Article pubs.acs.org/IC
Accurate Gas Phase Formation Enthalpies of Alloys and Refractories Decomposition Products Yury Minenkov,*,† Valery V. Sliznev,‡ and Luigi Cavallo*,† †
Physical Sciences and Engineering Division, King Abdullah University of Science and Technology, KAUST Catalysis Center, Thuwal 23955-6900, Saudi Arabia ‡ Ivanovo State University of Chemistry and Technology, Research Institute for Thermodynamics and Kinetics of Chemical Processes, 153460 Ivanovo, Russian Federation S Supporting Information *
ABSTRACT: Accurate gas phase formation enthalpies, ΔHf, of metal oxides and halides are critical for the prediction of the stability of high temperature materials used in the aerospace and nuclear industries. Unfortunately, the experimental ΔHf values of these compounds in the most used databases, such as the NIST-JANAF database, are often reported with large inaccuracy, while some other ΔHf values clearly differ from the value predicted by CCSD(T) methods. To address this point, in this work we systematically predicted the ΔHf values of a series of these compounds having a group 4, 6, or 14 metal. The ΔHf values in question were derived within a composite Feller− Dixon−Peterson (FDP) scheme based protocol that combines the DLPNO−CCSD(T) enthalpy of ad hoc designed reactions and the experimental ΔHf values of few reference complexes. In agreement with other theoretical studies, we predict the ΔHf values for TiOCl2, TiOF2, GeF2, and SnF4 to be significantly different from the values tabulated in NIST-JANAF and other sources, which suggests that the tabulated experimental values are inaccurate. Similarly, the predicted ΔHf values for HfCl2, HfBr2, HfI2, MoOF4, MoCl6, WOF4, WOCl4, GeO2, SnO2, PbBr4, PbI4, and PbO2 also clearly differ from the tabulated experimental values, again suggesting large inaccuracy in the experimental values. In the case when largely different experimental values are available, we point to the value that is in better agreement with our results. We expect the ΔHf values reported in this work to be quite accurate, and thus, they might be used in thermodynamic calculations, because the effects from core correlation, relativistic effects, and basis set incompleteness were included in the DLPNO−CCSD(T) calculations. T1 and T2 values were thoroughly monitored as indicators of the quality of the reference Hartree−Fock orbitals (T1) and potential multireference character of the systems (T2).
1. INTRODUCTION Fundamental thermodynamic characteristics, such as formation enthalpies, entropies, and heat capacities for both condensed and gas phases, are the building blocks for the scientific prediction of the stability of high temperature materials.1,2 Above 1000 °C chemical equilibrium is readily reached, reducing the role of kinetics factors, and due to experimental difficulties in carrying out measurements in such conditions, thermodynamic modeling is sometimes the only source of information on chemical reactivity at high temperatures. Traditionally, there are two major industries interested in this kind of information, which can be used in the design of high temperature materials: aerospace engineering,3−5 where the main source of heat is the gas streams (either directly from the engine or from air in supersonic flights), and nuclear power engineering, where the heat is produced in the bulk of the nuclear fuel.6−8 Alloys and refractories of Ti, Zr, Mo, W, Sn, Ge, and Pb, or even the pure metals, are often used in these applications,9 and thermodynamic functions of the possible © XXXX American Chemical Society
products of these materials after corrosion and/or decompositions are required. Needless to say, in these areas, thermodynamic data are of particular importance and demand an exceptional degree of reliability. Indeed, since the 1950s highly accurate thermodynamic functions with uncertainties of only about a fraction of a kcal/ mol were experimentally measured for many transition and heavy metal complexes to satisfy the growing demand in the space and nuclear industries. The results have been tabulated in many (online) thermochemical databases,10,11 with the NISTJANAF tables12 and Glushko/Gurvich13,14 and Krasnov15 handbooks being the most important in the field. It is not rare, however, that high quality gas phase thermodynamic data are either still missing in the literature or have larger uncertainties; exemplary cases are MoCl6, MoOF4, and WOF4, with experimental uncertainties of 20, 30, and 15 Received: October 12, 2016
A
DOI: 10.1021/acs.inorgchem.6b02441 Inorg. Chem. XXXX, XXX, XXX−XXX
Article
Inorganic Chemistry
main peculiarities used in the present work. First, in our approach we do not calculate atomization energies but rather use reaction enthalpies as in studies of Bross and Peterson40 and Grant et al.41 The reason is that formation enthalpies from atomization energies might contain some error because the electron correlation energy in atoms and molecules is very different and no error compensation can be expected, since practically affordable correlated electronic-structure methods such as CCSD(T) capture at best 99% of the correlation energy.42 Thus, highly accurate atomization energies can be calculated only for very small molecules containing few bonds. In addition, as was demonstrated in the work of Grant et al.,41 the FPD protocol applied on isodesmic reactions might account for some potential multireference issues. Second, even though the FPD formulation includes the complete basis set (CBS) extrapolation of the core−valence correlation when such calculations are feasible,39 in many practical FPD calculations, valence and core−valence contributions to the CCSD(T) energy are usually calculated separately, with complete basis set (CBS) extrapolation calculated only on the valence contribution, while the core−valence contribution is evaluated with a modest basis set (often at triple-ζ quality) to reduce the computational cost. In contrast, in this work we extrapolated both valence and core valence correlation energy to the CBS limit, and we used correlation consistent polarized core valence cc-pwCVnZ basis sets43−53 which should result in better accuracy. Finally, using the DLPNO−CCSD(T) method allowed us to study relatively large molecules containing heavy elements with core−valence correlation included, and quadruple-ζ basis sets for which canonical CCSD(T) calculations would require too large computational costs. However, we are aware that the approach we used relies on available accurate formation enthalpies of some transition metal complexes used as reference, whereas the FPD approach based on atomization energies is capable of providing accurate formation enthalpies using a minimal amount of empirical data (atomic formation enthalpies) and is free from systematic errors in the experimental formation enthalpies. For this reason, we carefully choose these reference formation enthalpies among those for which experimental values with small uncertainty are in good agreement with theoretical predictions from very accurate methods. Another downside of our approach is that its application is only limited to the singlet ground states until the open-shell version of the DLPNO−CCSD(T) method will appear.
kcal/mol. Finally, it is not uncommon that different values for the formation enthalpies of the identical species are tabulated in different sources; an exemplary case is the formation enthalpy of WOCl4, which is −137.1 ± 5 kcal/mol according to NISTJANAF,12 while in the handbook of Krasnov and co-workers15 this value is set to −151.3 kcal/mol. Combined with the experimental complexity of gas phase thermochemistry measurements, highly accurate quantum mechanics calculations are increasing their role in the prediction of these fundamental thermochemical data, with an accuracy approaching 1−2 kcal/mol16,17 in the absence of strong multireference and spin−orbit effects. In this respect, it is important to remark that benchmark studies have shown that high level wave function theory methods, such as those of the coupled cluster method with single, double, and perturbative triple excitations, CCSD(T),18 in connection with a large basis set, are known to provide the most accurate reaction enthalpies,16,19 while density functional theory (DFT) methods are less accurate or can result in unpredictable poor performance in specific cases, at least for some functionals.20−27 The possibility to accurately calculate formation enthalpies with quantum mechanics methods, essentially using the composite Feller−Peterson−Dixon (FPD) approach based on the calculations of atomization energies,16,17,19,28 evidenced large discrepancies between the calculated and the measured formation enthalpies. For example, the calculated formation enthalpies of PbF4 and PbCl4 turned out to be lower by ca. 70− 100 kcal/mol29 compared to the experimental estimates given in the NIST-JANAF database, suggesting the experimental values to be severely wrong.12,29 Given this background, we decided to calculate poorly defined experimental formation enthalpies in the NIST-JANAF database. We focused on closed-shell complexes of Ti, Zr, Mo, W, Ge, Sn, and Pb, since the underlying CCSD(T) method employing the domain based local pair natural orbital approximation, DLPNO,30,31 was validated to reproduce highly accurate experimental reaction enthalpies for some of these metals with a mean absolute error of 2−3 kcal/mol.20,32−36 For these metals, we calculated the enthalpies of a set of 56 (mostly) isogyric reactions, which are reactions in which the same number of electrons pairs is preserved on both sides.37,38 In this approach, the chemical reactions are built in such a way that the formation enthalpy of one species, for example A in eq 1, is known with poor accuracy, while the formation enthalpies for all the other species, B, C, and D in eq 1, are known with great accuracy. At this point, the formation enthalpy of species A, ΔHf(A), is calculated via ΔHf of B, C, D and the reaction enthalpy of eq 1. A + B → C + D,
ΔHr(298)
2. COMPUTATIONAL DETAILS 2.1. Geometry Optimization. All geometry optimizations were performed with the GGA PBE54,55 functional and all electron quadruple-ζ quality basis sets λ356 as implemented in PRIRODA 13 suite of programs.57 Scalar relativistic effects (for Pb, Mo, W, Zr, Br, I) were taken into account via the Dyall Hamiltonian.58 The default adaptively generated PRIRODA grid, corresponding to an accuracy of the exchange-correlation energy per atom (1 × 10−8 hartree), was decreased by a factor of 100 for more accurate evaluation of the exchange-correlation energy. Default values were used for the selfconsistent-field (SCF) convergence and the maximum gradient for geometry optimization criterion (1 × 10−4 au), whereas the maximum displacement geometry convergence criterion was decreased to 0.0018 au. Geometries were characterized as true energy minima by the eigenvalues of the analytically calculated Hessian matrix. Translational, rotational, and vibrational partition functions for thermal corrections to arrive at total enthalpies were computed within the ideal gas, rigidrotor, and harmonic oscillator approximations. The temperature used
(1)
Due to extensive application of these metals in high temperature materials used in aerospace and nuclear industries, we believe that, apart from pure fundamental interest, the obtained gas thermodynamic functions of these metal oxides and halides might be important to predict stability at high temperatures and in aggressive atmospheres. We remark that the approach used in this work to calculate formation enthalpies is within the scheme of the general composite FPD approach.16,17,19,28,39 However, we expect it offers some practical advantages due to the replacement of the demanding CCSD(T) methods in the classical FPD scheme with very efficient DLPNO−CCSD(T) calculations. Since the FPD approach is not a fixed prescription, we introduce the B
DOI: 10.1021/acs.inorgchem.6b02441 Inorg. Chem. XXXX, XXX, XXX−XXX
Article
Inorganic Chemistry in the calculations of thermochemical corrections was set to 298.15 K in all the cases. 2.2. Single-Point Energy Evaluations. All single-point energy evaluations were performed with the DLPNO−CCSD(T) method30,31,59 as implemented in the ORCA suite of programs.60 For atoms heavier than Ti, the scalar relativistic effects were included via fully relativistic Stuttgart-type pseudopotentials.45,49,61,62 Since we found that core-correlation effects can be important for reactions involving transition metals species,20 all non-ECP electrons were included in the correlation treatment (“FC_NONE” ORCA option). The very tight PNO settings (TCutPairs = 10−5, TCutPNO = 1 × 10−7, TCutMKN = 10−3) were used to reduce any numerical noise in the calculations (for the core electrons, TCutPNO = 1 × 10−9). These settings should result in 1−2 kcal/mol deviation from the canonical CCSD(T) results.36 And indeed, the mean absolute error (MUE) and mean signed error (MSE) between canonical CCSD(T) and DLPNO− CCSD(T) absolute energies calculated with TZ quality basis sets for a selected set of 62 molecules studied in the present work turned out to be 2.3/-2.3 kcal/mol; see Table S1. However, even smaller MUE/MSE = 0.8/−0.2 kcal/mol between canonical CCSD(T) and DLPNO− CCSD(T) have been obtained when 45 reaction energies, built out of the selected 62 molecules and used in this work to predict the formation enthalpies, were compared, due to error cancellation; see Table S2. In fact, the deviation larger than 2.0 kcal/mol has been obtained only for 3 out of 45 reactions, validating the DLPNO− CCSD(T) method to calculate reliable reaction energies. The default SCF convergence criterion NormalSCF (Energy change 1 × 10−6 au) was replaced with tighter VeryTightSCF (energy change 1 × 10−9 au) to achieve a better converged wave function. The following triple and quadruple-ζ correlation consistent basis sets were used in the present work. Hydrogen was described with the cc-pVnZ basis sets of Dunning.63 Oxygen, fluorine, and chlorine were described with the all electron correlation consistent polarized core valence cc-pwCVnZ basis sets of Peterson and Dunning.44 Titanium was described with the all electron correlation consistent polarized core valence cc-pwCVnZ basis set of Balabanov and Peterson.47 Molybdenum, zirconium, hafnium, tungsten, bromine, iodine, lead, germanium, and tin were described with the pseudopotential based correlation consistent polarized core valence cc-pwCVnZ-PP basis sets of Peterson and co-workers.50−52 The 10 core electrons of Ge and Br, the 28 core electrons of Zr, Mo, I, and Sn, and the 60 core electrons of Hf, W, and Pb were described with Stuttgart-type fully relativistic effective core potentials.45,49,61,62 The correlation fitting basis sets def2qzvpp/C developed by Hättig,64,65 necessary for the resolution of the identity approximation as a part of DLPNO scheme, were used. All def2/C basis sets were downloaded from the official web page of Turbomole.66 To eliminate the effects of basis set incompleteness, we used the extrapolation schemes for HF and correlation energies of individual species suggested by Helgaker;67−69 see eqs 2 and 3. For two adjacent triple and quadruple-ζ basis sets: n ∞ EHF = EHF + αe−1.63n
(2)
n ∞ Ecorl = Ecorl + βn −3
(3)
In addition to the main protocol introduced above, few additional calculations with different basis sets and/or scalar relativistic methods have been performed. First, to explore the effect of diffuse functions, all reactions involving Hf complexes have been recalculated with augcc-pwCVnZ(-PP) basis sets. Second, to investigate errors related to using relativistic pseudopotentials on an Hf atom incorporating 4f electrons, which can mix with the valence orbitals of O and F, as discussed by Gong et al.,70 we performed tests of all electron calculations with the scalar relativistic Douglas−Kroll−Hess Hamiltonian and all electron relativistically contracted Sapporo basis sets.71 In addition, the formation enthalpies involving Ti have also been recalculated with the all electron scalar relativistic DKH Hamiltonian using the relativistically contracted basis sets cc-pwCVnZ-DK47 on Ti, cc-pCVnZ-DK63,72,73 on O, F, and Cl, and cc-pVnZ-DK on H63,72 to estimate the sensitivity of the predicted formation enthalpies to the relativistic effects. 2.3. Choice of Reference Formation Enthalpies. To compose the thermochemical equations to evaluate the formation enthalpy in question, the reference transition metal complexes have been carefully selected, since the experimental inaccuracy will affect the overall accuracy of the predicted formation enthalpies. For this reason, we only selected enthalpies that (a) have narrow experimental uncertainty (
