How To Arrive at Accurate Benchmark Values for Transition Metal Compounds: Computation or Experiment?

Y. A. Aoto, A. P. de Lima Batista, A. Köhn, A. G. S. de Oliveira-Filho, J. Chem. Theor. Comput., 2017
Contributed by Theo Keane

Copyright 2017 American Chemical Society 

When performing calculations of any kind, it is important to establish how accurate a method one intends to use is for a given application. Transition metals (TMs) are often problematic systems for computational chemists, because they exhibit “strong correlation”, i.e. either static or dynamic correlation is significant in systems that contain TMs (usually both). This paper adds to the existing literature of benchmark results for TM compounds by performing some rather high-level calculations on 60 diatomic TM compounds. I believe this is intended to improve upon the recent 3dMLBE20 set of Truhlar and co-workers,[1] which was criticised in a pair of papers published in early 2017.[2,3]

In this new benchmarking set, 43 molecules contain first row TMs, 7 contain Ru, Rh or Ag, and the remaining 10 contain Ir, Pt and Au. The multiplicities range from 1 to 7. For these molecules they have assembled experimental data, including bond length, harmonic vibrational frequency and bond dissociation energies. Single reference benchmark values were obtained via (RO/U)CCSD(T)/aug-cc-pwCVnZ-PP[4d, 5d metals]aug-cc-pwCVnZ[else], with n = T, Q, 5, and were extrapolated to the Complete-Basis-Set (CBS) limit. They also investigated the effect of core-valence correlation on the single-reference values. Furthermore, internally contracted Multi-Reference CCSD(T) (icMRCCSD(T)) calculations were performed in the aug-cc-pwCVTZ(-PP) basis, based on full-valence CASSCF reference wavefunctions to investigate the effect of static correlation – the full details of the chosen active spaces are provided in the SI (Table S6). Finally, relativistic effects were considered: scalar relativistic corrections were obtained by comparing frozen core CCSD(T)/aug-cc-pwCVTZ(-PP) calculations with and without the 2nd-order Douglas-Kroll-Hess (DKH2) Hamiltonian. For the 4d and 5d TM containing molecules, Spin-Orbit corrections were obtained from CASSCF calculations with full valence active spaces and the full, 2 electron Breit-Pauli operator. It is important to note that, with the exception of the SO correction, these corrections were not merely calculated for the equilibrium geometry, rather these were calculated at multiple points along the bond length. Overall, the authors have clearly spent a great deal of care ensuring that their ‘benchmark level’ calculations are truly deserving of the title.

An interesting thing to note is that multi-reference, spin-orbit and core-valence correlation corrections all appear to be very weak and sometimes do not improve the agreement with experiment (Table 3). CBS extrapolation is by far the major way to reduce error. This is very important to bear in mind when looking at previous benchmarking results. The authors also note that the usual ‘multireference’ diagnostics are practically useless: there is weak correlation between diagnostics and, more critically, there is very weak correlation between any of the diagnostics and the magnitude of any MR corrections. The M diagnostic[4] is the best performing one; however, it still fails for approximately 30% of cases and yields both false positives and false negatives. The authors also briefly investigate the effect of including 4f orbitals into the correlation treatment for Ir and Pt and find that this has a very weak effect on their results (SI, Table S5).

Finally, the authors use their new benchmark set to rank some functionals. Overall, at the DFT/aug-cc-pVQZ + DKH2 correction level, it appears that hybrid functionals performs on average the best for bond-dissociation energies and equilibrium distances, when compared to the fully corrected results (Table 7). On the other hand, pure functionals perform better for harmonic frequencies. In agreement with the conclusions of the original 3dMLBE20 paper, it is clear that many functionals beat plain CCSD(T)(FC)/aug-cc-pwCVTZ. This reinforces the critical need for CBS extrapolation when performing CC calculations.

(1) Xu, X.; Zhang, W.; Tang, M.; Truhlar, D. G. Do Practical Standard Coupled Cluster Calculations Agree Better than Kohn–Sham Calculations with Currently Available Functionals When Compared to the Best Available Experimental Data for Dissociation Energies of Bonds to 3 D Transition Metals? J. Chem. Theory Comput. 2015, 11 (5), 2036–2052 DOI: 10.1021/acs.jctc.5b00081.
(2) Cheng, L.; Gauss, J.; Ruscic, B.; Armentrout, P. B.; Stanton, J. F. Bond Dissociation Energies for Diatomic Molecules Containing 3d Transition Metals: Benchmark Scalar-Relativistic Coupled-Cluster Calculations for 20 Molecules. J. Chem. Theory Comput. 2017, 13 (3), 1044–1056 DOI: 10.1021/acs.jctc.6b00970.
(3) Fang, Z.; Vasiliu, M.; Peterson, K. A.; Dixon, D. A. Prediction of Bond Dissociation Energies/Heats of Formation for Diatomic Transition Metal Compounds: CCSD(T) Works. J. Chem. Theory Comput. 2017, 13 (3), 1057–1066 DOI: 10.1021/acs.jctc.6b00971.
(4) Tishchenko, O.; Zheng, J.; Truhlar, D. G. Multireference Model Chemistries for Thermochemical Kinetics. J. Chem. Theory Comput. 2008, 4 (8), 1208–1219 DOI: 10.1021/ct800077r.

Efficient DLPNO−CCSD(T)-Based Estimation of Formation Enthalpies for C‐, H‐, O‐, and N‐Containing Closed-Shell Compounds Validated Against Critically Evaluated Experimental Data

Eugene Paulechka and Andrei Kazakov (2017)
Highlighted by Jan Jensen

Copyright 2017 American Chemical Society

A computationally methodology is truly robust when it can be used independently and successfully by other groups.  So Frank Neese was understandably delighted when he saw this paper using his DLPNO-CCSD(T) method, as he mentioned during his talk at WATOC2017.

The paper shows tha that DLPNO-CCSD(T)/quadruple zeta//DFT-D3/triple zeta can be used to predict enthalpies of formation as accurate as you can measure them!  It is actually more accurate than G4, but considerably more computationally efficient. 

DLPNO-CCSD(T) cannot handle open shell systems so the energy of H, C, N, and O are replaced by empirical parameters.  This means that enthalpies of formation for molecules containing other elements cannot be computed without similar parametrisation, but usually atom energies/enthalpies of formation are used "only" to validate the method and are not needed to compute reaction energies.

The largest molecule considered is biphenyl and it is not clear to me that B3LYP-D3 is he optimum choice for more complex molecules requiring a conformer search. But, on the other hand, I also doubt the accuracy is very sensitive to the choice of functional for the small molecules used in the study.  It's easy enough to find out: the most time-consuming calculation (biphenyl) required only 10 hours using 10 CPUs. 


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Assessment of the accuracy of coupled cluster perturbation theory for open-shell systems. I. Triples expansions

Janus J. Eriksen, Devin A. Matthews, Poul Jørgensen, and Jürgen Gauss (2015)

Contributed by Jan Jensen

Figure 1: Normal distributions of the recoveries of (in percent (%), Figure 1a) and deviations from (in kcal/mol, Figure 1b) CCSDT–CCSD frozen-core/cc-pVTZ correlation energy differences for RHF, UHF, and ROHF references. (Taken from the paper)

This paper argues that we shouldn't expect the kind of accuracy we have come to expect from CCSD(T) for closed shell molecules, for open-shell molecules.  I have a slightly different take on the data.

But first it is fair to ask whether coupled-cluster is an appropriate standard for open shell systems in the first place because of spin-contamination (note there is also spin-contamination in ROHF- coupled-cluster).  If you get significantly different answers for UHF- and ROHF-based calculations it is not clear which one is the most reliable.

The first thing the paper shows - for 18 atoms and small open shell molecules - is that spin-contamination decreases by roughly an order of magnitude for each step in SCF > CCSD > CCSDT > CCSDTQ so that it is negligible for the latter.  For most of the molecules the spin-contamination is also negligible for CCSDT.  This is good to know.

The authors go on to show that CCSD(T) is a worse approximation for CCSDT for open shell systems (see Figure 1) and that CCSD(T) is always a worse approximation to CCSDTQ than CCSDT for open shell systems, in contrast to closed shell systems.  While that's true, the data also shows that the CCSDT correlation energy is closer to the CCSDTQ energy for open shell systems.  In fact the the mean error in CCSD(T) correlation energies relative to CCSDTQ is actually lower for open shell systems (0.80, 0.70, and 0.57 kcal/mol for RHF-, UHF-, and ROHF-based calculations).  So this is good news.

One note of caution in all of this is that the study uses cc-pVTZ (understandable as CCSDTQ calculations are performed). It remains to be seen whether the conclusions are true for the CBS.

On a related note Anacker, Tew, and Friedrich have recently presented an incremental scheme for estimating UHF-based CCSD(T) energies for larger systems.



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Exploring the Accuracy Limits of Local Pair Natural Orbital Coupled-Cluster Theory

Dimitrios G. Liakos, Manuel Sparta, Manoj K. Kesharwani, Jan M. L. Martin, and Frank Neese J. Chem. Theory Comput. 2015, 11, 1525−1539
Contributed by +Jan Jensen

Reprinted with permission from J. Chem. Theory Comput. 2015, 11, 1525-1539.
Copyright (2015) American Chemical Society.

Two years ago Neese and co-workers published the first CCSD(T) calculation on a protein (crambin), which, according to this latest paper remains "the largest coupled cluster calculations reported to date".  The calculations were made possible through the domain based local pair natural orbital (DLPNO) implementation of CCSD(T), which includes a variety of approximations with associated thresholds that control the accuracy.

The current paper defines three default thresholds termed "LoosePNO", "NormalPNO", "and TightPNO" recommended for rapid estimates, general thermochemistry and kinetics, and non-covalent interactions and conformational equilibria, respectively.  The latter two provide relative energies with 1 kcal/mol of conventional CCSD(T) calculations.

The study includes molecules up to 30 atoms including a "real-life" organometallic catalysis and a study of 51 conformers of melatonin.  Here the focus is very much on accuracy and little or no information was given on computational cost.  So I encourage any reader with practical experience using this method to leave a comment with their experiences.

Accurate CCSD(T) benchmark data sets for small complexes has been incredibly useful in parameterizing dispersion corrections and semi-empirical methods. It looks like DLPNO-CCSD(T) has made it practical to construct similar benchmark sets for chemical reactivity of organometallic and bio-catalytic models systems which currently are too big to be treated with conventional CCSD(T).

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Electron correlation, zero-point vibrational and temperature effects in Nuclear Magnetic Resonance spectroscopy

A.M. Teale, O.B. Lutnæs, T. Helgaker, D.J. Tozer, J. Gauss, Journal of Chemical Physics 2013, 138, 024111 (Pay-wall) and
J. Kaminský, M. Buděšínský, S. Taubert, P. Bouřa, M. Straka, Physical Chemistry Chemical Physics 2013, 15, 9223-9230 (Open Access)
Contributed by Marcel Swart

Last year has seen two important contributions in the field of determination of NMR chemical shifts by theoretical chemistry. More and more it is recognized that the computational prediction of ¹H and ¹³C chemical shifts is a useful tool for natural product, mechanistic, and synthetic organic chemistry.^[1] There are however doubts about how accurate these results are, and if any chemically relevant conclusions can be drawn from them.

The first paper^[2] compares the chemical shifts as obtained by both density functional theory and wavefunction theory (RHF, CCSD, CCSD(T), extrapolated) for a total of 28 molecules (for which previously already rotational g-tensors and magnetizabilities were computed^[3]). The authors also included zero-point vibrational effects on the computed chemical shieldings, and used extrapolation techniques to estimate uncertainties related to basis-set incompleteness^[4]. First, the authors established an accurate benchmark set of data, for which the accuracy was established by comparison with experimental data (including zero-point vibrational corrections). They found good agreement between CCSD(T)/aug-cc-pCVQZ and experiment (empirical equilibrium values), with a mean-absolute-error of 2.9 ppm for the chemical shieldings. Afterwards, these reference data were used to compare how well a variety of density functionals was able to reproduce them, with a sobering conclusion: "None of the existing approximate functionals provide an accuracy competitive with that provided by CCSD or CCSD(T) theory".^[2] The best performing functional (as shown before) was KT2 with a mean-absolute-error compared to the CCSD(T) data (both with the same aug-cc-pCVQZ basis set) of 10.2 ppm.

The second paper^[5] has a completely different approach, and deals with the characterization of fullerenes. For this purpose, computational chemistry might be used, but one again should be sure about the methods used. The authors used quantum vibrational averaging, a dielectric continuum model for the solvent (CPCM, 1,1,2-trichloroethane), classical (MM3) and first-principle (BP86/def-SVP) molecular dynamics simulations, and did experiments. These authors used a different set of density functionals, and found the best results for wB97X-D/IGLO-III (a root-mean-square deviation compared to experiment of only 0.4 ppm). However, surprisingly, in the analysis of the dynamical part they used either BP86 or BHandHLYP for the NMR chemical shifts, even though these showed larger RMSD values of respectively 1.7 and 0.8 ppm. Moreover, big differences were found in the chemical shifts from the snapshots of the 1 ns classical MD, and those of the 1.2 ps first-principles MD. For the five different atom types these differences were found in the range 1.8-7.9 ppm.^[6]

References and notes

[1] M.W. Lodewyk, M.R. Siebert, D.J. Tantillo, Chem. Rev. 2012, 112, 1839-1862 [DOI 10.1021/cr200106v]
[2] A.M. Teale, O.B. Lutnæs, T. Helgaker, D.J. Tozer, J. Gauss, J. Chem. Phys. 2013, 138, 024111 [DOI 10.1063/1.4773016]
[3] O. B. Lutnæs, A. M. Teale, T. Helgaker, D. J. Tozer, K. Ruud, J. Gauss, J. Chem. Phys. 2009, 131, 144104 [DOI 10.1063/1.3242081]
[4] These formulas have been developed for energies; hence, their use for the direct extrapolation of molecular properties is less well founded, and indeed can not be used with a two-point extrapolation for prediction of the basis set limiting value.
[5] J. Kaminský, M. Buděšínský, S. Taubert, P. Bouřa, M. Straka, Phys. Chem. Chem. Phys. 2013, 15, 9223-9230 [DOI 10.1039/C3CP50657F]
[6] Strangely enough, while the CPCM solvent model only had a modest effect on the chemical shifts of 0.2-0.3 ppm, the authors showed that first-principles MD (FPMD) simulations including solvent effects (COSMO) led to drastically different results for the chemical shifts of snapshots (0.1-4.1 ppm) from those from FPMD without them.


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Describing Noncovalent Interactions beyond the Common Approximations: How Accurate Is the “Gold Standard,” CCSD(T) at the Complete Basis Set Limit?

Řezáč, J.; Hobza, P.  J. Chem. Theor. Comput., 2013, 9, 2151
Contributed by Steven Bachrach.
Reposted from Computational Organic Chemistry with permission

The gold standard in quantum chemistry is the method that is considered to be the best, the one that gives accurate reproduction of experimental results. The CCSD(T) method is often referred to as the gold standard, especially when a complete basis set (CBS) extrapolation is utilized. But is this method truly accurate, or simply the highest level method that is within our reach today?

Řezáč and Hobza¹ address the question of the accuracy of CCSD(T)/CBS by examining 24 small systems that exhibit weak interactions, including hydrogen bonding (e.g. in the water dimer and the water^…ammonia complex), dispersion (e.g. in the methane dimer and the methane^…ethane complex) and π-stacking (e.g. as in the stacked ethene and ethyne dimers). Since weak interactions result from quantum mechanical effects, these are a sensitive probe of computational rigor.

A CCSD(T)/CBS computation, a gold standard computation, still entails a number of approximations. These approximations include (a) an incomplete basis set dealt with by an arbitrary extrapolation procedure; (b) neglect of higher order correlations, such as complete inclusion of triples and omission of quadruples, quintuples, etc.; (c) usually the core electrons are frozen and not correlated with each other nor with the valence electrons; and (d) omission of relativistic effects. Do these omissions/approximations matter?

Comparisons with calculations that go beyond CCSD(T)/CBS to test these assumptions were made for the test set. Inclusion of the core electrons within the correlation computation increases the non-covalent bond, but the average omission is about 0.6% of the binding energy. The relativistic effect is even smaller, leaving it off for these systems involving only first and second row elements gives an average error of 0.1%. Comparison of the binding energy at CCSD(T)/CBS with those computed at CCSDT(Q)/6-311G** shows an average error of 0.9% for not including higher order configuration corrections. The largest error is for the formaldehyde dimer (the complex with the largest biding energy of 4.56 kcal mol^-1) is only 0.08 kcal mol^-1. If all three of these corrections are combined, the average error is 1.5%. It is safe to say that the current gold standard appears to be quite acceptable for predicting binding energy in small non-covalent complexes. This certainly gives much support to our notion of CCSD(T)/CBS as the universal gold standard.

An unfortunate note: the authors state that the data associated with these 24 compounds (the so-called A24 dataset) is available on their web site (www.begdb.com), but I could not find it there. Any help?

References

(1) Řezáč, J.; Hobza, P. "Describing Noncovalent Interactions beyond the Common Approximations: How Accurate Is the “Gold Standard,” CCSD(T) at the Complete Basis Set Limit?," J. Chem. Theor. Comput.,2013, 9, 2151–2155, DOI: 10.1021/ct400057w.

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