additional experimental values and model assumptions to obtain the dissociation energies, as is the case for several hydrides.89,90 We also note that even for equilibrium distances surprisingly large deviations between the experimentally determined values (which are parameters from a Dunham expansion for fitting the observed spectral lines) and accurate computations of the Born−Oppenheimer potential energy minimum can occur, like for FeH for which an earlier study254 observed the same discrepancy as the present work. It is probable that non-Born−Oppenheimer effects (in particular curve crossings) seriously interfere with a straightforward interpretation of the experimental spectrum. The mentioned problems show that experimental reference values, which can be directly used to benchmark computational models, are often not easy to obtain. This suggests that sometimes highly accurate computations provide a more direct and simple route to such reference values. The different sets of reference data (experimental and computed) were then used to analyze a sequence of DFT functionals. Interestingly, we found that the relative performance of these functionals is very similar, irrespective of whether experimental data or accurate calculations were employed as reference. Thus, although neither the experimental data nor the high-level coupled-cluster calculations are free of errors, both lead to similar conclusions about the accuracy of DFT functionals. The analysis showed that exact-exchange corrected functionals like B97, B3LYP, and PBE0 provide the most accurate dissociation energies for molecules in the present data set. On the other hand, the local functionals TPSS, BP86, and PBE perform better for equilibrium distances and harmonic frequencies. We suggest that the presented collection of 60 transition metal containing diatomics is a good basis for further studies on DFT functionals and other approximate approaches to transition metal chemistry. An extension to include larger molecules is of course desirable, but it is also challenging as both experimental determination and computation of precise reference values become more involved. The proposed composite approach may

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.7b00688. Complete set of experimental data collected for this work and comparison to 3dMLBE20 database; details of active space selection in multireference computations and additional material for tests of multireference diagnostics; additional graphical representation of results and error distribution analysis; detailed analysis of the DFT results and complete set of results for all molecules and all methods (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Andreas Köhn). *E-mail: antoniogsof@ffclrp.usp.br (Antonio G. S. de OliveiraFilho). ORCID

Ana Paula de Lima Batista: 0000-0002-9675-6106 Antonio G. S. de Oliveira-Filho: 0000-0002-8867-475X Funding

A.P.d.L.B. and A.G.S.d.O.-F thank the São Paulo Research Foundation (FAPESP) for grants 2015/22203-6 and 2015/ 11714-0 and the support of the High Performance Computing of Universidade de São Paulo (HPC-USP)/Rice University (National Science Foundation Grant OCI-0959097). Y.A.A. and A.K. are grateful to the Deutsche Forschungsgemeinschaft (grant 2337/4-1). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are very grateful to Dr. Lan Cheng for his stimulating comments and suggestions on this manuscript. REFERENCES

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