Calibrating Reaction Enthalpies: Use of Density Functional Theory

Article pubs.acs.org/JPCA

Calibrating Reaction Enthalpies: Use of Density Functional Theory and the Correlation Consistent Composite Approach in the Design of Photochromic Materials Roger G. Letterman,† Nathan J. DeYonker,*,† Theodore J. Burkey,† and Charles Edwin Webster*,†,‡ †

Department of Chemistry and Computational Research on Materials Institute, The University of Memphis, Memphis, Tennessee 38152, United States ‡ Department of Chemistry and Center for Computational Sciences, Mississippi State University, Mississippi State, Mississippi 39762-9573, United States S Supporting Information *

ABSTRACT: Acquisition of highly accurate energetic data for chromium-containing molecules and various chromium carbonyl complexes is a major step toward calibrating bond energies and thermal isomerization energies from mechanisms for Cr-centered photochromic materials being developed in our laboratories. The performance of six density functionals in conjunction with seven basis sets, utilizing Gaussian-type orbitals, has been evaluated for the calculation of gas-phase enthalpies of formation and enthalpies of reaction at 298.15 K on various chromium-containing systems. Nineteen molecules were examined: Cr(CO)6, Cr(CO)5, Cr(CO)5(C2H4), Cr(CO)5(C2ClH3), Cr(CO)5(cis-(C2Cl2H2)), Cr(CO)5(gem-(C2Cl2H2)), Cr(CO)5(trans-(C2Cl2H2)), Cr(CO)5(C2Cl3H), Cr(CO)5(C2Cl4), CrO2, CrF2, CrCl2, CrCl4, CrBr2, CrBr4, CrOCl2, CrO2Cl2, CrOF2, and CrO2F2. The performance of 69 density functionals in conjunction with a single basis set utilizing Slater-type orbitals (STO) and a zeroth-order relativistic approximation was also evaluated for the same test set. Values derived from density functional theory were compared to experimental values where available, or values derived from the correlation consistent composite approach (ccCA). When all reactions were considered, the functionals that exhibited the smallest mean absolute deviations (MADs, in kcal mol−1) from ccCA-derived values were B97-1 (6.9), VS98 (9.0), and KCIS (9.4) in conjunction with quadruple-ζ STO basis sets and B97-1 (9.3) in conjunction with cc-pVTZ basis sets. When considering only the set of gas-phase reaction enthalpies (ΔrH°gas), the functional that exhibited the smallest MADs from ccCA-derived values were B97-1 in conjunction with cc-pVTZ basis sets (9.1) and PBEPBE in conjunction with polarized valence triple-ζ basis set/effective core potential combination for Cr and augmented and multiple polarized triple-ζ Pople style basis sets (9.5). Also of interest, certainly because of known cancellation of errors, PBEPBE with the least-computationally expensive basis set combination considered in the present study (valence double-ζ basis set/effective core potential combination for Cr and singly-polarized double-ζ Pople style basis sets) also provided reasonable accuracy (11.1). An increase in basis set size was found to have an improvement in accuracy for the best performing functional (B97-1).

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INTRODUCTION

functional groups, steric effects, and the proximity of each functional group to the metal center preceding chelation. Also, the bistability of these organometallic photochromes requires an adequate free energy of activation (ΔG⧧ > ∼30 kcal mol−1) for rearrangement such that the complex will not thermally isomerize between chelates at room temperature. Hence, the ground-state bond dissociation enthalpy (BDE) required to cleave a metal−ligand bond is of particular interest to researchers to aid in the rational design of new organometallic photochromes.3,4,9−16 Our lab and others have used photo-

The organometallic photochromic systems under development in our laboratories utilize light-driven linkage isomerizations of tethered bifunctional chelates to selectively cleave the metal− ligand bond and induce a subsequent rearrangement to form a chelate with a different functional group.1−8 In an ideal photochromic system, the photoisomerization will induce spectral changes in the organometallic complex when the second functional group chelates to the metal. Initiation of the photolinkage isomerization is, in part, dependent on the metal−ligand bond strength of the chelated functional group. For example, near-UV irradiation tends to break the weakest bond in many of these complexes.2 The thermodynamics of the linkage isomerization are dependent on several factors, including the relative metal−ligand bond strengths of tethered © XXXX American Chemical Society

Received: September 13, 2016 Revised: November 9, 2016

A

DOI: 10.1021/acs.jpca.6b09278 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

relativistic effects, and an increased complexity of the bonding, (e.g., π back-bonding from the metal to a CO ligand). Wilson, DeYonker, and co-workers suggest that the target accuracy for computed enthalpies of formation should be within 3.0 kcal mol−1 for transition-metal complexes. Jiang, DeYonker, and Wilson applied ccCA-TM toward the calculation of the ΔfH°gas of 225 molecules containing the third-row d-block (elements Sc to Zn).29,30 They reported that transition-metal chemical accuracy (MAD less than or equal to 3.0 kcal mol−1) was achieved for a subset of 70 of these molecules with experimental uncertainties less than or equal to 2.0 kcal mol−1. It was also observed that the MADs decrease monotonically with decreasing thresholds in experimental uncertainties. It should also be noted that the experimental uncertainties in the measured ΔfH°gas for many of the species in the 225 molecule test set are greater than 3.0 kcal mol−1 (and in some cases much greater) or are unreported. The ccCA is computationally expensive and scales as N7 (where N is the number of basis functions); thus, many of the organometallic photochromes studied in our lab are simply too large to examine via ccCA-TM. DFT methods based on a SCFtype algorithm formally scale as N4, while density-fitting reduces the scaling to N3, but the reliability of a given functional for computationally derived BDEs can vary greatly depending on the chemical environment. Numerous studies have been conducted in recent years to evaluate the performance of DFT methods toward the prediction of various properties of transition-metal-containing systems.31−41 However, previous studies differ in their recommendations of the ideal DFT functional. For example, Furche and Perdew reported the computed bond energetics of 64 3d transitionmetal compounds examined as part of an investigation into the performance of semilocal and hybrid DFT functionals. They concluded, in part, that TPSS is a general purpose functional which exhibits nearly uniform accuracy for their selected systems.32 Landis and co-workers reported a systematic computational study of transition metal−ligand bond enthalpies via B3LYP and CCSD(T) for a wide assortment of ligands in saturated transition-metal hydride species covering the entire d-block (including 79 M−H and M−CH 3 compounds). The reported differences in the metal−ligand bonding within a row were significant.33 Riley and Merz examined the performance of 12 DFT functionals in the calculation of 94 ΔfH°gas and 58 ionization potentials (IP). They reported that the TPSS1KCIS functional in combination with the TZVP basis set yielded the lowest average errors in the calculation of ΔfH°gas for the transition-metal-containing species in the test set. Cr-containing systems provided the worst results, and carbonyl-bearing compounds were shown to be problematic for functionals without exact exchange.34 Wilson, Cundari, and co-workers used correlation-consistent basis sets to assess the performance of 44 DFT functionals toward the prediction of ΔfH°gas for 19 3d TM complexes with small (0.05−3.3 kcal mol−1) experimental uncertainties, including carbonyl-containing compounds. They reported significantly larger average MADs for compounds containing carbonyls (e.g., the avg MAD (from experimental values) for all meta-GGA functionals tested was greater than 80 kcal mol−1).35 Wilson and co-workers in a separate study tested the performance of 13 single- and double-hybrid density functionals in combination with correlation-consistent basis sets in the calculation of ΔfH°gas for a set of 193 3d transition-metalcontaining molecules. They reported that B97-1 provided

acoustic calorimetry (PAC) to study the BDEs of various ligands in transition-metal (TM) complexes.1−4,9,10,17−19 Several aspects of PAC techniques have led our group to search for an alternate means to determine metal−ligand bond strengths. For example, ignoring the effects of reaction volumes (defined as the difference in molar volumes between products and reactants) can cause significant error in the determination of BDEs via PAC. Establishing the contribution of reaction volume to the BDE requires numerous PAC experiments for a single complex, wherein the elastic parameters of the solvent are altered.3 Our attempts to address the reaction volume component via computational methods have been met with limited success because the errors in the experimental (ranging from 5 to 59 mL mol−1) and computational molar volumes (22.8 mL mol−1 standard error in the estimated molar volume) are larger than the experimentally determined reaction volumes (absolute value ranging from 0.9 to 14.4 mL mol−1 for that study).2 For example, at room temperature, a change in reaction volume of 3 mL mol−1 in heptane leads to an error of 1 kcal mol−1 in the BDE derived from PAC.2 One of our goals is to alleviate the need to perform these particularly expensive and time-consuming experiments. Use of an appropriate computational method could provide an alternative approach to obtaining BDEs. In the current work, we seek to ascertain a less-expensive density functional theory (DFT) method calibrated to a composite method known to provide accurate energetic properties for transition-metal-containing species. The correlation consistent composite approach (ccCA) developed by Wilson, Cundari, and co-workers is one example of a highly accurate composite method for determining atomic and molecular energies.20−25 The ccCA-derived energies can be used to calculate the gas-phase enthalpy of reaction (ΔrH°gas), including BDEs and enthalpies of formation (ΔfH°gas). The ccCA is based on complete basis set (CBS) extrapolations of MP2 energies and additive CCSD(T) energies. Geometries and vibrational frequencies are obtained at the B3LYP/cc-pVTZ level of theory. Other than the inherent parameters in the B3LYP functional, ccCA is an entirely ab initio model chemistry. The ccCA has been shown to have a mean absolute deviation (MAD) from experiment of 1.0 kcal mol−1 for 454 energetic properties in the main group G3/05 training set for atoms and molecules.20−24,26 The prediction of ΔfH°gas for transition-metal containing species has also been explored via ccCA (by means of a slightly altered methodology called ccCA-TM). Many factors contribute to potential inaccuracies in the experimental determination of enthalpies, for example, material instability and incomplete combustion in the case of enthalpy of formation or the determination of reaction volume for photocalorimetric methods in the case of reaction enthalpy. Additionally, much of the primary literature of known gas-phase inorganic enthalpies of formation, where such information is available, is of questionable quality. For example, ΔfH°gas values for only two of the nine organometallic complexes examined in this work have been determined experimentally, and the reported errors for Cr(CO)5 and Cr(CO)6 are ±3.1 and ±20 kcal mol−1, respectively.27 In some cases, thermochemical information provided in the literature lacks an adequate assessment of experimental errors or exhibits large uncertainties, thereby hindering accurate benchmarking of theoretical methods.28,29 For theory, several factors lead to significant inaccuracies in prediction of energetic properties for species which contain transition metals: larger valence electron space, stronger B


Article

The Journal of Physical Chemistry A Table 1. Functionals Utilized for the Post-SCF Gradient Corrections to LDA Density Functional Energy GGA

meta-GGA

hybrid-GGA

hybrid-meta-GGA

OPerdew52,68,69 BOP49,73 BLYP48,49 BLYP-D79 OLYP48,68 OPBE53,68,86 XLYP88 BP8649,52,69 BP86-D79 HCTH/9351 HCTH/12097 HCTH/14797 HCTH/40799 PBE53,54 PBE-D79 mPBE104 revPBE105 RPBE107 mPW91 PW91110−112 FT97113 KT1114 KT2114 Becke88x+BR89c115−117

KCIS39 KCIS-modified74 mPBEKCIS76 PKZBx-KCIScor80 PKZB60,82 TPSS71,72 TPSS-D79 BmTau190 M06-L92 OLAP368,94 BLAP394 VS9898 Becke00100 τ-HCTH89

B9750,70 B97-151,70 B97-270,77 B97-D79 PBE070,83−85 OPBE053,54,68,86,87 τ-HCTH-hybrid70,89 mPW1PW70,91 mPW1K70,93 B1PW91(VWN5)70,95,96 B1LYP(VWN5)70,95,96 B3LYP-D79 B3LYP(VWN5)70,96,101 B3LYP*(VWN5)70,96,102 O3LYP(VWN5)103 X3LYP(VWN5)88 KMLYP(VWN5)70,96,106 BHandH48,70,108 BHandHLYP48,70,109

TPSSh70−72 M0575 M05-2X78 M0681 M06-2X81 mPBE0KCIS70,76 mPBE1KCIS70,76

experimental values. It is unclear how greater uncertainty in the calculation of ΔfH°gas for chromium-centered carbonyl-bearing compounds will affect the choice of DFT functional when examining the organometallic complexes of interest to our lab because calculating ΔfH°gas using the atomization scheme is a test of methodology much more stringent than calculating BDEs where chemical environments of the product and reactant are typically quite similar.45 Furthermore, we compare the DFT- and ccCA-derived enthalpies for systems where the experimental data are not available. This comparison will also be used to establish the fitness of various DFT methodologies for larger systems that are intractable for ccCA.

optimal results for Cr-containing species. They also noted that the results for Cr-containing species displayed the largest MADs (approximately 10−20 kcal mol−1) compared to all other metal-centered species, which is a conclusion also noted by Riley and Merz.34,36 Variations in the performance of DFT functionals, especially for Cr-centered organometallic complexes, may make the best choice of DFT method unclear in the absence of both reliable experimental data and benchmarking with highly accurate ab initio techniques. The atomization energies of CrO, CrO2, and CrO3 were evaluated by Chan, Raghavachari, Radom, and Karton via composite methods with post-CCSD(T) terms up to CCSDTQ(5)Λ and CCSDTQ(5). Heats of formation from best estimated atomization energies for CrO, CrO2, and CrO3 were determined to be 47.4 ± 1, −19.4 ± 1, and −68.5 ± 5 kcal mol−1, respectively.42 Zhang, Truhlar, and Tang reported the results of 42 exchange-correlation functionals (11 different types) used to predict the average bond energies of 70 (3d) transition-metalcontaining molecules, 19 of which were single-reference, and 51 were multireference. Larger errors on average were reported for middle transition metals (V, Cr, and Mn) than for either early or late transition metals.43 Xu, Zhang, Tang, and Truhlar tested and compared the results of coupled cluster (CC) and Kohn−Sham DFT methods on 20 3d transition-metal-containing diatomic molecules against the most reliable experimental data available. On average, nearly half of the exchange-correlation functionals tested performed better than CCSD(T) when utilizing the same extended basis set for a given molecule.44 In the current study, for mostly Cr-containing species with small experimental errors, we present a comparison of gasphase enthalpies of reaction (ΔrH°gas) [BDEs, exchange reaction enthalpies (ΔexH°gas), and enthalpies of formation (ΔfH°gas)] of ccCA and various DFT model chemistries to

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COMPUTATIONAL METHODS Density Functional Theory. Computations were carried out using the Gaussian0946 implementations of B3LYP (the B347 exchange and the LYP correlation48), BHandH (the Becke half and half exchange correlation),49 B97-1 (the Becke B9750 exchange and correlation modified by Handy, Tozer, and coworkers51), BVP86 (the Becke exchange (B)49 and the Perdew correlation (VP86)52), and PBEPBE (the PBE exchange and correlation53,54) functionals of density functional theory55 using the default pruned fine grids for energies (75, 302), default pruned course grids for gradients and Hessians (35, 110) (neither grid is pruned for chromium), and default SCF convergence for geometry optimizations (10−8). Single-point calculations on B3LYP/BS3 geometries were also carried out using the Amsterdam Density Functional program (ADF, versions 2008.01 and 2009.01).56−58 ADF uses Slater-type orbital (STO) basis sets rather than Gaussian type orbitals (GTO) and contains a density fitting procedure using auxiliary functions (fit functions) for the evaluation of the Coulomb potential and molecular density. All reported ADF calculations were post-SCF gradient corrections to the LDA59 (local density approximation) density functional energy, utilized a singleC


Article


correlation-consistent basis sets of Dunning and co-workers.124−128 BS7: The basis set for chromium was derived from the completely uncontracted Hay and Wadt basis set and ECP combination (LANL08).132,133,138,139 The basis set for all other atoms were the cc-pVTZ correlation-consistent basis sets of Dunning and co-workers.124−128 BS8 (used solely in single-point ADF calculations): The basis set for all atoms was the standard ADF ZORA QZ4P basis set (an uncontracted quadruple-ζ STO basis set with four polarization functions). ADF ZORA basis sets were optimized for use within ZORA relativistic calculations. The density fitting approximation140−143 for the fitting of the Coulomb potential was used for all BVP86/GTO and PBEPBE/GTO calculations; auxiliary density-fitting basis functions were generated automatically (by the procedure implemented in Gaussian 09) for the specified AO basis set.144 Spherical harmonic d functions were used throughout; that is, there are five angular basis functions per d function, etc. All structures were fully optimized, and analytical frequency calculations were performed on all structures to ensure a zeroth-order saddle point (a local minimum) was achieved. All DFT enthalpies reported were calculated using the respective functional with unscaled harmonic vibrational frequencies. All thermal corrections were performed at 298.15 K.

point grid accuracy factor (4.0), and used the default SCF convergence (10−8). The functionals utilized in the post-SCF gradient corrections to the LDA density functional energy are listed in Table 1.39,60 Relativistic effects were treated with the zeroth-order regular approximation (ZORA)61−66 with the default sum of atoms potential approximation (SAPA) potential.67 The frozen core approximation was used in all ADF computations. Description of the Correlation Consistent Composite Approach. The ccCA is a nonparameterized MP2-based model chemistry known to provide chemically accurate results. Unlike many other model chemistries, ccCA can predict reliable ΔfH°gas and BDEs for TM-containing molecules.20−22,28,30 The ccCA was developed to employ geometries optimized with B3LYP utilizing cc-pVTZ (BS3, defined in Table 2) followed by a series of single-point energy calculations for the general ccCATM formulation.28,29 A summary of the ccCA may be found in the Supporting Information. Table 2. Basis Set Combinations basis set

transition metal

other atoms

BS1 BS2 BS3 BS4 BS5 BS6 BS7 BS8

mod-LANL2DZ mod-LANL2DZ cc-pVTZ (Cr) mod-LANL2TZ(f) sc-stuttgart nonrel (Cr) sc-stuttgart fully relativistic LANL08 ZORA QZ4P

6-31G(d′) 6-31++G(d′,p′) cc-pVTZ 6-311++G(2df,2p) cc-pVTZ cc-pVTZ cc-pVTZ ZORA QZ4P

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RESULTS Standard Enthalpy of Formation. For the 32 molecules relevant to our study (the 19 chromium-containing molecules and 13 ligand “fragment” molecules, listed in Table 3), the gasphase enthalpy of formation (ΔfH°gas) was calculated in the manner described by Curtiss and co-workers145 from total electronic energies computed via ccCA and the temperaturedependent enthalpy correction computed via B3LYP/BS3. MAD is the Mean Absolute Deviation, MSD is the Mean Signed Deviation, and RMSD is the Root-Mean-Square Deviation. The ccCA-derived ΔfH°gas values are shown in Table 3 along with experimental measurements, where available. Next, the ΔfH°gas was calculated for these 32 molecules via GTO methods (B3LYP, BHandH, BVP86, PBEPBE, B97-1, and ωB97X-D functionals utilizing BS1 through BS7) and via STO methods (64 DFT functionals listed in Table 1 utilizing BS8 and the temperature-dependent enthalpy corrections computed via B3LYP/BS3). These DFT-derived ΔfH°gas values were compared to ccCA-derived values, and the complete sets of GTO and STO results may be found in the Supporting Information in Tables S1−S6 and S7−S18, respectively. Summaries of the MADs for the ΔfH°gas of each GTO and STO method from ccCA are shown in Tables 4 and 5, respectively. Standard Enthalpy of Reaction. The gas-phase standard enthalpy of reaction (ΔrH°gas) for each reaction shown in Table 6 was calculated as shown in eq 5 from ccCA-derived ΔfH°gas values.

Basis Sets. Several basis set combinations were utilized (see the following description and Table 2). In each case, where an all-electron main group cc-pVTZ basis set was used, the redundant functions were removed and then the basis set was linearly transformed, as suggested by Davidson.118 BS1: The basis set for chromium (341/341/41) was the Hay and Wadt basis set and effective core potential (ECP) combination (LANL2DZ)47 as modified by Couty and Hall, where the two outermost p functions have been replaced by a (41) split of the optimized chromium 4p function.119 The 631G(d′)120 basis sets were used for all other atoms. BS2: The metal basis set of BS1 was used for BS2. The 6-31+ +G(d′,p′)121−123 basis sets were used for all other atoms. BS3: The basis sets for all atoms were the cc-pVTZ correlation-consistent basis sets of Dunning and co-workers and Peterson and co-workers.124−130 BS4: The basis sets for chromium were the Hay and Wadt basis set and effective core potential (ECP) combination (LANL2TZ(f))131−133 that was uncontracted to [4s4p3d]. The 6-311++G(2df,2p)134,135 basis sets developed by Pople and coworkers were used for all other atoms. BS5: The basis set for chromium136 was the nonrelativistic contracted basis set developed by Peterson, Dolg, and Stoll and ECP (Stuttgart/Cologne group ECP10MHF118,137) combination. The basis sets for all other atoms were the cc-pVTZ correlation-consistent basis sets of Dunning and co-workers.124−128 BS6: The basis set for chromium136 was the fully relativistic contracted basis set developed by Peterson, Dolg, and Stoll and ECP (Stuttgart/Cologne group ECP10MDF118,137) combination. The basis sets for all other atoms were the cc-pVTZ

° = Δr Hgas

° ∑ ΔHgas

products

−

° ∑ ΔHgas

reactants

(5)

The ccCA-derived ΔrH°gas values are shown in Table 6 along with experimental measurements, where available. Next, the ΔrH°gas was calculated for each reaction via GTO methods (B3LYP, BHandH, BVP86, PBEPBE, B97-1, and ωB97X-D functionals utilizing BS1 through BS7) and via STO D



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Table 3. Experimental and ccCA-Derived ΔfH°gas with MSD, MAD, and RMSD Displayed for the Set of Species Presented (kcal mol−1)a species

experiment

ccCA

Cr(CO)5 Cr(CO)6 Cr(CO)5(C2H4) Cr(CO)5(C2H3Cl) Cr(CO)5(C2H2Cl2)b Cr(CO)5(C2H2Cl2)c Cr(CO)5(C2H2Cl2)d Cr(CO)5(C2HCl3) Cr(CO)5(C2Cl4) CO C2H4 C2H3Cl C2H2Cl2b C2H2Cl2c C2H2Cl2d C2HCl3 C2Cl4 CrBr2 CrBr4 CrCl2 CrCl4 CrF2 CrO2 CrO2Cl2 CrO2F2 CrOCl2 CrOF2 H2 O2 F2 Cl2 Br2

−153.9 (3.1) −240.0 (1.1)27,147 n.d. n.d. n.d. n.d. n.d. n.d. n.d. −26.4 (0.007)27 12.6 (0.036)148−150 5.3 (0.08)148−150 −0.7 (0.5)27 0.5 (0.33)27 −0.1 (0.5)27 −4.2 (0.5)27 −5.8 (1.0)27 −14.1 (4.3)151 −51.3 (4.5)151 −28.1 (0.4)152 −94.8 (3.3)152 −103.2 (3.0)152 −23.7 (1.2)153 −124.1 (1.0)152 −201.6 (5.0)152 −80.4 (5.4)152 −153.3 (3.4)152 0.0 0.0 0.0 0.0 7.4 (0.026)27 MSD MAD RMSD

−170.0 −239.5 −187.6 −192.1 −195.4 −193.6 −196.5 −198.0 −199.0 −26.3 12.7 5.3 −0.1 0.6 0.5 −3.4 −5.5 −9.1 −53.9 −32.1 −100.8 −107.4 −15.9 −124.1 −202.0 −79.7 −157.6 −0.5 0.8 0.2 0.2 8.0 0.8 2.3 4.2

27,146

Uncertainty values, when known, are given in parentheses; n.d. denotes no determined or known experimental value. bcis-Olefin. c Geminal-olefin. dtrans-Olefin.

Table 4. Summary of MADs (kcal mol−1) for ΔfH°gas from ccCA for Each GTO Method Listed B3LYP B97-1 BHandH BVP86 PBEPBE ωB97X-D

BS2

BS3

BS4

BS5

BS6

BS7

39.1 19.1 29.1 18.1 34.7 35.8

47.8 25.9 31.8 12.3 24.3 42.4

29.2 9.2 20.6 23.9 39.0 21.9

31.6 8.3 24.6 22.0 35.6 24.4

47.8 26.0 35.5 23.8 35.7 45.0

31.8 9.4 21.2 20.2 34.8 27.2

32.4 9.9 23.9 21.3 35.6 27.3

DISCUSSION

Standard Enthalpy of Formation. DFT methods often exhibit large errors in the exchange and correlation energies. Furthermore, the exchange energy may be underestimated, while the correlation energy is overestimated, leading to a cancellation of errors.154,155 The prediction of reaction enthalpies where the products and reactants exhibit very similar electronic structures is a less difficult task compared to predicting enthalpies of formation from standard state elements. The computation of ΔfH°gas from atomization energies is a daunting task for computational methods because ΔfH°gas is typically calculated from gas-phase atomization energies, and the electronic structures of the product and reactants are quite different. Therefore, inaccuracies in the prediction of this property are addressed first. While data on gas phase enthalpy of formation under standard conditions (ΔfH°gas) are unavailable for the Cr(CO)5olefin complexes investigated in the course of this work, ΔfH°gas values of −218 ± 20 and −153.9 ± 3.1 kcal mol−1 have been reported for Cr(CO)6 and Cr(CO)5, respectively.27,158 The value for Cr(CO)6 published in the NIST WebBook represents the average of nine ΔfH°gas values; however, the ccCA-derived ΔfH°gas for Cr(CO)6 (−239.5 kcal mol−1) compares favorably to the most exothermic of the nine measurements (−240.4 ± 1.1 kcal mol−1). This value is adopted as the experimental ΔfH°gas in this work. The ccCA-derived ΔfH°gas for Cr(CO)5 (−170.0 kcal mol−1) varies from experiment by 16.1 kcal mol−1 and represents the greatest deviation from experiment for any ccCA-derived ΔfH°gas examined in the course of this study. This significant deviation in the ccCA-derived and experimental values for Cr(CO)5 is a statistical outlier; however, the experimental ΔfH°gas for Cr(CO)5 published in the NIST WebBook relies on the ΔfH°gas for Cr(CO)6 of −217.1 ± 0.62 kcal mol−1 and, as previously noted, the ΔfH°gas for Cr(CO)6 may not be reliable.28,29 The MAD of ccCA-derived versus experimental ΔfH°gas shown in Table 3 for the 25 molecules where experimental values are known is 2.3 kcal mol−1. If the ΔfH°gas value for Cr(CO)5 is excluded, the MAD is improved to 1.7 kcal mol−1. The ccCA-derived and experimental ΔfH°gas values for each species (Table 3) were compared, and the MSD, MAD, and RMSD were determined to be 0.8, 2.3, and 4.2 kcal mol−1, respectively, when including Cr(CO)5. When Cr(CO)5 is removed from the data set, the MSD, MAD, and RMSD improved to 0.2, 1.7, and 2.8 kcal mol−1, respectively. As shown in Figure 1, the ccCA-derived and experimental ΔfH°gas values were well-correlated, as indicated by the R2 value of 0.997, a standard error in the y-estimate of 3.8 kcal mol−1, and a standard error in the y-intercept of 0.94 kcal mol−1. ΔfH°gas via Methods Employing GTO Basis Sets. Only the B97-1 functional produced a MAD below 10 kcal mol−1 when considering the entire test set with the best overall results corresponding to BS4, BS3, BS6, and BS7 having MADs of 8.3, 9.2, 9.4, and 9.9 kcal mol−1, respectively. Interestingly, Wilson and co-workers reported that the B97-1 functional produced a MAD of 8.0 kcal mol−1 at the B97-1/aug-cc-pVQZ//B97-1/ aug-cc-pVTZ level of theory for 25 complexes with coordinated bonds or organic ligands from the ccCA-TM/11 test set.36 They also reported a MAD of 6.9 kcal mol−1 for the B97-1 functional when paired with the cc-pVTZ basis set considering 19 TM species. The MAD was improved (3.1 kcal mol−1) when the size of the basis set was increased to cc-pVQZ.35 The next

a

BS1

Article

methods (64 DFT functionals, given in Table 1, utilizing BS8 and the temperature-dependent enthalpy corrections computed via B3LYP/BS3). These DFT-derived ΔrH°gas values were compared to ccCA-derived values. The GTO and STO results are shown in Tables S19−S24 and S25−S33, respectively. Summaries of the MADs for each GTO and STO method from ccCA are shown in Tables 7 and 8, respectively. E


Article

The Journal of Physical Chemistry A Table 5. Summary of MADs (kcal mol−1) for ΔfH°gas from ccCA for Each STO Method Listeda

a

GGA/BS8

MAD

meta-GGA/BS8

MAD

hybrid/BS8

MAD

revPBE RPBE HCTH/407 Becke88x+BR89c HCTH/93 OLYP OPBE XLYP HCTH/147 FT97 BLYP-D BLYP HCTH/120 BOP OPerdew KT2 mPW BP86-D BP86 mPBE PBE PBE-D PW91 KT1

8.5 8.7 9.0 10.1 11.0 11.3 12.4 13.2 13.7 13.9 14.3 14.3 14.6 15.0 18.8 28.6 28.6 32.4 32.4 35.2 46.7 46.7 48.1 56.2

KCIS BLAP3 VS98 PKZB TPSS-D TPSS BmTau1 Becke00 M06-L PKZBx-KCIScor KCIS-modified OLAP3 τ-HCTH mPBEKCIS

8.0 9.4 11.6 12.0 12.8 12.8 13.0 13.0 13.1 15.1 15.8 17.4 18.1 42.2

B97-1 B97 B97-2 B97-D B3LYP*(VWN5) M06 M05 PBE0 mPBE1KCIS mPBE0KCIS TPSSh τ-HCTH-hybrid B3LYP-D B3LYP(VWN5) O3LYP(VWN5) X3LYP(VWN5) mPW1PW B1PW91(VWN5) BHandH B1LYP(VWN5) M05-2X M06-2X KMLYP(VWN5) OPBE0 mPW1K BHandHLYP

5.5 10.1 10.3 11.3 14.5 15.4 16.6 18.5 20.0 20.8 21.0 21.2 22.8 23.2 25.9 26.4 28.3 32.2 33.8 41.5 43.2 48.1 52.0 55.2 65.8 90.8

All DFT/BS8(STO) computations utilized B3LYP/BS3 optimized geometries and temperature-dependent enthalpy corrections.

the hybrid functionals (B3LYP and BHandH) for the prediction of ΔfH°gas in the subset of small Cr-containing species, yet B97-1 provides the best performance.32 A comparison of the MSD, MAD, and RMSD for the hybrid functionals (B3LYP and BHandH) and pure functionals (BVP86 and PBEPBE) with regard to ΔfH°gas shows that the levels of theory utilizing pure functionals are more accurate. The hybrid functionals also display a bias toward underestimating ΔfH°gas, while the pure functionals display a bias for overestimating ΔfH°gas, as shown by the MSDs reported for each functional. The MADs of the computed ΔfH°gas for all DFT methods employing STO basis sets examined in this study are shown in Figure S3. The most accurate level of theory is B97-1/BS4 with a MAD of 8.3 kcal mol−1 compared to ccCA-derived values. Three other levels of theory are also competitive: B97-1/BS3, B97-1/BS6, and B97-1/BS7 with MADs of 9.2, 9.4, and 9.7 kcal mol−1, respectively. A plot of the ΔfH°gas computed via B97-1/ BS4 versus ccCA (Figure S4) demonstrates that these values are well-correlated, as shown by the R2 value of 0.994. ΔfH°gas via Methods Employing STO Basis Sets. Three GGA functionals (revPBE, RPBE, and HCTH/407), two metaGGA functionals (KCIS and BLAP3), and one hybrid functional (B97-1) produce MADs of less than 10.0 kcal mol−1 when considering the entire test set, with the lowest MAD (5.5 kcal mol−1) from the B97-1 functional. The differences between ΔfH°gas calculated via DFT methods utilizing STO basis sets and those calculated via ccCA for each species along with corresponding MSD, MAD, and RMSD statistics (Tables S7−S18) were compared. The metrics of energetic data for functionals utilizing the generalized gradient approximation are displayed in Figures S5

best levels of theory are BVP86/BS2 and BVP86/BS1 with MADs of 12.3 and 18.1 kcal mol−1, respectively. The least accurate level of theory is B3LYP/BS5 with MAD of 47.8 kcal mol−1. B3LYP results were unfortunately poor, considering this is arguably among the most widely used DFT functionals. Wilson, Cundari, and co-workers reported that hybrid functionals (which include various amounts of HF exchange) performed better in the determination of ΔfH°gas for the 19 TM species they examined.35 However, Jiang and Wilson have more recently shown that B3LYP exhibited the worst performance of the 13 functionals tested with correlation-consistent basis sets for the 34 Cr-containing species in their test set.36 Riley and Merz concluded that DFT methods used to calculate ΔfH°gas give errors for TM compounds much larger than those for organic ones, at least for the 6-31G** basis set.34 This conclusion proved true for each of the functionals tested in this work with the exception of the BHandH functional. This functional, perhaps fortuitously, provides MADs for the olefins larger than those for the organometallic complexes when paired with BS1 through BS7. The MSD, MAD, and RMSD for each level of theory shown in Tables S1−S6 are summarized as vertical bar charts; see Figure S1 for the hybrid functionals B3LYP, B97-1, and BHandH and Figure S2 for the pure functionals BVP86 and PBEPBE. For DFT methods paired with GTO-type basis sets, we find that MSDs vary from −47.8 to 37.8 kcal mol−1, MADs vary from 12.3 to 47.8 kcal mol−1, and RMSDs vary from 15.1 to 61.3 kcal mol−1. Furche and Perdew suggested that less-exact HF exchange is needed to describe systems with stronger correlation when determining ΔfH°gas. Therefore, it is interesting to note that the pure functionals (BVP86 and PBEPBE) in this study provide performance better than that of F


Article


The MADs of the computed ΔfH°gas for all DFT methods employing STO basis sets examined in this study are shown in Figure S10. The B97-1/BS8//B3LYP/BS3 level of theory is most accurate, exhibiting a MAD of 5.5 kcal mol−1 compared to ccCA-derived values. A plot of the ΔfH°gas computed via B971/BS8//B3LYP/BS3 versus ccCA (Figure S11) demonstrates that these values are well-correlated, as indicated by the R2 value of 0.995. Standard Enthalpy of Reaction. In the course of our development of photochromic organometallic complexes, we typically study isomerizations involving the reversible formation of two chelates via the breaking and formation of single bonds in the tethered bifunctional moiety of a polydentate ligand.1,4−6,8 Therefore, a comparison of reaction enthalpies (ΔrH°gas) may be a better metric than enthalpies of formation/ atomization energies for the determination of appropriate methods to model organometallic mechanisms and bond dissociation enthalpies in our lab. When computing reaction energies or reaction enthalpies, differences in the number of bonds and the nature of the bonding in products versus reactants may offset, thereby leading to cancellation of electronic structure errors. Wheeler et al. showed that, within the hierarchy framework of hydrocarbon reaction classes, the hyperhomodesmotic and homodesmotic reactions produced the least error for the closed-shell molecules examined.45 In other words, methodological dependencies are mostly eliminated via cancellation of error when products and reactants come from an increasingly similar chemical environment. A systematic hierarchy of reaction energies has not been derived for transition-metal species. However, the underlying principles pertaining to the number and type of bond in products and reactants as applied to closed-shell hydrocarbons holds true when the comparison of reaction enthalpies and bond dissociation enthalpies of Cr-containing species using ccCA and various DFT is made. The first step in the formation of these organometallic photochromic complexes is the photodissociation of a carbonyl ligand from the tricarbonyl parent, thereby allowing the coordination of a tethered ligand and the resultant formation of the desired chelate. This process is similar to the first photodissociation of CO from Cr(CO)6 and subsequent coordination of solvent or other ligand. The gas-phase BDE of CO from Cr(CO)6 was reported to be 36.8 ± 2 kcal mol−1 using laser pyrolysis by Smith and co-workers,156 and Fletcher and Rosenfeld have used time-resolved infrared laser absorption spectroscopy to estimate a BDE of 37.1 ± 5.0 kcal mol−1.157 The ccCA-derived ΔrH°gas for this reaction is 43.3 kcal mol−1, a difference of 6.5 kcal mol−1 from the experimental value with the smallest reported uncertainty. Experimental BDEs were reported for a series of Cr(CO)5ligand complexes [ligand = 1,1-dichloroethylene (1,1-C2H2Cl2), 1,1,2-trichloroethylene (C2HCl3), and 1,1,2,2-tetrachloroethylene (C2Cl4)] by Cedeño and co-workers via laser photoacoustic calorimetry (LPAC) in n-hexane.158 BDEs for these complexes were determined to be 14.3, 13.7, and 13.5 kcal mol−1, respectively. However, the authors did not utilize the correct quantum yield in the calculation of BDEs. The authors also erroneously assumed that displacement of solvent by olefin would result in no change in reaction volume. PAC results presented by Burkey and co-workers suggest that the reaction volume for displacement of heptane solvent by an olefin in Mo(CO)5-L is not zero.2 Lastly, the BDEs reported by Cedeño and co-workers lacked experimental uncertainties; therefore,

Table 6. Experimental and ccCA-Derived Gas-Phase Enthalpy of Reaction with MSD, MAD, and RMSD Displayed for the Set of Species Presented (kcal mol−1)a reaction

experiment

ccCA

Cr(CO)6 → Cr(CO)5 + CO Cr(CO)5(C2H4) → Cr(CO)5 + C2H4 Cr(CO)5(C2H3Cl) → Cr(CO)5 + C2H3Cl Cr(CO)5(C2H2Cl2) → Cr(CO)5 + C2H2Cl2b Cr(CO)5(C2H2Cl2) → Cr(CO)5 + C2H2Cl2c Cr(CO)5(C2H2Cl2) → Cr(CO)5 + C2H2Cl2d Cr(CO)5(C2HCl3) → Cr(CO)5 + C2HCl3 Cr(CO)5(C2Cl4) → Cr(CO)5 + C2Cl4 CrO2 → Cr + O2 CrF2 → Cr + F2 CrCl2 → Cr + Cl2 CrBr2 → Cr + Br2 CrO2Cl2 → Cr + O2 + Cl2 CrOCl2 → Cr + 1/2O2 + Cl2 CrOCl2 + F2 → CrOF2 + Cl2 CrO2F2 + Cl2 → CrO2Cl2 + F2 CrO2 + Br2 → CrBr2 + O2 CrO2 + F2 → CrF2 + O2 CrO2 + Cl2 → CrCl2 + O2 CrF2 + Cl2 → CrCl2 + F2 CrCl2 + Br2 → CrBr2 + Cl2 CrF2 + Br2 → CrBr2 + F2 CrCl2 + O2 + F2 → CrO2F2 + Cl2 CrF2 + O2 + Cl2 → CrO2Cl2 + F2

36.8 (3.0) 25.7 (1.0) n.d. n.d. 14.3 (1.0) n.d. 13.7 (0.5) 12.8 (1.6) 119.2 (2.2) 198.7 (4.0) 123.6 (1.4) 116.9 (5.3) 219.6 (2.0) 175.9 (4.6) −72.9 (8.8) 77.5 (6.0) 2.2 (5.5) −79.5 (4.2) −4.4 (1.6) 75.1 (3.4) 6.6 (4.7) 81.7 (7.3) −173.5 (5.4) −20.9 (4.0) MSD MAD RMSD

43.3 30.3 27.4 25.3 24.2 26.9 24.7 23.5 111.8 202.7 127.3 112.1 220.1 175.2 −78.0 78.0 −0.3 −90.9 −15.5 75.4 15.2 90.6 −170.8 −17.5 −0.6 4.9 6.3

a

Uncertainty values, when known, are given in parentheses; n.d. denotes no determined or known experimental value. bcis-Olefin. c Geminal-olefin. dtrans-Olefin.

Table 7. Summary of MADs (kcal mol−1) for ΔrH°gas from ccCA for Each GTO Method Listed B3LYP B97-1 BHandH BVP86 PBEPBE ωB97X-D

BS1

BS2

BS3

BS4

BS5

BS6

BS7

20.8 17.8 33.2 12.4 11.1 22.5

20.4 16.9 31.1 11.9 10.8 21.0

13.3 9.3 21.1 12.2 11.4 12.6

16.2 11.9 24.2 10.4 9.5 15.6

28.9 25.4 41.1 18.0 17.1 29.7

15.0 10.3 21.7 11.2 10.6 15.4

16.5 12.6 24.1 11.0 10.1 16.4

and S6. For DFT methods utilizing GGA functionals and STO basis sets, we find that the MSDs varied from −4.8 to 56.1 kcal mol−1, MADs varied from 8.5 to 56.2 kcal mol−1, and RMSDs varied from 11.8 to 69.2 kcal mol−1, as shown in Tables S8−S11. A similar comparison for functionals utilizing metageneralized gradient approximations and hybrid approximations was also performed. For DFT methods utilizing meta-GGA functionals and STO basis sets, we find that the MSDs vary from −13.2 to 42.0 kcal mol−1, MADs vary from 8.0 to 42.2 kcal mol−1, and RMSDs vary from 11.4 to 55.4 kcal mol−1. The metrics for meta-GGA functionals are summarized for each method in Figure S7. DFT methods utilizing hybrid functionals and STO basis sets exhibit MSDs, MADs, and RMSDs from −83.8 to 18.0, 5.5 to 90.8, and 7.4 to 124.1 kcal mol−1, respectively. The statistics for hybrid functionals are summarized in Figures S8 and S9. G


Article

The Journal of Physical Chemistry A Table 8. Summary of MADs (kcal mol−1) for ΔrH°gas from ccCA for Each STO Method Listed GGA/BS8

MAD

meta-GGA/BS8

MAD

hybrid/BS8

MAD

KT2 BP86-D PBE-D KT1 PW91 PBE HCTH/120 BLYP-D mPBE HCTH/147 BP86 mPW OPerdew RPBE revPBE Becke88x+BR89c HCTH/407 OPBE OLYP HCTH/93 BLYP XLYP BOP FT97

9.1 9.9 10.2 12.1 12.4 12.4 12.4 12.6 12.7 12.9 13.0 13.4 13.5 13.7 13.8 14.3 14.5 15.1 15.2 15.2 16.2 16.2 17.7 18.2

VS98 TPSS-D PKZB KCIS TPSS PKZBx-KCIScor Becke00 KCIS-modified M06-L τ-HCTH mPBEKCIS BLAP3 BmTau1 OLAP3

5.6 8.5 11.1 11.3 11.4 11.8 12.1 12.3 12.7 12.9 14.4 17.7 17.9 20.7

TPSSh B97-1 B3LYP-D M06 B97-D B97 B3LYP*(VWN5) M05 B97-2 mPBE1KCIS B3LYP(VWN5) X3LYP(VWN5) PBE0 O3LYP(VWN5) mPBE0KCIS mPW1PW B1LYP(VWN5) B1PW91(VWN5) BHandH OPBE0 M06-2X M05-2X mPW1K KMLYP(VWN5) BHandHLYP τ-HCTH-hybrid

7.8 8.9 9.5 9.5 10.1 10.4 10.6 10.8 11.8 11.9 13.1 13.2 14.8 15.9 16.4 16.6 16.7 17.6 22.1 24.0 26.6 27.4 27.9 29.8 32.3 220.8

respectively. It is not surprising that the gas-phase computed and solvated experimental results differ by 10 kcal mol−1 or more for aforementioned reasons. Moreover, our group recently performed a comparison of computed and experimental substitution enthalpies for a series of chromium arene tricarbonyl species in which a CO was substituted for a pyridyl ligand.1 The computed ΔrH°gas values were consistently greater (12 to 21 kcal mol−1) than the LPAC results in heptane and, in these experiments, the contributions of reaction volume, while recognized, were not determined. While the trends in the LPAC results were consistent with the reaction enthalpy trends, it was clear the absolute results were not an accurate representation of ΔrH°gas. It is necessary to consider the change in volume between reactants and products in the accurate determination of substitution enthalpies via LPAC. We concluded that the coordination of solvent plays an important role in the determination of the substitution enthalpy; therefore, ΔrH°gas cannot be compared directly with the LPAC experiments in the condensed phase. The aforementioned experimentally derived BDE of 13.5 kcal mol−1 for Cr(CO)5-(C2Cl4) determined directly from LPAC in n-hexane was in close agreement with a value of 12.8 ± 1.6 kcal mol−1 determined for gas phase BDE for this bond via transient infrared spectroscopy; however, this may be serendipitous.159 Gas-phase BDEs of 25.7 ± 1 and 24.7 ± 2.4 kcal mol−1 were reported for the Cr(CO)5-(C2H4) bond as identified via timeresolved infrared laser absorption spectroscopy by Grant and co-workers and Weitz and co-workers, respectively.160,161 We adopted the gas-phase values with the least experimental uncertainty, where available, for this study. A series of gas-phase decomposition and exchange reaction enthalpies were also calculated utilizing eight of the small Crcontaining species in the ΔfH°gas test set. These reactions are

Figure 1. Experimental ΔfH°gas versus ccCA-derived values.

Figure 2. Calculated values of ΔrH°gas derived via ccCA-derived versus experimentally determined values.

these values are used herein with reservation due to the paucity of experimentally determined BDEs for the organometallic complexes examined in this work. The ccCA-derived ΔrH°gas values for these reactions are 25.5, 25.8, and 24.3 kcal mol−1, H


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Figure 3. MADs from ccCA values (kcal mol−1) for ΔH°gas when all reactions are considered, computed via B3LYP, B97-1, BHandH, BVP86, PBEPBE, and ωB97X-D utilizing BS1−BS7.

Table 9. Summary of MADs (kcal mol−1) for ΔH°gas (All Reactions) from ccCA for Each GTO Method Listed B3LYP B97-1 BHandH BVP86 PBEPBE ωB97X-D

BS1

BS2

BS3

BS4

BS5

BS6

BS7

31.3 18.5 30.8 15.7 24.6 30.1

36.0 22.0 31.5 12.1 18.5 33.2

22.4 9.3 20.8 18.9 27.2 17.9

25.0 9.8 24.4 17.0 24.4 20.7

39.7 25.7 37.9 21.3 27.8 38.4

24.6 9.8 21.4 16.4 24.4 22.2

25.6 11.0 24.0 16.9 24.6 22.6

uncertainties in the ΔfH°gas were propagated to determine the total uncertainty for the ΔrH°gas. The average uncertainty for the test set of 16 reactions involving small Cr-containing species was 4.6 kcal mol−1 with the largest uncertainty (8.8 kcal mol−1) obtained from the substitution of Cl2 by F2 in CrOCl2. The ccCA-derived values of ΔrH°gas compare favorably with available experimental values for the reactions shown in Table 6, and the MSD, MAD, and RMSD for this set of reactions were −1.7, 5.0, and 6.6 kcal mol−1, respectively. Again, in the same manner as in Figure 1, the ccCA-derived versus experimental ΔrH°gas values shown in Figure 2 were wellcorrelated, as demonstrated by the R2 value of 0.995, a standard error in the y-estimate of 6.9 kcal mol−1, and a standard error in the y-intercept of 1.7 kcal mol−1

listed in Table 6 with the ccCA-derived and experimental ΔrH°gas. The experimental values for these reactions involving small Cr-containing species were calculated as the difference between the Δ f H° gas of products and reactants. The

Table 10. Summary of MADs (kcal mol−1) for ΔH°gas (All Reactions) from ccCA for Each STO Method Listed GGA/BS8

MAD

meta-GGA/BS8

MAD

hybrid/BS8

MAD

revPBE RPBE HCTH/407 Becke88x+BR89c HCTH/93 OLYP HCTH/147 OPBE BLYP-D HCTH/120 XLYP BLYP FT97 BOP OPerdew KT2 mPW BP86-D BP mPBE PBE-D PBE PW91 KT1

10.8 10.8 11.4 11.9 12.8 13.0 13.3 13.5 13.6 13.7 14.5 15.1 15.7 16.2 16.5 20.2 22.1 22.7 24.1 25.6 31.0 32.0 32.8 37.3

VS98 KCIS-original TPSS-D PKZB TPSS Becke00 BLAP3 M06-L PKZBx-KCIScor KCIS-modified BmTau1 TPSSh τ-HCTH mPBE1KCIS OLAP3 mPBEKCIS

9.0 9.4 10.9 11.6 12.2 12.6 12.9 12.9 13.7 14.3 15.1 15.3 15.9 16.5 18.8 30.3

B97-1 B97 B97-D B97-2 B3LYP*(VWN5) M06 M05 PBE0 B3LYP-D B3LYP(VWN5) mPBE0KCIS X3LYP(VWN5) O3LYP(VWN5) mPW1PW B1PW91(VWN5) BHandH B1LYP(VWN5) M05-2X M06-2X OPBE0 KMLYP(VWN5) mPW1K BHandHLYP τ-HCTH-hybrid

6.9 10.2 10.8 10.9 12.8 12.8 14.1 16.9 17.1 18.8 18.9 20.7 21.6 23.3 25.9 28.8 30.9 36.5 38.9 41.8 42.5 49.6 65.7 106.7

I


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Figure 4. MADs from ccCA values (kcal mol−1) for ΔH°gas of all reactions computed via methods employing the STO basis set (DFT/BS8// B3LYP/BS3). BHandHLYP and τ-HCTH-hybrid functionals were excluded due to poor performance.

functionals BVP86 and PBEPBE (Figure S13). The average MAD, calculated as the mean of the MADs for BS1−BS7 for each functional, shows that the pure functionals are more accurate on average than the hybrid functionals examined in this study. For example, the average MADs for the BVP86 and PBEPBE model chemistries are 11.5 and 12.4 kcal mol−1, respectively, compared to 18.7, 14.9, and 28.1 kcal mol−1 for the B3LYP, B97-1, and BHandH model chemistries, respectively. All model chemistries for DFT methods utilizing GTO basis sets display a bias toward overestimating the ΔrH°gas, as shown by a positive MSD. The MADs of all methods utilizing GTO basis sets employed to calculate ΔrH°gas are displayed in the vertical bar chart shown in Figure S14. B97-1/BS3 is the most accurate model chemistry that utilizes GTO basis sets and exhibits a MAD of 9.3 kcal mol−1 compared to ccCA-derived values. The accuracy of PBEPBE/BS4 with a MAD of 9.5 kcal mol−1 is also competitive. The MADs for all other model chemistries utilizing GTO basis sets range from 10.1 to 41.1 kcal mol−1. The PBEPBE/BS1 model chemistry is the most accurate of the model chemistries employing BS1 (containing the fewest number of basis functions) with a MAD of 11.1 kcal mol−1. For larger systems or in instances of limited computational resources, this model chemistry provides the best balance of efficiency and accuracy. The correlation between values of ΔrH°gas calculated via the B97-1/BS3 model chemistry versus ccCA values for the reactions displayed in Table 4 is shown in Figure S15. The R2 value of 0.988 shows that these data sets are well-correlated. ΔrH°gas via Methods Employing STO Basis Sets. ΔrH°gas values calculated via DFT methods utilizing STO basis sets were compared to those calculated via ccCA for each species, and the differences are shown along with statistical deviations in Tables S25−S33. We find that the MSDs vary from −2.9 to 11.4 kcal mol−1, MADs vary from 9.1 to 18.2 kcal mol−1, and RMSD vary from 11.1 to 21.3 kcal mol−1. The statistics for functionals utilizing the generalized gradient approximation are displayed in Figures S16 and S17.

Table 11. Best Performing Methods for the Calculation of ΔfH°gas, ΔrH°gas, and ΔH°gas (All Reactions) STOa

GTO ΔfH°gas B97-1/BS4 B97-1/BS3 B97-1/BS6 B97-1/BS7

ΔrH°gas B97-1/BS3 PBEPBE/BS4

all reactions B97-1/BS3 B97-1/BS4 B97-1/BS6

MAD (kcal mol−1) 8.3 9.2 9.4 9.7

MAD (kcal mol−1) 9.3 9.5

MAD (kcal mol−1) 9.1 9.8 9.8

ΔfH°gas B97-1/BS8 KCIS/BS8 revPBE/BS8 HCTH(407)/ BS8 BLAP3/BS8 ΔrH°gas VS98/BS8 TPSSh/BS8 TPSS-D/BS8 B97-1/BS8 KT2/BS8 B3LYP-D/BS8 M06/BS8 BP86-D/BS8 all reactions B97-1/BS8 VS98/BS8 KCIS/BS8

MAD (kcal mol−1) 5.5 8.0 8.5 9.0 9.4 MAD (kcal mol−1) 5.6 7.8 8.5 8.9 9.1 9.5 9.5 9.9 MAD (kcal mol−1) 6.9 9.0 9.4

a

All DFT/BS8(STO) computations utilized B3LYP/BS3 optimized geometries and temperature-dependent enthalpy corrections.

ΔrH°gas via Methods Employing GTO Basis Sets. ΔrH°gas was calculated via DFT methods utilizing GTO basis sets and ccCA for each reaction shown in Table 6. The differences between DFT- and ccCA-derived values are shown in Tables S19−S24. We find that the MSDs vary from 4.4 to 18.9 kcal mol−1, MADs vary from 9.3 to 41.1 kcal mol−1, and RMSDs vary from 11.7 to 61.3 kcal mol−1. These statistics are summarized as vertical bar charts for the hybrid functionals B3LYP, B97-1, and BHandH (Figure S12) and for the pure J


Article


The best performing STO methods (MAD < 10 kcal mol−1) are summarized in Table 10 when all reactions are considered (ΔfH°gas and ΔrH°gas). The MADs of the GGA, meta-GGA, and hybrid functionals utilizing STO basis sets examined in this study are shown in Figure 4; BHandHLYP and tau-HCTHhybrid were excluded from these figures due to poor performance (MADs of 65.7 and 106.7 kcal mol−1). The B97-1/BS8//B3LYP/BS3 level of theory exhibited the lowest MAD (6.9 kcal mol−1) of all STO methods investigated herein, and interestingly, the other functionals in the B97 family exhibited MADs from 10.2 to 10.9 kcal mol−1. The VS98 and KCIS functionals, similar to that of B97-1, exhibited MADs less than 10 kcal mol−1 but larger than 6.9 kcal mol−1. Conclusions and Summary. The performances of 6 functionals utilizing 7 GTO basis sets and 69 functionals utilizing an STO basis set were evaluated based on calculation of the standard gas-phase reaction enthalpy for a variety of reactions at 298.15 K via comparison to ccCA-derived values. These reactions include formation, exchange, and ligand dissociation reactions of Cr-containing species with organic ligands. A summary of the best performing methods for the calculation of ΔfH°gas, ΔrH°gas, and overall thermochemistry is shown in Table 11. When considering GTO basis sets, the B97-1/BS4 level of theory (MAD = 8.3 kcal mol−1) performs best for the calculation of ΔfH°gas. However, BS4 employs the largest number of basis functions of the GTO basis sets examined. The B97-1 functional with BS3, BS6, and BS7 were also competitive, and the accuracy of the B97-1 functional increases directly as the number of basis functions increases. The STO basis set (BS8) is the largest basis set employed in this study, and the B97-1/BS8 level of theory (MAD = 5.5 kcal mol−1) is also quite reliable for determining ΔfH°gas of Cr-containing molecules. Several other functionals utilizing BS8 are competitive for the calculation of ΔfH°gas, including KCIS, revPBE, HCTH(407), and BLAP3. For calculation of ΔrH°gas, the B97-1/BS3 level of theory (MAD = 9.1 kcal mol−1) is promising, while PBEPBE/BS4 obtained similar accuracy. No other functional/basis set combination employing a GTO basis set for the determination of ΔrH°gas achieved a MAD of less than 10 kcal mol−1. It is interesting to note that PBEPBE obtained reasonable accuracy (MAD of 11.1 kcal mol−1) when paired with the smallest basis set (BS1); therefore, this level of theory may be acceptable if computational cost is a concern. In contrast, B97-1/BS1 is much less accurate (MAD of 18.1 kcal mol−1). Of the methods utilizing an STO basis set (BS8), the VS98 functional performed best (MAD of 5.6 kcal mol−1) for the calculation of ΔrH°gas. Several other functionals employing BS8 are competitive with MADs of less than 10 kcal mol−1. These functionals include TPSSh, TPSS-D, B97-1, KT2, B3LYP-D, M06, and BP86-D. When all reactions of Cr-containing compounds are considered, the B97-1/BS3 level of theory is optimal (MAD = 9.1 kcal mol−1) for GTO methods with a similar accuracy achieved by B97-1/BS4 and B97-1/BS6. No other level of theory utilizing a GTO basis set achieved a MAD of less than 10 kcal mol−1 overall. For STO methods employing BS8 (and B3LYP/BS3 optimized geometries), the B97-1 functional has the best accuracy (MAD = 6.9 kcal mol−1). The VS98 and KCIS functionals are competitive with MADs of 9.0 and 9.4 kcal mol−1, respectively. Our laboratories will be using the most accurate levels of theory shown here to explore ligand design

A similar comparison for functionals employing metageneralized gradient approximations and hybrid approximations was also performed. For DFT methods utilizing meta-GGA functionals and STO basis sets, we find that the MSDs vary from −0.2 to 17.0 kcal mol−1, MADs vary from 5.6 to 20.7 kcal mol−1, and RMSDs vary from 7.1 to 25.8 kcal mol−1. The statistics for meta-GGA functionals are summarized for each method as vertical bar charts in Figure S18. DFT methods utilizing hybrid functionals and STO basis sets, excluding τ-HCTH-hybrid/BS8//B3LYP/BS3, exhibited MSDs that vary from 2.0 to 16.6 kcal mol−1, MADs that vary from 5.5 to 32.3 kcal mol−1, and RMSDs that vary from 7.8 to 43.8 kcal mol−1. The statistics for hybrid functionals are summarized in Figures S19 and S20. A chart of the MADs for all DFT methods utilizing BS8 is provided for ease of comparison (Figure S21). ΔrH°gas computed via VS98/BS8//B3LYP/BS3 was the most accurate model chemistry of the DFT methods employing STO basis sets examined in this study and exhibited a MAD of 5.6 kcal mol−1 compared to ccCA-derived values. Seven other competitive model chemistries were identified with MSDs of less than 10.0 kcal mol−1. TPSSh/BS8//B3LYP/BS3, TPSS-D/ BS8//B3LYP/BS3, B97-1/BS8//B3LYP/BS3, KT2/BS8// B3LYP/BS3, B3LYP-D/BS8//B3LYP/BS3, M06/BS8// B3LYP/BS3, and BP86-D/BS8//B3LYP/BS3 displayed MADs of 7.8, 8.5, 8.9, 9.1, 9.5, 9.5, and 9.9 kcal mol−1, respectively. A plot of the ccCA-derived ΔrH°gas values versus VS98/BS8//B3LYP/BS3 is shown in Figure S22. Again, the R2 value of 0.994 indicates that these data sets are well-correlated. Overall Method Performance When All Reactions Are Considered. As discussed previously, the enthalpy of formation is a particularly difficult property to calculate accurately. However, ΔfH°gas is fundamentally a bond forming reaction. Therefore, we evaluate the performance of each method employed in this study by considering all reactions, including ΔfH°gas and ΔrH°gas, in a single data set. The best performing GTO methods (MAD < 10 kcal mol−1) are summarized in Table 5 when all reactions are considered. The MADs of the six functionals utilizing GTO basis sets examined in this study are shown in Figure 3. The B97-1/BS3 level of theory exhibited the lowest MAD (9.1 kcal mol−1) of all GTO methods investigated. The B97-1/BS4 and B97-1/BS6 levels of theory also exhibited similar MADs of 9.8 and 9.8 kcal mol−1, respectively. When averaging MADs of all basis sets from a specific functional, the average MADs of B97-1/BS1−BS7 and BVP86/BS1−BS7 are 15.2 and 16.9 kcal mol−1, respectively. The basis set type-averaged MADs of the other functionals examined with the suite of GTOs are all greater than 20 kcal mol−1. BS3 consistently achieved the best results for the hybrid functionals B3LYP, B97-1, and BHandH, while BS2 provided the best results for the pure functionals BVP86 and PBEPBE. By way of comparison, the MADs for B97-1/BS3 and BVP86/ BS2 are 9.8 and 12.1 kcal mol−1. While no direct correlation exists between basis set size and accuracy for a given functional, in general, hybrid functionals perform better with larger basis sets, and the pure functionals perform better with smaller basis sets. For example, a qualitative basis set dependency is shown for the best performing functional (B97-1) in Table 9, where BS3, BS4, and BS6 are more accurate than BS1 and BS2. As expected, the basis set employing a nonrelativistic effective core potential (BS5) consistently provided the worst results for any given functional, as shown in Figure 3. K


Article


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and catalysis of Cr-containing compounds. When these levels of theory are used to elucidate proposed mechanisms where the chemical environment is not changing drastically, cancellation of errors will improve the accuracy tremendously. The calibration of methodologies increases confidence in the results from investigations of novel transition-metal chemistry.

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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.jpca.6b09278. Supporting tabulated data and illustrative figures are provided in Tables S1−S35 and Figures S1−S22 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Charles Edwin Webster: 0000-0002-6917-2957 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by the National Science Foundation under grants CHE 0911528 and OIA 1539035 and the Mississippi State University Office of Research and Economic Development. Computational work was performed using resources at the University of Memphis High-Performance Computing Facility and the Computational Research on Materials Institute at the University of Memphis (CROMIUM).

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REFERENCES

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