Energy Barriers of Vinylidene Carbene Reactions from the Anti

We apply the anti-Hermitian contracted Schrödinger equation (ACSE) [Mazziotti, D. A. Phys. Rev. Lett. 2006, 97, 143002] to compute two-electron reduce...
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J. Phys. Chem. A 2010, 114, 583–588

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Energy Barriers of Vinylidene Carbene Reactions from the Anti-Hermitian Contracted Schro¨dinger Equation Loren Greenman and David A. Mazziotti* Department of Chemistry and The James Franck Institute, The UniVersity of Chicago, Chicago, Illinois 60637 ReceiVed: August 14, 2009; ReVised Manuscript ReceiVed: October 3, 2009

Computational studies of carbenes must take into account the possibility of multireference correlation because the highest occupied and lowest unoccupied molecular orbitals can be nearly energetically degenerate. We apply the anti-Hermitian contracted Schro¨dinger equation (ACSE) [Mazziotti, D. A. Phys. ReV. Lett. 2006, 97, 143002] to compute two-electron reduced density matrices (2-RDMs) and their energies for two carbene reactions: (i) the acetylene-vinylidene rearrangement and (ii) the rearrangement of pent-1-en-4-yn-3-one to acryloylvinylidene, which then cyclizes to cyclopenta-2,4-dienone. The ACSE has some unique advantages in the treatment of carbene reactions and more general families of reactions in which the importance of multireference correlation is not known a priori: (i) the ACSE is more reliable than single-reference methods for confirming the presence or absence of multireference correlation and (ii) in the absence of multireference correlation, unlike multireference second-order perturbation theory (MRPT2), the ACSE recovers more singlereference correlation energy than similarly scaling coupled-cluster methods. Because MRPT2 does not recover as much single-reference correlation as the coupled-cluster or ACSE methods, it tends to underestimate reaction barriers within the carbene reactions. For example, in the rearrangement of pent-1-en-4-yn-3-one, the ACSE and CCSD(T) methods produce cyclization barriers of 18.9 and 18.7 kcal/mol with the 6-31G(d) basis set, whereas MRPT2 predicts this barrier to be 12.1 kcal/mol; furthermore, both the ACSE and CCSD(T) determine the energy of the transition state for acryloylvinylidene formation to be 6.6-6.7 kcal/mol above that of the carbene, and yet, MRPT2 does not predict a transition state. I. Introduction Carbenes frequently appear as intermediates or transition states in organic and organometallic chemistry.1 Transition-metal carbene complexes,1 especially with stable carbenes such as N-heterocyclic carbenes,2,3 have replaced phosphane transitionmetal complexes as catalysts in a number of reactions including polymerizations.2 The highest occupied molecular orbital (HOMO) for carbenes is the σ orbital, and the lowest unoccupied molecular orbital (LUMO) is the pπ orbital. Because these orbitals can be very close in energy, carbenes have been identified with singlet and triplet ground states.4 Computational studies must therefore take into account multireference correlation, that is zeroth-order contributions from multiple Slater determinants.5 Traditionally, multireference correlation is treated by (i) correlating a set of active orbitals by the complete-active-space self-consistent-field (CASSCF) method6 and then (ii) correlating the inactive orbitals by second-order perturbation theory (MRPT2 or CASPT2)7,8 or single-double configuration interaction (MRCI).9 Reduced density matrix (RDM) methods10-27 use the 2-electron RDM (2-RDM), defined in eq 1, as the primary variable instead of the N-electron wave function.

D(r1, r2 ;r1′, r2′) )

2

∫ Ψ(r1, r2, r3, ..., rN)Ψ/(r1′, r2′, r3, ..., rN) dr3...drN

(1)

One method for obtaining the 2-RDM without solving for the N-particle wave function is the solution of the anti-Hermitian * Corresponding author. E-mail: [email protected].

contracted Schro¨dinger equation (ACSE).11-22 The ACSE is the contraction of the Schro¨dinger equation onto the anti-Hermitian part of the two-electron space.11,12,19-22 The dependence of the ACSE on the 3-RDM is removed by its approximate cumulant reconstruction as a functional of the 2-RDM.11,12,28,29 The ACSE is solved by a set of differential equations which can be initiated from any guess 2-RDM, including (i) a Hartree-Fock 2-RDM11,13 or (ii) a CASSCF 2-RDM.14,16-18 As seen in previous applications to the electrocyclic reactions of bicyclobutane16 and the sigmatropic shifts in 1-propene and acetone enolate,17 this flexibility in the initial 2-RDM guess enables the ACSE to produce a balanced description of single- and multireference correlation. In this paper, we study two different carbene reactions by using the ACSE,11-22 the acetylene-vinylidene rearrangement, and an R-alkynone rearrangement. In addition to examining the energetics of these reactions, a central goal of this paper is to show that the ACSE has some unique advantages in the treatment of carbene reactions and more general families of reactions in which the importance of multireference correlation is not known a priori. First, when the ACSE is initiated with a correlated 2-RDM as from a CASSCF calculation,14,16-18 it is more reliable than single-reference methods for confirming the presence or absence of multireference correlation. Second, unlike MRPT2, the ACSE recovers single-reference correlation energies between those from coupled-cluster single-double excitations (CCSD) and CCSD with perturbative triple excitations [CCSD(T)], and hence, even if substantial multireference effects are not present, the ACSE generates the accuracy of the best single-reference methods with a similar computational scaling.

10.1021/jp907890d  2010 American Chemical Society Published on Web 11/05/2009

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1 p,q Ds,t ) 〈Ψ|a†pa†qatas |Ψ〉 2

2

(4)

The ACSE depends on elements of the 2- and 3-RDMs. However, the 3-RDM can be approximately reconstructed from the 2-RDM by its cumulant expansion,11,12,28,29 i,j,k i,j i,j,k Dp,q,s ) 1Dip ∧ 1Djq ∧ 1Dsk + 32∆p,q ∧ 1Dsk + 3∆p,q,s

3

Figure 1. Two carbene reactions are shown: (a) the acetylenevinylidene rearrangement and (b) the R-alkynone rearrangement. In panel b, a hydrogen migration from the triple-bonded A creates the vinylidene carbene B, and another hydrogen migration closes the ring to form C.

(5) where i,j i,j ∆k,l ) 2Dk,l - 1Dik ∧ 1Dlj

2

The acetylene-vinylidene rearrangement, shown in Figure 1a, is a simple carbene reaction which has been studied computationally.30-34 The earliest studies showed that vinylidene is a shallow local minimum on the C2H2 potential energy surface.30,31 Later attempts to quantify the barrier to acetylenevinylidene rearrangement32-34 found that methods lacking electron correlation predicted a barrier that was too high in comparison to experiment,35 whereas methods containing significant electron correlation often predicted a barrier that was too low.31 Density functional theory provided a range of results as varied as wave function methods;34 coupled-cluster methods with perturbative triple excitations [CCSD(T)] were required to reproduce experimental results.33 The ACSE predicts barrier heights for this reaction which agree with CCSD(T), and it also provides diagnostic tools for assessing the role of multireference effects. The rearrangement of pent-1-en-4-yn-3-one (A) to acryloylvinylidene (B) and subsequent cyclization to cyclopenta-2,4-dienone (C) (Figure 1b) is classified as an R-alkynone rearrangement,36,37 which has been the subject of a number of computational studies.37-39 Like the acetylene-vinylidene rearrangement, calculations of the barrier height of the R-alkynone rearrangement can vary widely depending on the amount of correlation in the method. Highly correlated perturbation theory methods39 predict low barriers for the first step of this reaction in comparison to methods which include multireference correlation.37 A method such as the ACSE, which treats both single- and multireference correlation, is ideal for carbene reactions as well as more general reactions in which the importance of static correlation is ambiguous. II. Theory The contracted Schro¨dinger equation (CSE) is the contraction of the density-matrix formulation of the N-electron Schro¨dinger equation onto the space of two electrons.19-22,40 The ACSE, which is the anti-Hermitian part of the CSE, can be expressed as an expectation value of the commutator between the twoelectronreduceddensityoperator(2-RDO)andtheHamiltonian,11-16,41

ˆ ]|Ψ〉 ) 0 〈Ψ|[ai†aj†alak, H

(2)

where

ˆ ) H

∑ 2Ks,tp,qa†pa†qatas

(3)

p,q,s,t

The symbol 2K denotes the reduced Hamiltonian which contains the one- and two-electron integrals. The 2-RDM (2D) is the matrix of expectation values of the 2-RDO,

(6)

and ∧ is an antisymmetric tensor product known as the Grassmann wedge product.21 The 3-RDM is approximately reconstructed by setting its cumulant (or connected) part to zero.19,21,28,42 By seeding the ACSE with a 2-RDM from a CASSCF calculation, we can account for both single- and multireference correlation. The ACSE scales in floating-point operations as r2a (r - ra)4 where r is the number of orbitals and ra is the number of active orbitals. This computational scaling is similar to that of the coupled-cluster method with single-double excitations, which scales as ro2(r - ro)4 where ro is the number of occupied orbitals. Convergence of the ACSE algorithm to a given 2-RDM and its energy is achieved when either (i) the total electronic energy or (ii) the least-squares error of the ACSE increases. Additional details of the multireference formulation of the ACSE are given in ref 14. III. Numerical Applications Geometries were optimized by the complete-active-space selfconsistent-field method6 with an (m, n) active space where m electrons are spread among n orbitals. The double-ζ 6-31G(d)43 and triple-ζ 6-311G(d,p)44 basis sets were considered. For the acetylene-vinylidene rearrangement, we employed the full valence (10,10) active space; for the R-alkynone rearrangement, we utilized the (8,8) space of ref 37. The energies from multireference perturbation theory (MRPT2)7 and the ACSE11,14 were obtained by using the CASSCF wave functions and/or 2-RDMs as starting points; energies from the single-reference coupled-cluster methods CCSD45 and CCSD(T)46 were also computed for comparison. We calculated the occupation numbers from CASSCF, CCSD, and the ACSE to assess the amount of multireference correlation obtained by each method. The GAMESS electronic structure package47 was employed to TABLE 1: Correlation Energies (mH) of Acetylene, Vinylidene, and the Transition State for their Rearrangement (TS), Given in the 6-31G(d) Basis Set from the CASSCF, MRPT2, CCSD, CCSD(T), and ACSE Methodsa energy (H)

correlation energy (mH)

RHF

CASSCF MRPT2 CCSD ACSE CCSD(T)

acetylene -76.8141 -153.1 -275.1 -277.1 -283.7 -287.5 TS -76.7366 -141.1 -271.3 -273.6 -281.7 -284.3 vinylidene -76.7610 -128.8 -255.1 -262.2 -268.3 -271.0 a The ACSE correlation energy, which is between those from CCSD and CCSD(T), is substantially lower than MRPT2. A (10,10) active space was employed. The 6-311G(d,p) basis set gave similar results.

Energy Barriers of Vinylidene Carbene Reactions from the ACSE

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TABLE 2: CASSCF, ACSE, and CCSD Occupation Numbers of Acetylene, Vinylidene, and their Rearrangement Transition State (TS) Given for the 6-31G(d) Basis Seta acetylene

TS

vinylidene

orbital

CASSCF

CCSD

ACSE

CASSCF

CCSD

ACSE

CASSCF

CCSD

ACSE

1 2 3 4 5 6 7 8 9 10 11 12 13 14

2.00 2.00 1.98 1.98 1.98 1.93 1.93 0.07 0.07 0.03 0.02 0.01 0.00 0.00

2.00 2.00 1.97 1.97 1.97 1.93 1.93 0.06 0.06 0.03 0.02 0.01 0.01 0.01

2.00 2.00 1.97 1.97 1.96 1.91 1.91 0.07 0.07 0.03 0.03 0.01 0.01 0.01

2.00 2.00 1.99 1.98 1.97 1.94 1.94 0.07 0.05 0.03 0.02 0.01 0.00 0.00

2.00 2.00 1.98 1.97 1.95 1.94 1.93 0.06 0.05 0.03 0.02 0.01 0.01 0.01

2.00 2.00 1.97 1.96 1.95 1.92 1.92 0.08 0.06 0.03 0.03 0.02 0.01 0.01

2.00 2.00 1.98 1.98 1.97 1.94 1.93 0.07 0.07 0.02 0.02 0.02 0.00 0.00

2.00 2.00 1.98 1.96 1.96 1.93 1.92 0.07 0.05 0.02 0.02 0.02 0.01 0.01

2.00 2.00 1.97 1.96 1.96 1.91 1.91 0.08 0.07 0.03 0.02 0.02 0.01 0.01

a An active space of 10 electrons in 10 orbitals was used for the CASSCF and ACSE calculations. The 6-311G(d,p) occupation numbers are similar.

TABLE 3: Energies (kcal/mol) of the Transition State and Vinylidene Relative to Acetylene, Obtained by Using the 6-31G(d) and 6-311G(d,p) Basis Setsa relative energy (kcal/mol) basis set 6-31G(d) 6-311G(d,p)

RHF CASSCF MRPT2 CCSD ACSE CCSD(T) 0.0 0.0

0.0 0.0

Acetylene 0.0 0.0

0.0 0.0

0.0 0.0

0.0 0.0

6-31G(d) 48.7 6-311G(d,p) 46.5

Transition State 56.2 51.0 50.9 53.7 47.4 47.3

49.9 46.3

50.7 47.0

6-31G(d) 33.3 6-311G(d,p) 34.8

48.6 49.5

43.0 42.8

43.7 43.5

Vinylidene 45.9 46.0

42.7 42.4

a An active space of 10 electrons in 10 orbitals was used for the CASSCF, MRPT2, and ACSE energies.

perform RHF, CASSCF, MRPT2, CCSD, and CCSD(T) calculations and to generate electron integrals for the ACSE calculations. A. Acetylene-Vinylidene Rearrangement. Figure 1a shows the rearrangement of acetylene to vinylidene, which proceeds through a transition state in which a hydrogen atom forms a three-membered ring with the two carbon atoms. Table 1 compares the correlation energies calculated by the multireference CASSCF, MRPT2, and ACSE methods with a (10,10) active space and the single-reference CCSD and CCSD(T) methods. Even with a full-valence active space, the CASSCF method misses a significant portion of the single-reference (dynamic) correlation energy. MRPT2 recovers much of the dynamic correlation but not as much as CCSD, the ACSE, or CCSD(T). The ACSE calculates more correlation than CCSD and almost as much as CCSD(T). Even though, as the occupation numbers in Table 2 show, the acetylene-vinylidene rearrangement does not have a significant amount of multireference correlation, the ACSE ensures that the multireference correlation that often appears in the context of carbene chemistry is not neglected. Table 2 also shows that the ACSE occupation numbers are slightly more correlated than those of CCSD. The energies of the transition state and vinylidene relative to acetylene’s energy are given in Table 3. In a given basis set, the MRPT2, CCSD, ACSE, and CCSD(T) transition-state energies vary by at most 2.3 kcal/mol, whereas the transition-

Figure 2. Energies (kcal/mol) of (a) the transition state and (b) vinylidene, relative to the energy of acetylene, are given as a functions of the active-space size, ranging from 4 electrons in 4 orbitals (4,4) to 10 electrons in 10 orbitals (10,10), in the 6-31G(d) basis set. Whereas the CASSCF and MRPT2 energies vary unpredictably with active space size, the ACSE energies remain relatively constant.

state energies in the 6-311(d,p) basis set are 3.6-3.7 kcal/mol smaller than those in the 6-31G(d) basis set. CASSCF predicts transition-state and vinylidene energies that are 2.7-7.4 kcal/ mol higher in either basis set than those from the other methods. For both basis sets, the MRPT2 vinylidene energy of about 46 kcal/mol is higher than the 43 kcal/mol result from the ACSE and coupled-cluster methods. Unlike MRPT2, the ACSE captures as much (or more) correlation as the CCSD method. Although MRPT2 is limited to second order of perturbation theory, the ACSE through its cumulant expansion of the 3-RDM includes second-order plus many higher-order correction effects.11 Importantly, the inclusion of many high-order pertur-

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TABLE 4: Energies (Hartrees) from RHF and Correlation Energies (mH) from the Other Methods, Reported in the 6-31G(d) Basis Set for the r-Alkynone Rearrangementa energy (H) RHF A ABTS B BCTS C

-266.4311 -266.3568 -266.3732 -266.3069 -266.4803

correlation energy (mH) CASSCF MRPT2 CCSD -132.5 -113.6 -128.3 -130.1 -115.0

-834.8 -826.7 -806.8 -853.8 -835.2

-855.0 -850.7 -845.5 -877.0 -856.7

ACSE CCSD(T) -875.0 -869.5 -863.6 -899.8 -876.3

-886.8 -882.6 -876.8 -913.3 -888.5

a

An active space of 8 electrons in 8 orbitals was used for the CASSCF, MRPT2, and ACSE methods. The correlation energy from the ACSE is greater than that of CCSD but less than that of CCSD(T).

bative diagrams in the 3-RDM when it is reconstructed from the 2-RDM is a general property of the cumulant RDM reconstruction that applies to all types of many-particle quantum systems.10,28 In addition, MRPT2 calculates the reverse barrier to be 1.5 kcal/mol in 6-311G(d,p) which is lower than the barriers of 3.5 kcal/mol from the ACSE and CCSD(T). The prediction of low barriers is a characteristic of MRPT2 that will also be evident in the R-alkynone rearrangement. The ACSE and CCSD(T) agree within 0.8 and 0.1 kcal/mol for the forward and reverse barrier heights of the acetylene-vinylidene rearrangement. Figure 2 shows how the size of the active space affects the CASSCF, MRPT2, and ACSE (a) transition-state and (b) vinylidene energies. In comparison to CASSCF and MRPT2, the ACSE is much less sensitive to the size of the active space. This stability is useful for problems in which (i) the size of the active space is not clear from chemical principles or (ii) larger active spaces are not readily accessible because of computational limitations from CASSCF. Generally, the ACSE captures more dynamic correlation than MRPT2 and as much as the coupledcluster methods. Furthermore, through its ability to capture multireference correlation in contrast to single-reference methods, the ACSE demonstrates that this carbene reaction is dominated by a single Slater determinant. B. r-Alkynone Rearrangement. The rearrangement of pent1-en-4-yn-3-one (A) to acrylovinylidene (B) and the cyclization of B to cyclopenta-2,4-dienone (C) are shown in Figure 1b. The ABTS transition state represents the hydrogen shift between A and B, whereas BCTS represents the cyclization transition state where a hydrogen transfers from the double-bond carbon to the vinylidene carbon and the ring simultaneously closes. Table 4 compares the correlation energies predicted by the different methods for this reaction. The results are similar to those in Table 1 in that the ACSE predicts correlation energies between those of CCSD and CCSD(T), and MRPT2 predicts correlation energies less than those of CCSD. The ACSE results are closer to those from CCSD(T) than to those from CCSD. Occupation numbers for the R-alkynone rearrangement from the ACSE demonstrate that this is not a multireference carbene reaction. Figure 3 and Table 5 show the potential energy surface of the R-alkynone rearrangement, with all energies relative to B. The CASSCF surface is the highest relative energy surface, whereas the MRPT2 surface is the lowest energy surface. The ACSE, CCSD, and CCSD(T) surfaces are similar; they predict a cyclization barrier of around 20 kcal/mol, whereas CASSCF predicts a 40 kcal/mol barrier, and MRPT2 predicts a 12 kcal/ mol barrier. MRPT2 does not predict ABTS to be a transition state. For problems which are not multireferenced, MRPT2 generates barrier heights which are too low, and in this particular

Figure 3. The potential energy surface of the R-alkynone rearrangement is shown in the 6-31G(d) basis set from the CASSCF (squares), MRPT2 (triangles), ACSE (circles), and CCSD(T) (solid line) methods. An active space of 8 electrons in 8 orbitals was used for the CASSCF, MRPT2, and ACSE methods. The ACSE and CCSD(T) surfaces are similar, whereas the MRPT2 surface is too low without a transition state (ABTS) between A and B. The barriers from the CASSCF surface are substantially too high.

TABLE 5: Energies (kcal/mol) of the Different Species in the r-Alkynone Rearrangement, Relative to B, Given in the 6-31G(d) Basis Set from the RHF, CASSCF, MRPT2, CCSD, CCSD(T), and ACSE Methodsa relative energy (kcal/mol) A ABTS B BCTS C

RHF

CASSCF

MRPT2

CCSD

ACSE

CCSD(T)

-36.3 10.3 0.0 41.6 -67.2

-38.9 19.6 0.0 40.5 -58.8

-53.8 -2.2 0.0 12.1 -85.0

-42.3 7.1 0.0 21.8 -74.2

-43.5 6.6 0.0 18.9 -75.2

-42.6 6.7 0.0 18.7 -74.5

a An active space of 8 electrons in 8 orbitals was used for the CASSCF, MRPT2, and ACSE methods. MRPT2 greatly underestimates the barrier heights, even predicting the ABTS not to be a transition state, whereas CASSCF significantly overestimates the barrier heights. The ACSE, CCSD, and CCSD(T) methods yield similar energies, the ACSE barrier heights agreeing most closely with those from CCSD(T).

case, it misses the barrier entirely. In general, for problems with an unclear amount of multireference correlation, CASSCF misses dynamic correlation, whereas MRPT2 does not adequately capture dynamic correlation. The ACSE is a better choice for many problems, because it provides a balanced description of both single- and multireference correlation. IV. Conclusion The ACSE has been applied to study two carbene reactions: (i) the acetylene-vinylidene rearrangement and (ii) an R-alkynone rearrangement. Reactions within carbene chemistry often will contain significant multireference correlation because of the possible degeneracy of the σ and pπ orbitals; the ACSE has the ability to detect and treat the multireference correlation in these and other general classes of reactions.14,16-18 From its computed 1- and 2-RDMs, the ACSE can determine the amount of multireference character in a system from a number of diagnostics including (i) the deviation of the 1-RDM (natural) occupation numbers from 0 and 2 and (ii) the Frobenius norm of the cumulant (connected) 2-RDM.48,49 Whether significant multireference correlation is present or not, these diagnostics provide critical evidence for characterizing the extent and nature of the electron correlation. Equally effective or reliable diagnostics do not exist within single-reference wave function methods, where multireference correlation can remain unde-

Energy Barriers of Vinylidene Carbene Reactions from the ACSE tected. Although multireference perturbation theories such as MRPT2 could use the initial CASSCF wave function to diagnose multireference correlation, they often neglect a sizable portion of the single-reference correlation. In contrast, the ACSE offers a more balanced treatment of single- and multireference correlation where single-reference correlation is captured at the accuracy of coupled-cluster methods. For the acetylene-vinylidene rearrangement the ACSE, CCSD, CCSD(T), and MRPT2 produce similar energy barriers of 46.3-47.4 kcal/mol in the 6-311G(d,p) basis set. For the reverse barrier, however, the ACSE and CCSD(T) yield 3.5 kcal/ mol, whereas MRPT2 underestimates the reverse barrier at 1.4 kcal/mol. A similar trend emerges in the R-alkynone rearrangement. Whereas the ACSE and CCSD(T) methods give similar barriers of 18.9 and 18.7 kcal/mol, MRPT2 predicts a cyclization barrier of 12.1 kcal/mol. Moreover, the ACSE and CCSD(T) methods yield a transition state ABTS that is 6.6-7.1 kcal/mol above the energy of carbene B, but the MRPT2 method, because its barrier is below the energy of the carbene, does not predict a transition state. Energies from the ACSE are also less sensitive to the size of the active space than those from MRPT2. The results of the carbene reactions demonstrate the importance of multireference correlation methods that not only treat multireference correlation but also capture single-reference correlation at an accuracy near that of CCSD(T). The ACSE method achieves its balance of single- (dynamic) and multireference (static) correlation through a synergistic combination of at least two key factors: (i) the independence of the ACSE from a mean-field reference wave function or RDM,14,16 which permits the initial 2-RDM to be selected from a CASSCF calculation, and (ii) the cumulant reconstruction of the 3-RDM as a functional of the 2-RDM,28,29 which incorporates second- and many higher-order correlation effects into the computed energy and 2-RDM. For carbene reactions, where it is difficult to determine a priori the energy separation of the σ and pπ orbitals, as well as for more general classes of reactions in which the degree of multireference correlation is unknown, the ACSE provides an important tool for the direct calculations of 2-RDMs that accurately model all types of electron correlation. Acknowledgment. D.A.M. gratefully acknowledges support from the NSF, the David and Lucile Packard Foundation, the Henry-Camille Dreyfus Foundation, and Microsoft Corporation. References and Notes (1) Frenking, G.; Sola, M.; Vyboishchikov, S. F. J. Organomet. Chem. 2005, 690, 6178–6204. (2) Herrmann, W. A. Angew. Chem., Int. Ed. 2002, 41, 1290–1309. (3) (a) Bourissou, D.; Guerret, O.; Gabbai, F. P.; Bertrand, G. Chem. ReV. 2000, 100, 39–91. (b) Herrmann, W. A.; Elison, M.; Fischer, J.; Kocher, C.; Artus, G. R. J. Angew. Chem., Int. Ed. 1995, 34, 2371–2374. (c) Crudden, C. M.; Allen, D. P. Coord. Chem. ReV. 2004, 248, 2247–2273. (4) (a) Pliego, J. R.; De Almeida, W. B.; Celebi, S.; Zhu, Z. D.; Platz, M. S. J. Phys. Chem. A 1999, 103, 7481–7486. (b) Baron, W. J.; DeCamp, M. R.; Hendrick, M. E.; Jones, M.Jr., Levin, R. H.; Sohn, M. B. In Carbenes; Jones, M., Moss, R. A., Eds.; Wiley: New York, 1973; Vol. 1, p 1; (c) Eisenthal, K. B.; Turro, N. J.; Sitzmann, E. V.; Gould, I. R.; Hefferon, G.; Langan, J.; Cha, Y. Tetrahedron 1985, 41, 1543–1554. (5) (a) Bernardi, F.; Bottoni, A.; Canepa, C.; Olivucci, M.; Robb, M. A.; Tonachini, G. J. Org. Chem. 1997, 62, 2018–2025. (b) Pliego, J. R.; DeAlmeida, W. B. J. Chem. Phys. 1997, 106, 3582–3586. (c) Pliego, J. R. Chem. Phys. Lett. 2000, 318, 142–148. (6) Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. M. Chem. Phys. 1980, 48, 157–173. (7) Hirao, K. Chem. Phys. Lett. 1992, 190, 374–380. (8) Andersson, K.; Borowski, P.; Fowler, P. W.; Malmqvist, P. A.; Roos, B. O.; Sadlej, A. J. Chem. Phys. Lett. 1992, 190, 367–373. (9) Werner, H. J.; Knowles, P. J. J. Chem. Phys. 1988, 89, 5803–5814.

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