5690
J. Phys. Chem. 1996, 100, 5690-5696
Basis Set Effects in Density Functional Calculations on the Metal-Ligand and Metal-Metal Bonds of Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 A. Rosa,† A. W. Ehlers,‡ E. J. Baerends,*,‡ J. G. Snijders,‡ and G. te Velde‡ Dipartimento di Chimica, UniVersita` della Basilicata, Via N. Sauro, 85, 85100 Potenza, Italy, and Afdeling Theoretische Chemie, Vrije UniVersiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands ReceiVed: NoVember 17, 1995; In Final Form: January 23, 1996X
The basis set superposition error (BSSE) for the Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 bonds is calculated for a large variety of STO basis sets. All investigated metal basis sets, being at least TZ for 3d and DZ for 4s are adequate. Ligand basis sets of TZD quality or better are required in order to have a low BSSE (6-8 kJ/mol or less) for both the metal-ligand and the metal-metal bond. When the ligand s, p basis set is of double-ζ quality, the BSSE is significant for the metal-ligand bond (16-22 kJ/mol depending on the polarization functions), but it is partly canceled by a basis set incompleteness error of opposite sign. For the metal-metal bond, the BSSE for DZ s, p ligand bases is unacceptably large (33-57 kJ/mol), leading to much too high bond energies if no correction for BSSE is applied. In general, the bond energies after correction for BSSE are rather stable. It is remarkable that for the metal-metal bond, but not for the metal-ligand bond, there is for all pure s, p ligand bases after correction for the BSSE a discernible basis set incompleteness error (ca. 15 kJ/mol), which only disappears after adding at least one polarization function. Agreement of the converged results for both geometries and bond energies with experiment is excellent.
Introduction The breaking and forming of M-CO and M-M bonds are recurrent reactions in organometallic chemistry, and accurate estimates of the corresponding bonding energies are essential for the understanding of many processes, ranging from organometallic synthesis to catalysis, surface chemistry, photophysics, and photochemistry (see, for example, refs 1-3 and references therein). Unfortunately, the first bond dissociation energies (FBDE’s) of a carbonyl ligand as well as the metal-metal bond energies are hard to determine experimentally. Even for basic compounds such as the hexacarbonyls M(CO)6 (M ) Cr, Mo, W), the FBDE is still controversial.4-6 For Mn2(CO)10, which is generally taken as a prototype for organometallic compounds containing a metal-metal bond, the experimental7 values for D(Mn-Mn) published between 1970 and today range from 79 ( 14 to more than 176 kJ/mol. Accurate theoretical data for the energetics of M-CO and M-M bonds could thus constitute a much needed supplement to the scarce experimental data. Theoretically, M-CO FBDE’s can be calculated successfully by highly correlated ab initio methods such as the coupled cluster method (CCSD(T))8 or the modified coupled-pair functional (MCPF)9 method. However, the calculations are already rather expensive for first-row transition-metal complexes and would be prohibitive if one attempts to compute the M-M bonding energies in organometallic complexes, such as Mn2(CO)10. Density functional theory (DFT), on the other hand, has proven to be a powerful tool in studies of organometallic thermochemistry,10 especially after the development of nonlocal DFT methods.11 The performance of various functions with density-gradient corrections in predicting geometrical and thermodynamic properties has been extensively tested for a number of organometallic systems,10,12,13 providing insight into the accuracy they can reach. The effects of basis set incom†
Universita` della Basilicata. Vrije Universiteit. X Abstract published in AdVance ACS Abstracts, March 15, 1996. ‡
0022-3654/96/20100-5690$12.00/0
pleteness on these properties have not received, on the contrary, any attention, despite the fact that analysis of these effects is relevant for the calibration of a theoretical method. That the basis set size may have a large influence on bonding energies recently became evident to us when calculating the metal-metal dissociation energy in Mn2(CO)10. We found, in fact, a value significantly different14 from that previously computed by Folga and Ziegler using the same DFT scheme but a larger basis set.12 Therefore, it seems timely to consider the question of basis set effects more closely. With this aim, here we investigate the effects of basis set incompleteness, including basis set superposition error (BSSE), on the energetics of Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 bonds in Cr(CO)6 and Mn2(CO)10, which are taken as prototypes for mononuclear and binuclear carbonyl complexes, respectively. The effects of basis set size on the predicted geometries of these complexes and of the related pentacarbonyl fragments, Cr(CO)5 and Mn(CO)5, are also investigated. Method and Computational Details The calculations reported in this paper have been carried out by using the new parallel version of the Amsterdam Density Functional (ADF) program.15-17 The computational scheme is characterized by a density-fitting procedure to obtain the Coulomb potential15a and by sophisticated 3D numerical integration techniques16 for the evaluation of the Hamiltonian matrix elements, including those of the exchange-correlation potential. The inner cores of chromium and manganese (1s2s2p) as well as carbon and oxygen (1s) were treated by the frozen core approximation.15a The frozen core orbitals have been taken from atomic calculations in a very large basis set. To make the valence orbitals orthogonal to these cores, the valence basis sets have been augmented by a single-ζ STO for each atomic core orbital. In our computational scheme, the accuracy of the calculations is determined, apart from the basis set quality, by several computational approximations such as the number of fit functions, the level of frozen core approximation, the numerical integration accuracy, and convergence criteria (for © 1996 American Chemical Society
Basis Set Effects in Density Functional Calculations
J. Phys. Chem., Vol. 100, No. 14, 1996 5691
TABLE 1: Orbital Expansion Bases for Carbon and Oxygen (All STO’s) carbon
2s 2s′ 2s′′ 2s′′ 2p 2p′ 2p′′ 2p′′′
DZ 1.24 1.98
TZ 1.28 2.10 4.60
0.96 2.20
0.82 1.48 2.94
3d 4f
oxygen s, p Basis QZ 1.06 1.52 2.68 4.20 0.98 1.44 2.60 6.51
DZ 1.70 2.82
TZ 1.72 2.88 7.58
1.30 3.06
1.12 2.08 4.08
Polarization Functions DF D 2.20 2.00 2.06
D 2.20
TABLE 4: Charge Density Fitting Bases for Chromium and Manganese (All STO’s)
QZ 1.70 2.48 4.31 5.87 1.14 1.82 3.45 7.56
DF 2.00 2.50
TABLE 2: Orbital Expansion Bases for Chromium and Manganese (All STO’s) chromium 3s 3s′ 3p 3p′ 3d 3d′ 3d′′ 4s 4s′ 4s′′ 4p 4p′ 4p′′ 4f 4f′
manganese
II
IV
IVA
IVB
II
IV
IVA
IVB
3.25 5.00 2.85 4.65 1.24 2.70 5.70 1.00 1.75
3.45 5.30 2.85 4.65 1.24 2.70 5.70 0.85 1.30 2.10 1.30
3.45 5.30 2.85 4.65 1.24 2.70 5.70 0.85 1.30 2.10 0.85 1.30 2.10
3.45 5.30 2.85 4.65 1.24 2.70 5.70 0.85 1.30 2.10 0.85 1.30 2.10 0.75 1.20
3.50 5.50 3.00 4.90 1.32 2.85 5.95 1.00 1.80
3.45 5.30 3.00 4.90 1.32 2.85 5.95 0.90 1.35 2.15 1.35
3.45 5.30 3.00 4.90 1.32 2.85 5.95 0.90 1.35 2.15 0.90 1.35 2.15
3.45 5.30 3.00 4.90 1.32 2.85 5.95 0.90 1.35 2.15 0.90 1.35 2.15 0.80 1.25
1.30
1.35
TABLE 3: Charge Density Fitting Bases for Carbon and Oxygen (All STO’s) 1s 2s 2s 2s 3s 3s 3s 3s 3s 2p 2p 3p 3p
C
O
10.80 12.11 8.22 5.58 5.55 4.00 2.89 2.08 1.50 10.00 6.04 5.39 3.52
14.72 15.69 10.22 6.66 6.38 4.44 3.09 2.15 1.50 11.44 6.70 5.81 3.70
3p 3p 3d 3d 3d 3d 3d 4f 4f 4f 5g 5g
C
O
2.30 1.50 10.00 6.22 3.87 2.41 1.50 8.00 4.90 3.00 5.50 3.50
2.36 1.50 10.36 6.39 3.94 2.43 1.50 8.00 5.29 3.50 5.50 3.50
the SCF procedure as well as for geometry optimization). With respect to these computational approximations, the level of the accuracy of the calculations reported in this study is very high. We used a very extensive density fit basis, close to saturation, and very high accuracy for the integration scheme. We specify in Table 1 the basis sets used for carbon and oxygen. The basis sets used for the chromium and manganese atoms are described in Table 2. The fit functions are reported in Tables 3 and 4. The metal bases II and IV are those available in the atomic database of the ADF program package,15-17 and the ligand bases DZ, DZD, TZD, and TZDF are from the same database where they are notated as II, III, IV, and V, respectively. The exponents of the Slater-type orbitals (STO’s) of these orbital basis sets have been obtained by a least-squares fit to numerical atomic orbitals from a Herman-Skillman-type numerical atomic calculation.18 However, the QZ s, p basis for carbon and oxygen
1s 2s 3s 3s 4s 4s 5s 5s 5s 6s 6s 6s 7s 7s 7s 7s 2p 3p 4p 5p 5p 6p
Cr
Mn
37.40 39.04 36.80 25.26 22.97 16.48 14.76 10.93 8.10 7.20 5.46 4.15 3.67 2.84 2.20 1.70 28.60 21.97 16.77 12.84 8.20 6.32
39.60 41.34 38.97 26.74 24.32 17.45 15.63 11.58 8.58 7.62 5.78 4.39 3.89 3.01 2.33 1.80 30.20 23.17 17.66 13.51 8.61 6.63
6p 6p 7p 3d 4d 5d 5d 6d 6d 6d 7d 4f 5f 5f 5f 6f 5g 5g 5g 5g 5g
Cr
Mn
4.18 2.77 2.15 24.40 18.89 14.66 9.47 7.39 4.95 3.31 2.60 15.60 10.79 6.25 3.62 2.54 11.40 7.79 5.32 3.63 2.48
4.39 2.90 2.25 25.75 19.88 15.39 9.92 7.72 5.16 3.45 2.70 16.35 11.32 6.56 3.80 2.67 11.90 8.17 5.60 3.85 2.64
is due to Clementi and Roetti.19 The polarization functions cannot be determined by optimization to numerical atomic solutions, since, e.g., the unoccupied C and O numerical 3d orbitals, and Cr and Mn 4f, are too diffuse. The exponent of the 4f polarization functions on the metal atoms has been chosen such that the function attains its maximum value at somewhere between 1/3 and 1/2 times the bond length. The choice of 3d and 4f polarization functions on C and O has been guided by the optimizations of McLean and Yoshimine20 on linear molecules. The present investigation makes full use of the self-consistent implementation21 of the density gradient (or nonlocal) corrections in the geometry optimization21b as well as energy calculation.21a The nonlocal corrections adopted were based on Becke’s functional for exchange11b and Perdew’s functional for correlation.11a The effects of ligand and metal basis sets are considered separately. Ligand basis set effects are analyzed using for chromium and manganese atoms the large IVA basis (see Table 2), while metal basis set effects are analyzed using for carbon and oxygen a TZD basis. Results and Discussion Geometries of Cr(CO)6 and Mn2(CO)10. The geometry of the chromium hexacarbonyl was optimized within Oh symmetry constraints. Table 5 shows the Cr-C and C-O bond lengths obtained by DFT calculations using different basis sets, together with the experimental values. With all basis sets, the Cr-C bond is predicted in close agreement with the experiment,22 with a maximum deviation from the experimental estimate of 0.01 Å. Comparison with the values obtained by the CCSD(T)23 and MP224 methods, 1.883 and 1.939 Å, respectively, indicates that DFT provides, in this case, a better fit to experiment. Although there are only minor differences for the optimized interatomic Cr-C distance among the ligand basis sets, the change due to adding a d polarization function on C and O, namely, a lengthening of the bond, is much larger than the change occurring upon saturating the s, p basis. Adding a 4f function only leads to a negligible shortening of the bond. The Cr-C bond distance shows only marginal changes among the metal basis sets II, IV, IVA, and IVB, which means that the effects of saturating the 4s and 4p basis as well as of adding a 4f polarization function appear to be negligible.
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TABLE 5: Theoretical and Experimental Bond Lengths (Å) of Cr(CO)6 and CO basis M
C, O
IVA IVA IVA IVA IVA IVA IVA IVA IVA II IV IVA IVB expta
DZ DZD DZDF TZ TZD TZDF QZ QZD QZDF TZD TZD TZD TZD
a
Cr(CO)6 r(M-C) r(C-O) 1.913 1.927 1.925 1.914 1.929 1.928 1.914 1.932 1.929 1.927 1.928 1.929 1.927 1.918
CO r(C-O)
1.177 1.154 1.152 1.179 1.155 1.151 1.179 1.155 1.151 1.155 1.154 1.155 1.155 1.141
1.158 1.137 1.135 1.161 1.137 1.133 1.159 1.137 1.133
1.128b
Reference 22. b Reference 25.
As for the optimized C-O bond length, the basis set effects are identical on going from Cr(CO)6 to free CO. The calculated r(C-O) values indicate that there is hardly any difference among basis sets DZ, TZ, and QZ. Adding one 3d has a sizable effect in the right direction, by shortening the C-O bond by ca. 0.02 Å (-2%). Extension of the polarization set with a 4f is less effective than adding the d polarization function but does improve the calculated C-O bond length. Analogous results were obtained in a previous density functional (Hartree-FockSlater) study on basis set effects on the spectroscopic properties of CO.26 We now turn to the Mn2(CO)10 system. The geometry of Mn2(CO)10 has been investigated by electron diffraction in the gas phase27 and by X-ray diffraction at room temperature28 and at 74 K.29 All these experiments lead to nearly the same geometry, which is consistent with a near-D4d symmetry. Density functional calculations by Folga and Ziegler12 give the eclipsed structure (D4h) higher in energy by 142.4 kJ/mol than the staggered one. For the above reasons, only the conformation of Mn2(CO)10 corresponding to the staggered structure of D4d symmetry has been considered in this study. Table 6 collects the results of our calculations along with the structural data obtained by X-ray diffraction at 74 K29 and by electron diffraction in the gas phase.27 The experimental parameters shown in the table are averaged over bond lengths and bond angles that would be equivalent in perfect D4d symmetry. The uncertainties are given in brackets.
According to the data listed in Table 6, the calculated bond angles appear to be insensitive to basis sets. The trend of the calculated Cax-Oax and Ceq-Oeq bond lengths vs the basis sets parallels that of the C-O interatomic distances in Cr(CO)6 and free CO and does not need further comment. The Mn-Cax and Mn-Ceq bond distances also show the same basis set dependency as r(Cr-C) in Cr(CO)6. They are theoretically predicted in very good agreement with the experimental values even with pure s, p ligand basis sets. Addition of d polarization functions on the CO ligands has a small but noticeable lengthening effect of ca. 0.01 Å (+0.8%). Adding a 4f function has less effect than adding a d function. Independently of the basis set used, the calculated Mn-Mn bond length is slightly longer compared to the gas-phase experimental value of 2.977 Å.27 The other experimental determinations of the Mn-Mn bond length by X-ray diffraction at room temperature28 and at 74 K29 give shorter bonds, i.e., 2.923 and 2.895 Å, respectively. Previous theoretical studies stressed that the Mn-Mn bond has a very shallow potential well. Therefore, small differences in the electrostatic repulsion between the Mn(CO)5 fragments may have a large influence on the Mn-Mn bond distance. The Mn-Mn theoretical values listed in Table 6 show that increasing the size of the ligand bases leads to a longer Mn-Mn bond. In going from DZ to TZ and to QZ basis, the lengthening of the Mn-Mn bond upon saturating the ligand s, p basis is clearly distinguishable but not uniform. Addition of a d polarization function to the unsaturated DZ and TZ ligand s, p bases also increases the MnMn bond length but has little effect with the QZ basis. The same effects, although less pronounced, are observed upon extension of the polarization set with one 4f function. Geometries of M(CO)5 (M ) Cr, Mn). Previous experimental30 and theoretical31 studies have shown that the M(CO)5 species with M ) Cr, Mn possess a square-pyramidal structure of C4V symmetry, and this conformation has been adopted in the present study. Low-temperature matrix IR spectroscopic studies30 reveal that the axial M-CO distance should be shorter than the equatorial M-CO bonds and, based on intensity ratio measurements, suggest that the bond angles R(Cax-M-Ceq) in Cr(CO)5 and Mn(CO)5 are in the range 90-95° and 93-99°, respectively. The calculated geometries for chromium and manganese pentacarbonyls are reported in Table 7. The optimized interatomic distances show that the basis set effects on M-C and C-O bonds in the pentacarbonyl fragments are the same as in
TABLE 6: Calculated and Experimental Structural Parameters for Mn2(CO)10 basis
interatomic distances, Å
bond angles, deg
M
C, O
Mn-Mn
Mn-Cax
Mn-Ceq
Cax-Oax
Ceq-Oeq
Cax-Mn-Ceq
Oeq-Ceq-Mn
IVA IVA IVA IVA IVA IVA IVA IVA IVA II IV IVA IVB expta exptb
DZ DZD DZDF TZ TZD TZDF QZ QZD QZDF TZD TZD TZD TZD
2.951 2.994 3.001 2.982 3.034 3.046 3.031 3.015 3.032 3.017 3.010 3.034 3.034 2.895(1) 2.977(11)
1.807 1.818 1.816 1.810 1.822 1.820 1.805 1.822 1.820 1.820 1.819 1.822 1.815 1.820(3) 1.803(3)
1.852 1.864 1.862 1.856 1.871 1.870 1.857 1.872 1.869 1.869 1.868 1.871 1.867 1.859(3) 1.873(5)
1.179 1.156 1.156 1.181 1.158 1.154 1.182 1.157 1.154 1.158 1.158 1.158 1.158 1.150(4) 1.147(2)
1.176 1.153 1.152 1.178 1.155 1.151 1.179 1.154 1.151 1.155 1.155 1.155 1.154 1.140(4) 1.147(2)
93.4 93.6 93.5 93.5 93.5 93.6 93.6 93.8 93.9 93.4 93.4 93.5 93.2 93.94(10) 93.4(5)
177.5 177.6 177.5 177.1 177.1 177.3 176.9 177.7 177.7 177.0 177.0 177.1 176.8 177.77(19) 178.6(1.1)
a
X-ray diffraction, ref 29. b Electron diffraction, ref 27.
Basis Set Effects in Density Functional Calculations
J. Phys. Chem., Vol. 100, No. 14, 1996 5693
TABLE 7: Calculated Bond Lengths (Å) and Angles (deg) of the Pentacarbonyls M(CO)5 (M ) Cr, Mn) DZ IVA Cr(CO)5
Mn(CO)5
DZD IVA
DZDF IVA
TZ IVA
TZD IVA
TZDF IVA
QZ IVA
QZD IVA
QZDF IVA
TZD II
TZD IV
TZD IVB
r(M-Cax) 1.833 1.845 1.845 1.833 1.847 1.845 1.836 1.847 1.846 1.843 1.845 1.839 r(M-Ceq) 1.910 1.924 1.926 1.908 1.926 1.925 1.911 1.927 1.926 1.924 1.924 1.922 r(C-Oax) 1.187 1.162 1.161 1.189 1.163 1.159 1.189 1.163 1.159 1.163 1.163 1.163 r(C-Oeq) 1.179 1.155 1.154 1.182 1.157 1.153 1.184 1.156 1.153 1.157 1.156 1.157 R(Cax-M-Ceq) 91.3 91.3 91.4 91.6 91.3 91.4 91.1 91.4 91.4 91.3 91.0 90.8 β(M-Ceq-Oeq) 178.7 179.3 178.9 179.4 179.4 179.4 178.6 179.3 179.3 179.4 179.0 178.5 r(M-Cax) 1.811 1.826 1.824 1.813 1.831 1.829 1.812 1.829 1.826 1.829 1.829 1.826 r(M-Ceq) 1.859 1.870 1.868 1.860 1.874 1.872 1.858 1.876 1.873 1.873 1.873 1.870 r(C-Oax) 1.181 1.157 1.157 1.183 1.158 1.155 1.184 1.158 1.155 1.158 1.159 1.159 r(C-Oeq) 1.178 1.154 1.153 1.180 1.155 1.151 1.180 1.155 1.151 1.155 1.156 1.155 R(Cax-M-Ceq) 95.9 95.7 95.8 95.8 95.8 95.9 95.7 95.8 95.8 95.7 95.8 95.8 β(M-Ceq-Oeq) 177.5 177.7 177.7 177.8 177.7 177.7 177.5 177.9 177.9 177.7 177.7 177.9
TABLE 8: Experimental Cr(CO)5-CO and (CO)5Mn(CO)5 Bond Dissociation Enthalpies (kJ/mol) methoda EI
E/EI
TE
Cr(Co)5-CO
KS
EI/PES
PAC
Mn(CO)5-Mn(CO)5
104 ( 3e 79 ( 14n
96 ( 13f
94g
161.9d g154h
LP 154 ( 8c
168.2b 171i
159 ( 21l
g176m
a Key: EI ) electron impact mass spectrometry; E/EI ) equilibrium studies using electron impact mass spectrometry; TE ) thermochemical estimate; KS ) kinetic studies in solution; EI/PES ) electron impact mass spectroscopy in conjunction with photoelectron spectroscopy; PAC ) photoacoustic calorimetry; LP ) laser pyrolysis. b Reference 4. c Reference 5. d Reference 6. e Reference 7a. f Reference 7b. g Reference 7c. h References 7d and 7e. i Reference 7f. l Reference 7g. m Reference 7h. n Reference 7b.
the overall molecules Cr(CO)6 and Mn2(CO)10. The values listed in Table 7 emphasize the importance of ligand basis effects, particularly of a 3d polarization function, over metal basis effects. It is worth noting that all bases predict the axial M-CO distance shorter than the equatorial M-CO bonds, in agreement with experiment. There are only marginal differences (less than 1°) for the optimized bond angles among the basis sets, which confirms the low sensitivity of these geometrical parameters to the quality of the basis set. Our calculated R(Cax-M-Ceq) bond angles are in Cr(CO)5 and Mn(CO)5 slightly larger than 91° and 95° respectively, and all in the experimental ranges. In the case of chromium pentacarbonyl, this bond angle is calculated as slightly smaller than 90° at the MP2 level.24 Energetics of the Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 Bonds. The Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 bond dissociation energies are defined as the reaction enthalpies corresponding to the process 1 and 2, respectively.
Cr(CO)6 f Cr(CO)5 + CO
(1)
Mn2(CO)10 f 2Mn(CO)5
(2)
Table 8 summarizes the experimental enthalpies of processes 1 and 2 obtained by different experimental techniques. There are three experimental data available for the FBDE of Cr(CO)6; two are estimates based on kinetic studies on solution,4,5 and the third is an accurate gas-phase value based on pulsed laser pyrolysis obtained by Lewis et al.6 As for the Mn-Mn bond enthalpies, the spread in the experimental estimates is apparent from the data of Table 8. The experimental values range from 79 ( 14 kJ/mol obtained using electron impact spectrometry techniques7b to more than 176 kJ/mol obtained by laser pyrolysis.7h The fragment-oriented approach of our DFT computational scheme allows us to evaluate the energies of reactions 1 and 2 following two different routes. These are illustrated in Schemes 1 and 2 for the Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 bond dissociation energies, respectively. According to the first route, the bonding dissociation energies are calculated by just subtract-
SCHEME 1
SCHEME 2
ing the energies of the optimized structures of the overall molecules (Cr(CO)6 and Mn2(CO)10) and of the polynuclear fragments (Cr(CO)5 and CO, and two Mn(CO)5). According to the alternative route, the Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 bond dissociation energies are computed in two steps. First, we compute the energy gained when “snapping” the bond, i.e., the snapping energies E(Cr-CO) and E(Mn-Mn) obtained by building Cr(CO)6 from Cr(CO)5 and CO and Mn2(CO)10 from two Mn(CO)5 fragments, the fragments being in the conformation they assume in the optimized structures of the overall molecules. In the next step, we compute the energy gained when the isolated fragments relax from the conformation taken up in the overall molecules to their optimal ground-state structures. In this study, the second route has been used, since the BSSE’s associated with the Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 bond energies can be computed directly using the appropriate polynuclear fragments, Cr(CO)5 and CO and Mn(CO)5, respectively, via the counterpoise method.32 It also
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TABLE 9: Energy Contributions to the First Bond Dissociation Energy (kJ/mol) of Cr(CO)6 for Different Basis Setsa basis set M C, O IVA IVA IVA IVA IVA IVA IVA IVA IVA II IV IVA IVB a
DZ DZD DZDF TZ TZD TZDF QZ QZD QZDF TZD TZD TZD TZD
Ee(Cr-CO)
ER1
ER2
De(Cr-CO) w/o correction for BSSE
196.9 192.3 193.7 203.6 186.5 186.2 192.5 184.2 185.3 184.1 183.8 186.5 (186.6) 186.2
-8.4 -12.7 -8.1 -8.9 -8.6 -8.6 -8.6 -10.0 -8.5 -8.5 -9.5 -8.6 (-8.5) -9.5
-1.2 -1.1 -1.1 -1.1 -1.2 -1.1 -1.5 -1.6 -1.5 -1.1 -1.1 -1.2 (-1.3) -1.2
187.3 178.5 184.5 193.6 176.7 176.5 182.4 172.6 175.3 174.5 173.2 176.7 (176.8) 175.5
BSSE CO + Cr(CO)5 (ghost) Cr(CO)5 + CO (ghost) -13.0 -10.0 -9.4 -15.6 -3.0 -2.5 -7.1 -1.0 -0.9 -2.2 -2.8 -3.0 (-3.0) -3.3
-8.9 -6.0 -6.7 -5.8 -3.0 -0.8 -2.2 -1.2 -0.7 -2.5 -3.1 -3.0 (-2.9) -1.6
De(Cr-CO) corrected for BSSE 165.4 162.5 168.4 175.2 170.7 173.2 173.1 170.4 173.6 169.8 167.3 170.7 (170.9) 170.6
The values in parentheses are calculated using a somewhat smaller fit set; see text.
TABLE 10: Energy Contributions to the Mn-Mn Bond Dissociation Energy (kJ/mol) of Mn2(CO)10 for Different Basis Sets M IVA IVA IVA IVA IVA IVA IVA IVA IVA II IV IVA IVB
basis set C, O DZ DZD DZDF TZ TZD TZDF QZ QZD QZDF TZD TZD TZD TZD
Ee(Mn-Mn)
2ER1
De(Mn-Mn) w/o correction for BSSE
2BSSE Mn(CO)5 + Mn(CO)5 (ghost)
De(Mn-Mn) corrected for BSSE
165.4 154.9 156.5 126.6 126.6 120.4 114.7 120.4 119.7 126.5 126.0 126.6 129.7
-8.0 -5.4 -5.4 -7.6 -5.6 -4.8 -6.8 -4.8 -4.2 -5.6 -5.4 -5.6 -5.8
157.4 149.5 151.1 119.0 121.0 115.6 107.9 115.6 115.5 120.9 120.6 121.0 123.9
-56.6 -33.0 -34.0 -18.8 -8.0 -2.0 -6.6 -3.4 -1.6 -7.2 -6.4 -8.0 -7.8
100.8 116.5 117.1 100.2 113.0 113.6 101.3 112.2 113.9 113.7 114.2 113.0 116.1
allows for the analysis of the basis set effects on both components of the bonding energy, the snapping energy and the relaxation energy. The calculated Cr(CO)5-CO and (CO)5Mn-Mn(CO)5 bond energies are given in Tables 9 and 10 where we break down the various contributions, i.e., snapping energy, relaxation energies, and BSSE’s, for different basis sets. Our computed bonding energies represent, however, only the electronic contribution to the reaction enthalpies of eqs 1 and 2. Thus, thermal corrections and contributions due to the vibrational zeropoint corrections are not included. For reactions 1 and 2, these corrections have been estimated to be very small, less than 2 kJ/mol for reaction 123 and about 1 kJ/mol for reaction 2.12 Therefore, our bond energies should be directly comparable to the experimental ∆H298 estimates. We will first discuss basis set effects on the Cr(CO)5-CO bond energy. As for the snapping energy, our data indicate that a metal basis set beyond II has virtually no effect; the largest difference in the calculated values only amounts to 2.7 kJ/mol. Ligand basis effects appear to be more important. By observing the energy snapping trends along the purely s, p basis sets DZ, TZ, and QZ as compared to the series DZD, TZD, and QZD, one will notice that the difference between TZ and QZ is still large (11 kJ/mol), but it is much reduced between TZD and QZD (2.3 kJ/mol). Addition of a 4f function on C and O always has a negligible effect. When we compare TZD basis with our largest ligand basis, QZDF, it appears that both the s, p sets and the polarization set are practically converged in the TZD basis. Since the Cr(CO)5 fragment is only slightly distorted from its equilibrium geometry, the calculated ER1 value is small compared to the bond energy. The CO relaxation energy, ER2, is almost negligible in all bases. The bonding energy, De(CrCO), obtained combining ER1 and ER2 with the snapping energy
Ee(Cr-CO), shows approximately the same basis set dependence as Ee(Cr-CO). The De(Cr-CO) data (without BSSE correction) of Table 9 clearly show that both the s, p basis set and polarization set are practically converged in the TZD basis and a metal basis set beyond II only leads to minor changes in the Cr(CO)5-CO bond energy. Looking now at the basis set superposition error associated with the Cr(CO)5-CO bonding energy along the basis sets, we note a significant reduction of both CO and Cr(CO)5 contributions to the BSSE upon addition of a 3d polarization function to notably the larger (TZ and QZ) s, p bases (for the DZ s, p basis, the BSSE never gets small). Inclusion of a 4f function reduces the BSSE very little. These effects emphasize, on one hand, the incompleteness of pure s, p bases and, on the other hand, the minor importance of a 4f function in the ligand bases. In going from DZD to TZD and QZD, the total BSSE is reduced from 16 to 6 and 2 kJ/mol. Thus, while a DZD basis still suffers of incompleteness, a TZD appears to be already quite good, in line with the above-noted convergence of the s, p basis and polarization set in the TZD basis. The basis set superposition error changes very little among metal basis sets II-IVB and amounts to a maximum of 6.0 kJ/mol. This clearly indicates that the BSSE is largely determined by ligand basis set incompleteness. The BSSE is significant (in the order of 1020 kJ/mol) in the case of pure s, p bases and all bases of DZ quality, but it is small for basis sets with polarization functions of at least TZ quality. After inclusion of the BSSE, the FBDE is fairly constant, 165-175 kJ/mol, with the basis sets of TZD level, and beyond it is in the range 171-174 kJ/mol. As mentioned before, the accuracy of the calculations not only depends on the size of the basis set but also on the number of auxiliary functions used to fit the molecular density in order to obtain the Coulomb potential. The standard fit set provided,
Basis Set Effects in Density Functional Calculations along with the basis set, in the atomic database18 of the ADF program package15-17 is presumably adequate, but we have used a more extensive fit set (cf. Tables 3 and 4). To investigate the convergence of the fit sets, we repeated one calculation of the FBDE of Cr(CO)6 for a given basis set with the standard fit set of the ADF atomic database. The convergence in fit set is demonstrated by the less than 0.1 kJ change (see Table 9) when extending the fit set to its present size. In the second place, we may quote certain correction terms in the energy that are of first order in the fit error δF ) Fexact - Ffit (these terms are calculated so that only second-order errors remain). The firstorder correction terms are less than 1 kJ/mol, which is consistent with the neglected second-order terms being of order 0.1 kJ/ mol. It is interesting to compare our results with previous DFT calculations by Li et al.13b By using the same DFT scheme we have been using in this study and the DZD-IVA basis, they computed for Cr(CO)6 a FBDE of 192.0 kJ/mol, to be compared with our value of 178.5 kJ/mol with the same basis. The discrepancy between these estimates may be due to the use of an insufficient fit set by these authors. Before gauging our results for the FBDE for Cr(CO)6 against experimental data, it is useful to compare them with the values obtained from two highly sophisticated ab initio studies,23b,24 at the CCSD(T) level of theory. CCSD(T) calculations by Barnes et al.23b lead to a De value of 178.7 kJ/mol before correcting for BSSE. However, the large BSSE they computed, 23 kJ/mol, shows that their Gaussian basis is significantly more incomplete than almost all of our STO basis sets. Ehlers and Frenking24 reported a value of 191.6 kJ/mol. Because they assumed this value to be too high due to the short bond lengths of the MP2 geometries used, they repeated this calculation taking the experimental geometry of the hexacarbonyl and an assumed structure for the pentacarbonyl, leading to a decrease of the FBDE to 136.0 kJ/mol. The values reported here calculated using the optimized geometries lie between these limits. The experimental values of the first bond dissociation energy of Cr(CO)6 are reported in Table 8. They are derived from solution as well as gas-phase experiments and vary in the range of 154-168 kJ/mol. The gas-phase value of 154 ( 13 kJ/mol is derived from an estimated log A value for the factor A in the Arrhenius equation and is assumed to be too low.21 The values from solution experiments represent estimates based on the activation energy in the CO substitution process with L ) P(nC4H9)3 (168.2 kJ/mol, ref 4) and the exchange process with L ) CO (161.9 kJ/mol, ref 6). The values of the FBDE of Cr(CO)6 reported in this study show excellent agreement with the experimentally derived values, the converged theoretical results giving a value slightly above 170 kJ/mol. We shall now turn to the (CO)5Mn-Mn(CO)5 bonding energy. As may be inferred from Table 10, the basis set dependency of the snapping energy, Ee(Mn-Mn), is qualitatively similar to that observed for Ee(Cr-CO). In the case of Mn2(CO)10, however, in going from basis sets of DZ quality to basis sets of TZ quality, the snapping energy decreases much more than in Cr(CO)6. The difference between bases of TZ and QZ quality is, as in Cr(CO)6, rather small. The relaxation energy contribution to the Mn-Mn bonding energy, 2ER1, is quite small in all bases, since the (CO)5Mn• radical, similarly to the Cr(CO)5 pentacoordinated fragment, is only slightly distorted from its equilibrium geometry. The resulting MnMn bonding energy, (Ee(Mn-Mn) + 2ER1), shows a trend along the basis sets that parallels that of the snapping energy. De(Mn-Mn) (without BSSE) decreases upon improving the basis set quality, especially in going from bases of DZ quality to bases
J. Phys. Chem., Vol. 100, No. 14, 1996 5695 of TZ quality. Again, metal basis effects are much less relevant than ligand basis effects. The trend of basis set superposition error associated with the Mn-Mn bonding energy does not differ qualitatively from that shown by the BSSE associated with the Cr-CO bonding energy. For basis sets of DZ quality, it is, however, much larger, indicating that the deficiencies of bases of DZ quality are very critical in this case. From the BSSE values listed in Table 10, it becomes clear that, if no BSSE correction is applied, bases sets of at least TZD quality are needed to compute a reliable Mn-Mn bonding energy, whereas all metal basis sets considered in this study appear to be of good quality. By using the same DFT scheme we have been using in this investigation and a DZD-IVA basis, Folga and Ziegler12 computed a (CO)5MnMn(CO)5 bonding energy (without BSSE) of 174.2 kJ/mol, a value that is sensibly different from our estimate of 149.5 kJ/ mol. This discrepancy is again attributable to an insufficient fit set used by these authors. In fact, by using another fit set, these authors have recently computed a Mn-Mn bonding energy of 158.8 kJ/mol33 that compares well with our value obtained using a fit set very close to the saturation limit. After correcting for BSSE, the De(Mn-Mn) is dramatically reduced for all bases of DZ quality. The reduction is much less, but still significant, in the case of pure s, p bases of TZ and QZ quality. For basis sets with polarization functions and at least s, p sets of TZ quality, the BSSE correction has very little effect on De(MnMn). The same holds for all metal bases used in the present study. According to our calculations, the most reliable dissociation energy value for the Mn-Mn bond in Mn2(CO)10 is predicted in the range 112-114 kJ/mol. This favors the experimental estimates based on equilibrium studies7b and on electron impact measurements by Junk and Svec7b of 96 ( 13 and 104 ( 3 kJ/mol, respectively. It is worth noting that our theoretical prediction seems to support experimental arguments by Connor et. al.7c who emphasize that the value obtained by equilibrium studies, 96 ( 13 kJ/mol, compares well with independent electron impact measurements, such as the one by Junk and Svec. Conclusions Our investigation clearly confirms (cf. refs 10, 12, 13, and 21) that GGA-DFT methods can provide very accurate geometries and bond energies for organometallic systems. Cr-C and C-O bond lengths can be computed fairly accurately using a DZ or better s, p STO ligand basis set if at least one 3d polarization function has been added. The polarization function shortens the C-O bond by ca. 0.024 Å (-2%) and lengthens the Cr-C bond by ca. 0.015 Å (+0.8%). As a matter of fact, for C-O, the effect of saturating the s, p basis appears to be much less relevant than the effect of adding a 3d polarization function. This is not the case for the Mn-Mn bond length, where the quality of the ligand s, p basis is also very important. Improving the quality of the basis in general results in a lengthening of this bond. For all bond parameters, 4f polarization functions either on the metal or on the ligands seem to play a minor role. Improving the metal basis set beyond the level of our basis II has very little effect on both metal-CO and Mn-Mn bond lengths. As far as the bonding energies are concerned, there are some common conclusions for the Cr-CO and Mn-Mn bonds but also some clear differences. A common conclusion is that ligand basis sets that have TZ quality or better for the s, p orbitals and at least one d polarization function are virtually converged. They have a sufficiently small BSSE to warrant its neglect in most situations. A different conclusion has to be
5696 J. Phys. Chem., Vol. 100, No. 14, 1996 drawn for the “small” basis sets of DZ s, p quality. We note that for these basis sets, the basis set incompleteness error (BSIE), defined as the difference between the BSSE corrected bond energy and the converged value, causes for Cr-CO the BSSE corrected bond energy to be some 5-10 kJ/mol below the converged value. Since the BSSE increases bond energies and since the BSSE is on the order of 16-22 kJ/mol for these basis sets, it somewhat overcompensates for the BSIE. The result is that the bond energies without correction for BSSE are somewhat too large but not grossly in error. For Mn-Mn, however, the BSSE is so large for the DZ ligand s, p basis sets that the bond energies with BSSE are far too large. It is remarkable that for the Mn-Mn bond, but not for the Cr-CO bond, there is for all pure s, p ligand bases even after the BSSE correction a discernible basis set incompleteness effect (ca. 15 kJ/mol), which only disappears after adding at least one polarization function. It should be realized that the basis set effects “for the Mn-Mn bond” in fact do not refer to just the metal-metal contact but, in particular, reflect the many COCO and Mn-CO contacts, hence, the sensitivity of the “MnMn bond energy” to the quality of the ligand basis. We note that, after correcting for BSSE, the bonding energies computed in a small basis are reasonably close to the converged values. This implies that, although a TZD basis appears to be recommendable in terms of performance compared to cost, basis sets of DZD quality could still be a reasonable alternative, with the disadvantage that counterpoise correction for the BSSE has to be taken into account, at least for metal-metal bonds. The fact that a basis set is converged cannot be deduced from a small remaining BSSE alone, since a small BSSE may be accidental (for instance, when the ghost basis simply does not improve the fragment basis). However, the observed systematic decrease of the BSSE with increasing basis set size makes us confident that our largest basis sets are close to saturation. Acknowledgment. A.R. has been supported by the Euronetwork “Quantum Chemistry on Transition Metal Complexes”, No. ERBCHRXCT 930156. A.W.E. thanks the Deutsche Forschungsgemeinschaft for a stipendium. The foundation NCF is acknowledged for computer time and the Holland Research School for Molecular Chemistry for a grant that enabled us to carry out calculations on a parallel computer. References and Notes (1) Kirtley, S. W. In ComprehensiVe Organometallic Chemistry; Wilkinson, G., Ed.; Pergamon: Oxford, 1982; Vol. 3, Chapter 26, pp 753951. (2) Mingos, D. M. P. In ComprehensiVe Organometallic Chemistry; Wilkinson, G., Ed.; Pergamon: Oxford, 1982; Vol. 3, Chapter 19, pp 1-88. (3) Tyndall, G. W.; Jackson, R. L. J. Chem. Phys. 1988, 89, 1364. (4) (a) Angelici, R. J. Organomet. Chem. ReV. A 1968, 3, 173. (b) Covey, W. D.; Brown, T. L. Inorg. Chem. 1973, 12, 2820. (c) Centinis, G.; Gambino, O. Atti Acad. Sci. Torino 1 1963, 97, 1197. (d) Werner, H. Angew. Chem., Int. Ed. Engl. 1968, 7, 930. (e) Graham, J. R.; Angelici, R. Inorg. Chem. 1967, 6, 2082. (f) Werner, H.; Prinz, R. Chem. Ber. 1960, 99, 3582. (g) Werner, H.; Prinz, R. J. Organomet. Chem. 1966, 5, 79. (5) Bernstein, M.; Simon, J. D.; Peters, J. D. Chem. Phys. Lett. 1983, 100, 241.
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