Rotational isomerism and charge density properties of vinylcyclopropane (VCP)

Journal of Molecular Structure (Theochem), 122 (1985) 75-94 Elsevier Science Publishers B.V., Amsterdam -Printed in The Netherlands

ROTATIONAL ISOMERISM AND CHARGE DENSITY PROPERTIES OF VINYLCYCLOPROPANE (VCP)

BRUNO KLAHN and VOLKER DYCZMONS Theoretical Chemistry Group, University of Giittingen, D-3400 Gijttingen (Federal Republic of Germany)

Tammannstrale

6,

(Received 25 August 1984)

ABSTRACT Ab initio SCF calculations of more than double zeta quality have been carried out for nearly fully optimizing the s-trans, gauche and s-cis structures of VCP. The potential curve of internal rotation, which is obtained under (partial) consideration of the structural parameter relaxation, confirms the three minimum curve of preceding experimental and theoretical investigations. As was conjectured in a microwave study, the dipole moment of gauche VCP (0.29 D) proves to be considerably smaller than the s-trans value (0.48 D). The different CC bonds (the ring bonds, the double bond, and the bond between ring and vinyl group) are mutually compared on the basis of bond lengths, force constants, population analyses (Mulliken and Davidson-Roby analyses), and bond path considerations. Especially, a n-conjugative interaction between the cyclopropyl ring and the vinyl group is thoroughly studied as a function of internal rotation. The occurrence of two gauche structures is interpreted as a compromise between the attempt to achieve maximal r-conjugative interaction (which favours the s-cis structure) and to avoid steric interaction of the endstanding hydrogen atoms (which disfavours the s-cis structure). INTRODUCTION

The molecular properties of vinylcyclopropane (VCP), such as the number of rotarners and their energetic differences, the energy barrier heights of interconversion, the dipole moments and the structures of the conformers, have been explored in a competition between experimental and theoretical studies over recent years [l-lo]. At first the question was debated as to whether VCP has two (s-trans, s-cis), three (s-truns, two equivalent gauche), or even four equilibrium geometries. In Hehres’s early ab initio SCF calculations [ 41, which were mainly performed at the minimal basis set level, a four minimum potential curve was predicted, whereas the electron diffraction measurements of de Meijere and Liittke [3] as well as the NMR studies of de Mare and Martin [2] and of Giinther et al. [lo] favoured a threefold torsional potential. This discrepancy must be ascribed to the restricted computer facilities of that time, which did not allow a consequent use of the rather reliable double zeta basis set quality and a full geometry optimization. De Mare and Peterson 0166-1280/85/$03.30

o 1985 Elsevier Science Publishers B.V.

76 [8] demonstrated that the VCP torsional potential curve does not fit the experiments, if it is obtained from (STO-3G) minimal basis set calculations, no matter whether relaxation is considered or not. In the first case they obtained a two minimum and in the second case a four minimum curve. Even with a split valence shell basis set a stable s-cis VCP structure can occur as an artefact of the calculation, if relaxation is ignored (cf. Figs. 5 and 6 of ref. 8). This result may underline the necessity to consider relaxation effects in the present case, where steric interaction plays a role, as was advocated by Random and Pople [ 271. Today almost all recent theoretical and experimental investigations of VCP in the gasphase agree that the s-tram form (Fig. 1) is the most stable one, that there are two equivalent gauche rotamers but that an s-cis form does not exist [2,3,6-lo]. To our knowledge, the only exceptions are the microwave study of Codding and Schwendeman [ 51, which failed to reveal the presence of the gauche form, and the Raman study of Salares et al. [ 111, who explained their spectra by a twofold potential. The present ab initio SCF study considers the different structures of VCP anew using an improved basis set including polarization functions, which exceeds the basis sizes of the former papers. In contrast to the work of De Mare and Peterson [8], who used the rigid rotor approximation in the vicinity of each critical point, our potential curve was obtained under direct consideration of the structural parameter relaxation. The calculation confirms the presence of three equivalent geometries and finally yields the spectrum of torsional vibrations and the microwave spectrum [ 121 in reasonable agreement with the experimental data [ 5,6]. An outstanding feature of the electron density is the variation of the partial double bond character of the C-C4 bond as a function of the torsional angle. Skancke and Boggs [7] have already shown this expected conjugative n-interaction by means of a Mulliken population analysis only,

c5 4

Fig. 1. The s-trans structure of VCP and the numbering of atoms.

77

however, for a fixed molecular structure. Our calculated overlap populations (Mulliken population analysis), shared electron numbers (Davidson-Roby population analysis) and especially the features of the total charge density at the “bond critical points” very clearly illustrate the variation of conjugative interaction during rotation. The partial n-bond character of the Cl-C4 bond was previously observed in experimental and theoretical NMR studies [13] and by the interpretation of photoelectron spectra [14] ; a theoretical description of this conjugation in terms of Walsh orbitals [15] was first given by Hoffmann [16]. For s-truns and s-&s VCP this picture predicts a strengthening of the C2-C3, a weakening of the other ring bonds and, of course, a strengthening of the C1-C4 bond. This is just what our structural investigations and electron density analyses will show. METHOD

OF CALCULATION

The SCF structure optimizations were carried out using contracted Huzinaga Gaussian lobes [17]. The basis size is 7s/3p + 31111/21 for the carbon and 3s + 21 for the hydrogen atoms. Its quality is slightly better than double zeta thus surpassing the basis qualities of refs. 4,7 and 8, which use, at best, double zeta size for the valence shells. Our basis promises to give good bond lengths and reliable information about the stability of different conformers [ 181. When determining bond angles and the torsional potential curve, the basis was augmented by a full set of long range d-polarization functions [19] on the carbon atoms C1 and Cq of the inner bond (cl-orbital exponent vd = 0.3). The bond properties of VCP as a function of internal rotation are investigated by a Mu&ken [ 451 and a Davidson-Roby [ 46,471 population analysis (MPA and DRPA). The latter is expected to yield more reliable results, since it is based on atomic occupation numbers as obtained from projecting the one-particle density operator onto the space spanned by the atomic orbitals (AOs) of the atoms considered. Instead of the conventional AOs, modified atomic orbtials (MAOs), as introduced by Heinzmann and Ahlrichs [48], were generated from the molecular density and overlap matrices by an iterative procedure, i.e., MAOs are AOs, which already include information about the molecular bonds. A population analysis, which is based on these MAOs, yields gross charges (MPA) and effective electronic charges (DRPA) that add up to the total nuclear charge, apart from a small unassigned residual charge. This unassigned charge is about 0.03 of an electron charge in the present calculations and thus smaller by a factor of roughly 10 compared with an analysis, which is performed with AOs. STRUCTURE

OPTlMI!ZATIONS

Starting with the molecular geometry of Skancke and Boggs [7] all structural parameters of S-tram, gauche, and s-cis VCP were individually optimized

78

except those of the ring methylene groups. Their C-H bond lengths were assumed to have a common value and their orientations in space were kept fixed according to the s-trans structure of ref. 7 (LHCH = 114.6”). Thus a total of 15 parameters had to be optimized for s-truns and s-c&, whereas 22 parameters of gauche VCP were determined (Table 1). Some of the transition state parameters in Table 1 may indicate how the structure varies between the s-trans and the gauche form*. The ring proves to be weakly distorted compared with cyclopropane, whose C-C bond length was experimentally and theoretically determined to be r. = 1.511 W [20] or re = 1.513 A [21], respectively. The ring of s-trans and s-cis VCP looks, as if the carbon atom C1 were slightly pulled outwards. This result coincides with the former calculations [7,8] and is in qualitative TABLE 1 Structural parameters of VCP (bond lengths in A, angles in degrees, SCF energies ESCF in atomic units) Parameter

EDa

Cl-G Cl-c,

1.522

C-C, C-G c,=c,

‘G-H, G-H, G-H, ‘G-H, (G-H;

1.475 1.334

1.099 G-H),

!

~C,C,C,

LC,C,C, LC.C,C. LC;C;H; LC,C,H, LC$,H; LCSC,H, LC,C,H, LC.C.H. LH;C;H, LH,C,H, LH,C,H, -J%CF

120.1 f 0.9 126.2 f 1.4 I

(116.8 f 1.5)C

119.7 f 1.5

s-tram 0” 1.522 1.522 1.512 1.474 1.317 1.077 1.082 1.077 1.079 1.076 119.6 119.6 124.7

Transition state (65.2”) 1.514 1.519 1.491

124.4

ii;.; 116.2 119.1 121.7 121.9 116.4 (114.6)d 193.71960

193.71462

gaucheb 117.8”

s-cis 180”

1.521 1.514 1.515 1.485 1.316 1.078 1.081 1.077 1.078 1.076 119.6 122.0 126.1 115.0 115.9 115.0 118.9 121.7 122.2 116.1

1.522 1.522 1.510 1.485 1.318 1.076 1.080 1.076 1.079 1.072 121.1 121.1 126.8 115.2 115.2 114.8 118.4 121.6 122.5 115.9

( 114.6)d

( 114.6)d

193.71763

193.71536

aElectron diffraction results of ref. 3 (model II). bThe vinyl group is clockwise rotated around the axis C, + C, by 117.8’. CAssumed value. dAssumed value according to ref. 7. *Cartesian coordinates of the optimized VCP structures are deposited with B.L.L.D., Boston Spa, Wetherby, Yorkshire, England as Supplementary Publication number SUP 26277 (1 page).

79

agreement with the concept of a net charge transfer from the ring bonds CI--C2 and CI-C3 into the Cl-C4 single bond, where this charge contributes to a n-conjugative interaction between ring and vinyl group [ 161. Moreover, in accordance with the predictions of Hoffmann [16] the Cz--C3 bond lengths of s-tians (1.512 A) and s-cis VCP (1.510 A) are slightly shorter than the value of cyclopropane (1.513 A, cf. Table 2). A recent theoretical study on cyclopropanecarboxaldehyde [22] and microwave measurements on cyanocyclopropane and cyclopropaneacetylene [23 ] showed that the same distortion occurs in agreement with the X-ray diffraction data, surveyed by Allen [24]. The ED measurements of Traetteberg [ 91, on the other hand, failed to distinguish the lengths of the individual C-C ring bonds. The ring distorsion of gauche VCP differs significantly from the s-trans and s-cis conformers. Compared to cyclopropane, in gauche VCP only the CI-C2 bond is clearly lengthened, i.e. weakened, whereas the bond lengths of Cl-C3 (1.514 W) and Cz-CS (1.515 A) nearly coincide with the C3H6 value (1.513 A). There is only a relatively small but still remarkable structural difference between the vinyl group and ethylene. The C4=C5 bond length of VCP (1.316 A-l.318 A, cf. Table 1) exceeds the C=C bond length of ethylene, which was calculated to be 1.314 A (Table 2), i.e., the C!4=C5bond of VCP is slightly weaker than the “free component”. The s-tram and s-cis structures have longer C4=CSbonds (1.317 A and 1.318 A). At a torsional angle of 90”) where the n-conjugation between ring and vinyl group is essentially removed, the WC5 bond adopts its minimal value of 1.315 A, which nearly coincides with the ethylene value (1.314 A). The Cl-C4 bond of VCP (s-tram: re = 1.474 A, gauche: r, = 1.485 A) proves to be remarkably shorter than the C-C bond of ethane (e” = 1.528 A [25], r:” = 1.541 A). The considerable extension of the C1-C4 bond length by 0.017 A, when internally rotating the vinyl group from the s-tram form to the transition state, and its slight reduction by 0.006 A, when approaching the gauche form, constitutes a powerful argument for a conjugation between C1 and Cd, which is partially removed under internal rotation dependent on the dihedral angle. The C-C4 bond length should naturally TABLE 2 CC bond properties of ethylene, ethane, and cyclopropane as obtained from our SCF calculations without polarization functions. f = valence stretching force constant, n = overlap population (MPA), D = shared electron number (DRPA) Molecule

re (A)

f (mdyn a-l)

n

0

‘7-L

1.314 1.541 1.513

11.3 4.5 -

1.266 0.663 0.554

2.185 1.324 1.304

CJK CJ-&

80

adopt its maximal value for a torsional angle of about 90”, an assumption, which the calculation actually confirmed (r, = 1.497 A). The same trend was recently observed also for some comparable molecules, e.g., for the central bonds of cyclopropanecarboxaldehyde [ 221, for 1,3-butadiene [29], and for bifuran [ 301. The relaxation of the remaining structural parameters such as LCZCIC4, LC3C1C4,LC1C4C5,LC4C5HB,and the C-H bond lengths of the ring methylene groups can apparently be explained by an avoided steric interaction between Hz, H3, and H8, when internally rotating the vinyl group from s-tram to s-cis VCP. This interpretation is mainly suggested by the considerable increase of the angles L&C& and LX&C, by 1.9” and 1.8”, respectively, where X denotes an axis bisecting the angle LCICICJ. The interpretation of avoided steric interaction is also in agreement with the increase of the angle LC4CSH8 by 0.6” and the decrease of the C-H lengths at C, and C3 from 1.076 A to 1.072 A. For comparison, the angle L&&H, varies only by maximally 0.1” and the other C-H lengths by at most 0.002 A. The structure relaxation thus totally effects an increase of the angle between the C5H8 direction and the ring plane by 4.5” for s-cis VCP in comparison to a hypothetical s-cis structure as obtained from a rigid rotation of the vinyl group from s-truns to s-cis, using the optimized structural parameters of s-tmns VCP. Since the angle L CC& increases, the sum of LC1C4H6and LC5C4Hsmust naturally diminish. Evidently, C1C4H6 decreases twice as much (1.4”) as C&H6 (0.7”), since the vinyl group “tries to maintain” its structure. The bond parameters of our calculations and of the ED measurements of Traetteberg [9] agree reasonably well except for the C=C double bond and the C-H bond lengths, whose theoretical values are smaller by 1.7% or (on average) 2.2%. The maximal angle deviation is 1.4” for s-tmns (LC4C5H8)and 2.1” forgauche VCP (LC$C&), most of the angles agree considerably better. Just as in this theoretical investigation, Traetteberg also observed a considerable increase of the angles LXCICll and LC1C4C5,when going from s-trans to gauche VCP. The comparison with the older ED data of de Meijere and Liittke [3] , which are listed in Table 1, lead essentially to the same valuation. Here the maximal angle deviation occurs also for L&&H8 (2.2”). Moreover, the relaxation of our parameters corresponds essentially to the trends as obtained by the ab initio SCF study of De Mare and Peterson [8]. Their C=C bond lengths are also rather short compared to the experimental values. It is typical of ab initio SCF calculations that they tend to underestimate bond lengths, if the basis set is sufficiently saturated. This shortcoming can be removed only by using correlated wave functions. Potential curve of internal rotation For any torsional angle the structural parameters were linearly interpolated between those of the optimized geometries. This kind of interpolation, which

81

is sometimes called “linear internal coordinate path method” [ 261, is certainly not optimal. However, by contrast to the rigid rotor approximation, this method yields, immediately, one continuous potential curve, which connects the minimal energies of all optimized structures (Fig. 2). A refinement by additionally optimizing the transition state structure might decrease the barrier height by less than lo%, as was estimated by determining some of the transition state parameters. The rotational potential curve (Fig. 2) shows s-trans and gauche VCP to be stable conformers with the s-tram one to be more stable by 5.18 kJ mol-‘, but the existence of a stable s-c& conformer is denied. The energy differences between the considered structures are given in Table 3 together with the present theoretical and experimental values. The s-truns/guuche difference of 5.18 kJ mol-’ lies between the ED value of 4.60 f 0.84 kJ mol-’ [3] and the Raman value of 5.98 + 0.60 kJ mol-’ [6]. Also the s-truns + gauche interconversion energy of 13.1 kJ mol-’ agrees satisfactorily with the estimated Raman value of 16.41 kJ mol-’ and is very similar to the calculated energy difference (13.3 kJ mol-‘) of de Mare and Peterson [8]. The equilibrium torsional angle of the gauche structure (117.8”) deviates only by 0.8” from the recent ED angle of Traetteberg [9] and corresponds to the interval 110-120”, which was predicted by the early measurements of de Meijere and Liittke [ 31. The near coincidence of all results with those obtained from recent theoretical and experimental work suggests, that the deficiencies of the early theoretical studies [4] have essentially been overcome. Subsequent, more refined investigations certainly might modify slightly the numbers of Tables 1

Fig. 2. Potential energy curve of VCP as a function of the internal rotational angle 7 relative to the s-trans SCF-energy level. (Basis set augmented with d-functions on C, and C 4, rid = 0.3).

s-tranmauche s-transtr. state s-tram-s-&

6.17a 13.3s 9.84a 120.2 0.403s 0.19ga 0.257a O.4O2a

no 17d

This work rld’0.3 5.18 13.1 11.2 117.8 65.2 0.483 0.252 0.286 0.437

,,d= 0.8

8.06 14.7 120.2 67.8 0.434s 0.215 0.288 -

6.86 13.3 10.79 121.2 67.4 0.446 0.276d 0.421d

SCF ref. 8 11.3 0.43 0.39

SCF ref. 7 4.60 * 0.84 110-120 -

ED ref. 3

-

_b

0.498 f 0.007 -

_b

b >7.5 -

MW ref. 5

5.98 f 0.60 16.41c 12.7c 132c 7sc -

RaIllaIr ref. 6

The SCF energies of s-tram VCP are -193.69007 au. (no r)d) and -193.73214 a.u. (qd = 0.8). ‘Structural parameters as used for the potential curve (Fig. 2). bgauche minimum not observed. CEstimated values obtained from the potential constants of ref. 6. dRef. B(b).

@r.skac ps-tmns ptr.St& Clgauche Ir&&

@gauche

AE

Property

Energy differences AE in kJ mol-I, rotational angles in degrees, and dipole moments in debye as obtained from different calculations and measurements

TABLE 3

._ _

g

83

and 3 but not the fundamental properties of the potential curve as shown in Fig. 2. In order to give a feeling for the remaining basis set dependence of the results, Table 3 lists also the data as obtained from calculations with another d-orbital exponent (nd = 0.8”) and without using d-basis functions. Especially the s-trundguuche energy difference reacts relatively sensitively to the choice of the basis, and the rotational angles Ggoucheand $tr.stahvary by more than 2”) a fact that is based on the small torsional force constant (cf. Table 5). The occurrence of a stable gauche minimum can presumably be explained as a compromise between two competitive effects. The n-conjugative interaction between C1 and C4 as a function of the dihedral angle clearly favours a stable s-cis structure (apart from the s-trans structure) rather than a gauche form, whereas the repulsion between Hz, HB, and H8 as suggested by the angle relaxation tries to prevent the s-cis form. The same arguments should also apply to similar molecules, for example to 1,3-butadiene. Indeed Durig et al. [ 281 explained the Raman spectrum of 1,3-butadiene by a potential curve of internal rotation, which has also two equivalent gauche minima and no s-cis minimum. The relatively strong conjugational effect in the central C-C bond causes an s-trans + gauche interconversion energy (24.7 kJ mol-‘), which is nearly twice as big as the VCP value, whereas the steric interactions in both molecules might be of comparable importance. An increasing prevalence of the conjugational effect over the steric interaction evidently suggests a shift of the transition state and gauche equilibrium dihedral angles towards 90” or 180”) respectively. In fact, compared to VCP these shifts were actually obtained in a recent ab initio SCF study on 1,3-butadiene by de Mare and Neisius 1291. The s-truns:gauche rotamer population of VCP has been estimated from the calculated energy levels of torsional vibrations, where only molecules with angular momentum J = 0 have been considered [ 121. This ratio amounts to about 12: 1 at dry ice temperature (195 K) and to about 4:l at room temperature (293 K). De Meijere and Liittke explained their ED experiments, which were performed at room temperature, by a ratio of 3:l [ 31. The considerable abundance of s-truns VCP over thegauche rotamer at 195 K might essentially be responsible for the fact that gauche VCP was not observed in the MW study of Codding and Schwendeman [ 51. Dipole moments Table 3 shows the dipole moments p of VCP in comparison with the available theoretical and experimental data. The s-truns value of 0.483 D is remarkably close to the experimental dipole moment (0.498 D, ref. 5). The * t-,d = 0.8 is approximately the exponent of the carbon atom, which minimizes the SCFenergy of methane. It gives also deeper SCF-energies for VCP than qd = 0.3 does (cf. Tables 1 and 3).

a4

considerably smaller dipole moment of gauche VCP (0.286 D) constitutes another argument, why gauche VCP was not observed in the microwave spectrum [5] . Of course, the experimental conditions, i.e., measurements at dry ice temperature, are certainly the major reason. The dependence of the calculated dipole moments on the basis set is also illustrated in Table 3. The value of gauche VCP proves to be relatively stable as against the choice of qd, a fact that demonstrates the significance of our value. Additional test calculations, which were performed for the smaller propylene molecule using different basis sets, suggest that the error of the dipole moment should remain below 0.1 D. Force constants Some important (harmonic) valence stretching and valence bending force constants of VCP were obtained from the curvatures of the SCF energy as functions of the bond lengths or, respectively, the bond angles. The data, which are collected in Table 4, do not lay claim to high accuracy, but they still give a rough idea of the bond strengths. The force constants of different rotamers are not distinguished, since their precision would not suffice to correlate them to structural relaxation effects. Such a discussion is much better done in terms of the population analysis infra. The C=C force constants of ethylene (Table 2) and of the vinyl group (Table 4) have the common value of 11.3 mdyn W-l. The numbers are not precise enough to decide the question, whether the ethylene bond is really the stronger one as suggested by the bond length considerations supra. For comparison, the calculations of Blom et al. [35] on the propylene molecule yielded a force constant of 11.63 mdyn A-’ for the C=C bond, a value, which in fact is slightly exceeded by their ethylene value of 11.75 mdyn A-‘. The C-C single bonds (Tables 2 and 4), on the other hand, show somewhat more significant differences among each other. The Cl-C4 bond of VCP (5.0 mdyn A-‘) proves to be stronger than the C-C ethane bond (4.5 mdyn A-‘), which is another hint at a conjugation between C1 and C+ A similar TABLE 4 Valence stretching and bending force constants, Bond (C-W,, (c-c)ri,

CI-C, c,=q

av

f, of gauche VCP

Stretchinga

Angle

Bendine

6.2 4.7 5.0 11.3

%-G--H* (cc=$-!&

0.84 1.2 1.7 1.4 0.07

c:-&& @torsion

aForce constant measured in mdyn 1$-I. bX denotes an axis in the ring plane bisecting the angle LC,C,C,.

85

result was obtained for the propylene molecule again by Blom et al. [ 35,621. Their propylene C-C force constant of 5.16 mdyn A-’ clearly surpasses their ethane value of 4.78 mdyn A-‘. Moreover, the corresponding force constant of 1,3-butadiene (5.93 mdyn A-’ [40] ) illustrates that the partial double bond character of the C-C bond in 1,3-butadiene is much more pronounced than in propylene and VCP. The VCP force constant of G-C4 is also bigger than the average of the three ring bonds (4.7 mdyn A-l), a result that is reflected in the bond lengths in Table 1. The force constants of the individual ring bonds are not distinguished, since their differences were not reliable enough to be explained in terms of the ring distorsion. Table 4 shows the torsional force constant to be smaller by one order of magnitude than the bending force constants of the H-C-H, C=C-H, and C-C=C bending vibrations. This illustrates how easily the vinyl group can be rotated internally. The significance of the data in Table 4 can be seen by comparing them to the presently available theoretical and experimental values of similar bonds such as the values of et&me, cyclopropane, ethylene, propylene, and 1,3butadiene [31-441. Apart from the H-C-H value even our bending force constants agree fairly well with the literature data. Population analysis The discussion of this section is dedicated to the features of the carboncarbon “bond orders”, i.e., the overlap populations n(CC) of the MPA (Mulliken Population Analysis) and the shared electron numbers a(CC) of the DRPA (Davidson-Roby Population Analysis). The rather unconventional shared electron numbers can be considered as a measure of covalent bond strength. To obtain an idea of their size in some cases the reader is referred to the literature [63]. The bond orders of VCP, which were calculated for some selected torsional angles (@ = 0”, 65.2”, 90”, 118’, and 180”) without including polarization functions, are collected in Table 5 and should be contrasted with the corresponding data of cyclopropane, ethylene, and ethane given in Table 2. The bond orders as obtained from the augmented basis set (qd = 0.3) are additionally shown in Figs. 3 and 4. At a torsional angle of 4 = 90”) where conjugation between C 1 and C4 is essentially suppressed, the shared electron number of Cz-C3 (u = 1.309) is practically identical with the cyclopropane value (a = 1.304), whereas u( C2-CJ) is slightly increased in the s-trans and s-cis structures at the expense of the otherring bond orders, u(Cl-Cz) and u(Cl-C3). Moreover, (by contrast to the MPA) the DRPA actually exhibits one “strong” (C,-C3) and two “weak” ring bonds (Cl-C2 and Cl-&), especially in s-trans and s-cis VCP. The detailed consideration of the data (Tables 1 and 5) shows that the shared electron numbers completely parallel the bond lengths relaxation. This applies also the the CL+=& double bond as well as to the central Cl-C, single bond. The shared electron number u(C&,) varies between 2.120

86 TABLE 5 Bond orders of VW (basis set without polarization functions) Bond

Torsional angle (0 65.2”

90”

118”

180”

0.566 0.561 0.559 0.722 1.288

0.563 0.569 0.557 0.725 1.287

0.557 0.572 0.555 0.736 1.287

0.555 0.555 0.564 0.743 1.281

numbers (I (DRPA) 1.259 1.279 1.259 1.275 1.319 1.309 1.376 1.331 2.120 2.142

1.274 1.280 1.309 1.341 2.141

1.267 1.286 1.309 1.347 2.140

1.266 1.266 1.322 1.357 2.129

0”

Overlap populations n (MPA) CC* 0.558 CC, 0.558 C,C, 0.557 CC, 0.752 C,C, 1.278 Shared electron GIG CC, C,C, CC, C&

\\:1 (Cl

1.16

072

fb) 0.71

,_ 0.70

.,rzi%Lj 0’

60’

120”

160”

Fig. 3. Overlap populations of C-C, (a), C-C, with d-functions on C, and C,, nd = 0.3).

60”

0”

1204

160”

(b), and C,=C, (c) (Basis set augmented

Fig. 4. Shared electron numbers of C,-C, (a), C,-C, mented with d-functions on C, and C,, vd = 0.3).

(b), and C,=C, (c) (Basis set aug-

87

and 2.142 adopting its maximum near 9 = 80” and keeping strictly below the ethylene value of (I = 2.185. The bond orders a(C1C4) and n(C&,), on the other hand, clearly show that the C1 -C4 single bond of s-trans, gauche, and s-cis VCP is stronger than the C-C bond of ethane (n = 0.663, (I = 1.324) and also stronger than the VCP ring bonds. This fact was first established by the MPA of Skancke and Boggs [7] and is also reflected in the force constants of Table 4. For a dihedral angle near the transition state structure (c = 1.331) the shared electron number approaches the ethane value (1.324) from above. The variation of the C1C, bond orders (Figs. 3, 4 and Table 5) illustrates very convincingly the degree of partial x-character between C1 and C+ The bond is maximally stabilized at $J = 0” and 4 = 180” and is progressively weakened, when rotating the vinyl group to about 9 = 90”. Simultaneously, charge is transferred from the central C1-C4 bond into the adjacent bonds CL+=&,C1-C2, and Cl-C3 ($ = 0” + 90”, or @ = 180” + go”), whereas Cz-C3 also feeds its adjacent bonds C1-C2 and C1-C3, but to a lesser degree. The extrema in Figs. 3 and 4 vary somewhat about the dihedral angle of 90”) dependent on the bond and also on the population analysis considered. We do not want to comment on such details, since to some extent the data are still subject to the convergence of the MAO iteration procedure and also to the basis set used. However, these technically conditioned uncertainties do not influence the principal trends. These are essentially the same, no matter which basis set has been used and whether they are obtained from the MPA or the DRPA (cf. Figs. 3 and 4). Only in the case of rather subtle properties such as the ring distortion or the relative size of the C=C bond orders in ethylene and VCP do discrepancies between both methods occur. Critical points and bond paths An alternative investigation of the charge density p(r) is obtained by considering its critical points and bond paths. This method is extensively discussed in refs. 49-53 and in the literature cited therein, to which we refer for the details. In the diction of Bader and coworkers [50-531 the saddle points of the molecular density p(r) between two nuclei A and B bound to each other are bond critical points rc = r,(AB), which are defined by the position re of vanishing density gradient (vp(r,) = 0). Since any bond critical point is the origin of two density gradient paths, which terminate at the nuclei of the considered bond, these nuclei are connected by a path of maximal charge density*, which passes through the appertaining saddle point r,(AB). Such a “bond path” directly reveals the bend of a bond, in particular in the cyclopropyl ring. *At any point of this path the charge density is maximal in comparison with the density values in the vicinity of this point on a plane perpendicular to the path.

In Fig. 5, the extent of bond strain is illustrated by the perpendicular distance d(AB) between the bond critical points and the internuclear axis AB of any two carbon atoms A and B. Other measures of bond strain are the “critical angles” between the axis AB and the directions A - r,(AB) and B - r,(AB) as well as the “path angle deviations”, i.e., the angle deviations between the bond axis AB and the bond path directions at the nuclei A and B. For the C-C bonds of cyclopropane, e.g., Runtz et al. [49] report a critical angle of 4.46”) our own (unaugmented) basis set yielded 5.1” instead. The critical angles of the VCP three-membered ring amount to 5.1-5.4”, whereas the appertaining path angle deviations are about twice as large (cf. Fig. 5), i.e. these bonds are considerably bent. When the vinyl group rotates from s-tram to s-cis, the bend of the bonds at C1 undergo some noteworthy changes. Whereas the distance d(C&) is steadily 0.068 A just as for cyclopropane, the values of d(CIC2) and d(CICJ)

(a)

Fig. 5. Critical points (solid circles), critical angles, bond paths (-o-e-), path angle deviations (in part shown in parentheses), CC bond lengths and bond path lengths of s-trans (a), gauche (b), and s-cis (c) VCP. The curvature of the bond paths is exaggerated. Lengths are given in A.

89

are first increased by 0.003 ft for the gauche rotamer and then again reduced to 0.068 in s-cis VCP. Also the C1-C4 bond shows more bond strain in the gauche and even in the s-cis form than in s-trans VCP (d(C,C,) = 0.004 A (s-trans), 0.010 A (gauche), and 0.011 A (s-c&)). This fact is still more dramatically described by the path angle deviations of C1-CL at C1, which are 1.3” for s-truns and not less than 2.7” (2.6”) in the case of gauche (s-cis) VCP. We contrast these deviations with the simultaneous variation of the bond angles LC4C1CZand LC4C1C3(119.6”-122.0”, cf. Table 1). Since both effects partially “cancel out”, the angles between the CC bond paths at C1 are nearly independent of the dihedral angle. In fact, the path directions of C-C4 and C1-Cz/C1-C3 include angles of 114.4” (s-Pans), 113.8”/114.8” (gauche), and 114.5” (s-cis). A similar cancellation can be observed by considering the angle variations at Cd. Whereas the bond angle LCIC4CS varies from 124.7” via 126.1” to 126.8” (Table l), the angle between the appertaining bond paths is again practically constant and has the values 125.8”, 125.9”, and 126.2”. In brief, the VCP molecule manages to avoid steric interaction between the hydrogen atoms H8 and H2,3 by adequate bond angle relaxation without seriously changing the angles between the bond paths at C1 and C+ This statement followed qualitatively when the calculations were repeated using the augmented basis set (qd = 0.3) of the preceding sections. The CC ring bond lengths (1.510 A-l.522 A) are relatively short in comparison to the experimental ethane bond length (1.528 R [25] ). The ring path lengths (1.518 A-l.533 A), on the other hand, vary in fact in the range of the ethane value and exceed the appertaining bond lengths by fully 0.01 A. The difference between bond and bond path lengths of the other bonds is typically less than 0.001 A. At the carbon atoms C1, Cz, and C3 the (ring) bond paths include angles of 78.8”-80.4”. They are definitely smaller than 90” and should not be identified with the angles between hybrids, which exceed 90”. This phenomenon is repeatedly described in the literature [49, 56, 571. For instance, Stevens et al. [57] calculated an angle of 77” between the bond paths of cyclopropane, whereas the’same wavefunction yielded 102” (or, respectively, 116”) between the hybrid directions. Similar hybrid angles were obtained in a classic paper of Coulson and Moffitt [54] by virtue of maximal overlap arguments (104”, C,H,) and by Eckart-Maksic and Maksic [55] in a semi-empirical study on VCP (105.2”-106.2”). Difference densities, which have been studied theoretically and by X-ray diffraction [57-60] , accentuate bond bending rather strongly. Its degree corresponds closely to the degree obtained from the hybrids [58]. Features of the electron density at the bond critical points The three principal curvatures of p(r) at r = rc(hl < A2 < A,), i.e. the eigenvalues of the Hessian matrix V v p(rC) [ 52, 531, show how quickly the

density increases along the bond path towards the nuclei (hg > 0) and how quickly it decreases perpendicular to this direction (hl < X2 < 0). If the density has axial symmetry at rc, the two negative curvatures X1 and hz coincide, and the “ellipticity” E = Al/h2 - 1 [53] becomes zero. An example of this case is given by the C-C bond of ethane (Table 6). However, if the contour plot of p in a plane perpendicular to the bond path at rc is elliptically distorted, a case that occurs especially for double bonds, then the principal curvatures A1 and hz differ from each other, and the less negative value (hz) describes the decrease of p in the n-bond direction. Thus, the ellipticity is positive and measures the degree of elliptical distorsion of the contour lines. Ethylene possesses a value of e(C=C) = 0.219 (Table 6). The ellipticity can also be interpreted as a measure of ?r-electron delocalization and of hyperconjugation and is therefore suitable for directly ‘investigating the conjugative n-interaction between the cyclopropyl ring and the vinyl group.

The eigenvectors v1 and v2 of the Hessian matrix belonging to the eigenvalues h1 and h2, which span the plane perpendicular to the bond path at rc, point into the directions of maximally and minimally decreasing p(r), i.e., their directions coincide with the principal axes of the elliptic density contour lines. For instance, at the bond critical point r,(C=C) of ethylene the eigenvector of X2 coincides with the n-bond direction. Moreover, the value of V2p(r,) = h1 + A2 + h3, i.e., the trace of the Hessian matrix, characterizes the covalence of a bond. According to the discussion of Bader et al. [53], the value is positive for strictly ionic bonds and for van TABLE 6 Properties of the charge density at the bond critical points (SCF-wavefunctions polarization functions) Bonda

p(r,) (a-?

H,C-CH, C-C in C,H, (C1-G )W (CI-G), (G-C, )W s- trans C,-C, gauche s-cis s-trans C,=C, gauche s-cis H,C=CH,

1.486 1.484 1.465 1.475 1.489 1.733 1.690 1.687 2.359 2.360 2.349 2.374

h, (A“) -9.56 -9.61 -9.32 -9.41 -9.62 -11.61 -11.20 -11.21 -17.29 -17.10 -17.05 -17.73b

% (A-5) -9.56 -8.28 -7.94 -8.04 -8.30 -11.42 -11.08 -11.02 -14.33 -14.14 -14.14 -14.55b

& (A-5) 6.00 5.89 5.95 5.90 5.82 4.83 5.08 5.10 4.69 3.60 4.20

V”&-J (a+) -13.1 -12.0 -11.3 -11.5 -12.1 -18.2 -17.2 -17.1 -26.9 -27.6 -27.0

without

E 0.000 0.161 0.174 0.170 0.159 0.016 0.011 0.017 0.207 0.209 0.206 0.219

a‘cav’r = mean value of s-trans, gauche, and s-cis. bValues calculated from the Hessian matrix at the bond centre. A weak artificial density maximum was obtained at the ethylene C=C bond centre between two saddle points. The listed p(r,) value of ethylene is the saddle point density.

91

der Waals bonds, and it is negative in covalent bonds. A C=C double bond typically reveals a more negative asp-value than a C-C single bond, since the former is radially more contracted than the latter, which results in more negative curvatures h I and AZ. As expected, the saddle point density p( r,) of C&--C3in Table 6 nearly coincides with the corresponding value of cyclopropane and ethane, whereas the bonds Cl-Cz and CI-C3 have only slightly smaller densities thus reflecting the ring distorsion. The curvatures along the bond paths and perpendicular to the ring plane (h3 and X1) illustrate also the similarity of these bonds among each other. The curvature in the ring plane (hz) naturally exceeds X1 (<0) and thus exhibits substantial ellipticities of about e = 0.17 (cf. Table 6 and ref. 53). The discussion about the relative strength of the double bonds in the vinyl group and in ethylene (supra) is also corroborated by Table 6. The density p(rJ of ethylene slightly exceeds the vinyl group values of ah rotamers, and the curvatures h 1 and AZ of both molecules exhibit the less contracted electron charge cloud in the vinyl group. Moreover, the VCP n-orbital causes a considerable ellipticity of 0.21, which is comparable with the ethylene value (0.22). The Cl-C4 bond parameters (Table 6) vary between those of a standard single and a standard double CC-bond, thereby more resembling the single bond. The relatively small ellipticity of Cl-C4 displays essentially cylindrical bond symmetry. However, the charge density is distinctly contracted towards the bond path compared with the density of ethane (cf. X1 and AX). This contraction results in values of p (r,) and V’p, which considerably surpass those of ethane. The conjugative interaction between ring and vinyl group is clearly revealed by the ellipticities as a function of the torsional angle. The C1-C4 bond takes its largest ellipticity for s-tram and s-cis VCP (e = 0.016/0.017) and has a definitely reduced value for gauche VCP (E = 0.011). This variation is still more impressively illustrated by the full curve ~(4) of Fig. 6, which was obtained from our augmented basis set (qd = 0.3). The C4=C5 bond shows a countercurrent trend just as the bond orders of Table 5 and Figs. 3 and 4. Corresponding trends were also observed for the C-C and C=C bonds of internally rotated 1,3-butadiene [ 611. The concept of a conjugative n-interaction between C1 and C4 is finally very conclusively manifested by considering the eigenvector v2 of the Hessian matrix, whose direction in C=C bonds coincides with the n-bond direction. In s-tram and s-cis VCP the vector v2(C1-C4) proves to be parallel to the C4=C5 n-bond direction. In gauche VCP v2(C1-C4) includes an angle of 14.8” with v2(C4=CS) and 44.6” with the C2-C3 bond direction. On the other hand v2(C4=C5) and the C2-C3 direction include an angle of 59.2”. Consequently, the conjugative n-bond direction at the saddle point of the Cl-C4 bond is a compromise direction between C2-C3 and the n-bond direction of C4=C5, i.e. we have a distorted conjugative n-bond between C1 and C!+

I 0"

60"

Fig. 6. Ellipticity of the C,-C, C, and C,, qd = 0.3.)

dihedral angle I 120'

180'

bond in VCP. (Basis set augmented with d-functions on

CONCLUSIONS

The occurrence of three equilibrium geometries in VCP (one trans and two equivalent gauche structures) can be explained by two competing main effects, the A-conjugation between ring and vinyl group and the steric interaction of the endstanding hydrogen atoms Hz, HB, and Hs. The first effect, which favours the s-tram and s-cis geometries, has been shown by structural relaxation and force constant considerations as well as by investigating the CC bond orders and the bond critical point properties of the total electron density as a function of the dihedral angle. The effect of steric repulsion on the other hand, which disfavours the s-cis structure, was observed in the relaxation of structural parameters. The considerable opening of the angles LC&C1C4 and LC1C4C5, when internally rotating from s-tram to s-cis VCP, gives evidence of the repulsion. The investigation of the electron density by population analyses and bond critical point properties, however, failed to reveal the steric interaction. ACKNOWLEDGEMENTS

We are very grateful to Prof. W. Liittke (Gottingen) for thorough and stimulating discussions and for valuable assistance concerning the literature search. We thankprof. M. Traetteberg (Trondheim) for making her ED results available to us prior to publication, P. Reynders (Gottingen) for providing us with the computer plot of Fig. 1, which was generated by the plot program “Schakal” of E. Keller (Freiburg), and G. R. De Mare (Bruxelles) and

93

D. Cremer (KSln) for comments on the manuscript. All computations were performed on the SPERRY 1100 of the Gesellschaft fiir wissenschaftliche Datenverarbeitung Gijttingen. The financial support of the Fonds der Chemischen Industrie is gratefully acknowledged. REFERENCES 1 W. Liittke and A. de Meijere, Angew. Chem. Int. Ed., 5 (1966) 512. 2 G. R. De Mare and J. S. Martin, J. Am. Chem. Sot., 88 (1966) 5033. 3 A. de Meijere and W. Liittke, Tetrahedron, 25 (1969) 2047. 4 W. J. Hehre, J. Am. Chem. Sot., 94 (1972) 6592. 5 E. G. Codding and R. H. Schwendeman, J. Mol. Spectrosc., 49 (1974) 226. 6 L. A. Carreira, T. G. Towns and T. B. Malloy Jr., J. Am. Chem. SOC., 100 (1978) 385. 7 A. Skancke and J. E. Boggs, J. Mol. Struct., 51(1979) 267. 8 (a) G. R. De Mare and M. R. Peterson, J. Mol. Struct., Theochem, 89 (1982) 213. (b) G. R. De Mare, private communication. 9 M. Traetteberg, private communication. 10 (a) H. Giinther, H. Klose and D. Wendisch, Tetrahedron, 25 (1969) 1531; (b) H. Gunther, H. Klose and D. Cremer, Chem. Ber., 104 (1971) 3884. 11 V. R. Salares, W. F. Murphy and H. J. Bernstein, J. Raman Spectrosc., 7 (1978) 147. 12 B. Klahn, to be published. 13 (a) G. Schrumpf, Tetrahedron Lett., (1970) 2571. (b) L. Ernst and T. Schaefer, Chem. Ber., 105 (1972) 2368. 14 P. Bischof, R. Gleiter, E. Heilbronner, V. Hornung and G. Schriider, Helv. Chim. Acta, 53 (1970) 1645. 15A. D. Wa.ish,Trans. Faraday Sot., 45 (1949) 179. 16 R. Hoffmann, Tetrahedron Lett., (1965) 3819; (1970) 2907. 17 S. Huzinaga, Preprint 1, University of Alberta, 1971. 18 W. Kutzelnigg, Einftihrung in die Theoretische Chemie, Band 2: Die chemiache Bindung, Verlag Chemie, Weinheim, 1978, p. 531. 19 F. Driessler and R. Ahirichs, Chem. Phys. Lett., 23 (1973) 571. 20 J. L. Duncan, J. Mol. Spectrosc., 60 (1976) 225. 21 P. Reynders, private communication. Thevalue is calculated under the same conditions as the VCP structure of this paper. 22 G. R. de Mare and M. R. Peterson, J. Mol. Struct., Theochem, 104 (1983) 115. 23 M. D. Harmony, R. N. Nandi, J. V. Tietz, J.-I. Choe, S. J. Getty and S. W. Staley, J. Am. Chem. Sot., 105 (1983) 3947. 24 F. H. Allen, Acta Crystallogr., Sect. B, 37 (1981) 890. 25 J. L. Duncan, D. C. McKean and A. J. Bruce, J. Mol. Spectrosc., 74 (1979) 361. 26M. C. Flanigan, A. Kormonicki and J. W. McIver, Jr., in G. A. Segal (Ed.), Semiempirical Methods of Electronic Structure Calculations, Part B, Plenum Press, New York, 1977, p. 19;A. Kormonicki and J. W. McIver, Jr., J. Am. Chem. Sot., 96 (1974) 5798. 27 L. Random and J. A. Pople, J. Am. Chem. Sot., 92 (1970) 4786. 28R. J. Durig, W. E. Bucy and A. R. H. Cole, Can. J. Phys., 53 (1975) 1832. 29 G. R. de Mare and D. Neisius, J. Mol. Struct., Theochem, 109 (1984) 103. 30 E. Orti, J. Sanchez-Marfn and F. Thomas, J. Mol. Struct., Theochem, 108 (1984) 199. 31 L. S. Bartell, S. Fitzwater and W. J. Hehre, J. Chem. Phys., 63 (1975) 4750. 32 P. Pulay and W. Meyer, Mol. Phys., 27 (1974) 473. 33 G. E. Hansen and D. M. Dennison, J. Chem. Phys., 20 (1952) 313. 34 M. Spiekerman, D. Bougeard and B. Schrader, J. Mol. Struct., 60 (1980) 55. 35 C. E. Blom, P. J. Slingerland and C. Altona, Mol. Phys., 31 (1976) 1359. 36 J. L. Duncan and G. R. Burns, J. Mol. Spectrosc., 30 (1969) 253. 37 P. Pulay and W. Meyer, J. Mol. Spectrosc., 40 (1971) 59.

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