ARTICLE pubs.acs.org/JPCA
On the Unusual Weak Intramolecular C...C Interactions in Ru3(CO)12: A Case of Bond Path Artifacts Introduced by the Multipole Model? Louis J. Farrugia*,† and Hans Martin Senn† †
WestCHEM, School of Chemistry, University of Glasgow, Glasgow G12 8QQ , Scotland
bS Supporting Information ABSTRACT: In a recent publication in this journal, an experimental charge density analysis on the triruthenium cluster Ru3(CO)12 showed unusual C...C bond paths linking the axial carbonyl ligands [Gervasio, G.; Marabello, D.; Bianchi, R.; Forni, A. J. Phys. Chem. A 2010, 114, 9368, hereafter GMBF]. These were also observed in one theoretical DFT calculation, and are associated with very low values of F(rb) and r2F(rb). Our independent experimental charge density analysis on Ru3(CO)12 is entirely consistent with GMBF and confirms the presence of these apparent weak interactions in the multipole model density. However, we conclusively demonstrate that these unusual C...C bond paths between the axial carbonyl ligands are in fact artifacts arising from the Hansen-Coppens multipole model, which is used to analyze the experimental data. Numerous relativistic and nonrelativistic gas-phase DFT calculations, using very extensive basis sets and with corrections for dispersion effects, uniformly fail to reproduce these intramolecular features in the QTAIM topology of the electron density. Moreover, multipole fitting of theoretical static structure factors computed from these quantum electron densities results in the reappearance of the C...C bond paths between the axial carbonyl ligands in the derived molecular graphs. On the other hand, using the experimental structure factors to generate “experimental” X-ray constrained DFT wave functions once again yields molecular graphs which do not show these secondary C...C bond paths. The evidence therefore strongly implicates the multipole model as the source of these spurious features and in turn suggests that great caution should be applied in the interpretation of bond paths where the values of F(rb) and r2F(rb) are very low.
’ INTRODUCTION The electronic structures of low-valent transition metal cluster compounds are of great interest, mainly because of the unusual nature of the metalmetal bonds which even now are not completely understood.1 Due to this interest, numerous experimental charge density studies on metalmetal bonded species have been undertaken, initially using the deformation density methodology,2 and more recently3 the Quantum Theory of Atoms in Molecules4 (QTAIM) approach. The application of QTAIM to metalmetal bonding has been reviewed by Farrugia and Macchi.5 An archetypal example of a metalmetal bonded transition metal complex is Ru3(CO)12 (1). The structure of 1 in the crystal phase is well-known,6 and 1 is isomorphous and isostructural with the heavier congener Os3(CO)12.7 The nearly ideal D3h molecular geometry results in rather short Cax...Cax contacts, slightly greater than 2.8 Å and significantly shorter than the sum of van der Waals radii. The reason why cluster 1 adopts this geometry in the crystal phase is still an unresolved problem. Calculations which range in rigor and accuracy from molecular mechanics to quantum DFT all show that the distorted D3 structure is the lowest-energy gasphase structure, albeit by a very small energetic margin.8 The D3 distortion in molecules of the type M3(CO)12 (M = Fe, Ru, Os) involves a concerted twist of all M(CO)4 units relative to the metal triangle, and the only molecule which actually adopts this structure in the solid state is FeRu2(CO)12.9 Molecules such as 1 containing several heavy atoms probably represent the current r 2011 American Chemical Society
limits in terms of experimental charge density studies. In a recent such study on 1 by Gervasio et al.10 (hereafter referred to as GMBF), unusual C...C bond paths between the axial carbonyls were observed in the experimental structure but only in one theoretical all-electron DFT calculation using the B97-D functional with a minimal 3-21G basis set. GMBF attribute these bond paths to genuine, albeit weak, stabilizing C...C interactions which could be responsible for the observed D3h geometry. As far as we are aware, there are no other reports of such interactions in metalcarbonyl complexes, though as pointed out by GMBF, they have been observed between the CO groups in the organic molecule syn-1,6,8,13-biscarbonyl[14]annulene by Destro and Merati.11 In our independent experimental study on 1 reported herein, we also observe such C...C bond paths, but, as we demonstrate, these are almost certainly artifacts which arise from the multipole model which is used to analyze the experimental results.
’ EXPERIMENTAL AND COMPUTATIONAL PROCEDURES Data Collection, Processing, and Spherical Atom Refinement. Details of data collection procedures are given in Table 1. Received: October 18, 2011 Revised: November 30, 2011 Published: November 30, 2011 738
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factors of 0.7380.529. A second semiempirical correction13 (without a θ-dependent correction) was applied to remove any residual absorption anisotropy due to the mounting medium and accounts for other errors such as machine instabilities. The data were sorted and merged using SORTAV,14 giving 17416 independent data with a mean redundancy of 12.3. The data set is essentially 100% complete for 0 < θ e 50° with only six highorder reflections missing to θmax. A spherical atom refinement using SHELXL97-215 was initially undertaken, with full-matrix least-squares on F2 and using all the unique data. All atoms were allowed anisotropic thermal motion. Thermal ellipsoid plots were obtained using the program ORTEP-3 for Windows.16 All calculations were carried out using the WinGX package of crystallographic programs.17 Multipole Refinement. The multipole formalism of Hansen and Coppens18 as implemented in the XD-2006 program suite19 was used
Table 1. Experimental Crystallographic and Refinement Dataa parameter compound formula
C12O12Ru3
Mr space group
639.33 P21/n
crystal system
monoclinic
a/Å
7.9847(3)
b/Å
14.6576(6)
c/Å
14.3889(6)
β/deg
100.521(2)
V/Å3
1655.72(12)
Z Dcalc/g cm3
4 2.56
F(000)
1200
λ/Å
0.71073
μ(Mo-Ka)/mm1
2.77
FðrÞ ¼ Pcore Fcore ðrÞ þ Pval k3 Fval ðkrÞ
crystal size/mm
0.246 0.166 0.133
transmission coefficients (range)
0.5290.738
θ range/deg
2.0050.06
sin(θmax)/ λ no. of data used for merging
1.0785 213835
þ
no. of unique data
17416
hkl range
17 f 17; 31 f 24; 30 f 31
Rint
0.0251
Rσ
0.0133 17416
no. of refined parameters final R [I > 2σ(I)] (all data)
245 0.0164 (0.0187)
Rw2 [I > 2σ(I)] (all data)
0.0385 (0.0391)
goodness of fit S
1.15
largest remaining feature
0.897(max) 0.691(min)
in electron density map/eÅ3 max shift/esd in last cycle
0.01
Multipole refinement no. of data in refinement no. of refined parameters
16427 321
final R [I > 2σ(I)] (all data)
0.0141(0.0174)
Rw [I > 2σ(I)]
0.0135
goodness of fit S
1.458
maximum features in electron
0.546, 0.425, 0.076(rms)
density map/eÅ3 (all data) (data truncated to
0.428, 0.341, 0.043(rms)
resolution of 0.8 Å1) max shift/esd in last cycle
0.00002
TðhÞ ¼
R = Σ(Fo Fc)/Σ(Fo). Rw = {Σ(w(Fo Fc) )/Σ(w(Fo) )} . = {Σ(w(Fo2 Fc2)2)/Σ(w(Fo2)2)}1/2. Rσ = Σ[σ(Fo2)]/Σ[Fo2]. Rint = Σ{n/(n 1}1/2|Fo2 Fo2(mean)|/ΣFo2 (summation is carried out only where more than one symmetry equivalent is averaged). a
2
2
1/2
l
∑ k03Rl ðk0 rÞ m∑¼ 0 Plm( dlm( ðΩÞ l¼0
ð1Þ
The function minimized in the least-squares procedure was Σw(|Fo| k|Fc|)2, with only those reflections with I > 3σ(I) included in the refinement. The multipole expansion was truncated at the hexadecapole level for all atoms. Each pseudoatom was assigned a core and spherical-valence scattering factor derived from the relativistic 4-component DiracFock wave functions of Volkov and Macchi.20 The radial fit of these functions was optimized by refinement of the expansion-contraction parameter k. The valence deformation functions for the C and O atoms used a single-ζ Slater-type radial function multiplied by the density-normalized spherical harmonics. The radial fits for the chemically distinct axial and equatorial C and O atoms were optimized by refinement of their expansion-contraction parameters k, k0 . The radial terms used for the Ru atoms were the relevant order Fourier-Bessel transforms of the Volkov-Macchi20 wave functions. Several models for the refinement were examined, and we here report the results from the D3h restricted model, which has one independent Ru atom and two independent carbonyl ligands (axial and equatorial). The fit using this highly restricted model is only slightly worse than for a model with no symmetry restrictions at all (RF is 1.41% vs 1.36%, with 321 vs 920 parameters being refined). A significant improvement in fit was observed when third and fourth order anharmonic Gram-Charlier coefficients were included for the Ru atoms. The anharmonic thermal parameter T(h) is expressed as a series expansion in terms of the harmonic thermal parameter T0(h)
Spherical atom refinement no. of data in refinement
lmax
Rw2
4 2 1 π3 iCijk hi hj hk þ π4 Dijkl hi hj hk hl þ ::: T0 ðhÞ 3 3
ð2Þ
The coefficients Cijk and Dijkl are the third and fourth order Gram-Charlier quasi-moments of the probability density function (pdf) and are refinable parameters. The acceptability of the resulting pdf for all Ru atoms was verified as described previously.21 The successful deconvolution of thermal motion was judged by the Hirshfeld rigid bond criterion.22 The mean Δ-msda for the CO bonds was 1.75 104 Å, with the highest individual value of 5 104 Å for C(21)O(21). As we have previously observed for transition metalligand bonds,23 the Δ-msda
A single crystal specimen of 1 was cooled from ambient temperature to 100 K over a period of 1 h. Data were collected on a Bruker-APEX-II diffractometer. A total of 213835 intensity measurements were harvested from the 23 j or ω-scan sets. An absorption correction by Gaussian quadrature12 was then applied to the reflection data, giving a range of transmission 739
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Table 2. Experimental and Theoretical Topological Properties at Bond Critical Pointsa bond RuRu
RuCax
RuCeq
Ax CO
Eq CO
C....C
d1b
d2b
F(rb)c
r2F(rb)d
λ1d
λ2d
λ3d
ε
G(rb)e,f
G(rb)/F(rb)
V(rb)e
E(rb)e
1.431
1.431
0.15
1.57
0.30
0.27
2.13
0.12
0.11
0.72
0.10
0.00
1.427
1.427
0.27
1.39
-0.63
-0.57
2.59
0.10
0.14
0.50
-0.17
-0.04
1.428
1.428
0.28
1.00
-0.59
-0.54
2.12
0.09
0.13
0.48
-0.20
-0.06
1.035
0.910
1.04
8.89
5.58
5.41
20.15
0.03
1.24
1.22
1.86
0.62
1.029
0.917
0.94
10.02
-3.67
-3.64
17.32
0.01
1.13
1.20
-1.55
-0.42
1.030
0.916
0.94
10.46
-3.56
-3.54
17.76
0.00
1.13
1.21
-1.52
-0.39
1.025
0.900
0.91
10.81
4.52
4.35
19.67
0.04
1.20
1.31
1.65
0.45
1.037 1.038
0.889 0.888
1.00 1.00
9.72 10.46
-4.08 -4.06
-4.05 -4.01
17.86 18.52
0.01 0.01
1.17 1.18
1.17 1.18
-1.67 -1.63
-0.49 -0.45
0.398
0.744
3.67
26.95
40.58
40.27
53.90
0.01
5.77
1.57
13.41
7.65
0.382
0.757
3.31
16.42
-36.90
-36.76
90.07
0.00
7.33
2.21
-13.51
-6.18
0.387
0.753
3.29
10.01
-36.11
-36.05
82.17
0.00
6.80
2.07
-12.92
-6.11
0.404
0.738
3.53
27.00
37.24
36.62
46.36
0.03
5.33
151
12.56
7.22
0.382
0.756
3.32
17.37
-37.02
-36.84
91.23
0.00
7.40
2.23
-13.59
-6.19
0.386
0.752
3.30
10.75
-36.21
-36.19
83.15
0.00
6.87
2.08
-12.99
-6.12
1.411
1.430
0.11
0.57
0.29
0.23
1.09
0.26
0.05
0.43
0.05
0.00
a
The top line gives experimental values averaged over D3h symmetry, the second line gives theoretical values from B97-D/def2-TZVP calculation (expt. geometry), and the third line from PBE0/cc-pVTZ calculations (expt. geometry). b d1, d2 are distance of the bcp to the nuclei in units of Å. c In units of e Å3. d In units of e Å5. e In units of Hartree Å3. f Estimated by the approximation of Abramov.25
for the RuC bonds is slightly higher (mean value = 1.4 103 Å). A scatter plot of the scale factor Fobs/Fcalc against sinθ/λ showed no significant trends, and difference Fourier maps and a residual density analysis24 implied no unmodeled features remained in the data (Figures S1S3, Supporting Information). The kinetic energy densities at the bcp's, G(r), given in Table 2 for the experimental densities, were estimated using the approximation of Abramov25 eq 3 GðrÞ ¼ ð3=10Þð3π2 Þ2=3 FðrÞ5=3 þ ð1=6Þ∇2 FðrÞ
contracted and diffuse densities respectively, compared with the atomic densities. The largest ΔF features were generally in the region of the Ru atoms, possibly due to residual core-polarization effects (see below). The “molecule from a crystal” approach in XDPROP19 was used to mimic a gas-phase analysis, and the criterion for the acceptance of a critical point in F(r) at position vector r was set to |rF(r)| < 1010 eÅ4 to ensure comparability with QTAIM analysis of the wave functions. Theoretical Studies. Gas-phase DFT calculations were undertaken with the programs TURBOMOLE28 and ORCA,29 at the experimental (D3h-idealized30) and D3h-optimized geometries, using several functionals and basis sets. A single point calculation at the D3h-idealized experimental geometry was also undertaken with the program GAUSSIAN0931 using the extensive all-electron cc-pVTZ basis32 and the PBE0 functional.33 The optimized structures typically gave RuRu distances up to 0.1 Å longer than experiment. Topological analysis was undertaken using the AIMPAC,34 AIM2000,35 or AIMAll36 codes. To explore the effect of changing the central RuRu distance in 1, scans were performed along this coordinate. At each point the RuRu distance was fixed, but all other atoms were allowed to optimize under strict D3h molecular symmetry. These all-electron nonrelativistic DFT calculations were done using the PBE,37 PBE+D,38 and B97-D39 functionals with def2-TZVPP40 (C, O) and TZVPPalls241 (Ru) basis sets with the program TURBOMOLE.28 Relativistic allelectron calculations were done with the ZORA scalar-relativistic Hamiltonian as implemented in ORCA,29 using the ANO-RCC basis for Ru42 (fully decontracted, g- and h-primitives deleted) and ZORA-relativistically recontracted def2-TZVPP basis40,43 for the other atoms. Further computational details and additional computational results can be found in the Supporting Information. X-ray constrained wave functions were calculated by the method of Jayatilaka et al.44 using the program TONTO.45 Experimental structure factors corrected for extinction and dispersion were obtained using SHELXL15 (LIST 3 option) and of these 4693 satisfying the conditions F > 3σ(F) and sinθ/λ < 0.7 Å1 were used in the optimization of the wave function. The BLYP46 functional
ð3Þ
Corresponding potential energy densities at the bcp’s, V(r), were obtained from eq 4 V ðrÞ ¼ ð1=4Þ∇2 FðrÞ 2GðrÞ
ð4Þ
Similar multipole refinements were undertaken using theoretical static complex structure factors obtained analytically using the program DENPROP26 from gas-phase wave functions (see below) for reciprocal lattice points corresponding to a pseudocubic unit cell of a = 30 Å . Both the experimental molecular geometry and optimized geometries were used in the wave function computation, and structure factors were calculated to a similar resolution as the experiment, i.e. sinθ/λ e 1.1 Å1. The same level of multipole expansion as used for the experimental data was applied to all the models, but all thermal parameters were set to zero and the positional parameters were not refined. The space group is effectively P1, and in view of the well-known problems27 of refinement of the multipole model in noncentrosymmetric space groups, phase-constrained refinement as implemented in XD-200619 was used to avoid any phase ambiguity. The refinements all converged well, with maximum shifts less than 1.0 106. In all cases excellent fits were obtained, typically with R(F) < 0.2% and ΔF ≈ (0.2 eÅ3. Calculations using the Volkov-Macchi atomic scattering factors20 gave uniformly slightly better fits than the other scattering factors in the XD databank. The k0 parameters for the O atoms were typically ∼1.1, while those for the C atoms were typically 0.70.8, indicating 740
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Figure 2. Experimental molecular graph of 1 obtained from the pseudoatom multipole model. (3,-1) bond critical points are shown as small red spheres, (3,+1) ring critical points as small yellow spheres, and (3,+3) cage critical points as small green spheres. The viewpoint is the same as in Figure 1. Note the significantly curved Cax...Cax bond paths and the inwardly curved RuRu bond paths.
Figure 1. Ortep plot of 1 showing the atomic labeling scheme, with thermal ellipsoids drawn at the 50% probability level. Important metrical parameters: Ru(1)Ru(2), 2.84671(5); Ru(1)Ru(3), 2.84761(5); Ru(2)Ru(3), 2.85918(6); C(11)...C(21), 2.8339(7); C(11)...C(31), 2.7847(7); C(21)...C(31), 2.8033(8); C(14)...C(24), 2.8297(7); C(14)...C(34), 2.8091(7); C(24)...C(34), 2.8268(7) Å.
that we observe two additional pairs of ring and cage cp’s associated with the C...C bcp’s, which fulfill the Poincare-Hopf criterion. The values of F(rb) and r2F(rb) in Table 2 are similar to those reported by GMBF10 and indicate strong covalent RuC interactions involving the carbonyl ligands, where the agreement between theory and experiment is rather good. This agreement also extends to the Laplacian r2F(rb) and to the individual eigenvalues λ of the Hessian matrix, which are generally much more difficult to reproduce. As usually observed, the agreement is less exact for the highly polar CO bonds, especially for their Laplacian values, in particular λ3. Of particular interest are the curved bond paths between all pairs of axial carbon atoms. These have very low average values for F(rb) (0.11 eÅ3) and r2F(rb) (0.57 eÅ5) and relatively high ellipticity, consistent with the very flat electron density in the region. Similar values are reported by GMBF.10 While the values of F(rb) and r2F(rb) for these bonds paths are similar in magnitude to those for the RuRu bcp’s (Table 2), there is no instance where the latter have been absent. These secondary Cax...Cax bond paths therefore form a special case, as discussed below. The estimated value of zero for the total energy density is suggestive of a very weak and hardly stabilizing interaction. Nevertheless, these bond paths are quite consistently observed, regardless of the complexity of the multipole model used. Indeed they are also observed in the pro-molecule density (i.e., the sum of atomic spherical densities) constructed from the multipole model. However, in all the gas-phase DFT calculations on 1 that we have undertaken, using all-electron basis sets, with and without corrections for dispersion or relativistic effects, and at either the optimized or experimental geometries, we find absolutely no evidence for these C...C interactions. A representative plot of the theoretical molecular graph is shown in Figure 3. This result is quite surprising, as it is normally the case that theoretical calculations show the same molecular graphs, even to the extent of including weak interactions observed in experiment.48 GMBF10 have stressed the importance of using dispersion-corrected DFT functionals to reproduce these features, but in our study using reasonably large all-electron basis sets we do not find this has any effect. It should also be noted that the dispersion correction38 is a purely classical term added to the DFT electronic energy, and as such has no effect on the wave function or the density at a
was used, with the bases def2-TZVPP40 for C, O and TZVPPalls241 for Ru. Values of the Lagrange multiplier λ up to 0.3 (for which χ2 = 2.704, R(F) = 0.016) were chosen to ensure a significant incorporation of experimental information into the wave function. As reported by Jayatilaka et al.,44 convergence of the wave function proved slow.
’ RESULTS AND DISCUSSION An ORTEP view of 1 is shown in Figure 1. The structure is well-known,6 and we here reiterate only the important features relevant to this study. The molecule has close to ideal D3h symmetry (continuous symmetry measure47 = 0.50, molecular rms deviation 0.077 Å), but there is a slight distortion toward C2v symmetry, since the Ru(2)Ru(3) distance is ∼0.01 Å longer than the other two RuRu distances. There are no significant geometric differences between our spherical atom and multipole refinements, and we report here the slightly more precise metric parameters from the multipole refinement. The mean RuCaxOax angle of 173.2 ((0.4)° is significantly further from linearity than the mean RuCeqOeq angle of 178.9 ((0.6)°. As a result, the average Oax...Oax distance of 3.03 ((0.06) Å is greater than the average Cax...Cax distance of 2.82 ((0.03) Å. It is worthwhile to note that while the axial C atoms are relatively close, the mean cis-Cax...Ceq distance [2.75 ((0.05) Å] is in fact slightly shorter. Topological Analysis of the Electron Density. We have examined the charge density in 1, reconstructed from the experimental and theoretical atomic multipole populations, and from DFT calculations on the gas phase molecule. The topological characteristics of the bond critical points in the electron density derived from the experimental multipole model and representative theoretical calculations are given in Table 2. The experimental and theoretical integrated atomic charges and atomic volumes, d-orbital populations, and the individual experimental critical point data are given in the Supporting Information (Tables S1S3 respectively). The experimental molecular graph, shown in Figure 2, is similar to that reported by GMBF,10 except 741
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Figure 3. Theoretical molecular graph of 1 obtained from the PBE0/ccpVTZ wave function. (3,-1) bond critical points are shown as small red spheres and (3,+1) ring critical points as small yellow spheres. The viewpoint is the same as in Figure 1. Figure 4. The variation in geometrical parameters along the RuRu scan coordinate. The experimental values are marked by asterisks (*) at RuRu = 2.851 Å.
given structure. Moreover, GMBF10 have suggested that these C...C interactions may provide a rationalization for the D3h structure of 1 in the solid state, and we therefore thought it important to establish the veracity of these features. In a number of DFT scan calculations, a set of fixed RuRu distances between 2.69 Å (very much shorter than the experimental distance of 2.85 Å) and the optimized RuRu distance for the respective functional (typically ∼2.94 Å) was chosen, while allowing the remaining coordinates to optimize under exact D3h molecular symmetry. The variation in geometrical parameters is shown in Figure 4. As the RuRu distance is reduced from the optimized bond length down to 2.69 Å, the Cax...Cax distance is correspondingly reduced in a linear fashion (Figure 4b), but in no case are any Cax...Cax bond paths observed in the molecular graph, even at the shortest RuRu distance. It is clear that simply reducing the Cax...Cax separation is not sufficient to manifest these bond paths. We next analyzed the effect of projecting the densities obtained from the above DFT calculations into static structure factors calculated to a similar resolution as the experiment (sinθ/λ e 1.1 Å1). The density implicit in these theoretical structure factors was modeled by a least-squares fit to a similar set of multipole parameters as used for the experimental structure factors (i.e., strict D3h chemical equivalence was maintained). In all cases excellent fits were obtained, typically with R(F) < 0.2% and ΔF ≈ ( 0.2 eÅ3. The best fit was for structure factors obtained from the nonrelativistic PBE0/cc-pVTZ wave function, where R(F) = 0.09% and ΔF ≈ ( 0.1 eÅ3. The difference Fourier map (Fψ FMM) from this fit is shown in Figure 5a. It is quite featureless and demonstrates that potentially an excellent fit is possible for these medium-resolution data sets, even using the (relatively restricted) multipole model elaborated by eq 1 above (but see below the point regarding ultrahigh resolution data). The topology of the electron density obtained in this way is effectively filtered through a multipole model, and we call this the indirect theoretical topology, in contrast to the direct theoretical topology obtained directly from the wave function. Despite the apparently excellent least-squares fits, the indirect theoretical topologies are, in most cases, not the same as the direct theoretical topologies. Specifically, in most indirect theoretical topologies (see Table S5, Supporting Information), secondary Cax...Cax bonds paths were observed in the molecular graph similar to those observed in our experimental study and that of GMBF.10
It seems highly likely, therefore, that the use of the multipole modeling is introducing these artifacts, which are strikingly absent in the parent quantum electron density. In all cases, the ellipticities of these Cax...Cax bcp’s were high (typically >0.5), and, in some cases, the associated ring critical points, with similarly low F(rb) values, were physically very close to their corresponding bcp’s, indicative of an impending catastrophe situation4 in the molecular graph. It is worth noting that, in the case of the best fit mentioned above (for the PBE0/cc-pVTZ wave function), the indirect topology is the same as the direct topology, i.e. no Cax...Cax bonds paths were observed. The standard Hansen-Coppens multipole model as elaborated in eq 1 has well-known deficiencies49 which could give rise to these discrepancies. For instance, as recently demonstrated by Fischer et al.,50 this standard model is quite inadequate to provide a satisfactory fit to ultra high-resolution theoretical structure factors, primarily because of core polarizations. The core density described in eq 1 (and as implemented in XD200619) is treated as an isolated spherical atomic density, with no polarization or expansion/contraction allowed. To get an adequate fit, Fischer et al.50 found it necessary to employ a more flexible model, where the core density is allowed to expand/contract or undergo polarization. In order to examine whether an indirect theoretical topology obtained from a more elaborate multipole model provides more consistent results with the direct theoretical topology, we have carried out refinements using ultra highresolution structure factors for 1 (calculated from the PBE0/ cc-pVTZ wave function to sinθ/λ e 1.8 Å1). In concurrence with the previous work of Fischer et al.50 we find that the difference Fourier map (Fψ FMM) from a best fit using the standard single pseudoatom multipole model shows very significant residuals (ΔFmax ≈ (0.5 eÅ3) in the vicinity of the Ru atoms, see Figure 5b. The application of a more elaborate multipole model involving three pseudoatoms per Ru atom (three shells comprising 4d, 3d and remaining core) and two pseudoatoms for the O and C atoms results is a substantially better fit (see Figure 5c) with ΔFmax < (0.1 eÅ3. Nevertheless, despite this increased sophistication in the multipole model, the derived molecular graph still displays secondary Cax...Cax bond paths. 742
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Figure 5. Difference Fourier map (Fψ FMM) from multipole refinements against theoretical structure factors for 1 using (a) medium resolution data (sinθ/λ e 1.1 Å1) and standard Hansen-Coppens pseudoatoms, (b) ultrahigh resolution data (sinθ/λ e 1.8 Å1) with standard Hansen-Coppens pseudoatom model, and (c) ultrahigh resolution data (sinθ/λ e 1.8 Å1) with extended pseudoatom model (see text) allowing core polarization. Contours are drawn at 0.05 eÅ3 with positive contours in blue and negative contours in red.
r2F(r) in the plane of these bcp’s, given in Figure 6. Only in the case of (a) and (c) are bcp’s observed, and it is particularly striking to compare parts (c) with (d) of Figure 6. These figures (and the corresponding Laplacian profiles along the Cax-Cax vector) are virtually indistinguishable, yet (c) shows the Cax...Cax bonds paths while (d) does not. Delocalization Indices. Since the bond path cannot be simply equated with a chemical bond,7 we have considered another QTAIM indicator, the delocalization index, which does not suffer from catastrophic changes. The delocalization index, δ(ΩA, ΩB), provides a measure of the Fermi correlation shared (and hence electrons shared) between two atomic basins ΩA and ΩB.51 These basins need not share a common interatomic surface, so that this index may be computed for any pair of atoms, regardless of whether they are formally bonded. At the HartreeFock (HF) level, and for homopolar bonds, the index δ(ΩA, ΩB) straightforwardly provides the number of pairs of shared electrons,51 but for other cases there is no direct relationship between this index and chemical concepts of bond order, as discussed by Farrugia and Macchi.5 At the DFT level the interpretation of δ(ΩA, ΩB) is even more ambiguous, but despite this, the DFT delocalization
In fact, additional Oeq....Oeq bond paths are observed, see Figure S4, Supporting Information. Finally, and in order to utilize the information present in the experimental structure factors without recourse to a multipole model, we have followed the method of Jayatilaka et al.44 to obtain an X-ray constrained “experimental” wave function. A constraint is applied in the SCF procedure to provide the best fit to the experimental structure factors. This approach allows us to perform a QTAIM analysis using standard QTAIM tools3436 without involvement of a multipole model. A topological analysis of the density from an “experimental” nonrelativistic BLYP/def2TZVPP(C, O), TZVPPalls2(Ru) wave function failed to reveal any secondary Cax...Cax bcp’s in the density. The resulting molecular graph is homeomorphic with those obtained from the purely theoretical wave functions reported above, see Figure 3. This important observation provides yet further evidence that these secondary bcp’s are artifacts arising from the multipole model used to analyze the density and are not inherent in the experimental data. It should be emphasized that there are no obviously visible features in the density or its Laplacian which could be used to rationalize the presence or otherwise of these bcp’s. This is shown by plots of the negative Laplacian function 743
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Figure 6. Maps of the negative Laplacian function r2F(r) in the plane of the Cax...Cax bond paths, (a) from experimental multipole model, (b) from optimized PBE0 DFT wave function, (c) from theoretical multipole model by refinement against static structure factors calculated from PBE-ZORA wave function at experimental geometry, using scattering factors from SCM database, (d) exactly the same as (c) but using scattering factors from VM database. Contours are drawn at 1.0 10‑3, (2.0 10n, (4 10n, (8 10n (n = 3, 2, 1, 0, +1, +2) e Å5, negative contours are shown in red, positive in blue.
Table 3. Delocalization Indices δ(ΩA,ΩB)a for Compound 1 atom pairs, AB
Table 3 gives the delocalization indices for specific atomic A-B interactions in 1, obtained from the ZORA-relativistic PBE wave function and also from the “experimental” X-ray constrained BLYP wave function The values of δ(ΩRu, ΩC) for the Ru-carbonyl bonds are close to unity, which is typical9 for this type of covalent interactions. Likewise the value of δ(ΩRu, ΩRu) is typical of the lower values observed for metalmetal bonds,5 commensurate with them having “partial covalency” in the sense that less than one electron pair is associated with each such bond. Of particular interest are the values for δ(ΩC, ΩC) for the C...C contacts. As expected, these are all very small, consistent with little or no chemical interaction, and this is especially so for the Cax...Cax contact. This has a smaller value than Cax...Ceq or Ceq...Ceq within the same Ru(CO)4 unit (for which C...C bond paths have never been observed), though it is about three times the size of δ(ΩC(eq), ΩC(eq)) between adjacent Ru(CO)4 units. Table 3 also gives, to our knowledge, the first reported “experimental” values of δ(ΩA, ΩB), obtained from the X-ray constrained wave function. These are very similar to the purely theoretical values and give no credence at all to the idea of any significant Cax...Cax bonding interactions.
δ(ΩA,ΩB)
RuRu
0.462 (0.457)
RuCax
1.056 (1.012)
RuCeq
1.197 (1.126)
RuOax
0.167 (0.158)
RuOeq
0.203 (0.189)
CaxOax
1.602 (1.641)
CeqOeq
1.612 (1.683)
CaxCax CaxCeq
0.047 (0.051) 0.085 (0.086)
CeqCeqb
0.058 (0.054) 0.016 (0.020)
The first value is from the PBE-ZORA wave function; the second value in parentheses is from the mean values from X-ray constrained “experimental” BLYP wave function. b The top value is for the C...C index within the same Ru(CO)4 unit; the second value is for the C...C index between adjacent Ru(CO)4 units. a
indices are in general quite similar numerically to those computed at the HF level.52 744
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’ CONCLUSIONS In this study, we have seen that the unusual Cax...Cax bond paths in Ru3(CO)12 are only observed in the density derived from a multipole model fit to experimental or theoretical structure factors. They are never observed by direct QTAIM analysis of theoretical or “experimental” wave functions.53 Despite the fact that the presence of the bond paths would provide an attractive rationale for the observed D3h structure of Ru3(CO)12, we must conclude that they are artifacts. The exact reason why the multipole model has a high proclivity to give rise to such artifacts is currently unclear. It should be noted that GMBF10 used the somewhat different Stewart54 formalism of the multipole model and reported similar results. The issues presented here are somewhat reminiscent of the controversy concerning the presence or otherwise of non-nuclear attractors in hcp beryllium crystals,55 which is highly dependent on the method used to analyze the density. Our results suggest that great caution should be applied in the interpretation of bond baths close to a catastrophe point, where the values of F(rb) and r2F(rb) are very low.
Mynott, R. J. Organomet. Chem. 1978, 160, 139. (k) Benard, M.; Coppens, P.; DeLucia, M. L.; Stevens, E. D. Inorg. Chem. 1980, 19, 1924. (3) (a) Macchi, P.; Proserpio, D. M.; Sironi, A. J. Am. Chem. Soc. 1998, 120, 13429. (b) Macchi, P.; Garlaschelli, L.; Martinego, S.; Sironi, A. J. Am. Chem. Soc. 1999, 121, 10428. (c) Macchi, P.; Schultz, A. J.; Larsen, F. K.; Iverson, B. B. J. Phys. Chem. 2001, 105, 9231. (d) Scherer, W.; Hieringer, W.; Spiegler, M.; Sirsch, P.; McGrady, G. S.; Downs, A. J.; Haaland, A.; Pederson, B. Chem. Commun. 1998, 2471. (e) Abramov, Y. A.; Brammer, L.; Klooster, W. T.; Bullock, R. M. Inorg. Chem. 1998, 37, 6317. (f) Bianchi, R.; Gervasio, G.; Marabello, D. Inorg. Chem. 2000, 39, 2360. (g) Bianchi, R.; Gervasio, G.; Marabello, D. Helv. Chim. Acta 2001, 84, 722. (h) Bianchi, R.; Gervasio, G.; Marabello, D. Acta Crystallogr., Sect. B 2001, 57, 638. (i) Farrugia, L.; Mallinson, P. R.; Stewart, B. Acta Crystallogr., Sect B 2003, 59, 234. (j) Farrugia, L. J.; Evans, C. C. R. Chimie 2005, 8, 1566. (k) Macchi, P.; Donghi, D.; Sironi, A. J. Am. Chem. Soc. 2005, 127, 16494. (l) Macchi, P.; Garlaschelli, L.; Sironi, A. J. Am. Chem. Soc. 2002, 124, 14173. (m) Overgaard, J.; Clausen, H. F.; Platts, J. A.; Iversen, B. B. J. Am. Chem. Soc. 2008, 130, 3834. (n) Overgaard, J.; Platts, J. A.; Iversen, B. B. Acta Crystallogr., Sect. B 2009, 65, 715. (o) Platts, J. A.; Evans, G. J. S.; Coogan, M. P.; Overgaard, J. Inorg. Chem. 2007, 46, 6291. (p) Ortin, Y.; Lugan, N.; Pillet, S.; Souhassou, M.; Lecomte, C.; Costaus, K.; Saillard, J.-Y. Inorg. Chem. 2005, 44, 9607. (4) (a) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; International Series of Monographs in Chemistry, Vol. 2; Oxford University Press: Oxford, 1990. (b) Popelier, P. Atoms in Molecules: An Introduction; Prentice Hall: Harlow, 2000. (c) The Quantum Theory of Atoms in Molecules - From Solid State to DNA and Drug Design; Matta, C. F., Boyd, R. J., Eds.; Wiley-VCH: Weinheim, Germany, 2007. (5) Farrugia, L. J.; Macchi, P. Struct. Bonding (Berlin) 2011, xx, xxxx, DOI: 10.1007/430_2010_30. (6) (a) Mason, R.; Rae, A. I. M. J. Chem. Soc. A 1968, 778. (b) Churchill, M. R.; Hollander, F. J.; Hutchinson, J. P. Inorg. Chem. 1977, 16, 2655. (c) Braga, D.; Grepioni, F.; Tedesco, E.; Dyson., P. J.; Martin, C. M.; Johnson, B. F. G. Transition Met. Chem. 1995, 20, 615. (d) Slebodnick, C.; Zhao, J.; Angel, R.; Hanson, B. E.; Song, Y.; Liu, Z.; Hemley, R. J. Inorg. Chem. 2004, 43, 5245. (e) Gervasio, G.; Bianchi, R.; Marabello, D. Chem. Phys. Lett. 2005, 407, 18. (7) (a) Corey, E. R.; Dahl, L. F. Inorg. Chem. 1962, 1, 521. (b) Churchill, M. R.; DeBoer, B. G. Inorg. Chem. 1977, 16, 878. (8) (a) Lauher, J. W. J. Am. Chem. Soc. 1986, 108, 1521. (b) Sironi, A. Inorg. Chem. 1996, 35, 1725. (c) Peng, B.; Li, Q.-S.; Xie, Y.; King, R. B.; Schaefer, H. F., III Dalton Trans. 2008, 6977. (9) Braga, D.; Farrugia, L. J.; Gillon, A. L.; Grepioni, F.; Tedesco, E. Organometallics 1996, 15, 4684. (10) Gervasio, G.; Marabello, D.; Bianchi, R.; Forni, A. J. Phys. Chem. A 2010, 114, 9368. (11) Destro, R.; Merati, F. Acta Crystallogr., Sect. B 1995, 51, 559. (12) Coppens, P.; Leiserowitz, L.; Rabinovich, D. Acta Crystallogr. 1965, 18, 1035. (13) Blessing, R. H. Acta Crystallogr., Sect. A 1995, 51, 33. (14) Blessing, R. H. J. Appl. Crystallogr. 1997, 30, 421. (15) Sheldrick, G. M. Acta Crystallogr., Sect. A 2008, 64, 112. (16) Farrugia, L. J. J. Appl. Crystallogr. 1997, 30, 565. (17) Farrugia, L. J. J. Appl. Crystallogr. 1999, 32, 837. (18) Hansen, N. K.; Coppens, P. Acta Crystallogr., Sect. A 1978, 34, 909. (19) Volkov, A.; Macchi, P.; Farrugia, L. J.; Gatti, C.; Mallinson, P. R.; Richter, T.; Koritsanszky, T. XD-2006 - a computer program for multipole refinement, topological analysis of charge densities and evaluation of intermolecular energies from experimental or theoretical structure factors, 2006. (20) Volkov, A.; Macchi, P. VM databank of scattering factors in XD2006, unpublished results. (21) Farrugia, L. J.; Khalaji, A. D. J. Phys. Chem. A 2011, 115, 12512. (22) Hirshfeld, F. L. Acta Crystallogr., Sect. A 1976, 32, 239. (23) Smith, G. T.; Mallinson, P. R.; Frampton, C. S.; Farrugia, L. J.; Peacock, R. D.; Howard, J. A. K. J. Am. Chem. Soc. 1997, 119, 5028. (24) Meindl, K.; Henn, J. Acta Crystallogr., Sect. A 2008, 64, 404.
’ ASSOCIATED CONTENT
bS
Supporting Information. Final refined parameters, observed and calculated structure factors in CIF format for the multipole refinements, computational details; supplementary figures (five) of plots of scale factors versus sinθ/λ, residual density maps and distribution plots, energy profiles; supplementary tables (five) of integrated atomic charges and volumes, d-orbital populations, critical point properties, optimized structural parameters. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was supported by the University of Glasgow. ’ REFERENCES (1) (a) Summerville, R. H.; Hoffmann, R. J. Am. Chem. Soc. 1979, 79, 1501. (b) Heijser, W.; Baerends, E. J.; Ros, P. Faraday Symp. 1980, 14, 211. (c) Bauschlicher, C. W. J. Chem. Phys. 1986, 84, 872. (d) Rosa, A.; Baerends, E. New J. Chem. 1991, 15, 815. (e) Bo, C.; Sarasa, J.-P.; Poblet, J.-M. J. Phys. Chem. 1993, 97, 6362. (f) Reinhold, J.; Hunstock, E.; Mealli, C. New J. Chem. 1994, 18, 465. (g) Reinhold, J.; Kluge, O.; Mealli, C. Inorg. Chem. 2007, 46, 7142. (h) Ponec, R.; Cooper, D. L. Faraday Discuss. 2007, 135, 31. (i) Ponec, R.; Lendvay, G.; Chaves, J. J. Comput. Chem. 2008, 29, 1387. (j) Ponec, R.; Gatti, C. Inorg. Chem. 2009, 48, 11024. (2) (a) Low, A. A.; Kunze, K. L.; MacDougall, P. J.; Hall, M. B. Inorg. Chem. 1991, 30, 1079. (b) Freund, H.-J.; Hohneicher, G. Theor. Chim. Acta (Berlin) 1979, 52, 145. (c) Freund, H.-J.; Dick, G.; Hohneicher, G. Theor. Chim. Acta (Berlin) 1980, 53, 181. (d) Leung, P. C. W.; Coppens, P. Acta. Crystallogr., Sect. B 1983, 39, 535. (e) Martin, M.; Rees, B.; Mitschler, A. Acta Crystallogr., Sect. B 1982, 38, 6. (f) Clemente, D. A.; Biagini, M. C.; Rees, B.; Herrmann, W. A. Inorg. Chem. 1982, 21, 3741. (g) Mitschler, A.; Rees, B.; Lehmann, M. S. J. Am. Chem. Soc. 1978, 100, 3390. (h) Mitschler, A.; Rees, B.; Wiest, R.; Bernard, M. J. Am. Chem. Soc. 1982, 104, 7501. (i) Wang, Y.; Coppens, P. Inorg. Chem. 1976, 15, 1122. (j) Geibel, W.; Wilke, G.; Goddard, R.; Kr€uger, C.; 745
dx.doi.org/10.1021/jp2100039 |J. Phys. Chem. A 2012, 116, 738–746
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ARTICLE
(25) Abramov, Y. A. Acta Crystallogr., Sect. A 1997, 53, 264. (26) Volkov, V.; Koritsanszky, T.; Chodkiewicz, M.; King, H. F. J. Comput. Chem. 2009, 30, 1379. (27) (a) El Haouzi, A.; Hansen, N. K.; Le Henaff, C.; Protas, J. Acta Crystallogr., Sect. A 1996, 52, 291. (b) Spackman, M. A.; Byrom, P. G. Acta Crystallogr., Sect. B 1997, 53, 553. (28) (a) TURBOMOLE, version 6.3, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH, 19892007, TURBOMOLE GmbH, since 2007; 2011. Available from http:// www.turbomole.com/ (accessed December 13, 2011). (b) Ahlrichs, R.; B€ar, M.; H€aser, M.; Horn, H.; K€olmel, C. Chem. Phys. Lett. 1989, 162, 165. (c) H€aser, M.; Ahlrichs, R. J. Comput. Chem. 1989, 10, 104. (d) Treutler, O.; Ahlrichs, R. J. Chem. Phys. 1995, 102, 346. (e) Sierka, M.; Hogekamp, A.; Ahlrichs, R. J. Chem. Phys. 2003, 118, 9136. (f) Ahlrichs, R. Phys. Chem. Chem. Phys. 2004, 6, 5119. (29) ORCA, version 2.8, 2011. (30) Pilati, T.; Forni, A. J. Appl. Crystallogr. 2000, 33, 417. (31) GAUSSIAN09, revision A.02, Frisch, M. J. et al. Gaussian Inc.: Wallingford, CT, 2009. (32) (a) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007. (b) Peterson, K. A.; Figgen, D.; Dolg, M.; Stoll, H. J. Chem. Phys. 2007, 126, 124101. (33) Perdew, J. P.; Ernzerhof, M.; Burke, K. J. Chem. Phys. 1996, 105, 9982. See also ref 37. (34) Biegler-K€onig, F. W.; Bader, R. F. W.; Tang, T.-H. J. Comput. Chem. 1982, 3, 317. (35) Biegler-K€onig, F. J. Comput. Chem. 2000, 12, 1040. (36) Keith, T. A. AIMAll, version 10.07.01, 2010. http://aim.tkgristmill.com/ (accessed December 13, 2011). (37) (a) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. (b) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1997, 78, 1396. (38) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. J. Chem. Phys. 2010, 132, 154104. (39) Grimme, S. J. Comput. Chem. 2006, 27, 1787. (40) (a) Weigend, F.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2005, 7, 3297. (b) Weigend, F. Phys. Chem. Chem. Phys. 2006, 8, 1057. (41) Ahlrichs, R.; May, K. Phys. Chem. Chem. Phys. 2000, 2, 943. (42) Roos, B. O.; Lindh, R.; Malmqvist, P. A.; Veryazov, V.; Widmark, P. O. J. Phys. Chem. A 2005, 109, 6575. (43) Pantazis, D. A.; Chen, X.-Y.; Landis, C. R.; Neese, F. J. Chem. Theory Comput. 2008, 4, 908. (44) (a) Jayatilaka, D Phys. Rev. Lett. 1998, 80, 798. (b) Jayatilaka, D.; Grimwood, D. J. Acta Crystallogr., Sect. A 2001, 57, 76. (c) Grimwood, D. J.; Jayatilaka, D . Acta Crystallogr., Sect. A 2001, 57, 87. (d) Bytheway, I.; Grimwwod, D. J.; Jayatilaka, D . Acta Crystallogr., Sect. A 2002, 58, 232. (e) Bytheway, I.; Grimwood, D. J.; Figgies, B. N.; Chandler, G. S.; Jayatilaka, D . Acta Crystallogr., Sect. A 2002, 58, 244. (f) Grimwood, D. J.; Bytheway, I.; Jayatilaka, D. J. Comput. Chem. 2003, 24, 470. (g) Jayatilaka, D.; Grimwood, D. J. Acta Crystallogr., Sect. A 2004, 60, 111. (h) Hudak, M.; Jayatilaka, D.; Perasínova, L.; Biskupic, S.; Kozísek, J.; Bucinsky, L. Acta Crystallogr., Sect. A 2010, 66, 78. (45) Jayatilaka, D.; Grimwood, D. In Computational Science ICCS 2003; Sloot, P., Abramson, D., Bogdanov, A., Gorbachev, Y., Dongarra, J., Zomaya, A., Eds.; Lecture Notes in Computer Science, Vol. 2660; Springer: Berlin, 2003; p 142. (46) (a) Becke, A. D. Phys. Rev. A 1988, 38, 3098. (b) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785. (47) Zabrodsky, H.; Pele, S.; Avnir, D. J. Am. Chem. Soc. 1993, 115, 8278. (48) Farrugia, L. J.; Kocovsky, P.; Senn, H. M.; Vyskocil, S. Acta Crystallogr., Sect B 2009, 65, 757. (49) The deficiencies of the Hansen-Coppens multipole model relate to the simplicity and appropriateness of the Slater-type radial functions and the truncation of the multipole expansion at l = 4. These problems have been recently summarized. See: Koritsanszky, T. S. Hydrogen Bonding - New Insights; Grabowski, S. J., Ed.; Springer: Dordrecht, 2006; pp 441470.
(50) Fischer, A.; Tiana, D.; Scherer, W.; Batke, K.; Eickerling, G.; Svendsen, H.; Bindzux, N.; Iversen, B. B. J. Phys. Chem. A 2011, 115, 13061. (51) (a) Bader, R. F. W.; Stephens, M. E. J. Am. Chem. Soc. 1975, 97, 7391. (b) Fradera, X.; Austen, M. A.; Bader, R. F. W. J. Phys. Chem. A 1999, 103, 304. (c) Matito, E.; Sola, M. Coord. Chem. Rev. 2009, 253, 647. (52) (a) Poater, J.; Sola, M.; Duran, M.; Fradera, X. Theor. Chem. ngyan, J. G.; Sannigrahi, A. B. J. Phys. Acta 2002, 107, 362. (b) Kar, T.; A Chem. A 2000, 104, 9953. (53) The single exception reported in ref 10 refers to a B97-D calculation using the 3-21G basis for all atoms. In our opinion, the authors place far too much emphasis on this single result, especially as the density obtained from such a minimal basis set must be subject to significant error. (54) Stewart, R. F. Acta Crystallogr., Sect A 1976, 32, 565. (55) (a) deVries, R. Y.; Briels, W. J.; Fiel, D. Phys. Rev. Lett. 1996, 77, 1719. (b) Iversen, B. B.; Larsen, F. K.; Souhassou, M.; Takata, M. Acta Crystallogr., Sect B 1995, 51, 580. (c) Jayatilaka, D. Phys. Rev. Lett. 1998, 80, 798. (d) Silvi, B.; Gatti, C. J. Phys. Chem. A 2000, 104, 947.
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