J. Phys. Chem. 1996, 100, 10547-10553
10547
Topological Analysis of Experimental Electron Densities. 1. The Different C-C Bonds in Bullvalene T. Koritsanszky,*,† J. Buschmann, and P. Luger Institute for Crystallography, Free UniVersity of Berlin, Takustrasse 6, 14195 Berlin, Germany ReceiVed: August 23, 1995; In Final Form: January 15, 1996X
The static electron density based on the rigid-pseudoatom model is extracted from low-temperature, highresolution X-ray diffraction data of bullvalene (tricyclo[3.3.2.02.8]deca-2,5,8-triene). A comparative analysis of the anisotropic displacement amplitudes based on this and the earlier neutron diffraction study reveals bias in the neutron parameters possibly due to inadequate treatment of extinction. The topological properties of the experimental charge distribution are compared to those obtained from wave functions at different levels of theory. The agreement reached is excellent, especially for the values of the electron density at the bond critical points. The four different C-C bond types can clearly be established. All ring and cage critical points can also be located, confirming the reliability of the experimental density even far away from the nuclei. The model dependence of the topological properties is also discussed, and the importance of the proper treatment of the hydrogen atoms is shown.
Introduction The electron distribution (ED) is of fundamental importance in modern quantum chemistry; it is a physical observable, and from its topological analysis all static and reactive properties of a molecular system can be characterized.1 The “atoms in molecules” concept has proven to be a feasible approach in describing chemical bonds,2 nonbonded interactions,3 molecular4 and crystal structures,5 and, last but not least, chemical reactivity.6 The quantitative characterization of the chemical bond and atomic interactions provided by the topological analysis of the total electron density relies on simple parameters such as the location of the bond critical points (rb), the value of the ED (Fb) and its Laplacian (Lb) at this point, the ellipticity (), and the length of the bond path (lb).7,8 A critical point (CP), where grad F vanishes, is characterized by its rank (the number of nonvanishing principal curvatures) and its signature (the algebraic sum of the signs of these curvatures). The bond or (3,-1) CP (from the three nonvanishing principal curvatures two are negative) is the point where the ED takes its minimum value along the bond path, the line of the maximum density linking two bonded atoms. The magnitude of the ED at the bond CP correlates with the bond distance and the bond order. The length of the bond path can differ significantly from the geometrical bond distance (the internuclear separation); thus, it gives a numerical value for the bent character of a bond. The Laplacian of the ED (the sum of the principal curvatures) describes areas where charge is concentrated or depleted, and its value at the bond CP increases with the number of electrons participating in the bond. The bond ellipticity ( ) λ1/λ2 - 1, where λ1 and λ2 are the two negative principal curvatures of F at rb) is the measure for the deviation of the ED at the bond CP from axial symmetry; that is, it is an indicator of the π character of the bond. Such analyses based on ab initio wave functions are of growing importance with an increasing number of applications. Recent experimental advances and computer program developments make it possible to perform similar studies on the ED † Permanent address: Central Research Institute for Chemistry of the Hungarian Academy of Sciences, H-1025 Budapest, Hungary. X Abstract published in AdVance ACS Abstracts, June 1, 1996.
S0022-3654(95)02466-X CCC: $12.00
extracted from X-ray diffraction data. Such a procedure involves modeling the thermally averaged electron density and refining its parameters to a set of carefully measured structure factors. The success of the method is subject to experimental accuracy and model adequacy. The information obtained has often been interpreted only qualitatively, in terms of deformation electron density maps. Only a few studies were so far concerned with evaluating topological properties of the experimental charge distribution, and applications remained at a preliminary stage.9-12 In a series of papers we intend to report on several studies undertaken to extract static electron densities from highresolution X-ray diffraction data and to explore the applicability of the topological analysis of these densities. The following are to be discussed: (a) how the experimental topological characteristics of typical covalent bonds compare to those obtained from wave functions, (b) how transferable these properties are and whether the effects of second neighbors and of substituents are detectable; (c) how reliable the experimental Laplacian function is in describing the effect of the crystal field on the nonbonded charge concentrations and in revealing “closed shell” interactions. The most important aspect of such studies would be to learn about the correlation between details of the model applied in the refinement and details of the topology of the electron density obtained. Such information would help to reveal model ambiguities and to control the refinement. Since the bond topological properties mentioned above bear significance only in relation to idealized bonds, their analysis should include the comparison of results obtained for different systems. A comparison based on wave functions is more straightforward than one based on experimental observation. In the latter case the data collection for a reference system could require considerably different experimental conditions. Moreover, it is expected that properties characteristic of the isolated molecules are not preserved in the crystal due to intermolecular forces. The effect of the crystal field is less important for neutral molecules lacking nonbonded electrons. This is the case if hydrocarbons are considered and the comparison is limited to different C-C bonds. An ideal situation occurs when the different bond types to be compared are formed within the same molecule. Bullvalene is such an example, having four types of © 1996 American Chemical Society
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Figure 1. Stereo plot of the bullvalene molecule. The atoms considered to be chemically different are labeled.
TABLE 1: Crystal Data and Experimental Conditions for Bullvalene formula crystal system space groupa Z cell dimensionsb a (Å) b (Å) c (Å) β (deg) cell volume, V (Å3) F(000) density, dcalc (g/cm3) crystal size (mm) radiation wavelength, λ (Å) linear absorption coefficient, µ (cm-1) temperature, T (K) scan method, step (deg) scan width (deg) scan rate (ω, deg/min) data collection range 2θ (deg) h, k, l standard reflections no. of reflections collected no. of unique reflections
C10H10 monoclinic B21/c 8 6.480(4) 20.584(13) 10.370(6) 90.17(2) 1383.16 560 1.250 0.66 × 0.30 × 0.48 Mo KR, Zr filter 0.710 68 0.76 110 ω-2θ, 0.02-0.04 ∆ω ) 1.06 ( 0.52 tan ω 0.67-4.0 6.0-105.0 -13 < h < 12, -29 < k < 43, -20 < l < 20 -4, 0, 4 -2, -14, -1 9740 6479
Transformation of the primitive space group P21/n to B21/c: -1 0 -1, 0 -1 0, -1 0 1. b Cell parameters in P21/n: a ) 6.106(2) Å, b ) 20.584(13) Å, c ) 6.122(5) Å, β ) 116.00(2)°, V ) 691.58 Å3. a
C-C bonds: single, double, and bent bonds and additional single bonds with a strong conjugation from the neighboring π bonds. It possesses a wide range of chemical reactivity and structural isomerism13 which is attributable to this variety in the C-C bond formation and the interactions between the different bonds. In this introductory paper we present a topological study on the bullvalene molecule (Figure 1) based on both the experimental and the theoretical ED. Experimental Conditions, Structure Refinement Single crystals for bullvalene were grown in petroleum ether solution. The X-ray diffraction experiment was carried out with a SIEMENS four-circle single-crystal diffractometer. A collimator 8 cm long and 1 mm in diameter was used. Before starting the data collection a scan of the primary beam was made to make sure that a homogeneous plateau of about 1 mm was present at the crystal site. This allowed the use of a relatively large crystal. A nitrogen gas stream device designed by Dietrich
and Dierks14 was used for cooling the sample. The temperature was kept at 110(1) K during the entire measurement. The cell dimensions and the orientation matrix used during the data collection were obtained by least-squares refinement on 38 centered reflections with 40.0° < 2θ < 49.2°. These reflections were automatically re-centered and a new orientation matrix was determined once the intensity of one of the two standard reflections dropped by more than 3% with respect to its highest previous value. This happened about once a day during the 2 week long experiment, making up for only slight changes in the orientation matrix which were not accompanied by significant changes in the cell parameters. Since the intensities of the reflections decreased sharply at high Bragg angles, the measurement could not be extended beyond 95° in 2θ. Several hundred reflections were collected up to 105° in 2θ, but all were found unobserved. The data were scaled according to the change in the intensities of the standard reflections and corrected for the Lorentz and polarization effects. An analytical absorption correction15 was also applied. The internal merging index for averaging the symmetry equivalent reflections was Rint ) 0.0155. We note here that the measurement was performed using the space group P21/n, but the cell parameters and the reflection indices were transformed to the nonstandard space group B21/c to be in compliance with the single-crystal neutron diffraction study published earlier.16 More details on crystal data and experimental conditions are given in Table 1. The starting parameters of the conventional refinement were those obtained by neutron diffraction. Density Model, Multipole Refinements The experimental static deformation ED is described in terms of atom-centered density units (rigid-pseudoatom model) being a product of Slater-type radial functions and real spherical harmonics.17 The Slater exponents are composed from those of the single-ζ wave functions.18 The spherical core and valence densities are represented by Hartree-Fock radial functions over an extended Slater-type basis.18 Beside the conventional crystallographic parameters, multipole populations and radial screening constants are fitted to the X-ray data. In the model applied the C3V molecular symmetry was adopted and four types of chemically different carbon atoms (C(r), C1(sp2), C2(sp2), and C(sp3)) were considered up to the hexadecapolar level of the expansion with appropriate constraints dictated by the corresponding local atomic site symmetries (m, m, m, C3V). For the atoms of the cyclopropane ring the y and for the sp2 atoms the z local axis was chosen to be perpendicular to the mirror plane assumed. For the sp3 carbon atom the xz plane was taken as a mirror plane with the local z axis along the pseudo-3-fold
The Different C-C Bonds in Bullvalene
J. Phys. Chem., Vol. 100, No. 25, 1996 10549
TABLE 2: Summary of Least-Squares Refinements spherical-atom model no. reflections (F > 3σ(F)) no. variables R Rw (w ) 1/σ(F)) S
I
multipole models II III
TABLE 4: Libration (L), Translation (T) and Screw-Rotation (S) Tensors in the Cartesian Crystal Framea neutron (I)b
IV R
4480
κ
C(r)
132 0.53 0.044 3.27
153 0.033 0.022 1.63
156 0.032 0.021 1.55
193 0.032 0.021 1.57
365 0.032 0.021 1.58
Cl(sp2)
C2(sp2)
C(sp3)
0.9563
0.9563
0.9563
0.9544 1.2
4.153
4.233
4.243
4.207
11 1 -1 10
-0.06
20 21 22 2 -2
0.05 0.03 0.06
30 31 3 -1 32 33 3 -3
0.24 0.03
40 41 42 4 -2 43 44 4 -4
0.03 -0.06
0.04 -0.05
-0.03
-0.10 -0.13 0.03 0.02 0.01 0.05 -0.02
0.0356
0.0200
1.2
1.2
0.791 0.801
0.801
0.11
0.09
0.09
0.07 -0.19
-0.18
0.03 0.00
0.04 0.03
0.07 0.00
0.06 0.00
0.28 0.03
0.27 0.03
0.24
0.06
0.04
0.05
0.02 0.03
0.01 -0.02
-0.03 0.01
0.00 0.00
L11 L22 L33 L12 L13 L23
192 (9) 235 (9) 204 (9) 64 (6) 2 (7) 4 (7)
L 193 (11) 233 (10) 207 (11) 67 (7) 1 (8) -5 (8)
223 (8) 276 (7) 208 (8) 59 (5) -6 (6) -8 (6)
T11 T22 T33 T12 T13 T23
1062 (19) 1177 (18) 1119 (19) -87 (14) -5 (14) -25 (14)
T (Å2) 1046 (22) 1154 (21) 1121 (23) -86 (16) -7 (17) -31 (17)
1391 (16) 1465 (15) 1338 (16) 99 (12) 2 (12) -43 (12)
S11 S12 S13 S21 S22 S23 S31 S32 S33
-18 (6) 33 (6) 29 (6) -1 (6) -32 (6) -4 (6) 46 (6) 31 (6) -14 (6)
S (Å rad) -16 (7) 24 (6) 24 (7) -4 (7) 33 (7) -4 (7) 45 (7) 30 (7) -17 (7)
-22 (5) 26 (5) 38 (5) -4 (5) 39 (5) -4 (5) 45 (5) 33 (5) -16 (5)
H(r) H(sp2) H(sp3)
1m 00
0.0309
X-ray multipole (I) 0.0114d
(rad2)
TABLE 3: Multipole Populationsa atoms
neutron (II)c
0.03
0.28
a Entries are multiplied by 104. b Based on published parameters. Parameters after anisotropic extinction correction. d The model includes a local libration around the molecular 3-fold axis (L ) 0.0019(2) rad2). Changes in the components of the T, L, and S tensors are less than 3 times their esd’s. c
-0.10
a Esd’s for the Hartree-Fock radical screening constants (κ) and for the valence charges (P00) are indicated as upper indices. For the multipole populations (Plm, l > 0) σ(Plm) < 0.01.
rotation axis. In these local frames the following restrictions are to be applied to the indices (l, m) of the spherical harmonics: (l, m)l-j,+), (l, m)l-2j,(), and (l, m)3j,+) for C(r), C(sp2), and C(sp3) carbon atoms, respectively, where j is an integer. The positional and thermal parameters of the hydrogen atoms were deduced from the neutron measurement (see below), and their valence deformation was represented by a bond-directed dipole (l)1, m)1). This parameterization (model I) with fixed Slater exponents leads to an observationsto-variables ratio of 29. In refinement II the screening parameters of the deformation radial functions were also optimized, while in model III isotropic thermal parameters were assigned to all hydrogen atoms, which were refined together with their positional parameters. This latter parametrization corresponds to the usual treatment when no neutron data are available. The implementation of the C3V molecular symmetry into the refinement is justified by the neutron geometrical parameters,16 the inspection of which shows that the related bond distances and angles are in agreement within three standard deviations. Nevertheless, a refinement (model IV) without symmetry restrictions on the multipole populations was also performed. In all cases the refinement of an isotropic extinction parameter (type I, Lorentzian distribution of mosaic spread)19 was included. The statistical figures of merit of the different refinements and the multipole populations based on model I are summarized in Tables 2 and 3, respectively. It is important to note that model I with only 22 additional variables considerably improved the fit in comparison to the
spherical atom refinement. A further increase in the number of variables does not lead to a better fit. This is especially pronounced for models III and IV which perform not better than as model I despite the, respectively, 40 and 211 additional parameters refined. Thermal Motion Analysis The multipole refinement can lead to an unbiased estimation of the total, thermally averaged electron density. This does not necessarily hold for properties derived from it. The static distribution can seriously be biased by inadequacies in the thermal motion model, and similarly atomic displacement parameters (ADPs) do not represent only vibrational motion if valence deformations due to chemical bonding are poorly described. The ADPs of the carbon atoms are analyzed in the framework of the rigid-body-motion model,20,21 and the residual amplitudes (Uobs - Umodel) are visualized with the computer graphics program PEANUT.22 The values obtained from multipole refinement against X-ray data (model I) are, on average, 20% higher than those given by the neutron measurement published earlier.16 The difference, as found for many other systems,23,24 can mainly be described by the translation component of the rigid-body motion (Table 4). This indicates that either the X-ray or the neutron intensity data contain uncorrected components, most likely due to thermal diffuse scattering, which are accounted for by the ADPs, but the bias introduced appears to be the same for all atoms. In Figure 2 the residual ADPs (∆U ) Uobs - UTLS) are displayed in terms of root-mean-square displacement surfaces (RMSDs) represented by the ∆U tensors. Figure 2a, based on the X-ray data, shows systematic features in the cyclopropane ring. At each C atom the positive/negative residuals are parallel/ perpendicular to the opposite bond formed by the other two
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Figure 2. Graphical representation (PEANUT) of the residual ADPs of the carbon atoms in terms of their root-mean-square displacement surfaces. (The atom numbering is chosen as in the neutron structure analysis.16) A surface is defined by the equation 〈∆u2(e)〉1/2 ) (e′∆Ue)1/2 ) const, where e is a unit vector pointing from the atomic center to an arbitrary direction. (a) ADPs from X-ray multipole refinement: ∆U ) Ux - Ux(TLS). (b) ADPs from the published neutron data: ∆U ) Un - Un(TLS). (c) ADPs based on the neutron diffraction data corrected for extinction: ∆U ) Un′ - Un(TLS)′. (d) ADPs representing intramolecular vibration calculated from the ab initio force field: ∆U ) Uc - Uc(TLS).
carbons. This residual could be interpreted as an internal libration around the 3-fold axis of the molecule, but this hypothesis is not supported by the neutron measurement (Figure 2b). If the molecule is nearly rigid, as suggested by the neutron data, then the systematic trend revealed by the X-ray ADPs is due to inadequacy in the static density model. If the residual RMSDs, calculated from the X-ray ADPs, represent an internal contribution to the molecular motion, then the neutron data must be biased. To further examine this dilemma, we have proceeded in two directions: (i) the neutron structure refinement was repeated with the inclusion of different extinction models; (ii) the contribution of the intramolecular motion to the ADPs was calculated from the ab initio force field. The anisotropic extinction model of “crystal type I” with Lorentzian distribution of mosaic spread19 turned out to improve the fit significantly (Rw(F2) ) 0.032, S ) 1.17 compared to the published values of 0.065 and 0.96, respectively). As a result, the ADPs and their esd’s were decreased, and the RMSD surfaces (Figure 2c) show the features characteristic of that obtained from the X-ray data. The lowest vibrational frequency (ν ) 272.3 cm-1 at the HF/6-31G** level) can be assigned to a “twisting” mode, with the main component being the libration of the cyclopropane ring around the molecular 3-fold axis. The rigid-body fit to the intramolecular ADPs leads to RMSD surfaces shown in Figure 2d. The residuals here represent intramolecular contributions not absorbed into the T, L, and S tensors. The arrangement of the surfaces corresponds to the pattern found for the
ADPs from the multipole refinement, supporting their physical significance. The ADPs of the hydrogen atoms based on the extinctioncorrected neutron data (Un(H)) were used to generate ADPs directly applicable in the multipole refinement (Ux(H)) in the following way:
Ux(H) ) Un(H) + Tx(C) - Tn(C) where Tx(C) and Tn(C) are the translation tensors obtained by the rigid-body fit of the X-ray and neutron ADPs of the carbon atoms. Theoretical Calculations Ab initio calculations were performed at the HF level using 6-31+G** and 6-311++G(3df,3pd) standard basis sets.25 The electron correlation was included at the MP2/6-31+G** level. Default options were used for the SCF convergence and threshold limits for the final changes in the maximum forces and displacements in the geometry optimizations. The experimental geometrical data16 averaged to C3V symmetry were taken as starting parameters. The inspection of the bond distances obtained (Table 5) shows the need either for the MP2 level or for the use of diffuse and multiple polarization functions in the HF optimization to account for the observations. The excellent agreement between theory and experiment (the stationary and the thermally averaged geometry) is important in view of the fact that the charge accumulation in the bond increases as the
The Different C-C Bonds in Bullvalene
J. Phys. Chem., Vol. 100, No. 25, 1996 10551
TABLE 5: Ab Initio Optimized and Experimental C-C Bond Distances (Å) in Bullvalene
TABLE 8: Topological Paramaters for the C-C Bonds in Bullvalenea HF/ HF/6-311++GMP2 X-ray 6-31+G** (3df,3pd) 6-31+G** multipole
HF/ HF/6-311++GMP2 X-ray 6-31+G** (3df,3pd) 6-31+G** multipole neutron CdC C(r)-C C-C C(r)-C(r)
1.3255 1.4831 1.5192 1.5178
1.3369 1.4845 1.5244 1.5366
1.3529 1.4677 1.5102 1.5326
1.3450(2) 1.4727(2) 1.5157(2) 1.5352(2)
1.342(1) 1.473(1) 1.516(1) 1.533(1)
TABLE 6: Ab Initio Optimized (MP2/6-31+G**) C-C Bond Distances (Å) in Selected Hydrocarbons C-C ethane ethylene acetylene 1,3-butadiene propene cyclopropane vinylcyclopropane methylvinylcyclopropane
CdC
CtC C(r)-C(r)
1.5293 1.3390 1.2162 1.4603 1.5185
1.3471 1.3427
1.4822 1.3205 1.5023 1.5099, 1.4769 1.3473 1.5147
1.4997 1.5023 1.5147
Fb Lb n C(r)-C Fb Lb n C-C Fb Lb n C(r)-C(r) Fb Lb n CdC
a
2.44 -28.2 0.44 2.28 1.86 -18.5 0.07 1.20 1.76 -16.9 0.02 1.07 1.62 -11.7 0.50 0.92
2.30 -25.9 0.41 1.96 1.81 -17.2 0.07 1.14 1.69 -15.3 0.02 0.99 1.51 -9.5 0.54 0.81
2.31 -25.2 0.43 1.98 1.91 -19.6 0.08 1.27 1.79 -17.5 0.02 1.11 1.57 -11.0 0.47 0.87
2.36(2) -26.0(1) 0.29 2.09 1.92(2) -19.3(1) 0.09 1.33 1.78(1) -16.0(1) 0.04 1.11 1.54(1) -7.3(1) 0.90 0.89
Fb (/Å-3), Lb (e/Å-5).
TABLE 7: Bond Topological Parameters (MP2/6-31+G**) in Selected Hydrocarbonsa C-C Fb Lb n
CdC
CtC
Ethane 1.69 -16.0 0.0 1.00 Ethylene 2.33 -26.6 0.41 2.01
Fb Lb n
Acetylene
Fb Lb n
2.69 -27.9 0.0 2.97
Fb Lb n
1.94 -20.9 0.08 1.25
1,3-Butandiene 2.31 -26.1 0.41 1.99
Fb Lb n
1.74 -16.8 0.01 1.05
Propene 2.32 -26.1 0.43 2.00 Cyclopropane
Fb Lb n
1.66 -12.8 0.52 0.96
Fb Lb n
1.86 -19.1 0.07 1.20
Vinylcyclopropane 2.42 -28.5 0.43 2.22
Fb Lb n
1.76 -17.3 0.04 1.07
Methylvinylcyclopropane 1.87 2.30 -19.4 -25.6 0.07 0.45 1.22 1.96
a
C(r)-C(r)
1.65, 1.66 -12.6, -12.8 0.51, 0.52 0.96, 0.96 1.62, 1.62 -12.1, -12.1 0.51, 0.51 0.92, 0.92
Fb (e/Å-3), Lb (e/Å-5).
bond distance decreases, which makes the direct comparison of local topological properties, obtained by different methods, troublesome. A simple relation between the bond order (n) and the charge density at the bond critical point has been used to describe
Figure 3. Cyclopropane ring. (a) Map of the gradient vector field of the theoretical charge density displayed in terms of trajectories of grad F. Bond paths and lines denoting the intersections of the interatomic surfaces with the plane of the ring are drawn as heavy lines. (b) Experimental deformation density. (c) Theoretical deformation density (HF/6-31+G**). Contour intervals are 0.1 e/Å3.
different C-C bonds and the effect of conjugation.7,8 Ethane, ethylene, and acetylene can be selected as reference molecules for typical single, double, and triple bonds. Table 6 contains results of their geometry optimization (MP2/6-31+G**) in terms of the geometrical bond distances. For comparison, additional calculations were carried out on propene, 1,3-butadiene, cyclopropane, and its vinyl and methylvinyl substitutes, each of which has structural elements in common with bullvalene. Results and Discussion Chemical bonds and their interaction can be described in terms of bond topological properties of the ED. In this respect, the values of the ED and its Laplacian at the bond CP and the ellipticity are of great significance. The molecular orbital theory predicts π charge transfer from unsaturated bonds to the adjacent single bond (hyperconjugation). This effect results in an
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TABLE 9: Ring and Cage Critical Points in Bullvalenea HF/ 6-31+G**
HF/6-311++G(3df,3pd)
MP2 6-31+G**
X-ray multipole
F L
1.33 1.61
Cyclopropane Ring 1.25 1.29 2.31 1.69
1.34(1) 2.1(1)
F L
0.05 1.25
7-Membered Ring 0.05 1.29
0.07 1.44
0.12(1) 1.4(1)
F L
0.06 1.25
0.05 1.21
0.11(1) 1.2(1)
a
Cage 0.05 1.29
F (e/A-3), L (e/A-5).
increase/decrease in the ED, as well as in its Laplacian, at the CPs of the adjacent single/double bonds relative to their magnitude in the “isolated” bonds. The π character of this charge delocalization manifests itself in nonzero ellipticity induced in the single bond by the double bonds. These observations are well illustrated by the bond parameters of 1,3butadiene in Tables 6 and 7. The results for vinyl- and cis-methylvinylcyclopropane indicate that the cyclopropyl group is an effective π-donor like a double bond in forming conjugated systems. The latter compound together with propene is also an example of hyperconjugation. In both cases the CH3-C bond exhibits partial double-bond character transmitted from the adjacent π-donor. The first and second analytic derivatives of both the theoretical and experimental EDs of bullvalene were evaluated, and all stationary points were located (AIMPAC,26 XD27). Table 8
gives a quantitative comparison of the results in terms of different bond properties. The theoretical parameters show only a moderate basis set dependence, and also only a slight change occurs if electron correlation is included at the MP2 level. The quantities deduced from the experimental ED (model I) are averages calculated according to the assumed symmetry. The general agreement between theory and experiment is good. It is especially excellent for Fb, reflecting the well-known feature that the experimental method has its highest accuracy in the bonding areas. The trend shown in the strength of the different bonds is as expected on the basis of simple orbital theory. The double bond is the strongest with the highest charge concentration, while the bonds forming the cyclopropane ring are the weakest with the least charge accumulation in the bonding area. The latter bond is more elongated perpendicular to the interatomic vector than the former. There are significant differences in the topological properties of the two formal single bonds; the bond next to the cyclopropane ring appears to be stronger and it exhibits more π character than that formed by the C(sp2) and C(sp3) atoms. This is in accordance with theoretical and experimental28 observations on the conjugation effect which is due to the unsaturated character of the cyclopropane ring. Table 8 also contains bond orders calculated from the ED at the bond critical points, making use of the relation:
n ) exp{A(Fb - B)} proposed earlier.7 The parameters A and B are obtained by leastsquares fitting of this function to the idealized single, double, and triple bonds in ethane, ethylene, and acetylene, respectively, using the MP2/6-31+G** values for Fb (Table 7). The bond path lengths are practically equivalent to the
Figure 4. Relief maps of the negative Laplacian functions in the plane of the cyclopropane ring (a, b) and of the ethylenic wing (c, d) obtained from the wave function (top) and from the experimental static electron density (bottom).
The Different C-C Bonds in Bullvalene geometrical bond lengths for all bonds except that of the C(r)C(r) in the cyclopropane ring, indicating its bent character. Even in this case there is only a slight difference between the two quantities, which is well illustrated by the trajectory map in Figure 3a. The displacement of the experimental deformation density peak from the bond center (Figure 3b) is found to be 0.12 Å, which gives an exaggerated estimation for the bent character of the bond if compared to the corresponding distance of the bond CP in the total ED (0.02 Å). Table 9 contains the characteristic data for the ring and cage critical points. The satisfactory agreement between theory and experiment seems to confirm that density features far away from the nuclei can also be adequately described by model functions fitted to a limited number of X-ray diffraction data. Figure 4 displays relief plots of the negative Laplacian in the section through the cyclopropane ring and in the plane containing the atoms of the ethylenic wing. These plots demonstrate how the experimental Laplacian function compares, in detail, with that derived from theory. Characteristic of covalent interaction is the continuous negative area between a bonded atom pair, which is clearly reproduced by the experimental function for all bonds. The locations of bond charge concentrations (the maxima in the negative Laplacian) are in good agreement, especially for the C-C bonds. The experimental Laplacian function appears to be slightly underestimated in the bonding regions, indicating that the single Slater-type radial functions are not “structured” enough to represent fine details of the ED. The same analysis based on the results of refinement II overestimates the bond topological parameters for all bonds by about 20% and gives no difference between the single bonds. It is important to mention that model III provides a basically correct picture of the bonding situation in terms of bond topological parameters, although their values are about 10% below the theoretical ones. A weakness of this model appears in predicting the location of the bond CPs. This refinement leads to zero net atomic charges, i.e. unpolarized bonds, which makes no sense, especially when C-H bonds are considered. The “full parameter” fit (model IV) resulted in nonsignificant changes in the conventional variables. Despite the changes in the individual multipole populations, the total static ED remained unchanged; that is, the averaged bond topological parameters were the same as those obtained by the constrained model. Conclusion The results presented in this paper show that the experimentally obtained electron density can be accurate enough, especially in the bond regions, to provide fine details of chemical bonds and their interactions. Beside the experimental accuracy the careful modeling of the static density deformations as well as the thermal motion of the molecule are essential. The model incorporating the neutron parameters, standard Slater exponents for the deformation radial functions, and the expected chemical symmetry accounts well for all topological features of the different C-C bonds considered. Not only the strengths (bond order) and the π character of the bonds but also the ED far away from the atomic centers are given correctly (ring and cage CPs). A promising result is the almost quantitative agreement between the theoretical and experimental Laplacian functions. The attempt to optimize the extent of expansion or contraction of the deformation radial functions, although it results in stable convergence, seems to fail, as the static ED obtained is considerably overestimated in the bonding areas. This study also demonstrates the importance of the proper treatment of the hydrogen atoms if quantitative estimations of static electronic properties are to be obtained from X-ray diffraction data. In hydrocarbons, as in most of the hydrogen-
J. Phys. Chem., Vol. 100, No. 25, 1996 10553 containing organic molecules, most of the charge transfer occurs at the expense of charge of the hydrogen atoms. Without adequate modeling of their displacement parameters there is hardly any chance to obtain reasonable values for their valence charge. This indeterminacy, due to the electroneutrality constraint usually applied, spreads over all parameters refined. It can limit the reliability not only of the experimental ED but, even more serious, that of the electrostatic potential. Experimental charge density studies should be extremely careful not only about the accuracy of the data collection but, with equal importance, about the selection of the refinement model and strategy applied to interpret the data. The topological analysis of the experimental ED and its Laplacian can become a sensitive and efficient method for locating model ambiguities. Acknowledgment. This research was supported by Bundesminister fu¨r Forschung und Technologie (Germany), Grant PFR-AN204-03B27A460, and by Fonds der Chemischen Industrie. Supporting Information Available: Tables of X-ray positional and anisotropic displacement parameters (2 pages) and listing of observed and calculated structure amplitudes (6 pages). Ordering information is given on any current masthead page. References and Notes (1) Bader, R. F. W. Atoms in Molecules, A Quantum Theory; Calendron Press: Oxford, 1990. (2) Bader, R. F. W.; Esse`n, H. J. Chem. Phys. 1984, 80, 1943. (3) Carroll, M. T.; Cheeseman, J. R.; Osman, R.; Weinstein, H. J. Phys. Chem. 1989, 93, 5120. (4) Bader, R. F. W.; Nguyen-Dang, T. T.; Tal, Y. J. Chem. Phys. 1979, 70, 4316. (5) Zou, P. F.; Bader, R. F. W. Acta Crystallgr. 1994, A50, 714. (6) Bader, R. F. W.; Chang, C. J. Phys. Chem. 1989, 93, 2946. (7) Bader, R. F. W.; Slee, T. S., Cremer, D.; Kraka, E. J. Am. Chem. Soc. 1983, 105, 5061. (8) Cremer, D.; Kraka, E.; Slee, T. S.; Bader, R. F. W.; Lau, C. D. H.; Nguyen-Dang, T. T.; MacDougall, P. J. J. Am. Chem. Soc. 1983, 105, 5069. (9) Lau, C. O. H.; Bader, R. F. W.; Hermansson, K.; Berkovitch-Yellin. Chem. Scr. Abstr. 1986, 26, 476. (10) Kappkhan, M.; Tsirel’son, V. G.; Ozerov, R. P. Dokl. Phys. Chem. 1989, 303, 1025. (11) Destro, R.; Bianchi, R.; Gatti, C.; Merati, F. Chem. Phys. Lett. 1991, 186, 47. (12) Flensburg, C.; Larsen, S.; Stewart, R. F. J. Phys. Chem. 1995, 99, 10130. (13) Graham, J. D.; Santee, E. R., Jr. J. Am. Chem. Soc. 1966, 88, 3453. (14) Dietrich, H.; Dierks, H. Messtechnik 1970, 78, 184. (15) Alcock, N. W. Acta Crystallogr. 1974, A30, 332. (16) Luger, P.; Buschmann, J.; Mc Mullan, R. K.; Ruble, J. R.; Matias, P.; Jeffrey, G. A. J. Am. Chem. Soc. 1986, 108, 7825. (17) Hansen, N. K.; Coppens, P. Acta Crystallogr. 1978, A34, 909. (18) Clementi, E.; Roetti, C. At. Nucl. Data Tables 1974, 14, 177. (19) Becker, P. J.; Coppens, P. Acta Crystallogr. 1975, A31, 417. (20) Schomaker, V.; Trueblood, K. N. Acta Crystallogr. 1968, B24, 63. (21) Koritsanszky, T.; Bu¨rgi, H.-B. UBSBA: University Bern Segmented Rigid Body Analysis Program, University Bern, 1991. (22) Hummel, W.; Hauser, J.; Bu¨rgi, H.-B. J. Mol. Graphics 1990, 8, 214-220. (23) Koritsanszky, T.; Stru¨mpel, M. K.; Buschmann, J.; Luger, P.; Hansen, N. K.; Pichon-Pesme, V. J. Am. Chem. Soc. 1991, 113, 9148. (24) Koritsanszky, T.; Buschmann, J.; Denner, L.; Luger, P.; Kno¨chel, A.; Haarich, M.; Patz, M. J. Am. Chem. Soc. 1991, 113, 8388. (25) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foresman, J. B.; Johnson, B. G.; Schlegel, H. B.; Robb, M. A.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian 92, Revision C; Gaussian, Inc.: Pittsburgh, PA, 1992. (26) Biegler-Ko¨nig, F. W.; Nguygen-Dang, T. T.; Tal, Y.; Bader, R. F. W.; Duke, A. J. J. Phys. B: At. Mol. Phys. 1981, 14, 2739. (27) Koritsanszky, T.; Howard, S. T.; Mallinson, R. P.; Su, Z.; Richter, T.; Hansen, N. K. XD, A Computer Program Package for Multipole Refinement and Analysis of Charge Densities from X-ray Diffraction Data, Free University of Berlin, 1995. (28) Nilveldt, D.; Vos, A. Acta Crystallogr. 1988, B44, 289.
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