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Thermodynamics of Titanium and Vanadium Reduction in NonAqueous Environment Calculated at Various Levels of Theory Zygmunt Flisak Faculty of Chemistry, University of Opole, Oleska 48, 45-052 Opole, Poland S Supporting Information *

ABSTRACT: Reduction of titanium and vanadium compounds is a process accompanying the activation of coordinative olefin polymerization catalysts. Four density functional theory (DFT) functionals, coupled cluster with single, double, and perturbative triple excitations method CCSD(T) as well as complete active-space second-order perturbation theory method CASPT2 with a complete activespace self-consistent field CASSCF reference wave function were applied to investigate the thermodynamics of titanium and vanadium reduction. The performance of these theoretical methods was assessed and compared with experimental values. The calculations indicate that vanadium(IV) chloride is more easily reduced by trimethylaluminum than the corresponding titanium compound; the energies of reaction calculated at the CCSD(T) level are equal −57.21 and −33.10 kcal/mol, respectively. The calculations deal with the redox reactions of metal chlorides in the gas phase, rather than solvated ions in the aqueous solution. This approach may be more appropriate for olefin polymerization, usually carried out in nonpolar solvents.

2. INTRODUCTION Redox reactions of the group-4 and group-5 metals play an important role in modern chemistry and chemical industry. The reactions involving changes in oxidation states, apart from being aesthetically pleasing (cf. colors of inorganic compounds containing vanadium at different oxidation states), are also vital in industrial processes. These include (but are not limited to), for example, corrosion processes, electrochemical energy generation and storage, and catalysis, including activation of small molecules and olefin polymerization. Within the latter, the question of the oxidation state in the catalytic active site has pervaded the academia and industry for many decades. As a result of extensive studies in this area, it is generally assumed that the classical Ziegler−Natta systems exhibit catalytic activity due to the presence of titanium(III) generated by the action of a cocatalyst on the precursor,1,2 whereas soluble metallocene and postmetallocene systems are composed of the ion pairs with the cations containing titanium(IV) or zirconium(IV) species.3,4 Based on these experimental findings, a number of theoretical works dealing with both heterogeneous5−11 and homogeneous catalysts9,12−18 have been published within recent five years. However, redox chemistry of early transition metals in coordinative polymerization is dealt with less often in theoretical works. Whereas changes in the oxidation state of titanium have already been studied and are relatively well understood (see, e.g., refs 5 and 19−22), still comparatively little is known about the vanadium oxidation states in olefin polymerization catalysts. This is presumably due to the limited number of experimental studies devoted to such systems and © 2012 American Chemical Society

more complex redox chemistry of this element. In this area, only periodic trends in polymerization over catalysts based on first-row transition metals (Ti−Mn and Sc−Co, respectively) were studied theoretically.23,24 Significant efforts have been undertaken experimentally to protect titanium and vanadium active sites against excessive reduction to form inactive low-valent species in the medium of coordinative olefin polymerization.25,26 One may speculate that vanadium catalysts, usually less frequently studied and utilized than the titanium counterparts, pose more serious problems related to inactivation via reduction,27,28 because only a trace of actual catalytically active species is present due to their low stability.29 Theoretical studies devoted to transition metal redox processes concentrate mainly on the correct reproduction of experimental standard potentials for ions in aqueous solution, either by the design of the correct models that describe this phenomenon (including the number of solvating molecules)30 or by development of computational methodology (e.g., DFT +U approach).31,32 Although useful for predicting the operation of electrochemical cells, the results of these studies can provide only a quantitative reference for the processes that occur in nonaqueous environment, typical for the low pressure olefin polymerization. The purpose of this work is to benchmark various theoretical methods against experimental data related to titanium(IV) and Received: December 5, 2011 Revised: January 13, 2012 Published: January 20, 2012 1464

dx.doi.org/10.1021/jp211690y | J. Phys. Chem. A 2012, 116, 1464−1468

The Journal of Physical Chemistry A

Article

hydrogen as well as hypothetical intramolecular redox reactions proceeding in the gas phase according to the following equations: 1 MCl 4 + H2 → MCl3 + HCl (1) 2

titanium(III) reduction in gas phase and then apply selected methods for analogous vanadium(IV) and vanadium(III) processes, where no quantitative experimental data are available.

3. COMPUTATIONAL DETAILS DFT calculations were carried out by using the ADF 2010.01 program33−37 with the four selected density functionals, i.e., one generalized gradient approximation functional (GGA): BP86,38−40 one meta-GGA: TPSS,41,42 one hybrid-GGA: B3LYP,43−45 and one meta-hybrid: M06.46,47 Valence triple-ζ Slater-type orbital basis set with no frozen core approximation (denoted as TZP) was applied to all atoms. The molecular density and the Coulomb and exchange potentials were fitted with an auxiliary s, p, d, f, and g set of Slater-type orbital functions corresponding to valence quadruple-ζ with four sets of polarization functions, centered on each nucleus. For the exchange potential, the true density was used. In most of the cases, no symmetry constraints were applied; however, they were occasionally used to help convergence. The geometry convergence criteria were 1.0 × 10−5 for energy and gradients. The integration parameter was set to 7.0. Linear dependency of the function sets was controlled with the DEPENDENCY keyword for hybrid and meta-hybrid functionals. Analytical frequencies were calculated in the case of BP86 functional. For CCSD(T) calculations, the MOLPRO package was employed.48−53 These calculations were done with the aug-ccpVnZ basis set for hydrogen,54 aug-cc-pV(n+d)Z for chlorine55 and the small-core effective core potential based aug-cc-pVnZPP basis set for titanium and vanadium,56−58 where n = D and T; this combination of basis sets will be referred to as aVnZ, see ref 59. No core−valence correlation corrections were calculated. The CCSD(T) calculations for the open shell species were based on restricted open shell Hartree−Fock (ROHF) wave function due to program limitations. Full geometry optimization was carried out for all structures with the exception of Cl2Ti(μ2Cl)2Al(CH3)2 and Cl2V(μ2Cl)2Al(CH3)2, where only a single point calculations on geometries optimized with the BP86 density functional were performed. CASSCF and CASPT2 energy calculations were performed with the Molcas version 7.6 suite of programs60−62 on the structures optimized at the CCSD(T) level with the aVTZ basis set. Valence triple-ζ atomic natural orbital basis set with a polarization functions (ANO-L-VTZP)63−65 was used for all elements.

MCl3 +

1 H2 → MCl2 + HCl 2

MCl 4 → MCl3 +

(2)

1 Cl2 2

(3)

1 Cl2 (4) 2 where M stands for titanium or vanadium. Additionally, the most plausible reaction of metal tetrachloride reduction with trimethylaluminum, as suggested in ref 22, together with the decomposition of the product to yield metal(III) chloride were taken into account: 1 MCl 4 + Al(CH3)3 → Cl2M(μ 2Cl)2 Al(CH3)2 + C2H6 2 MCl3 → MCl2 +

(5)

Cl2M(μ 2Cl)2 Al(CH3)2 → MCl3 + Al(CH3)2 Cl

(6)

4.1. Reduction of Metal Chlorides with Hydrogen. Experimental gas phase heats of reactions 1−4 (Table 1) were Table 1. Energy of Titanium(IV) and Titanium(III) Chloride Reduction with Hydrogen and the Intramolecular Redox Process, kcal/mol reaction no.

a

method

1

2

3

4

BP86/TZP TPSS/TZP M06/TZP B3LYP/TZP CASPT2/ANO-L-VTZP CCSD(T)/aVDZ CCSD(T)/aVTZ experiment

43.95 44.86 46.06 33.61 48.05 34.71 35.13 31.44

56.01 57.36 61.46 49.63 a 49.89 50.68 50.14

65.63 65.09 67.57 56.13 72.05 61.29 58.94 53.50

77.69 77.59 82.97 72.15 a 76.47 74.49 72.20

Not calculated.

calculated by means of Hess’s law from the corresponding heats of formation at standard conditions published in ref 67 and made available on the National Institute of Standards and Technology (NIST). By NIST definition, standard conditions are T = 293.15 K and p = 101.325 kPa; therefore, no extrapolation to 298.15 K was performed. The data presented in Table 1 indicate that the CCSD(T) results approach the experimental values for the reactions of titanium compounds. Gas phase heats of reactions 1−4 are all positive and the CCSD(T) calculations overestimate them by a few kcal/mol. It turns out that improving the basis set from n = D to n = T only slightly deteriorates the results for reactions 1 and 2, whereas it significantly improves the accuracy of calculations for reactions 3 and 4. Such a behavior could be attributed to favorable error cancellation for n = D caused by higher errors in a less complete and accurate basis set. It should be stressed that the CCSD(T) calculation can provide a benchmark for the reactions involving vanadium, where no experimental data are available (Table 2).

4. RESULTS AND DISCUSSION The experimental data comparing redox properties of titanium and vanadium is relatively scarce, especially when nonaqueous environment (characteristic for olefin polymerization) is taken into account and most of the works deal with standard electrode potentials in water solutions. Still, there is a clear indication that vanadium at the oxidation state of 4 and 3 is more easily reducible than the titanium at the corresponding oxidation states. For example, standard electrode potential (SEP) of the MO2/M2O3 system in basic solution are equal −1.383 V for M = Ti and −0.527 V for M = V, respectively.66 Similarly, the SEP of the M3+/M2+ system in the acidic solution is more negative for titanium than for vanadium (an estimated value of −0.9 V versus −0.527 V for vanadium).66 In our theoretical study, we have considered two types of processes: reduction of metal(IV) and metal(III) chlorides with 1465



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electrons on fourteen orbitals. The summary of active spaces selected is given in Table 3. Thermodynamics of vanadium reduction calculated with the CASPT2 method is in very good agreement with the CCSD(T) results; on the other hand, its performance for the titanium(IV) chloride reduction is even worse than this of any density functional. It should be stressed that the application of larger active spaces does not improve the results. It would be interesting to develop a generalized model that could potentially predict, or even replace, the theoretical calculations of the thermodynamic parameters for the reactions discussed above. Redox reactions are invariably associated with the change in the number of electrons and can be treated with the so-called conceptual DFT.70 The simplest approach links the reduction process with electron affinity and the oxidation process with ionization potential. The former can be approximated by the energy of the LUMO orbital and the latter by the energy of the HOMO orbital, i.e., the lower the energy of the LUMO orbital, the higher the standard potential of the reduction reaction. This qualitative dependency was demonstrated by cyclic voltammetry experiments on certain titanocenes71 and ferrocenes72 and it also works exceptionally well for metal chlorides discussed in this paper (Figure 1). It

Table 2. Energy of Vanadium(IV) and Vanadium(III) Chloride Reduction with Hydrogen and the Intramolecular Redox Process, kcal/mola reaction no.

a

method

1

2

3

4

BP86/TZP B3LYP/TZP CASPT2/ANO-L-VTZP CCSD(T)/aVDZ CCSD(T)/aVTZ

28.84 16.59 18.95 15.69 17.07

35.85 28.50 b 24.06 26.51

50.52 39.11 43.18 42.27 40.89

57.52 51.03 b 50.64 50.32

No experimental data available. bNot calculated.

Among density functionals chosen, B3LYP provides the most accurate results for the heats of reactions, probably due to fortuitous cancellation of errors. This feature of the B3LYP functional is widely discussed in the literature. On the other hand, the results obtained for other functionals, although not as accurate as for B3LYP, lie within a close range and are almost independent of the type of the functional. It should be noted that all the methods selected provide a qualitative indication that vanadium(IV) is more easily reducible compared with titanium(IV). Further decreasing the oxidation state from +3 to +2 encounters higher thermodynamic penalties for both titanium and vanadium, but the latter is still more privileged in this process. Highly positive energies of reactions suggest that hydrogen as a reducing agent is not potent enough to bring neither vanadium nor titanium to a lower oxidation state. Intramolecular redox reactions 3 and 4 are even more endoenergetic. The role of multireference character in the wave function was estimated from the T1 diagnostic values68 for metal chlorides calculated at the CCSD(T)/aVDZ level, which did not exceed 0.026 for titanium and 0.037 for vanadium. These moderate values mean that the wave function is dominated by a single electron configuration. It should be mentioned that despite the fact that for closed-shell systems the T1 values greater than 0.02 might suggest unreliable single reference coupled cluster wave function,68 the values twice as large are acceptable for openshell species.69 Nevertheless, the DFT and CCSD(T) results were supplemented with CASPT2 calculations performed for reaction 1 to check for any discrepancies. The active spaces for hydrogen, chlorine, and hydrogen chloride molecules were composed of one occupied and one virtual orbital (two electrons in two orbitals). For titanium tetrachloride, the active space included five occupied orbitals with the 3d contribution and five corresponding virtual orbitals, which in total gives ten electrons in ten orbitals. Titanium trichloride exhibits more ionic character as compared with titanium tetrachloride; therefore, it was treated as a single configuration state function and the active space consisted of one 3d0 orbital populated with one electron. For vanadium chlorides, the active space was composed of all occupied orbitals with the 3d character, including those singly occupied, and the corresponding number of virtual orbitals. Thus the active space for vanadium tetrachloride was composed of eleven electrons on twelve orbitals and for vanadium trichloride twelve

Figure 1. ΔH of reactions 1 and 2 as a function of the LUMO orbital energy. Calculated for B3LYP (filled dots) and BP86 (empty dots). R2 values are equal 0.952 and 0.928, respectively.

should be mentioned that other measure of redox potentials, electrophilicity index,73 a quantity that depends on the ionization potential I and the electron affinity A according to the following equation: ω=

(I + A)2 8(I − A)

(7)

has been devised and successfully applied for the redox potentials of transition metal ions in an aqueous solution.74 4.2. Reduction of Metal(IV) Chlorides with Trimethylaluminum. Trimethylaluminum, unlike hydrogen, is a good reducing agent for titanium(IV) and vanadium(IV) chlorides. All heats of reactions are negative (Table 4). It is expected that

Table 3. Active Spaces for CASSCF and CASPT2 Calculations active space

H2

Cl2

HCl

TiCl4

TiCl3

VCl4

VCl3

(2,2)

(2,2)

(2,2)

(10,10)

(1,1)

(11,12)

(12,14)

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Table 4. Energy of Metal Chloride Reduction with Trimethylaluminum, kcal/mol 5

6 M = Ti

BP86/TZP B3LYP/TZP CCSD(T)/aVDZ

−15.90 −26.71 −33.10

E-mail: [email protected]. 5

Notes

6

The authors declare no competing financial interest.

■

M=V 26.40 25.73 34.98

−23.36 −55.60 −57.21

30.62 30.86 40.04

ACKNOWLEDGMENTS The author thanks Kirk A. Peterson for providing the titanium and vanadium basis sets as well as helpful discussions; David A. Dixon for his assistance and comments on electron correlation; Erik van Lenthe for his suggestions regarding DFT calculations; Roland Lindh for his remarks on multiconfigurational methods; W. Andrzej Sokalski, Szczepan Roszak, and Teobald Kupka for helpful discussions. Wroclaw Supercomputing and Networking Centre as well as Academic Computer Centre CYFRONET AGH (Grant No. MNiSW/SGI3700/UOpolski/126/2006) are acknowledged for generous allotment of computer time and software.

the entropic contribution to the Gibbs free energy should not counterbalance strongly negative energies of reactions and the reduction processes (5) will be spontaneous. The results of calculations reported in Table 4 neglected the influence of organoaluminum compound dimerization according to the following reaction: 2Al(CH3)3 ⇆ [(Al(CH3)3 )]2

(8)

■

Its energetic effect determined both experimentally and theoretically75−77 is about −20 kcal/mol; hence, even though it is taken into account, the reduction process remains still exoenergetic. The values presented in Table 4 indicate that B3LYP gives more accurate results than BP86, which is consistent with the results obtained for the reduction with hydrogen (Tables 1 and 2). The negative energy of vanadium(IV) reduction is almost twice as large as this for titanium(IV), which indicates that the former is more easily reducible. The dissociation of bridged compounds, described by the reaction eq 6 is strongly endoenergetic, independent of the kind of transition metal. The entropic contribution is expected to assist this reaction, but the Gibbs free energy is still expected to be positive.

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5. CONCLUSIONS Our calculations based on different levels of theory confirm that vanadium(IV) chloride undergoes the reduction process more easily than titanium(IV) chloride. In this reaction, trimethylaluminum is a better reducing agent than hydrogen. The results obtained with the CCSD(T) method approach the experimental values for titanium reduction and provide a reference for analogous reaction of vanadium, where no experimental data are available. Multireference method performs satisfactorily only for vanadium and not for titanium. Among density functionals chosen, the B3LYP gives the most accurate results. Theoretical results obtained in this work may serve as a benchmark of selected theoretical methods and a reference for the experimentalists who are aware of a more complicated nature of vanadium with respect to titanium in the redox processes. Further calculations dealing with the models that better describe the actual metallic active sites anchored to the support in the heterogeneous catalysts or surrounded by the ligands and solvent in homogeneous systems are necessary to extend the ideas reported in this paper.

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AUTHOR INFORMATION

Corresponding Author

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ASSOCIATED CONTENT

S Supporting Information *

Cartesian coordinates and absolute energies of selected optimized structures. This material is available free of charge via the Internet at http://pubs.acs.org 1467



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