J. Phys. Chem. 1995,99, 3465-3472
3465
Theoretical Study of the Chemical Bonding in Ni(C2b) and Ferrocene Kristine Pierloot Department of Chemistry, University of Leuven, Celestijnenlaan 200F, B-3001 Heverlee-Leuven, Belgium
B. Joakim Persson and Bj;drn 0. Roos* Department of Theoretical Chemistry, Chemical Centre, P.O.B. 124, S-221 00 Lund, Sweden Received: August 2, 1994; In Final Form: November 28, 1994@
The equilibrium structure and bond energies of the transition-metal complexes Ni(C2h) and ferrocene have been studied by using the complete active space (CAS)SCF method and second-order perturbation theory (CASPT2). It is shown that the major features of the electronic structure are properly described by a CASSCF wavefunction based on an active space comprising the bonding and antibonding orbitals directly involved in the metal-ligand bond. Remaining correlation effects are dealt with in the second, CASPT2, step. The computed energies have been corrected for BSSE and relativistic corrections have been added. Resulting bond distances and bond energies are in agreement with experimental data, when available. Ni(C2h): r(CC) = 1.443 A, r(Ni-C) = 1.868 A, De = 32.3 (36.4 f 5) kcal/mol. Ferrocene: r(Fe-ring) = 1.643 (1.66) A, De = 156 (157 f 3) kcal/mol (experimental values within parentheses).
1. Introduction In a recent study the complete active space (CAS) SCF method' in combination with multiconfigurationalsecond-order perturbation theory (CASPT2)2s3 was used to compute the structure and binding energies of a number of fiist-row transition-metal carbonyl^.^ The results showed that the method is capable of predicting metal to ligand bond distances with an accuracy of about 0.01 A and binding energies with error limits of f 5 kcdmol. These results are unique in the sense that no other method has been able to achieve the same accuracy. Multireference CI methods are severely hampered by the necessity to correlate a large number of electrons and singleconfiguration-based techniques (like coupled clusters) do not introduce enough flexibility in the reference function to be able to describe the important near-degeneracy effects in the 3d shell of the metal atom. It was shown that an accurate determination of the binding energy requires the use of extended basis sets, inclusion of relativistic corrections, correlation of the 3s,3p semicore electrons, and correction for zero-point energy differences. The BSSE corrections were found to be substantial even with the larger basis sets but could be successfully accounted for by the full counterpoise method. It was only with all these terms included in the geometry optimization and the energy calculation that agreement was established between the theoretical and experimental dissociation energies. In the present work we extend the same approach to a study of the metal-carbon bond. The two systems chosen for this study are the nickel-ethene complex, for which a number of earlier studies exist, and the ferrocene molecule. The latter molecule is chosen, since its structure has been a challenge for theory for a long time, and since no theoretical calculation of its binding energy exists. The basic idea of the CASSCFICASPT2 method is that if all important nondynamkal correction effects are included in a multiconfigurational, CASSCF type, reference wavefunction, remaining correlation effects can be treated by using (low-cost) second-order perturbation theory. The method has been applied @Abstractpublished in Advance ACS Absrracfs, February 15, 1995.
with success in studies of the geometric structure and bond energies of a series of molecules built from first-row atoms.5 The small size of these molecules made it possible to include all valence electrons into the active space of the CASSCF treatment. Obviously, this is no longer the case for the molecules considered here, and a careful choice of active electrons has to be made, based on the most important electronic structure features. Since the number of active orbitals in a CASSCF calculation is limited to 12-14, it must be possible to include the near-degeneracy effect within a CASSCF wavefunction based on an active space of at most this size. However, most first-row transition-metal compounds containing only one metal atom can be described qualitatively correctly by using an active space of modest size. It was shown in the carbonyl study that an active space comprising 10 electrons in 10 orbitals, CAS( lOIlO), describes the near-degeneracy effects in the metal 3d shell well and fonns a consistent basis for the CASPT2 treatment. We shall show that the same active space can be used for the molecules studies here. Other CASSCF/ CASFT2 studies of transition-metal compounds include some transition metal hydrides and dimers6 and the ligand field spectra of the fist-row hexacyanometalate complexes.' Both these studies have given results in agreement with experiment, which indicates that the approach will work well for transition-metal compounds. Further evidence is given by a series of studies of the electronic spectra of first-row transition-metal atoms and ion^.^^^ It was shown that the CASPT2 method could be used to accurately reproduce the energy splitting between different electronic states of the atom and ion, provided that an extensive active space was used, including radial correlation in the 3d shell, relativistic effects, and the differential correlation effects of the 3s and 3p shells, which can be substantial, especially for atoms or ions with low spin coupled open 3d shells.8 This is an important aspect of the approach, since good binding energies can only be obtained if the energy separation between the atomic states (especially d"s2, dn+'s, and d"+2) are reproduced with good accuracy. Much of the errors obtained in binding energies computed by using single-configuration based methods can be traced to an inability to compute correctly the a t o k c promotion energy.
0022-365419512099-3465$09.0010 0 1995 American Chemical Society
3466 J. Phys. Chem., Vol. 99,No. 11, 1995 2. Details of the Calculations The calculations are performed in two steps. First a CASSCF wavefunction is determined, which in the second step is used as reference function for the second-order perturbation calculation. The corresponding MO's (the natural orbitals of the CASSCF wavefunctions, suitably transformed) form the oneelectron basis for the CASPT2 calculation. The CASPT2 method computes the first-order wavefunction and the secondorder energy in the full space of configurations generated by this basis set. The zeroth-order Hamiltonian is constructed from a Fock-type one-electron operator that reduces to the MollerPlesset Hartree-Fock operator for a closed-shell case. In all calculations the so-called "non-diagonal" approach has been used, that is, the full Fock matrix (including the non-diagonal elements) is used in the construction of the zeroth-order Hamiltonian. All valence electrons are correlated in the CASPT2 calculation. It has been pointed out previouslylo~ll that such a treatment is needed in order to recover all correlation contributions to the binding energy. We have also considered the effect of correlating the 3s and 3p electrons. They are probably not of direct importance for the binding but affect the separation between the atomic ground state and the dn+2valence state.8 This separation has to be computed with great care in order to obtain reliable values for the binding energies. It is well-known that also relativistic effect are also of importance and the massvelocity and Darwin terms have been included in the present treatment by using frst-order perturbation theory at the CASSCF leve1.12J3 More details about basis sets etc. are given for each of the two molecules separately in the next sections. The calculations have been performed with the MOLCAS-2 quantum chemistry software,14either on a IBM RS/6000-550 workstation or on an IBM 3090-600VF.
3. Results and Discussion
3.1. The Ni(C&) Molecule. 3.1.1. Introduction and Details. The electronic structure of the complex between a nickel atom and the ethene molecule was first given a qualitative explanation in the CASSCF-CI study performed by Widmark et al.15 They showed that the ground state is a singlet with two covalent bonds formed between nickel and the carbon atoms of the ethene moiety. With the nickel and carbon atoms in the xz plane, and the z-axis pointing toward the midpoint of the CC axis, the following bonding scheme was obtained: one bond is formed between the 3dxzorbital and the z* orbital and the other between the n orbital and a 3d 4s hybrid orbital, which acts as an acceptor for the two z-electrons. Two metal electrons are moved into the corresponding 3d-4s hybrid constructed such that the density is located along an axis perpendicular to the line connecting the Ni atom and the middle of the CC bond in the NiCC plane. This hybridization minimizes the exchange repulsion between the ethene n-electron pair and the diffuse part of the nickel electron cloud. Although this bonding picture was qualitatively correct, the computed binding energy was 10 kcaymol, which is only about 30% of the experimental value, 35.5 f 5 kcaymol, according to a recent measurement.16 The use of a small basis set and inadequate treatment of the electron correlation are the main reasons for the too small computed dissociation energy. More recently, Blomberg et al.l7 have computed the binding energy by using larger basis sets and a variety of quantum chemical methods. Single-configuration-basedtechniques gave binding energies in the range 20-35 kcal/mol. The most accurate result was claimed to be the CCSD(T) value 27.6 kcdmol. These
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Pierloot et al. energies are, however, not corrected for the BSSE, which amounts to about 5 kcaYmol with the basis set used by Blomberg et al. Thus, a satisfactory theoretical determination of the bond energy between nickel and ethene have not yet been obtained. We shall show below that the CASSCF/CASPT2 method is able to give a quantitatively accurate description of the bond. The present study was performed with two different basis sets. One was the valence double-f plus polarization set used by Blomberg et al.: (Ni,14sllp6d3f), (C,9s5pld), (H,Sslp) contracted to [Ni,5s4p3dlfl, [C,3s3pldl, [H,3slpl. The results obtained with this basis set can be directly compared to their results. It will be labeled basis A below. The final calculations were performed with a larger AN0 basis set (basis B), based on a (Ni,21~15plOd6f4g),(C,14s9p4d3f), (H,8s4p) primitive set contracted to [Ni,6s5p4d2flgl, [C,4s3p2dlfl, [ H , 3 ~ 2 p ] . l ~ ~ l ~ The choice of the active space was made by using the experience obtained in the study of the carbonyl compounds! It was shown that the transition-metal ligand bonding could be described with an active space correlating 10 electrons in 10 orbitals. For doubly occupied 3d orbitals not taking part in the bonding, one pair of orbitals is needed: the 3d and a correlating 4d orbital (the 3d double-shell effectg). The inclusion of a correlating orbital for each 3d electron pair is necessary in order to obtain accurate energy separations for the d9s, dl0, and d8s2 energy levels in the metal atom, a prerequisite for a correct description of the chemical bond. For the 3d and ligand electrons directly involved in the bonding, the bonding and antibonding orbitals form the active pair. We obtain with this model, the following active space for the Ni(C2h) molecule: the 3d-4s hybrid plus a corresponding correlating orbital, the 3d2, 3d,, and 3d, orbitals with matching 4d orbitals and the bonding and antibonding combination between a 3dxz orbital and the n* orbital of ethene. We shall call this choice CAS(lO/lO). Surprisingly enough, the ethene n orbital is left inactive. It couples to the 3d 4s acceptor orbital, so one might expect that it should be important to include it among the active orbitals. This was done by extending the active space to 12 orbitals (with the n orbital active, a second orbital of b2 symmetry, which is a combination of 4dxzand n*,should also be included) and 12 electrons: CAS(12112). This extensions did not change the CASPT2 results to any appreciable degree. The total C A S E 2 energies differed only 0.51 kcaymol, illustrating once more that when the most important orbitals have been included in the active space is the CASPT2 energy stable with respect to further extensions of the active space. All electrons except the carbon 1s and the nickel Is, 2s, 2p were included in the CASPT2 correlation treatment. The correlation energy for the Ni 3s,3p shells is known to contribute to the binding energy (between 2 and 6 kcal/mol in the recent study of the metal carbonyls4). A geometry optimization was performed in the parameters r(Ni-C), r(C-C), and the angle, 8, between the CC bond and the HCH plane. The remaining two geometry parameters of ethene (r(C-H) and the HCH angle) were kept at their values for the free molecule. symmetry was assumed with equal bond distances between nickel and the two carbon atoms. The optimization was done at the CASPTZ level (using the CAS(10/10) reference function) by a fit of computed energies to a second-degree polynomial in the three degrees of freedom. The geometry optimization of Ni(C2H4) was preceded by a full optimization of the geometry of ethene performed with the two n orbitals active. The resulting values of the CH bond length and the HCH angle were used in the complex. It was found in the carbonyl study that the BSSE correction was essential for a correct description of the structure and
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. I . Phys. Chem., Vol. 99, No. 11, I995 3467
Chemical Bonding in Ni(C2b) and Ferrocene TABLE 1: CAS(lU12) Natural Orbital Occupation Numbers for Ni(GI-LP
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9al (n 3d 4s) loa1 ( 3 4 4 l l a l (3dy2-22 - 4s) 5b2 (3dx2 n*)
+
1.980 1.979 1.962 1.888
4b1 (3dy2) 2a2 ( 3 d d
1.978 1.978
13al(4d~-y2-2) 12al (4dy2-9 - 4s) 6b2 (3dx, - n*) 7b2 ( 4 x 2 ) 5bl (4dy2) 3a2 (4dJ
0.021 0.039 0.127 0.006 0.022 0.022
The dominant character of the orbitals is indicated within parentheses.
energetics of a transition-metal complex. All calculations were therefore performed both with and without this correction added. The CASPT2 energies of the nickel atom and the ethene molecule respectively were determined with and without the ghost basis set of the other moiety present. The energy difference gives the BSSE correction. Finally, relativistic corrections were added (the mass-velocity and Darwin terms) by using first-order perturbation theory. It has recently been shown that the presently used AN0 basis set gives very small contraction errors for the relativistic corrections.20 3.1.2. Results. In Table 1 we give an overview of the natural orbitals and their occupation numbers, obtained with basis B at the computed equilibrium geometry. The large active space was used. The dominant character of the orbitals is given in the table. There is only little delocalization of the n orbital onto the metal atom. This orbital and the three b2 orbitals form one group, occupied by almost exactly four electrons. The other orbitals occur in pairs, one strongly and one weakly occupied, with two electrons in each pair. The occupation number of the 4d orbital is larger than 0.02 in all cases, except for the one added in the same symmetry as the delocalized 3d,, orbital, where it is small. This illustrates why no extra correlation orbital is needed for 3d orbitals directly involved in the ligand bonding. The CAS(10/10) results are very similar to those given in Table 1 except that 9al is now inactive and 7b2 virtual. An analysis of the orbital populations at the CASSCF level shows the same similarity. The nickel populations are 3d9.094s0.40 with both active spaces, and the total number of electrons in the n orbitals is 2.16. The total Mulliken charge on the nickel atom is + O M . The CAS(10/10) calculation yields a dipole moment of -1.02 au, while CAS(12/12) gives a slightly smaller value, -0.94 au. In all, the two calculations result in very similar electron distributions. Calculated geometries and binding energies are presented in Table 2. The CASPT2 geometry parameters of the free ethene molecule are in agreement with experiment. It may be noticed that the CC bond distance decreases by 0.012 A, when the basis set is increased from A to B. This calculation was performed with the two n orbitals active. Complex formation increases the CC bond length by 0.10 8,. The same result was obtained in a previous cal~ulation'~although the CASSCF distances obtained were 0.01 8, longer. The optimal NiC distance is 1.876 8,, which on the other hand is 0.1 8, shorter than the old CASSCF value. Also the tilt angle, 8, is different: 32" compared to 21". These differences show that the CASSCF/ CASPT2 treatment gives much stronger bonding between the nickel atom and ethene. There is no experimental geometry to which the present results can be compared, but based on our experience with the present method, the error in computed bond distances are not larger than 0.02 8,. The largest deviation for a metal to carbon bond length found in the recent study of the carbonyls was 0.02 A, but somewhat smaller basis sets were used. The larger basis B gives here a NiC distance which is
0.025 8, shorter than that obtained with basis A. The effect of the BSSE is to increase the bond length by 0.008 8, and also decrease the tilt angle by 4". The dissociation energy, D,, was computed by using the procedure that was successfully used for the metal carbonyls; that is, De is obtained as the energy difference between ground state Ni(C2b) and the sum of the energies for ethene and Ni(d9s, 3F). The active space used in the calculation of the nickel atom was 3d, 4d, and 4s (1 1 orbitals). In the CAS(10/10) case the ethene energy was obtained from an SCF/MP2 calculation, while CASPT2 with two active orbitals (nand n*)was used in the CAS(12/12) case. De was computed by using both basis sets and both active spaces. The effect of increasing the active space is to lower the energy by about 1 kcdmol. The reason is the difference between the MP2 and CASPT2 energies of ethene. The CASPT2 energy of the complex is the same with both active spaces, but MP2 leads to a slight understimate of the correlation energy for ethene as compared to CASPT2. The change of the basis set from A to B increases De by 3.4 kcaymol. The final result is 32 kcaymol, of which 1.5 kcdmol is a relativistic correction and -4 kcaymol is due to BSSE. Brown et al.16 have recently reported a value of 35.5 & 5 kcdmol for the binding energy of Ni(C2b) at 0 K, which was obtained from a kinetic study of the association reaction of nickel atoms with ethene in combination with RRKM calculations, using vibrational data from a LSD calculation by Papai et aL21 The same data can be used to estimate the zero-point correction to the binding energy. It is found to be 0.9 kcdmol, yielding an experimental value for De of 36.4 & 5 kcdmol. The CASPT2 value is 4 kcaymol smaller but it within the error bars of the experiment. The corresponding calculation on NiCO gave a binding energy close to the experimental value4 and the present result is also expected to be in error by less than f 2 kcaYmol. 3.2. The Ferrocene Molecule. 3.2.I . Introduction and Details. Ferrocene is another prototype of metal-carbon bonding, which has been a challenge for ab initio quantum chemistry because of the difficulty in obtaining accurate structural results. Indeed, calculations performed at the Hartree-Fock level with good basis sets overestimate the vertical iron-ring distance by as much as 0.23 and correlation treatments including only a limited number of electron^^^-^^ did not have the expected success, leaving a remaining error of 0.07 A. Only recently, after it became clear that correlating all metal and ligand valence electrons is a prerequisite for obtaining quantitative accuracy, more accurate ab initio results have been obtained. Park and Alm10P7 performed MCPF and MP2 calculations. They emphasized the importance of correlating all the 66 valence electrons (this number indicates that also the Fe 3s,3p electrons were included) and the necessity to include the dispersion interaction between the rings, which they computed by using the MP2 approach. The calculated metalring distance was in good agreement with experiment. Thus it seems that the structure of the ferrocene molecule is quantitatively understood. There exists, however, no determination of the binding energy prior to the present study. Here we shall use the CASCF/CASPT2 method to compute the metal-ring distance and the homolytic as well as the heterolytic dissociation energy for ferrocene. In order to be able to compare with earlier single-reference-based treatments we shall also present results obtained with the SCF/MF2 method. The results will be compared to the recent MCPF and MP2 results and to the most recent results obtained by using the alternative density-functional approach.28
Pierloot et al.
3468 J. Phys. Chem., Vol. 99, No. 11, 1995
TABLE 2: Geometry Parameters (in A and Degrees) for C2H4 and Ni(C&), and Binding Energy (kcallmol) for Ni(C2H4), Computed with the CASSCF/CASPT2 Method basis A basis B PT2 +rca -BSSEb PT2 +rca -BSSEb exptl geometry for CZ& 1.354 1.339' 1.342 1.342 1.355 r(C-C) 1.086 1.082 1.082 1.085' 1.086 r(C-H) 117.2 117.2 117.3 117.8' 117.3 LHCH geometry for Ni(C2K) 1.886 1.876 1.901 1.887 1.868 r(Ni-C) 1.902 1.453 1.452 1.439 1.443 1.444 1.450 r(C-C) 24.1 24.2 33.5 36.0 31.9 e 23.9 dissociation energy 36.6 36.4 30.5 35.5 38.0 33.9 10 active orbitals 28.8 32.3 35.5 f 5 d 12 active orbitals other results' 35.1 MCPF 27.6 CCSD(T) 27.5 ACPF With relativistic correction added. With also the BSSE correction added. ' From the compilation in ref 38. From ref 16. e From ref 17. The AN0 basis sets used are based on a slightly smaller set of primitives than was used for the ethene complex. However, this recently developed basis set has proven to give results for structures and binding energies of almost the same quality as is obtained with the larger set.29 The starting primitive sets are (Fe,17s12p9d4f), (C,lOs6p3d), and (H,7s). The following combinations of contracted sets on Fe and C and H were used:
TABLE 3: CAS(10/10) Natural Orbital Occupation Numbers for Ferrocene
basis 1 Fe [6s4p3d] C [3s2pld] H [2s] yielding 193 contracted functions
orbitals, as well as for the radial 3d-4d correlation on the metal. The bonding involves a donation of charge from the doubly occupied ligand orbitals into the formally empty metal 3d orbitals, counteracted by back-donation from the occupied 3d orbitals into the low-lying antibonding orbitals on the ligands. At the SCF level, the covalent character of the bonding is manifested by the delocalization of the molecular orbitals within the symmetry representations containing 3d. However, the covalent strength of the bonding cannot be captured to its full extent by a single-determinant wavefunction. Indeed, the charge transfer from the metal to the ligands and vice versa is increased considerably when adding to the wavefunction excitations from the bonding to antibonding combinations of the metal 3d and the ligand orbitals. This is nicely illustrated by the CAS(10/ 10) natural orbital occupation numbers, shown in Table 3. The occupation of the antibonding orbitals is most certainly not negligible, 0.273 electrons. Within each representation, the occupation numbers of the bonding and antibonding orbitals roughly add up to twice the degeneracy, indicating that the antibonding orbitals indeed act as correlating orbitals for the corresponding bonding orbitals. Apart from its effect on the bonding, the CAS(10/10) calculation also includes the radial 3d-4d correlation on the metal as indicated by the considerable mixing between the antibonding ligand orbitals and the metal 4d orbitals. The bonding in ferrocene is mainly built from interactions between the iron 3d and cyclopentadienyl (Cp) n orbitals. Within D5h symmetry, the 10 p n orbitals of the two cyclopentadienyl rings yield the symmetry-adapted combinations ai, e;, e;, a?, ey, and e?, of which, of course, only the a{, e;', and e5 orbitals may interact with the 3d orbitals on the metal. A qualitative molecular orbital scheme of ferrocene is of the form (a{)2(a$')2(e;)4(e{')4(e5)4(a{)2,in which the 10 electrons from the two cyclopentadienyl rings and the eight Fe electrons are accommodated, so that the four lowest levels are predominantly ligand in character and the two highest occupied levels mainly metal 3d. This leaves room for charge donation from the occupied Cp n orbitals into the empty 3dn orbitals within the e? representation and from the doubly occupied 3dd orbitals
basis 2 Fe [6s4p3dlfl C [3s2pld] H [2s] yielding 200 contracted functions basis 3 Fe [6s5p4d2fl C [3s2pld] H [2s] yielding 215 contracted functions basis 4 Fe [6s5p4d2fl C [4s3pld] H [2s] yielding 255 contracted functions
CzVsymmetry was used in all calculations. However, since the ground state is nondegenerate, the wavefuncion automatically reflects the full symmetry of the molecule, D5h, without additional symmetry restrictions. Only the vertical iron-ring distance was varied, with 0.025 Q, whereas all other geome parameters were kept at their experimental values,30 1.440 for C-C bonds and 1.104 A for C-H bonds. The choice of the active space was made following the same criteria as was used for Ni(C2h) and the carbonyls. Thus the active space consists of the metal 3d orbitals and their antibonding (in case of a doubly occupied 3d orbital) and bonding (in case of an empty 3d orbital) counterparts. In ferrocene, the 3d orbitals are found in the 8ai (3da), 4e5 (3dd), and 5ey (3dn) shells (Cmvnotation) and the ground state, 'Ai, corresponds to 8a{24e$4. Thus the following orbitals were included in the active space: (8,9)a{, (4,5)ey, (4,5)e$, and again 10 electrons were correlated in the CASSCF step. This is the same CAS(10/10) active space as was used for Ni(C2h). All valence electrons, originating from the metal 3d, C 2s,2p, and H 1s orbitals, were correlated in the second, MP2 or CASPT2 step, thus including 58 electrons. The metal semicore 3s3p orbitals were kept frozen in the calculations on the complex. The correlation effect involving these electrons was instead included in the calculation of the ionization and excitation energies of iron. The CASSCF wavefunction accounts for the bonding and antibonding interactions between the metal 3d and the ligand
~~
8a; (3da) 4el(3d6) W(Cpn) a
1.964 3.869 3.895
~~
9ai (4da) 5e$(4dd-Cpn) 5eY(3&)
~~
~
0.031 0.104 0.138
The dominant character of the orbitals is shown within parentheses.
Chemical Bonding in Ni(C2b) and Ferrocene TABLE 4: Mulliken Populations of the Fe 3d and Cp Orbitals in Ferrocene. SCF and CASSCF Results Fez+(‘Ai) 2CpSCF CASSCF dif 1.807 1.749 -0.058 do 2.000 3.667 3.370 -0.297 d6 4.000 +0.550 0.389 0.939 dJc 0.000 29.517 29.543 +0.027 CPU 30.000 -0.142 5.405 5.263 6.000 CPZ
+
Figure 1. Total density difference plot between the CAS(10/10) wavefunction and the Hartree-Fock wavefunction for ferrocene. The figure describes the density shifts in one of the five symmetry planes passing through the central iron atom. Full contours correspond to an increase in electron density and dashed contours to a decrease. At the dotted lines AQ contours are f0.00025,f0.0005, f0.001,f0.002, iz0.004,f0.008,f0.016,f0.032,and 0.0e/au3. into the empty Cpn* orbitals within e;. All four levels are included in the active space of the CAS(10/10) calculation, and the natural orbital occupation numbers in Table 3 indicate that both bonding trends are strongly reinforced at the CASSCF level. Within the representation ai, the highest doubly occupied orbital, predominantly 3da, does not possess an empty Cpn counterpart, and the correlating orbital in this representation is an almost pure 4d orbital (apart from some small contributions from the C-H bonds on the Cp rings). The effect of correlation on the bonding can be illustrated by comparing the populations of the relevant orbitals at the SCF and CASSCF level (Table 4) or by comparing the total density obtained from both wavefunctions (Figure 1). We especially notice the large increase of the Fe d.n population in the CASSCF as compared to the SCF wavefunction. Its population is more than doubled, indicating that the SCF wavefunction in fact captures less than half of the charge donation from the Cp n orbitals into the empty 3d.n shell. This drift of ligand electrons into the metal is counteracted to a smaller extent by an increased back-donation from the doublyoccupied 3dd orbitals into Cpn*, and the Cpn populations in Table 4 reflect the sum of both bonding mechanisms. As could be expected, the Cp 4 population is much less affected by correlation. The 3d-4d excitations are of course not reflected by the populations in Table 4 but are manifested by the general expansion of the density around the metal in Figure 1. A similar figure has been published before,*’ based on CASSCF and contracted CI calculations. The present CASSCF results yield a larger charge flow from the ligands to the metal and also show the effect of including a second 3d shell in the active space. The interfragment contribution to the correlation included in the CASSCF wavefunction is of course bound to vanish at an infinite metal-ligand distance. In the heterolytic dissociation limit (Fe2+ 2 C p 3 the orbitals will localize such that two of
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J. Phys. Chem., Vol. 99, No. 11, 199.5 3469 the strongly occupied orbitals become pure Cp n orbitals, while three become pure 3d orbitals, corresponding to the electronic configuration (a{)2(ei)4,lA’1 for Fe2+. The basis set superposition errors (BSSE) on the CASPT2 energy were estimated by using the full counterpoise method. Due to technical limitations it was not possible to compute the BSSE for the rings in one step, adding ghost orbitals for the metal and the second ring. Instead a two-step procedure was used with contributions from the metal and the other ring computed separately. The two corrections were then added. It is inevitable that the resulting counterpoise correction will be too large. The size of the error was controlled for basis 2, where it was found to be 1.2 kcdmol. It is likely to be smaller for basis 4. The different configurations used in the BSSE calculation were thus chosen as follows: A calculation on Fe2+(’A’1) with and without ghost basis sets on the ligands, using CAS(6/8)/CASPT2 with an active space consisting of all 3d and three 4d orbitals (one for each doubly occupied 3d orbital). A calculation on one ligand, Cp-, with and without ghost basis sets on the metal, and using SCFMP2. The same calculation again, but now with ghost basis from the other ring. The BSSE for Fe plus twice the added contributions to the ligand BSSE give the total counterpoise correction. 3.2.2. The Metal to Ring Distance. The results obtained for the vertical iron-cyclopentadienyl distance in ferrocene are given in Table 5 . This table also includes for reference the corresponding calculated total energies. It is clear from these results that different levels of correlation treatment give very different bond distances. The large positive error (0.2 A) obtained at the SCF level is consistent with the earlier results22 and is obviously due to the poor description of the covalent character of the metal-Cp bond at the SCF level. As could be expected, MP2 is not at all capable of correcting for this, leading to very large (up to 0.18 8, using basis 3) errors in the opposite direction. A considerable improvement of the results is obtained at the CAS( 10/10)/CASPT2level. The CASSCF reference now takes care of the covalent bonding between the Fe 3d orbitals and the ligands, resulting in a shortening of the metal-ring distance by 0.14 A as compared to SCF, while CASPT2 offers an improvement of the same magnitude with respect to MP2. The final CASPT2 distances are, however, still too short by 0.01 -0.05 A, depending on the size of the basis set used. This is mainly due to BSSE, however, and after adding the counterpoise corrections the error becomes positive for basis 1, while for the larger basis sets, the iron-ring distance is slightly too short: 0.01-0.02 A. This is a satisfactory result; the remaining error is of the same size as the error on the CrCO distance obtained in our earlier study of in Cr(C0)6, which was about 0.01 A? The only other calculation that gives a result of similar accuracy comes from the MCPF study of Park and A l r ~ d o f . Their ~ ~ value is 1.68 8, and is thus slightly too long. They emphasize the importance of including all valence electrons into the correlation treatment in order to arrive at a correct description of the ring to metal distance. It should, however, be noted that no BSSE correction is included in this value, which if added would make the distance 0.02-0.03 A longer. Rosch and JorgZ6have performed density functional studies. Their results show that is imperative to add nonlocal corrections in order to arrive at a meaningful distance. The results strongly indicate that all major electronic features of ferrocene are included in the chosen 10-orbital active space. The final CASFT2 fiist-order perturbed wavefunction does not contain any excitation in or out of the reference space with a
3470 J. Phys. Chem., Vol. 99, No. 11, 1995
Pierloot et al.
TABLE 5: Fe-Cp Distance (in A) and Total Energy (in Ha) Calculated with Different Combinations of Contracted Basis Sets on Fe and C basis 1 basis 2 basis 3 basis 4 SCF MP2 CASSCF CASF'T2 CASPT2 BSSE
+
1.864 1.519 1.718 1.649 1.681
-1646.829 -1648.411 -1647.021 -1648.424 -1648.368
SCFb MP2b MCPFb LDA' L D W exptld
391 394 917 588 008
1.864 1.488 1.718 1.620 1.653 (1.648)"
- 1646.837 457 -1648.507 285 -1647.027 330 -1648.517 084 -1648.455 133 (-1648.458 860)" 1.894 1.58 1.684 1.585 1.648 1.66
1.859 1.479 1.712 1.609 1.647
-1646.843 524 -1648.550 487 -1647.033 746 - 1648.557 723 -1648.486 162
1.864 1.486 1.716 1.617 1.643
-1646.860 694 -1648.620415 -1647.049 890 -1648.629 861 -1648.581 585
Values within parentheses obtained with the ring BSSE computed in one step (see text). From ref 27. Basis set Fe(4s4p3dlf), C(4s3p), H(2s). Not corrected for BSSE. Fe 3s,3p included in the MP2 and MCPF correction treatment. DFT results from ref 28. LDA/NL with both exchange (Becke) and correlation (Perdew) nonlocal corrections. Experimental result from ref 30. TABLE 6: Basis Set Superposition Errors for Various Fe(Cph Fragments (kcdmol) basis 1 basis 2 basis 3 basis 4 (CP-) + Fe(ghost) 12.0 11.2 15.7 10.3 (CP-) (Cp)(ghost) 5.5 5.8 5.9 3.9 Fez+ (Cp)z(ghost) 2.0 3.2 1.6 1.9 sum 35.4 38.8 44.8 30.3
+ +
large coefficient (> 0.05), indicating that all near-degeneracies are indeed included in the CASSCF wavefunction. Another argument for the adequacy of the 10-orbital active space comes from the weight w of the CASSCF wavefunction in the CASPT2 first-order wavefunction. This weight gives a measure of how large a fraction of the wavefunction is treated variationally in the CASSCF calculation and how much is treated by perturbation theory. Obviously, w will decrease with an increasing basis set size and with an increasing number of correlated electrons. For ferrocene, w varies between 0.643 for basis 1 and 0.624 for basis 4. For Cr(C0)6 a smaller weight of 0.574 was obtained, with a basis set comparable to basis 4, consistent with the fact that a larger number of electrons was correlated in this case (from the formula w = (1 x)-"~, we find x = 0.0164 for ferrocene and x = 0.0170 for Cr(C0)6). This is an indication that the quality of the present treatment is indeed the same for ferrocene as it was for Cr(C0)6. The slightly larger error obtained for ferrocene should probably be connected to limitations in the one-particle basis, especially for the Cp rings, rather than to limitations in the correlation treatment. As the results in Table 5 show, the MP2 and CASPT2 bond distances are indeed strongly dependent on the basis sets used. This is not only true for the basis sets used in this work: using a slightly smaller basis set (with no d functions on C) Park and Almlof2' have calculated an MP2 distance which is considerably (up to 0.1 A) larger than the results obtained here. Another striking example is the fact that the CASPT2 bond distance becomes shorter by as much as 0.03 8, when going from basis 1 to basis 2, or by adding one f function to the Fe basis set, leaving the basis on the Cp rings invariant. Increasing the basis set further (bases 3 and 4) induces a further bond strengthening. Only part of the bond-length variations can be accounted for by an inspection of the basis set superposition errors. In Table 6 we have collected the value of the BSSE obtained with the different basis sets. The dominant part obviously comes from the Cp ligands (this was also the case for the carbonyls). As could be expected, the (Cp-)z part of the BSSE is drastically increased when going from basis 1 to basis 3, due to the increased flexibility of the Fe ghost basis. For basis 3, the
+
-
TABLE 7: Binding Energy (kcdmol) of Ferrocene Computed from the Heterolytic Dissociation Fe(Cp)2 FeZ+(lA;) 2Cpbasis 1 basis 2 basis 3 basis 4 SCF 619 619 622 622 MP2 751 781 796 778 CASSCF 657 658 661 660 CASPT2 715 750 764 748 CASPT2 BSSE 680 711 (712)" 719 717 CASPT2 BSSE + rc 685 716 (719)" 726 724 exptl 723 f lob Values within parentheses obtained with the ring BSSE computed in one step (see text). Experimental results from refs 32 and 33.
+
+ +
(Cp-)2 BSSE is as large as 43 kcaYmol, while the Fe2+ BSSE is small. This clearly represents an unbalanced situation. The balance can partly be restored by increasing the size of the carbon basis set. Yet, the numbers in Table 6 indicate that an even larger basis set on Cp, containing additional d and f polarization functions might be necessary to further improve the result for the iron-ring distance. For comparison it may be noted that the total BSSE obtained by Parks and Almlof was 31 kcaymol, which is about the same as the present basis 4 value.27 It is, however, not clear whether their correction also includes the ring-ring contribution, which amounts to 7 kcaY mol. 3.2.3. The Ferrocene Binding Energy. Charge distributions in molecules that are dominated by ionic bonds are more similar to the corresponding free ions than they are to the homolytic dissociation products. Consequently, it is easier to compute the binding energy corresponding to heterolytic dissociation, since in this case the changes in correlation and relativistic energies are small and less error prone (see, for example, ref 31). The correction to neutral dissociation products is then made by using experimental values for electron affinities and ionization potentials. We shall, however, show that with the present method the homolytic and heterolytic approach yields the same results. For ferrocene, the heterolytic dissociation path is: Fe(Cp)2 Fe*+(lA{) 2Cp-. The binding energy is obtained as the energy difference between ferrocene and the dipositive iron atom plus two cyclopentadienyl anions. Thus, the more difficult issue of computing the promotion energy of iron and the electron affinity of the cyclopentadienyl radical is avoided. The geometry of the anion was optimized by using the same basis set and procedure as was used for the cyclopentadienyl radical. CAS(6/8)/CASPT2 calculations were performed for Fe2+ and MP2 calculations for Cp-. The results are presented in Table 7.
-
+
J. Phys. Chem., Vol. 99, No. 11, 1995 3471
Chemical Bonding in Ni(C2b) and Ferrocene TABLE 8: Binding Energy (kcaVmol) of Ferrocene Com uted from the Heterolytic Dissociation Fe(Cp)z Fez+( D) 2Cp-
s +
~
basis 1 CASPT2 CASPT2 CASET2 CASPT2
+ BSSE
+ BSSE + rc + BSSE + rc + 3s,3p
617 582 588 591
basis 2
-
,LO86
~~~~
basis 3 basis 4
650 665 611 (613)" 620 617(619)" 626 620 (622)" 630 636 f 10
f i Z 3
648 618 624 628
The experimental value for the dissociation energy has been obtained from the compilation of Richardson et Their studies reported the heterolytic dissociation energy to 636 k c d mol. The value in the table has been corrected with the lAi(l1)5D energy difference for Fe2+.34 The experimental value for the heterolytic dissociation energy was obtained from the homolytic value, 157 f 3 kcaYmo1, using the electron affhity of Cp, 38.4 & 5 kcdmol, and the ionization energy from Fe(d6s2, 5D) to Fez+(d6, 5D), 555 f 2 kcdmol. The computed value is in agreement with experiment. This is to some extent fortuitous, since we have not included the zero-point energy corrections, which would probably decrease the theoretical value by a few kcdmol. On the other hand, the BSSE correction has been overestimated. A complete calculation would have given a slightly larger binding energy, as indicated by the values given within parentheses for basis 2. Thus there seems to be some cancellation of errors. It should also be emphasied that the sizable BSSE correction (30 kcaYmol with basis 4) lends some uncertainty to the computed binding energies. The results energies are, however, surprisingly insensitive to the basis set. It is only with basis 1 that a result is obtained which is far from the experimental value. Thus it seems that the most important extension of the basis set is the addition of the first f-type function to the iron basis. The SCF value for the binding energy is more than 100 kcdmol too small, while MP2 overestimates the binding with about 50 kcdmol. We can also compute the heterolytic dissociation to the 5D ground state of Fe2+. The results obtained are presented in Table 8. The same trends are seen here, but the error is now somewhat larger, even if the computed value is still within the error limits of the experiment. The difference can be traced to the energy separation between Fe2+(5D)and the promoted state Fe2+ (l A!). The computed value is 92.4 kcaYmo1, which is 5.6 kcdmol (0.24 eV) larger than experiment. This is a rather large, but not untypical, error in CASPT2 computed excitation energies for transition-metal ions.8 Note, that it is now essential to include core correlation effects, since they will influence the energy separation by about 3 kcdmol. Finally, we can also compute directly the energy for the homolytic dissociation to Fe(d6s2, 5D) 2Cp. This is more difficult and is often avoided, since it involves (indirectly) the computation of the electron affinity of Cp and the double ionization of Fe. However, it was shown in our earlier work on the carbonyls and also above for Ni(C&) that a direct ground state to ground state calculation of the dissociation energy is possible with the CASSCF/CASPT2 approach, since the atomic promotion energy is accurately reproduced. However, here double ionization is involved, making the test on the method even harder. For the homolytic dissociation we need to compute the energy of a Cp radical. This system is Jahn-Teller distorted and the calculation starts by a geometry optimization. This optimization was performed by pointwise calculations at the CASPT2 level. Basis 4 was used, but with an additional p-type function on the
+
1
.
~
~
e
4
7
t
6 0
3 1.473 8
3
1.368 CP-
exptl Values within parentheses obtained with the ring BSSE computed in one step (see text). Experimental results from refs 32 and 33.
~
,(1.081)
Cp(1)
Cp(I1)
Figure 2. Optimized geometries (CASPT2) for the cyclopentadienyl anion and the two forms of the cyclopentadienyl radical. Distances in angstroms.
+
TABLE 9: Homo1 tic Dissociation Energy (kcaymol) of Ferrocene to Fe(d6s1,SD) 2Cp 3
~
basis 1 basis 2 basis 3 basis 4
~
CASPT2 CASET2 CASPT2 CASPT2
exptl
+ BSSE + BSSE + rc + BSSE + rc + 3s,3p
176 140 140 123
206 218 167 173 168 174 150 157 157 f 3
203 173 173 156
hydrogens. All electrons (except the carbon 1s) were correlated (as in ferrocene) and the active space comprised the five n orbitals. The optimization was performed in two steps. First the D5h symmetry was kept and the CC and CH bond distances were optimized by using a two-dimensional grid. The resulting distances were R(C-C) = 1.425 A and R(C-H) = 1.081 A. The CH distance was fixed at this value in the second step, and the CCC angles were assumed to be 108". The CH bonds were assumed to bisect the CCC angles. This leaves two CC bond distances to be optimized and an implicit assumption of C2" symmetry for the radical. The 2EYstate splits up in two states in C2" of 2A2 and 2B1 symmetry. The CASPT2 optimization gave for 2A2 a structure (Cp(1) in Figure l), which can be described as allyl plus ethene, while the 2B1 state corresponds to methyl plus butadiene (Cp(II)). Figure 2 gives the bond distances. Also, Cp- was optimized by using the same procedure (in D5h symmetry). The resulting bond distances were R(C-C) = 1.423 8, and R(C-H) = 1.086 A, which is slightly shorter than the experimental bond distances in ferrocene. The two Cp radical geometries have almost identical total energies, the difference is only 0.05 kcaYmol and the energy lowering due to the Jahn-Teller distortion is 3.5 kcdmol. Qualitatively similar results have been obtained in earlier s t ~ d i e s ,although ~ ~ , ~ ~they were performed with too small basis sets (STO-3G) to be conclusive. As a byproduct of our studies, the electron affinity (EA) of the Cp radical was also obtained. Using the same basis set as in the geometry optimizations, a value of 37 kcdmol was obtained, to be compared with the experimental value 38 & 4 kcdmol (obtained from the enthalpies of formation3'). The slightly smaller basis sets used for ferrocene gives 35 kcaYmol (basis 4 gives a 0.3 kcdmol larger value than basis 1-3). These values were, however, not directly involved in the calculation of the homolytic dissociation energy, which instead used the total energy of the Cp radical computed at the equilibrium geometry by using the R O W / CASPT2 method in order to the be consistent with the active space used in ferrocene. The resulting binding energies are presented in Table 9. The CASPT2 energy for the iron atom was obtained from a CASSCF/CASPT2 calculation with 14 active orbitals. There is a sizable contribution to the binding energy from the 3s,3p semicore electrons. This was not computed directly in the complex but was estimated from its effect on the Fe(5D) to Fe2+(lAi)double ionization energy. It was assumed that a better estimate would be obtained from such an atomic
Pierloot et al.
3472 J. Phys. Chem., Vol. 99,No. 11, 1995
calcuation, which could be performed with a large basis set (1ls9p8d3f with functions in the core-valence region), than by a calculation on the complex, which would have to be performed with a smaller basis set. On the other hand, it is likely that such a procedure leads to an overestimate of the effect and therefore to a too small binding energy. The agreement with experiment is therefore to some extent fortuitous and is again a result of canceling errors (a zero-point energy correction would decrease the binding energy). Anyway, the calculations show that the method is capable in predicting the binding energy even for such a complex case as ferrocene.
4. Conclusions We have presented results from a CASSCF/CASPT2 study of the structure and bonding in Ni(C2I-b) and ferrocene. Resulting structural parameters and binding energies are in agreement with experiment: For Ni(C2h) the study gives a binding energy of 32 f 5 kcallmol, while for ferrocene a homolytic value of 156 f 5 kcallmol is obtained. The corresponding experimental estimates are 36 k 5 and 156 f 3 kcdmol, respectively. This accuracy can only be achieved with extended basis sets and a full correlation treatment, which includes also the 3s,3p semicore electrons. Their contribution to the binding energy is especially important in ionic compounds, where the metal atom has a considerable positive charge. The BSSE corrections are large even when the most extended basis sets are used. Accurate calculations of the bonding parameters must therefore be carried out with these corrections included, which makes the calculations complicated to perform. Metal to ligand bond distances computed without the BSSE corrections are 0.01-0.03 8, too short. Relativistic and core correlation contributions to the binding energies are not negligible and the estimation of these effects have rather large uncertainties. Since also the BSSE corrections are somewhat uncertain, the final estimations of the binding energies have sizable error bars, in spite of the fact that the CASPT2 method gives a good account of the correlation contribution to the binding energies. It is believed that the presented energies are accurate to about 5 kcdmol. The results, which are fully consistent with those obtained for the transition-metal carbonyls, reinforce the conclusion that the CASSCF/CASPT2method can be used to study the bond properties of transition metal complexes, and yield results that are superior to those which can be obtained with traditional single-configuration-based methods. Acknowledgment. The research reported on this article has been supported by a grant from the Swedish Natural Science Research Council (NFR) and by IBM Sweden under a joint study contract. K.P. thanks the Belgian National Science Foundation (NFWO) and the Belgium Government (DPWB) for a research grant. References and Notes (1) Roos,B. 0.The complete active space self-consistent field method and its applications in electronic structure calculations. In Advances in Chemical Physics; Ab Initio Methods in Quantum Chemistry-Il; Lawley, K. P., Ed.; John Wiley & Sons: Chichester, England, 1987; Chapter 69, p 399.
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