J . Phys. Chem. 1994,98, 2561-2510
2567
A Theoretical Investigation of the Fundamental Steps in Ziegler-Natta Catalysis: A Comparison of Density Functional, Hartree-Fock, and Second-Order Maller-Plesset Perturbation Theories Frank U. Axe’ Biosym Technologies, Inc., 9685 Scranton Road, San Diego, California 92121
James M. Coffin Biosym Technologies, Inc., 2443 Warrenville Road, Suite 600, Lisle, Illinois 60532 Received: December 7. 1993’
Local (LDF) and nonlocal (NLDF) density functional, Hartree-Fock (HF), and second-order Maller-Plesset perturbation (MP2) theories were used to study the fundamental reaction steps of Ziegler-Natta catalysis in the model TiClzCH3+ system. This comprised a detailed theoretical characterization of the ethylene binding and insertion reactions: TiC12CH3+ C2H4 TiClzCHs(CzH4)+ and TiClzCH&H4)+ TiClz(C3H7)+. The geometries of all species in the two reactions were fully optimized at all levels of theory, and the energy changes for the two reactions were calculated at all levels of theory. The basis set superposition error associated with the ethylene binding reaction was assessed by the counterpoise method at all levels of theory. The results show that the extended theories (NLDF and MP2) predict geometries and energetics that are substantially different from their respective parent theories (LDF and HF). This is due primarily to ways in which nonbonded interactions are handled by each of these methods. The overall agreements between the calculated geometries and energetics predicted by NLDF and MP2 theories are good.
-
+
Introduction Ziegler-Natta (ZN) catalysts comprise an important class of compounds that are capable of polymerizing olefins to high molecular weight, unbranched polymers, and in certain cases this is accomplished with stereoregularity.192 ZN catalysts are most commonly used in the commercialproductionof polyethylene and isotactic The most widely accepted mechanism for ZN polymerization is the one proposed by Cossee.l-3 In this mechanism (Figure l), a vacant site in the coordination sphere of a metal-alkyl complex binds an olefin to form a metal-alkyl-olefin complex (step 1). The olefin is then believed to insert directly into the metal-alkyl bond by passing through a four-centered transition (step 2). This creates a new metal-alkyl complex and regenerates the vacancy in the coordination sphere of the complex, so that the process of polymerization may continue. The Cossee mechanism formally consists of two fundamental reactions, olefin binding (step 1) and olefin insertion (step 2). Quantum mechanical methods can provide important insights concerningthe mechanistic details and thermodynamic properties of catalytic processes that are difficult to study by experimental means. Consequently,several first principlestheoreticalstudies“ of ZN catalysis have been reported. In a number of these papers,k7 TiClzCHs+ was treated as a model for a real homogeneous ZN catalyst, and the two fundamental reactions (steps 1 and 2) in the Cossee mechanism were investigated at both the Hartree-Fockk7 (HF) and second-order Morller-Plesset perturbation- (MP2) levels of theory by studying the following two reactions: TiCl2CH;
+ C,H4
+
TiC1,CH3(C2H4)+
TiC12CH3(C2H,)+
(1)
TiC12(C3H7)+
(2)
-
Similar results were reported for all ~tudies,’~ indicating consistency with the Cossee mechanism;moreover, somestudiesa showed that MP2 theory predicts energetic changes for both the ~~
0
~~~~
Abstract published in Advance ACS Absrracrs, February 1, 1994.
-
9
t #
P
I,,M-CH2CH2R
CH2-W
*
I
I ,
I,,M.
t
:
--. -R I
Figure 1. Schematic diagram of the Cossee mechanism showing the two fundamental steps in Ziegler-Natta catalysis.
ethylene binding (eq 1) and ethylene insertion (eq 2) reactions that are substantially different from what H F theory predicts. In all of these studies,a however, the MP2 results were always calculated at geometries that were optimized by the HF4*6or an approximate method? so that the geometric changes associated with MP2 theory were not determined. Also, the effect of basis set superposition error (BSSE) was never assessed for theethylene binding energy (eq 1) at either the HF or MP2 levels of theory. Density functional theory is a first principles quantum mechanical method for calculating accurate geometries and energetics of molecules with the inclusion of some electron correlation,9J0 which is particularly important for studying transition-metal complexes.l”l2 In the past, density functional theory was always performed in the local density functional approximation (LDF) for exchange and corre1ati0n;~JOhowever, more recently, nonlocal exchange and correlationfunctionals have been implemented within the density functional formalism9JO (NLDF), and many recent studies have demonstrated that nonlocal effects are important to properly predict geometries and energetics in m0lecules.’&1~To date, neither local nor nonlocal density functional theory has been applied to the study of ZN catalysis. In this paper, we present the application of LDF, NLDF, HF, and MP2 theories to the study of the two fundamental reactions
0022-3654/94/2Q98-2567s04.50~0 0 1994 American Chemical Society
2568 The Journal of Physical Chemistry, Vol. 98, No. 10, 1994
Axe and Coffin TABLE 1: Calculated Geometries for TiCl&HJ+ (in A and
den)
parameter T i 41 C-HI
C-H2 Ti41 H1-C-Ti Hd-Ti
C-Ti41 Cl-Ti-Cl
Figure 2. Conformationsof TiClZCHI+,TiC12CHa(CzH4)+,and TiClz(C3H7)+used in the calculations.
in the Cossee mechanism for the model TiC12CH3+system (eqs 1 and 2). The geometries of all species in the ethylene binding and insertion reactions (eqs 1 and 2) were fully optimized at all levels of theory, and the energy changes of the ethylene binding and insertion reactions (eqs 1 and 2) were calculated at all levels of theory. Also, the effect of BSSE was assessed for the ethylene binding energy (eq 1) at all levels of theory. The results of these calculations will be compared in order to determine the performance of each method for studying ZN catalysis. This paper represents the first in a series of contributions utilizing density functional theory to study ZN catalysis and related reactions.
Methods The conformations of TiC12CH3+, TiC12C%(CzH4)+, and TiC12(C3H,)+ used in this study are shown in Figure 2. For each molecule, C, symmetry was conserved. Although slightly lowerenergy structures have been found when C, symmetry is removeYJ* (particularly for TiC12(C,H,)+), the imposition of symmetry results in calculations that are more computationally tractable, while not substantially affecting the results.18 All density functional calcuIations were carried out with the deMon programof BiosymTechnologies Inc., which was originally developed in the laboratory ~ f S a l a h u b . ~The ~ JdeMon ~ program is capable of performing local density calculations using the exchange-correlation functional of Vosko-Wilk-Nusair.20 Nonlocal corrections to the exchangefunctionalas proposed by Perdew and Wang21 or BeckeZ2 and nonlocal corrections to the correlation functional of Perdew23 were combined with the local exchangecorrelation functional and computed self-consistently. (The combinationsof these two different nonlocal exchange functionals with the nonlocal correlation functionalwill be designated NLDFPP and NLDF-BP for the P e r d e ~ ~ ~ - P e r d eand w ~ Beckea~ Perdew23nonlocal functionals, respectively, throughout this text.) The molecular orbitals were expanded in terms of Gaussian basis ~ets2~~25 of polarized split-valence quality with the following contraction schemes: (6332 1/ 531/4 1), (632 1/ 5 2 1/ 1),(62 1/ 4 1/ l), and (41/1) for Ti, C1, C, and H, respectively. An auxiliary basis set of Gaussian functions25 was used to represent the total density for the purpose of calculating the Coulomb, exchange, andcorrelation terms. This basisset hadtheform2556of (5,5;5,5),
LDF
NLDF-PP
NLDF-BP
HF
MP2
1.96
2.00 1.12 1.10 2.17 94.9 113.9 109.0 114.5
2.00 1.12 1.10 2.14 94.0 114.8 104.9 109.3
1.94
1.94 1.12 1.09 2.11 88.3 118.0
1.13 1.10 2.1 1 92.3 115.7 104.1 108.0
1.11 1.08 2.14 94.3 111.9 106.3 117.9
107.1 109.4
(5,4;5,4),(4,3;4,3),and(3,1;3,1) forTi,Cl,C,andH,respectively. Atomic centered grid$’ consisting of 32 concentric spheres with 26 angular points each were used to evaluate the exchange and correlation functionals. (This number of grid points corresponds to the FINE option in the deMon program.) Nonrandom arrangements of the grid points were used in order to preserve C, symmetry. Geometries were optimized by analytical gradient methods and converged to a maximumgradient of 0.002 hartree/ bohr. The H F and MP2 calculations were performed with the Turbomole program of Biosym Technologies Inc., which was originally developed in the laboratory of Ahlrichs.28J9 The MP2 calculations included all double excitations of all electrons. The transition-metalbasis set consisted of Huzinaga’s MIDI-4* basis24 for the 5F state of titanium. This basis set was augmented4with two additional p functions with exponents of 0.083 and 0.028, giving an overall contraction of (43321/42111/31) in the calculations. The standard 6-31G** basis sets30 were used for the C1, C, and H atoms. This metal and ligand basis set is comparable to the one used in the density functional calculations. Geometries were optimized by analytic gradient methods and converged to a maximum in the gradient of 0.001 hartree/bohr. The BSSE was estimated for the ethylene binding reaction (eq 1) by the counterpoise method” at the optimized geometry of TiC12CH3(C2H4)+ for levels of theory. This comprised four separate calculations at each level of theory: (a) calculating the energy (Ea)of TiC12CH3+at the geometry in TiC12CH3(C2H4)+ including the basis functions on the C2H4 moiety, (b) calculating the energy (Eb)of C2H4 at the geometry in TiC12CH3(C2H4)+ including the basis functions on the TiClzCH3 moiety, (c) calculating the energy (Ec)of TiC12CH3at the geometry in TiC12CH,(CzH4)+, and (d) calculating the energy (Ed) of C2H4 at the geometry in TiC12CH3(CzH,)+. In the LDF and NLDF calculations the grid points associated with the ghost basis functions were also included. The BSSE is given by BSSE = E, + Eb Ec - Ed. Locating transition states requires calculating accurate first and second derivatives and algorithms to locate saddle points on the potential energy surface. The current version of the deMon program is not capableof calculatinganalyticalsecond derivatives. Also, thecurrent versions of deMon and Turbomoledo not contain routines for locating saddle points. Therefore, no attempt was made to determine the barrier for the ethylene insertion reaction (eq 2) at this time.
Results and Discussion Geometries. The geometries predicted by LDF, NLDF, HF, and MP2 theories for TiClzCHs+,TiClzCH,(C2H4)+, and TiC12(C3H7)+ are shown in Tables 1, 2, and 3, respectively. The calculated Ti-Cl bond lengths in TiC12CH3+range from 1.94 to 2.00 A. LDF theory predicts the T i 4 1 distance to be shorter thanwhat NLDF theories predict. HF and MP2 theories predict the same value for the T i 4 1 distance. NLDF theories predict the longest values of all theories for the Ti-CI distance. The T i 4 1 distancepredicted by LDF theory is in closer agreement with the result predicted by MP2 theory than NLDF theories. The predicted Ti-Cl distances range from 2.1 1 to 2.17 A. LDF
The Journal of Physical Chemistry, Vol. 98, No. 10, 1994 2569
Fundamental Steps in Ziegler-Natta Catalysis
TABLE 2 Calculated Geometries for TiCl&H~(cfi)+ (in & deal parameter
LDF
NLDF-PP
NLDF-BP
HF
MP2
Ti-CI Ti422 Ti-C, C&3 Ti-Cl HI-C-Ti H&-Ti
1.98 2.29 2.59 1.36 2.13 95.2 114.3
2.03 2.43 2.81 1.37 2.19 100.9 111.8
2.02 2.37 2.74 1.37 2.16 98.2 113.0
1.95 2.42 2.82 1.34 2.17 101.2 109.4
1.96 2.36 2.75 1.35 2.13 91.9 115.9
TABLE 3: Calculated Geometries for TiCl2(CJ-I7)+ (in A and den) Darameter LDF NDLF-PP NLDF-BP HF MP2 Ti-C1 2.15 2.23 2.21 2.28 2.15 Ti43 2.00 1.94 1.94 1.96 2.00 1.61 1.60 1.59 1.59 (21x2 1.58 1.57 1.57 1.55 1.56 C&3 1.54 Ti-CI 2.13 2.19 2.16 2.17 2.13 Ti-Hz 2.16 2.23 2.20 2.26 2.15 119.2 116.9 117.6 119.0 CI-C&~ 119.4 84.4 83.9 88.7 84.0 C&-Ti 82.7 theory predicts the T i 4 1 distance to be shorter than what NLDF theories predict. MP2 theory predicts a Ti-Cl distance shorter than what HF theory predicts. NLDF theories predict the longest values of all theories for the T i 4 1 distance. The LDF result is in close agreement with the MP2 result for the calculated Ti-Cl distance. All of these calculations predict a pyramidal structure for TiC12CH3+, as described by Jolly and Maryni~k.3~ Also, all methods examined predict that the in-plane methyl hydrogen (HI) is involved in an agostic interaction with the titanium atom as evidenced by the small HI-C-Ti angles, which range from 92.3O to 94.9O in the calculations and are similar to the experimental33 H-C-Ti angle of 93.5O in Ti(Cl)3(dmpe)(CH3) (where dmpe = 1,2-bis(dimethylphosphino)ethane), The calculated Ti-Cl bond lengths in TiC12CH&H4)+ are all longer than those values predicted for TiC12CH3+ for each theory studied. The relative agreements between all theories for the calculated Ti-CI distances are qualitatively the same as that found for the calculated Ti-CI distances in TiC12CH3+. Similarly, the predicted T i 4 1 bond lengths in TiC12CH3(C2H4)+are longer than those predicted for TiC12CH3+for each theory studied. The relative agreements between all theories for the calculated Ti-C1 distances are also qualitatively the same as the calculated Ti-Cl distances in TiC12CH3+. The calculated Ti422 distances in TiC12CH,(C2H4)+ range from 2.29 to 2.43 A. LDF theory predicts the shortest distance at 2.29 A, nearly 0.1 A shorter than all other theories. NLDF theory lengthens the Ti-Cz distance by 0.14 and 0.08 Ausing the PP and BP nonlocal functionals, respectively. The short Ti422 distance predicted by LDF theory relative to NLDF theory may be the result of underestimated nonbonded repulsions in the local approximation, which is demonstrated by calculations on hydrogen-bonded molecules14J4 and transitionmetal compounds.10-12 HF and MP2 calculations predict Ti422 distances of 2.42 and 2.36 A, respectively. Correlation effects reduce the Ti422 distance by 0.06 A, which may be due to HF theory overestimating the nonbonded repulsions' between the ligands in TiC12CH,(C2H4)+. The Ti-Cz distances predicted by NLDF-BP and MP2 theories are similar to one another, while the Ti422 distances predicted by NLDF-PP and HF theories are similar tooneanother. Allofthe theoriespredict that theethylene C-C bond lengthens slightly upon coordination by the TiC12CH3+moiety. Even with theethylenebound, however, the methyl H1 atom in TiC12CH,(C2H,)+ appears to be involved in an agostic interaction with the Ti atom as evidenced by the HI-Cl-Ti angle, which ranges from 91.1O to 100.9O in the calculations. The predicted Ti421 distances in TiC12(CaH7)+ range from 2.15 to 2.28 A. LDF theory predicts a value of 2.15 A for this distance, while NLDF-PP and NLDF-BP theories predict longer 1,35936
TABLE 4 kcal/mol)
Calculated Ethylene Binding Energies (in
TiC12CHj++ C2H4 method AEO LDF -45.7 NLDF-PP -37.9 NLDF-BP -35.2 HF -38.6 MP2
-44.2
+
TiC12CH3(C2H4)* ED
AEC
2.2 1.4 1.8 3.6 9.5
-43.5 -36.5 -33.4 -35.0 -34.7
No BSSE correction added. b BSSE. e BSSE correction added. values of 2.23 and 2.21 A, respectively. Again, LDF theory may be underestimating the nonbonded repulsions in TiC12(C3H7)+ by predicting a Ti421 distance that is more than 0.05 A shorter than those predicted by NLDF theories. HF theory predicts a Ti-Cl distance of 2.28 A, the longest of all theories, while MP2 theory reduces this distance by -0.1 A to 2.15 A. This is consistent with correlation effects reducing the nonbonded repulsions in TiC12(C3H,)+ relative to HF theory. The Ti-C1 distance predicted by MP2 theory is shorter than the values predicted by NLDF theory. In fact, theTi-CI distance predicted by MP2 theory is the same as that predicted by LDF theory. The short Ti-CI distancespredictedby LDF, NLDF, and MP2 theories are consistent with the existence of agostic interactions between the titanium atom and the methyl hydrogens. In TiC12(C,H7)+, the Ti-Hzdistances predicted by LDF, NLDF, and MP2 theories are all approximately the same as the T i 4 1 distances, which is further evidence that agostic interactions may be occurring between the titanium atom and the methyl group. All of the theories studied predict the Cl-Cz distances to be longer than the CTC3 distances in TiC12(C3H7)+. The longer C 1 4 2distances may also be the direct result of agostic interactions between the titanium and the terminal methyl group atoms. The predicted T i 4 3 distances in TiC12(C3H7)+are very similar to the predicted Ti-Cl distances in TiC12CH3+ for each theory. The calculated Ti-Cl distances in TiC12(C,H7)+ are essentially the same as those predicted for TiC12CH3(C2H4)+ for each theory. These results suggest that, for the geometry optimizations on TiC12CHp+,TiC12CH,(C2H4)+, and TiC12(CjH7)+, LDF theory predicts metal-ligand distances that are shorter than NLDF theory, and MP2 theory predicts metal-ligand distances that are about the same or shorter than HF theory. Also, the 'bonded" metal-ligand distances predicted by MP2 theory agree better with thosepredicted by LDF theory than thosepredicted by NLDF theory, and the -nonbonded" metal-ligand distances predicted by MP2 theory agree well with those predicted by NLDF theory. Energetics. The ethylene binding energies (eq 1) calculated by LDF, NLDF, HF, and MP2 methods with and without BSSE correction are shown in Table 4. The BSSE is small for both the LDF and NLDF calculations (-2 kcal/mol), which is consistent with other BSSE calculations at the LDF3' and NLDF3* level. The BSSE is larger for the HF calculations (-4 kcal/mol) than for the LDF and NLDF calculations, and the BSSE is the largest for the MP2 calculations ( 10 kcal/mol), which is similar to recent calculations39 on Cr(C0)s. LDF theory predicts an ethylene binding energy corrected for BSSE of -43.6 kcal/mol, while NLDF-PP and NLDF-BP predict less negative BSSE corrected binding energies of -36.5 and -33.4 kcal/mol, respectively. Nonlocal corrections decrease the exothermicity of the ethylene binding energy (eq 1) by -7-10 kcal/mol relative to the local approximation. The reduced exothermicity of the ethylene binding energy predicted by NLDF theory is consistent with the idea that NLDF theory increases nonbonded repulsions between the ligands relative to LDF theory, which should destabilize TiC12CH3(C2H,)+ and reduce the exothermicity of the ethylene binding reaction at the NLDF level relative to LDF level. HF and MP2 theories predict very similar BSSE corrected ethylene binding energies of -35.0 and -34.7 kcal/mol, respectively. The MP2 result in this study, which is not corrected for N
2570 The Journal of Physlcal Chemistry, Vol. 98, No. 10, 19s,4 TABLE 5: Cdculrted Ethylene Insertion Energies (in kcal/mol) TiCl&Hs(CzH4)+ method
-
TiClz(C3H7)+
AE -22.4 -1 1.4 -13,s -4.9 -14.2
LDF NLDF-PP NLDF-BP HF
MP2
BSSE, of - 4 4 . 1 kcal/mol is in good agreement with the MP2 calculations,not corrected for BSSE, of Kawamura-Kuribayashi e? a1.,4 Jolly and Marynick,sb and Pacansky and Kmg? which predicted ethylene binding energies of -45.3, -48.8, and -45.6 kcal/mol, respectively. The BSSE corrected ethylene binding energies predicted by NLDF and MP2 theories, which are the most sophisticated levels of theory studied here, are also in good agreement. However, in the absence of BSSE correction the agreement is not good. The ethylene insertion energies (q 2) calculated by LDF, NLDF, HF, and MP2 methods are shown in Table 5. LDF theory predicts an ethylene insertion energy of -22.3 kcal/mol, while NLDF-PP and NLDF-BP theories predict less negative values of -11.4 and -13.5 kcal/mol, respectively. Both nonlocal functionals reduce the exothermicity of the ethylene insertion reaction (q2) by 10 kcal/mol relative to the local approximation. This decrease in the exothermicity of the ethylene insertion reaction (q2) is also consistent with the idea that NLDF theory increases nonbonded repulsions between the ligands, which should destabilizeTiClZ(C3H,)+ and reduce the exothermicity of the ethylene insertion reaction at the NLDF level relative to the LDF level. H F theory predicts a small ethylene insertion energy of -4.9 kcal/mol, while MP2 theory predicts a value of -14.3 kcal/mol. The increase in the exothermicity of the ethylene insertion reaction (q2) at the MP2 level is consistent with the idea that correlation effects reduce the nonbonded repulsions between the ligands, which should stabilize TiC12(C3H7)+ and decrease the exothermicity of the insertion reaction at the MP2 level relative to the H F level. The MP2 result of -14.3 kcal/mol obtained in this study is in moderately good agreement with the MP2 calculationsby Kawamura-Kuribayashi et al.4and Pacansky and Kang? which predicted ethylene insertion energies of -1 1.1 and-10.1 kcal/mol, respectively. TheMP2result fortheethylene insertion energy reported in this study is - 4 kcal/mol more exothermic than these previous calculations,416which may be due to the fact that the geometry of TiC12(C3H,)+ was optimized at the MP2 level. It is encouraging that the MP2 result for the ethylene insertion energy is in close agreement with the NLDF results, because these are the most accurate theories used in this study.
-
Conclusions These results demonstrate that the extended theories (NLDF and MP2) predict geometries and energetics for the TiC12CHs+ system which are substantiallydifferent from what their respective parent theories (LDFand HF) predict. Based on thesedifferences in both the geometries and the energetics, the accuracy of LDF and H F theory is at best only qualitative for modeling systems of this kind. The overall agreements between the geometries and the energetics predicted by NLDF and MP2 theories are good, although some differences exist. NLDF and MP2 theories are in agreement on their predictionsof "nonbonded" distances,while MP2 theory predicts "bonded" distances that are shorter than what NLDF theory predicts. The ethylene binding energies predicted by NLDF and MP2 theory are in good agreement after BSSE corrections have been included, and the ethylene insertion energies predicted by NLDF and MP2 are also in good agreement. Since no direct experimental evidence is available for the TiC12-
Axe and Coffin CHI+system, it isdifficult todetermine whichof the twoextended theories is more accurate. Further studies, which includedifferent density functionals, comparisons of ethylene insertion barriers, and calculations which include cotrelation beyond the MP2 level, are needed to assess the underlying disparities between NLDF and MP2 theories. Several of these possibilities are currently being studied.18
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