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Out-of-plane deformations of cyclic polyenes
OUT-OF-PLANE DEFORMATIONS OF CYCLIC POLYENES A FORCE-FIELD INTERPRETATION OF STRUCTURAL DATA
H. B. BiiRcir and E.
SHEFTER?
Laboratories of Inorganic and Organic Chemistry, Swiss Federal Institute of Technology (E. T. H.), 80% Zurich, Switzerland (Received in UK 23 June 1975;Accepted forpublication IOJuly 1975)
AbstRct-Speetroscopicfame fields of benzene are shown to be compatible with the observed non-coplanarity of benzene rings with their H- or C- substitueats in strained compounds, e.g. cyclophanes. A simplified force field for annulenes and cyclophanes is discussed, and ratios of the twist and out-of-ptane bending forceconstants involved are derived from structural data. The resistance to twisting of partial double bonds is found to increase with decreasing bond length. Various formulae to calculate out-of-plane deformation coordinates are collected in an appendix. The n-electron systems of organic molecules tend to be planar. Consequently, their deformations, static or dynamic, may be classified into deformations in the molecular plane and deformations out of the molecular plane. From a survey of the literature it seems that studies of the nonplanar deformations of polyenes by vibrational spectroscopy are rare while structural studies are plentiful. Among the molecules which show ?r-systems highly deformed from planarity we find cyclophanes and polycyclic annulenes. We have studied these deformations systematically and related them to the few available spectroscopic force constants. The coordinate system. In order to describe these deformations in a consistent way we have chosen to put our molecules into a right-handed coordinate frame and to number the atoms sequentially. Two types of internal molecular coordinates are needed to describe nonplanar deformations: out-of-plane bending (oop) deformations and twist (tw) deformations. Both of them may be expressed in terms of torsion angles which describe the nonplanarity of a sequence of four atoms (Fig. 1). The torsion angle w(l,2,3,4) is defined as the angle between the planes of the atoms 1,2 and 3 on the one hand and atoms 2, 3 and 4 on the other. The sign conventions are those of Klyne and Prelog.’ Twist deformation coordinates are defined in terms of two torsion angles’-’ ~(23)=&(1234)+o(Rz23R1)) = n+;(~(123R1)+o(Rz234)).
(1)
Oop bending deformations may also be expressed in terms of two torsion angles,’ e.g. for atom 3 in Fig. 1. ~(23)=Jn-(w(l,2,3R,)-w(l,2,3,4))l.r
(2)
According to this definition, the sign of the quantity within the vertical bars is opposite for x(23) and x(43). The value of s( + I or - 1) then serves to define the sign of the oop tG. A. Pfeiffer Fellow (1972-73)of the American Foundation for Pharmaceutical education; present address School of Pharmacy, State University of New York, Buffalo.
2-3 /
\
1
.’ 2
1 i@% 4
’
4
./ !a
./
x
R3
T
d Fig. 1. bending at atom 3. The choice of s is arbitrary for at least one atom in a conjugated polyene; once an s value has been assigned to one atom, the next atom is given the same sign if-for T set back to 0”-the two atoms are on the same side of the plane defined by their four substituents; it is given an opposite sign otherwise. This procedure is continued all through the conjugated chain. The case where 7 = 90” is equivalent to breaking the conjugation and necessitates a new arbitrary choice. An alternative definition of x is in terms of the dihedral angle w(423R,) (4-2 does not correspond to a chemical bond!)“’ x(23) = Ia - o(423R,)I . s.
(3)
It should be noted that x(23), x(43) and x(R& are identical if and only if the three bond angles at atom 3 are the same.
2977
Out-of-planedeformationsof cyclicpolyenes
this additional i~o~tion he calculated a force Eeld that is remarkable because ail except one or possibly two of its interaction constants (f’, f,“‘) are very small (Table 1, d and e). All of the previously mentioned force fields may now be tested not only with respect to their capability to reproduce observed frequencies but also with respect to their ability to describe nonplanar defo~ations of benzene derivatives such as [3,3]paracyclophane, the diolefins of [2,2]paracyclophane’0 and [2,2lmetap~a~ycIoph~e.” These molecules have been studied carefully by X-ray d&action methods; H atom positional parameters have been determined with esd’s between 0.015 A and 0.04 A. All three molecules show the p~~ubstituted benzene rings distorted into a boat conformation. The aromatic H atoms are displaced out of the plane defined by their three nearest C atoms by -0-04 A to - 0.244 A. Although individual displacements +2f” 7 ~,(7,-2-n+,)t2fp~X.(r,-9-71*2) (4) are not always statistically significant they are all in the same direction (to negative z, Fig. 2) indicating a systematic trend. (The intermolecular II. . . H contacts, where i stands for i = 1,. . . . ,6. Only eight symmetry force cons~ts may be determined from the six being longer than 2.4 A, should not affect the hydrogen observable normal vibrations of C&and C& (two each positions very severely). of Bzp and E2. symmetry, one of AZ=and Et, symmetry). For the subsequent treatment we have assumed local This is not enough to fix al1eleven valence force constants CZ, symmetry of the benzene ring and averaged the in eqn (4). The common way out of this indeterminacy has structural parameters accordingly. We have anrtlysed the been to put j? = jP = jz = O.%’VFF’s obtained with this averaged H atom oop displacements with the assumption approximation by various authors are listed in Table that the structure of the carbon skeleton is more or less 1fa-c). The oop force constants and the oop-tw interaction rigid and determined mainly by ring closure conditions of constants are virtually the same in all force fields. The the polycyclic molecule. We have further assumed that value of the tw constant shows some variation, however; the H atoms, in this constrained situation, try to find it ranges from 0.236 to 0.272 mdgn &ad’, and the energetically optimal positions. The mathematical model corresponding or&o interaction constant varies between incorporating these assumptions is based on eqns (4), (1) - 0.078 and - 0*058mdyn &rad2. and (3). Torsion angles between C atoms are fixed by the ring constraints and are given the experimental values. The torsion angle w(R112H3 (and symmetry related ones, Table 1. Valenceforce fieldof benzene(see Eqo4) Fig 2) is free to assume a value which minimizes the total a C b d e energy according to the condition
Other oop bending coordinates are also used, especially the oop bending angle S. It is the angle between a bond, say 2-R*, and the plane formed by two other bonds involving atom 2 (the plane 123 in Fig. lb)? The sign of S may be defined in the same way as for ,y. Relationships between these and others types of oop coordinates are given in an Appendix. TIte benzene oop force fields and the cyc~~p~a~es. The general valence force field (VFF) for oop deformations of benzene is given by
H.
5 F f? fx F f'
0.236
O-259
- 0.078 _* 0,312 0.016 -0.016 - 0.018 -0.032 -0*008 1
-0465 _* 0.313 0.014 -0.016 -0~014 -0.035 odm3 _*
0*272&t) 0*33st -O*OS8(13) -0404 0.024 0.&~3010 0.295 0.015 om -0.013 -0m7 -0*014 -0m2 -0.032 -0.032 -O*OM o*OOs _* _*
0.33% -0m7 0.015 o-293 oaO6 -0+06
d@V) dw(R, 12f&) =’
Takingthe assumed CL symmetry into account, this yields (K-f:-
-&2 -0m8 _*
fJ” t
fP ~2~i2jp)(~(R,12~*)+
~~6123))~2
-~+j~-2j~-2j~m)(-n-o(R,123)t~{6123))
*Assumed.tEstimatcd from model calculation.“Ref.& based Whiien’s model[5]. ‘Ref. 6, from chlorinatedbenzenes. ‘Ref. 7, from alkylated benzene%dRef. 8. ‘Ref. 8.
+ w(R, 12Hz))= 0.
on
Kydd has estimated the tw constant of benzene* with the help of a relationship between tw constants, n-bond orders and overlap integrals of glyoxai and ethylene. With
(6)
On solving for o(R112H3 the results in Table 2 are obtained. The agreement between experimental and calculated values is satisfactory. This shows that the benzene force Geld accounts reasonably well for the static oop bending deformations of H atoms in a strained
Fii 2.
H. B. BORGIand E.
2978
benzene ring.t Indeed, the quality of the agreement is surprising because the deformations from planarity are by no means intinitesimal but amount to as much as 21” in x and 24” in T. It is not clear whether this implies that the quadratic approximation to the force field is still valid, or whether the agreement is fottuitious and due to the cancellation of various systematic errors in the data, e.g. in the hydrogen coordinates (which represent maxima of electron density rather than nuclear positions) or errors in the force field, e.g. the neglect of higher than quadratic terms. A simplified oop force field for cyclophanes and annulenes; ratios of force constants. In the previous paragmph Eqn (6) has been used to calculate o(R112H2) and thereby the oop displacements of H atoms in cyclophanes. The calculation was based on the known spectroscopic force constants of benzene and the well determined carbon skeleton torsion angles of the cyclophanes. An alternative use of Eqn (6) is to obtain a relationship between unknown force constants when all the torsion angles are known. If interaction constants are small, as in the benzene force field (Table 1), the expression (6) may be simplified to the approximate relationship H,7(12)tHZ~(12)-o.
(7)
Similar expressions may be derived for [lOlannulenes
%FfRR
0
30 0 .
2 \’
03
X
0
OoO . IQ
0
0
0
00
cl0 I35
1%
k4G
dOI)
A Fl. 3. Ratios of oop and tw deformation angles versus bondlength of twisted bonds (0). Standard deviations of individualratios x/r(ij) are about 0.3 for d(Q)between I.45 and I.~oA and O&l.0 for d(ij) between 140 and I.33 A (estimated from experimental standard deviations in torsion angles). Also given are ratios of spectroscopic force constants H,,,,JH, for cyclohexeneand benzene(0).
and were found to vary between - 0.0 and 3.0. The seemingly uncoherent set of numbers -_x/~(ij) was plotted against the bond lengths
observed torsion angles’z”
Table 2. Observed (average)and calculated torsion kk.5
[3.3]Paracyclophanes [2.2]Paracyclophanedienel” [2.2]Metaparacyclophanediene”
ti
ofcyclophanes
o(RJ23)
o(6123)
o(R112Hz)
- 168.1” - 146.4” - 137.8”
7.6” IS9 2l.P
6.7(20)=’ 21.3(25’) 269(10)”
ti(R112Hz)”
u(R112H8
7.3” 22.2” 27.3”
7.5” 22.6” 279
‘Calculated from force field c, Table I. bCalculated from force field e, Table I. ’ ESD’sobtained from
deviationsof individualmeasurementsfrom average. (assumed to have C2. symmetry (Fig. 3)) on the basis of a simplilied force field: 2v = 2zQZ(12) t 4HXcy2(12)+ x2(2])) t 4K(*sr*(l2) + 4&&(23).
d(ij) (Fig. 4). The scatterplot shows that large ratios -x,(ij)/r(ij) are found for short bonds d(ij) and smaller ratios for the longer ones. , A corresponding expression for a transcycloolefin with local C2 symmetry at the double bond is based on
(8) 2V = 2H,x= + HJ'.
Provided that the bond angles 12If2 and Zf33 are equal, x(12) equals x(37) and from a condition analogous to Eqn (5), one obtains KW(l2)+
H&(12) + x(23)) = 0
(9)
(12)
Ifthe CCCC-torsion angle is constrained by the ringclosure condition it may be shown that H,IH,=2x/r.
(13)
A numerical value for I/k, was obtained from transcyclooctene: ‘*2x17 = 48”/18”= 2.7, d(C=C) = 1.330 A. It is generally accepted that the torsional barrier of a (The relationships (9) and (10) also hold for half the annulene molecule.) After rewriting, Eqns (7,9 and 10)are partial double bond increases as the bond length shortens, i.e. as the bond order increases. It therefore seems seen to yield ratios of force constants reasonable to attribute the negative slope of the curve in Fig. 4 mainly to an increase in the tw force constant HTct,,. &dH, = It is possible to add two more points to the graph, both (11) of which are ratios H,&Hxc,,, obtained from spectroscopic These ratios have been calculated (Table 3) from data. One is obtained from the benzene force field discussed earlier, the other one from the oop force or cyclohexene.m The two Wrictly speaking, the above calculations only provide a test of constant of ethylem? additional points fit very well into the trend.indicated by four symmetry force constants because only an A,. and two &. symmetry displacement coordinates are involved in the observed the structural data and lend further support to the idea that the slope of Fig. 4 is a good measure of the boat conformation of the benzene ring. K~(23)
+ H&22) = 0.
(Cx(ij))h(ij).
(10)
2979
Out-of-planedeformationsof cyclic polycnes
Table 3. Average-sof s&&d torsionangksand bondkngths*for some annulenesand cyclophanes;ratiosof oop and tw deformation Lwrdinates(Eqa 11) o(~~ 123) @(IO123) 40 I l,ll-Dimethyl-l,Cmethano[lO] 53.3” annulene 51.2” 30.2” I I-Chloro-3,8-methano[ 1I] annuknone 7-Metboxycarbonyl-anti28.30 1.6:8,13dimethano[14] 9.5” annukne 1,6:8,13-Butane-1,424.8” diylidene[l4]annulene
12~~) 4234)
4123 H,) v
$j’$
d(12)
d(23)
Ref.
- 140.2” - 139.9” - 144.5”
36.3” 39.30 31.5”
7.90 a.30 11.5”
- 177.6” - 176.9’ - 178.1”
040
1.59 1.455A 1.335A 1.72 1.453A l.wA 2.21 lQl48A 1.341A
12
0.27 0.89
- 148*4” - 165.1”
3&s” 12.8”
8.6” 33.0
- 176.9” - 156.0”
0.42 2.00
1.73 1.459A 0.61 I+MA
IDA 1.442A
14
- 144.6’
29.7”
18.6”
- 168.6”
0.94
0.81
1.422 A 1.362A
I5
[22]Metapamcyclophediene see Table2 [2.2JF%racycbpbane&ne [3.3lparacyclophane 4$Methano[ 1I]annulenone 15.6” - 160*1” Anti-1,6:8,13_biswlhano[141 1OY - 160.2” annulenetricarlwnyl
16.9’ 12.7”
29.1” 33.P
- 166.0” - 156.6”
I .26t
1.405A
1.32t
14OOA 1.391A 1.367A 1.412A 1,347A 1.436A
1.47t 2.22 3.00
1.31 064
13
16 17
ClltWlliUm
SC.- or &-symmetry assumed, see text.
t-X(12)/W).
much larger errors in X/T if this quantity itself is large and if T is small. Unfortunately, T will tend to be small for pure double bonds, i.e. for small d(ij) and large X/T. (2) Some of the compounds crystallize as order-disorder structures, other contain heavier elements (Cr. Cl) and thus some of the torsion angles involving hydrogen atom position may well be in error. (3) The assumption that the geometry of the carbon skeleton is constrained to be rigid by the polycyclic nature of the compounds involved, is only an approximate one. Relaxation of other degrees of freedom may well affect the correlation in a way which Fig.4. will only be seen from more complete calculations. (4) The simplified force fields neglect all interaction condependence of the tw force constant on the corresponding stants. Finally, it should be noted, that X/T for d(23) of C-C bond 1ength.t butanediylidene[ 14]annulene” is hopelessly out of range. We cannot offer an explanation for this. The basic idea behind the arguments presented so far The correlation between x/~(ij) and d(ij) appears to be has been that the partition of torsional strain between signScantly better in the range 146A< d(ij)c l-39 A various degrees of freedom (twist and oop bending) is than in the range 1.39 A< d(ij)< 1.33 A. There are cheaper energetically than concentrating all strain into several possible reasons for this dilTerence: (1) The one degree of freedom (twist). An alternative point of uncertainty A in X/T is proportional to 7-l and, in part, to vie# is to argue that twisting of double bonds decreases the r-overlap between the two carbon-parbitals inX/T itself: volved. Modest rehybridazion ( = oop bending) of these C A~/T) - IT-‘I. t%) + ix/71 . A(T)}. (14) atoms restores at least part of the overlap lost. This latter argument is very effective in predicting the signs of oop bending deformations provided the torsion angle which is Therefore, errors of a given magnitude in x and T produce under a constraint is properly recognized. Many of the above ideas have been anticipated by tit shouldbe notedthat the oop force ccdanCs of MH, seem several authors. Dun@ has discussed AgNO,-adducts of to be systematicalty smaller than those of C_CH-C: H, cyclodecene derivatives showing nonplanar double (ethylene)” is less than If, (berrzene),’and for butadiene,H, (C, bonds. On the basis of thermochemical arguments he terminal) is than
H. B. BCnol and E.
2980
investigated by ab initio calculations? Oop deformations calculated for constrained cis- and trans-torsion angles agree quantitatively with structural data. In a study of the photoelectron spectra of polycyclic[l4]annulenes Heilbronner et al. have suggested that the use of skeletal torsion angles leads to an overestimate of the correction for the nonplanarity of the v-electron system.m We find the twist angles to be 75-100% of the skeletal torsion angles. As pointed out in the introduction oop vibrations of polyenic and aromatic compounds have been badly neglected, The results presented in this paper should provide most of the qualitative information needed to set up raw force fields which may then be retined on the basis of the experimentally observed frequencies. A large and consistent set of such force constants together with structural data would open the door to a description of the energy surfaces of polyenes based more directly on experimental data and therefore, less dependent on the approximations inherent in semiempirical molecular
!%lEFTEX
shown that 2d’(l-cosa)=2(d2-Ay(l-cos
Acknowledgement-The
A=(i!?k&~n~d For the less symmetrical cases of chemical interest the following approximation may be used a(l2R2)- a(Rz23)Y. WW+ WI+ 17.25 3
A = (360”- a(lU)-
A relationship between bond angles and oop coordinate 6 is obtained from (Al)
,J(2&).
d*(2R,)
= 1=
d(2R,).
d(12)d(nph a(123).
3-dimensionalcoordinate system. It is useful to define a reciprocal coordinate system d*(iix d*(jk), d*(s) first system by d(ij).d*(jk)=O
which is related to the
I
i.e. tbe vector d*(ik) is perpendicular to the vectors d(ij) and d%). The angle between d*(i)
(0
which leads to (AS)
There is a simpler relationship between bond-angles and oop coordinate x cos x(12) = cos a(123). cosWRJ . cosaW3) sina(123)cos(12R2) ’
w
The transformation from 6- to x- coordinates is achieved by sinx(ij) = sin G(k)/sin a(ijk).
(Alo)
j#k (Al)
d(ij).d*(ii)=
-8)
sin 6(R2) = D/sin a(l23).
This appendix gives some formulae relating bondlengths and angles with various oop bending coordinates. One may consider the three bonds radiating from a tri--coordinated atom as vectors (d(ij), d(jk), d(j&)) defining a
4%)) (Aa)
cos(W authors thank Dr. 0. Ermer and Prof. J.
(A4)
A4 is expanded to linear terms about a, = 120”to give
orbital methods.3’32 D. Dunitz for discussion and critisism.
120”)
and d*(ik) is equal to x(jl). The
two types of coordinate systems are well known in crystallography as the direct (d) and reciprocal(d*) coordinate systems. Detailed derivations of most of the following formulae are found in books on X-ray crystallography. The volume of the parallelepiped spanned by d(lz), d(23) and d(2R,) is
V= d(12). d(23). d(2RJ. R R’= I-cos’a(123)-cos*a(12Rz)-cos’a(Rz23) +2cosa(l23)cosa(l2R~)cosa(R223) (A2) The distance A of atom 2 from the plane of atoms 1.3and R, is A = {sin*a(R33)/d’(12) + sin’ a(12R,)/d2(23) + sin’ a(123)/d’(2R2) +(cos a(lZR,)cos a(l23)-cos a(R&U))/d(23). d(2R,) +(cosa(R,23)cos a(l2R2)-cos a(l23))/6(12). d(23) (A3) +(cos a(l23) cos a(R,23)-cos a(12Rz))/d(2R,) . d(l2)}-‘“. R. If the local environment of atom 2 shows C,. -symmetry, it can be
‘W. Klyne and V. Prelog, Experienlia 16, 521 (1960). ‘F. K. Winkler and J. D. Dunitz. I. MO/. BioL 59. 169 (1971). ‘A. Warshel, hf. Levitt and S. Lifson, 1. Mol. Spectrosc. 33; 84 (1970). ‘0. Ermer and S. Lifson, J. Am. Chem. Sot. 95,412l (1973).
‘For transformations between 6’s and Cartesian displacement coordinates see: D. Whiffen, Phil. Trans. Roy. Sec. A248, 131 (1955). “r. R. Scherer, Spectrochim. Acta 23A, 1489(1967). ‘C. La Lau and R. G. Snyder, Ibid. 27A, 2073 (1971). ‘R. A. Kydd, Ibid. 27A, 2067 (1971). ‘P. K. Gantzel and K. N. Trueblood, Acta Crystollogr. 18,958 (l%S). “C. L. Coulter and K. N. Trueblood, Ibid 16, 667 (1%3). “A. W. Hanson, Ibid B27, 197 (1971). “R. Bianchi, G. Morosi A. Mugnoli and M. Simooetta, Ibid B29, 11% (1973). “R. L. Beddoes and 0. S. Mills, Israel J. Chem. 10, 485 (1972). “C. M. Gramaccioli, A. S. Mimun, A. Mugnoli and M. Simonetta, I. Am. C&m. Sot. 95, 3149(1973). “C. M. Gramaccioli, A. Mugnoti, T. Pilati, M. Raimondi and M. Simonetta, Acto Crystal/ogr. BM, 2365 (1972). “0. S. Mills, private-comm-&cation. “hf. J. Barrow and 0. S. Mills. Ckem. Comm. 220 (1971). “0. Ermer, Angew. Chem. 86,672 (1974);Ibid. Internat. Edit. 13, 604 (1974). “R. L. Amett and B. L. Crawford, 1. Chem.Phys. 18,118(1950). “N. Neto, C. di Laura. E. Castelluci und S. Califano, Spectrochim. Acta Au. 1763(1967). “E. M. Popov and G. A. Kogan, 1. Opt. Speclr. 17, 362 (1964). =E. Heilbronner and H. Bock, DOS HMO-&dell und seine Anwendung, Vol. 3. Verlag Chemie, Weinheim (1970). =W. L. Mock, Tetmhedron Letters 475 (1972);L. Radon. J. A. Pople and W. L. Mock, Ibid 479 (1972).
Out-of-plane deformations of cyclic polyenes “J. D. Dunitz, Pcrspccf. Szruct. Gun. (Edited by J. D. Dunitz and J. A. lbers), Vol. 2, p. 57. Wiley, New York (1968). =R. McClure, G. &I, P. Cqgon and A. T. McPbail, Chm. Comm. 128(1970). “0. Kenoard, D. L. Wampkr, J. C. Coppola, W. D. S. Motberwell and D. G. Watson, Acto Ctystollogr. B27. 1116(1971). =H. fmgartinger, Chem. Bn. 105,2068 (W72). “D. N. J. White, Helu. Chim. Acta 56, 1347(1973).
Ti?TRAVOL.31NO.Z3-I
2931
“F. K. Winkkr and J. D. Dun&z, Acta Cryddlogr. B31, 251 (1975). -C. Baticb, E. Heilbronoa and E. Vogel, Heh. Chim. Acto 57, 2288 (1974). “N. L. Allinger aad J. T. Sprague, 1. Am. Chem. Sot. 95,3893 (1973). “A. Warsbel and hf. Karplus, Ibid 94, 5612 (1972).
H. B. BiiRcir and E.
SHEFTER?
Laboratories of Inorganic and Organic Chemistry, Swiss Federal Institute of Technology (E. T. H.), 80% Zurich, Switzerland (Received in UK 23 June 1975;Accepted forpublication IOJuly 1975)
AbstRct-Speetroscopicfame fields of benzene are shown to be compatible with the observed non-coplanarity of benzene rings with their H- or C- substitueats in strained compounds, e.g. cyclophanes. A simplified force field for annulenes and cyclophanes is discussed, and ratios of the twist and out-of-ptane bending forceconstants involved are derived from structural data. The resistance to twisting of partial double bonds is found to increase with decreasing bond length. Various formulae to calculate out-of-plane deformation coordinates are collected in an appendix. The n-electron systems of organic molecules tend to be planar. Consequently, their deformations, static or dynamic, may be classified into deformations in the molecular plane and deformations out of the molecular plane. From a survey of the literature it seems that studies of the nonplanar deformations of polyenes by vibrational spectroscopy are rare while structural studies are plentiful. Among the molecules which show ?r-systems highly deformed from planarity we find cyclophanes and polycyclic annulenes. We have studied these deformations systematically and related them to the few available spectroscopic force constants. The coordinate system. In order to describe these deformations in a consistent way we have chosen to put our molecules into a right-handed coordinate frame and to number the atoms sequentially. Two types of internal molecular coordinates are needed to describe nonplanar deformations: out-of-plane bending (oop) deformations and twist (tw) deformations. Both of them may be expressed in terms of torsion angles which describe the nonplanarity of a sequence of four atoms (Fig. 1). The torsion angle w(l,2,3,4) is defined as the angle between the planes of the atoms 1,2 and 3 on the one hand and atoms 2, 3 and 4 on the other. The sign conventions are those of Klyne and Prelog.’ Twist deformation coordinates are defined in terms of two torsion angles’-’ ~(23)=&(1234)+o(Rz23R1)) = n+;(~(123R1)+o(Rz234)).
(1)
Oop bending deformations may also be expressed in terms of two torsion angles,’ e.g. for atom 3 in Fig. 1. ~(23)=Jn-(w(l,2,3R,)-w(l,2,3,4))l.r
(2)
According to this definition, the sign of the quantity within the vertical bars is opposite for x(23) and x(43). The value of s( + I or - 1) then serves to define the sign of the oop tG. A. Pfeiffer Fellow (1972-73)of the American Foundation for Pharmaceutical education; present address School of Pharmacy, State University of New York, Buffalo.
2-3 /
\
1
.’ 2
1 i@% 4
’
4
./ !a
./
x
R3
T
d Fig. 1. bending at atom 3. The choice of s is arbitrary for at least one atom in a conjugated polyene; once an s value has been assigned to one atom, the next atom is given the same sign if-for T set back to 0”-the two atoms are on the same side of the plane defined by their four substituents; it is given an opposite sign otherwise. This procedure is continued all through the conjugated chain. The case where 7 = 90” is equivalent to breaking the conjugation and necessitates a new arbitrary choice. An alternative definition of x is in terms of the dihedral angle w(423R,) (4-2 does not correspond to a chemical bond!)“’ x(23) = Ia - o(423R,)I . s.
(3)
It should be noted that x(23), x(43) and x(R& are identical if and only if the three bond angles at atom 3 are the same.
2977
Out-of-planedeformationsof cyclicpolyenes
this additional i~o~tion he calculated a force Eeld that is remarkable because ail except one or possibly two of its interaction constants (f’, f,“‘) are very small (Table 1, d and e). All of the previously mentioned force fields may now be tested not only with respect to their capability to reproduce observed frequencies but also with respect to their ability to describe nonplanar defo~ations of benzene derivatives such as [3,3]paracyclophane, the diolefins of [2,2]paracyclophane’0 and [2,2lmetap~a~ycIoph~e.” These molecules have been studied carefully by X-ray d&action methods; H atom positional parameters have been determined with esd’s between 0.015 A and 0.04 A. All three molecules show the p~~ubstituted benzene rings distorted into a boat conformation. The aromatic H atoms are displaced out of the plane defined by their three nearest C atoms by -0-04 A to - 0.244 A. Although individual displacements +2f” 7 ~,(7,-2-n+,)t2fp~X.(r,-9-71*2) (4) are not always statistically significant they are all in the same direction (to negative z, Fig. 2) indicating a systematic trend. (The intermolecular II. . . H contacts, where i stands for i = 1,. . . . ,6. Only eight symmetry force cons~ts may be determined from the six being longer than 2.4 A, should not affect the hydrogen observable normal vibrations of C&and C& (two each positions very severely). of Bzp and E2. symmetry, one of AZ=and Et, symmetry). For the subsequent treatment we have assumed local This is not enough to fix al1eleven valence force constants CZ, symmetry of the benzene ring and averaged the in eqn (4). The common way out of this indeterminacy has structural parameters accordingly. We have anrtlysed the been to put j? = jP = jz = O.%’VFF’s obtained with this averaged H atom oop displacements with the assumption approximation by various authors are listed in Table that the structure of the carbon skeleton is more or less 1fa-c). The oop force constants and the oop-tw interaction rigid and determined mainly by ring closure conditions of constants are virtually the same in all force fields. The the polycyclic molecule. We have further assumed that value of the tw constant shows some variation, however; the H atoms, in this constrained situation, try to find it ranges from 0.236 to 0.272 mdgn &ad’, and the energetically optimal positions. The mathematical model corresponding or&o interaction constant varies between incorporating these assumptions is based on eqns (4), (1) - 0.078 and - 0*058mdyn &rad2. and (3). Torsion angles between C atoms are fixed by the ring constraints and are given the experimental values. The torsion angle w(R112H3 (and symmetry related ones, Table 1. Valenceforce fieldof benzene(see Eqo4) Fig 2) is free to assume a value which minimizes the total a C b d e energy according to the condition
Other oop bending coordinates are also used, especially the oop bending angle S. It is the angle between a bond, say 2-R*, and the plane formed by two other bonds involving atom 2 (the plane 123 in Fig. lb)? The sign of S may be defined in the same way as for ,y. Relationships between these and others types of oop coordinates are given in an Appendix. TIte benzene oop force fields and the cyc~~p~a~es. The general valence force field (VFF) for oop deformations of benzene is given by
H.
5 F f? fx F f'
0.236
O-259
- 0.078 _* 0,312 0.016 -0.016 - 0.018 -0.032 -0*008 1
-0465 _* 0.313 0.014 -0.016 -0~014 -0.035 odm3 _*
0*272&t) 0*33st -O*OS8(13) -0404 0.024 0.&~3010 0.295 0.015 om -0.013 -0m7 -0*014 -0m2 -0.032 -0.032 -O*OM o*OOs _* _*
0.33% -0m7 0.015 o-293 oaO6 -0+06
d@V) dw(R, 12f&) =’
Takingthe assumed CL symmetry into account, this yields (K-f:-
-&2 -0m8 _*
fJ” t
fP ~2~i2jp)(~(R,12~*)+
~~6123))~2
-~+j~-2j~-2j~m)(-n-o(R,123)t~{6123))
*Assumed.tEstimatcd from model calculation.“Ref.& based Whiien’s model[5]. ‘Ref. 6, from chlorinatedbenzenes. ‘Ref. 7, from alkylated benzene%dRef. 8. ‘Ref. 8.
+ w(R, 12Hz))= 0.
on
Kydd has estimated the tw constant of benzene* with the help of a relationship between tw constants, n-bond orders and overlap integrals of glyoxai and ethylene. With
(6)
On solving for o(R112H3 the results in Table 2 are obtained. The agreement between experimental and calculated values is satisfactory. This shows that the benzene force Geld accounts reasonably well for the static oop bending deformations of H atoms in a strained
Fii 2.
H. B. BORGIand E.
2978
benzene ring.t Indeed, the quality of the agreement is surprising because the deformations from planarity are by no means intinitesimal but amount to as much as 21” in x and 24” in T. It is not clear whether this implies that the quadratic approximation to the force field is still valid, or whether the agreement is fottuitious and due to the cancellation of various systematic errors in the data, e.g. in the hydrogen coordinates (which represent maxima of electron density rather than nuclear positions) or errors in the force field, e.g. the neglect of higher than quadratic terms. A simplified oop force field for cyclophanes and annulenes; ratios of force constants. In the previous paragmph Eqn (6) has been used to calculate o(R112H2) and thereby the oop displacements of H atoms in cyclophanes. The calculation was based on the known spectroscopic force constants of benzene and the well determined carbon skeleton torsion angles of the cyclophanes. An alternative use of Eqn (6) is to obtain a relationship between unknown force constants when all the torsion angles are known. If interaction constants are small, as in the benzene force field (Table 1), the expression (6) may be simplified to the approximate relationship H,7(12)tHZ~(12)-o.
(7)
Similar expressions may be derived for [lOlannulenes
%FfRR
0
30 0 .
2 \’
03
X
0
OoO . IQ
0
0
0
00
cl0 I35
1%
k4G
dOI)
A Fl. 3. Ratios of oop and tw deformation angles versus bondlength of twisted bonds (0). Standard deviations of individualratios x/r(ij) are about 0.3 for d(Q)between I.45 and I.~oA and O&l.0 for d(ij) between 140 and I.33 A (estimated from experimental standard deviations in torsion angles). Also given are ratios of spectroscopic force constants H,,,,JH, for cyclohexeneand benzene(0).
and were found to vary between - 0.0 and 3.0. The seemingly uncoherent set of numbers -_x/~(ij) was plotted against the bond lengths
observed torsion angles’z”
Table 2. Observed (average)and calculated torsion kk.5
[3.3]Paracyclophanes [2.2]Paracyclophanedienel” [2.2]Metaparacyclophanediene”
ti
ofcyclophanes
o(RJ23)
o(6123)
o(R112Hz)
- 168.1” - 146.4” - 137.8”
7.6” IS9 2l.P
6.7(20)=’ 21.3(25’) 269(10)”
ti(R112Hz)”
u(R112H8
7.3” 22.2” 27.3”
7.5” 22.6” 279
‘Calculated from force field c, Table I. bCalculated from force field e, Table I. ’ ESD’sobtained from
deviationsof individualmeasurementsfrom average. (assumed to have C2. symmetry (Fig. 3)) on the basis of a simplilied force field: 2v = 2zQZ(12) t 4HXcy2(12)+ x2(2])) t 4K(*sr*(l2) + 4&&(23).
d(ij) (Fig. 4). The scatterplot shows that large ratios -x,(ij)/r(ij) are found for short bonds d(ij) and smaller ratios for the longer ones. , A corresponding expression for a transcycloolefin with local C2 symmetry at the double bond is based on
(8) 2V = 2H,x= + HJ'.
Provided that the bond angles 12If2 and Zf33 are equal, x(12) equals x(37) and from a condition analogous to Eqn (5), one obtains KW(l2)+
H&(12) + x(23)) = 0
(9)
(12)
Ifthe CCCC-torsion angle is constrained by the ringclosure condition it may be shown that H,IH,=2x/r.
(13)
A numerical value for I/k, was obtained from transcyclooctene: ‘*2x17 = 48”/18”= 2.7, d(C=C) = 1.330 A. It is generally accepted that the torsional barrier of a (The relationships (9) and (10) also hold for half the annulene molecule.) After rewriting, Eqns (7,9 and 10)are partial double bond increases as the bond length shortens, i.e. as the bond order increases. It therefore seems seen to yield ratios of force constants reasonable to attribute the negative slope of the curve in Fig. 4 mainly to an increase in the tw force constant HTct,,. &dH, = It is possible to add two more points to the graph, both (11) of which are ratios H,&Hxc,,, obtained from spectroscopic These ratios have been calculated (Table 3) from data. One is obtained from the benzene force field discussed earlier, the other one from the oop force or cyclohexene.m The two Wrictly speaking, the above calculations only provide a test of constant of ethylem? additional points fit very well into the trend.indicated by four symmetry force constants because only an A,. and two &. symmetry displacement coordinates are involved in the observed the structural data and lend further support to the idea that the slope of Fig. 4 is a good measure of the boat conformation of the benzene ring. K~(23)
+ H&22) = 0.
(Cx(ij))h(ij).
(10)
2979
Out-of-planedeformationsof cyclic polycnes
Table 3. Average-sof s&&d torsionangksand bondkngths*for some annulenesand cyclophanes;ratiosof oop and tw deformation Lwrdinates(Eqa 11) o(~~ 123) @(IO123) 40 I l,ll-Dimethyl-l,Cmethano[lO] 53.3” annulene 51.2” 30.2” I I-Chloro-3,8-methano[ 1I] annuknone 7-Metboxycarbonyl-anti28.30 1.6:8,13dimethano[14] 9.5” annukne 1,6:8,13-Butane-1,424.8” diylidene[l4]annulene
12~~) 4234)
4123 H,) v
$j’$
d(12)
d(23)
Ref.
- 140.2” - 139.9” - 144.5”
36.3” 39.30 31.5”
7.90 a.30 11.5”
- 177.6” - 176.9’ - 178.1”
040
1.59 1.455A 1.335A 1.72 1.453A l.wA 2.21 lQl48A 1.341A
12
0.27 0.89
- 148*4” - 165.1”
3&s” 12.8”
8.6” 33.0
- 176.9” - 156.0”
0.42 2.00
1.73 1.459A 0.61 I+MA
IDA 1.442A
14
- 144.6’
29.7”
18.6”
- 168.6”
0.94
0.81
1.422 A 1.362A
I5
[22]Metapamcyclophediene see Table2 [2.2JF%racycbpbane&ne [3.3lparacyclophane 4$Methano[ 1I]annulenone 15.6” - 160*1” Anti-1,6:8,13_biswlhano[141 1OY - 160.2” annulenetricarlwnyl
16.9’ 12.7”
29.1” 33.P
- 166.0” - 156.6”
I .26t
1.405A
1.32t
14OOA 1.391A 1.367A 1.412A 1,347A 1.436A
1.47t 2.22 3.00
1.31 064
13
16 17
ClltWlliUm
SC.- or &-symmetry assumed, see text.
t-X(12)/W).
much larger errors in X/T if this quantity itself is large and if T is small. Unfortunately, T will tend to be small for pure double bonds, i.e. for small d(ij) and large X/T. (2) Some of the compounds crystallize as order-disorder structures, other contain heavier elements (Cr. Cl) and thus some of the torsion angles involving hydrogen atom position may well be in error. (3) The assumption that the geometry of the carbon skeleton is constrained to be rigid by the polycyclic nature of the compounds involved, is only an approximate one. Relaxation of other degrees of freedom may well affect the correlation in a way which Fig.4. will only be seen from more complete calculations. (4) The simplified force fields neglect all interaction condependence of the tw force constant on the corresponding stants. Finally, it should be noted, that X/T for d(23) of C-C bond 1ength.t butanediylidene[ 14]annulene” is hopelessly out of range. We cannot offer an explanation for this. The basic idea behind the arguments presented so far The correlation between x/~(ij) and d(ij) appears to be has been that the partition of torsional strain between signScantly better in the range 146A< d(ij)c l-39 A various degrees of freedom (twist and oop bending) is than in the range 1.39 A< d(ij)< 1.33 A. There are cheaper energetically than concentrating all strain into several possible reasons for this dilTerence: (1) The one degree of freedom (twist). An alternative point of uncertainty A in X/T is proportional to 7-l and, in part, to vie# is to argue that twisting of double bonds decreases the r-overlap between the two carbon-parbitals inX/T itself: volved. Modest rehybridazion ( = oop bending) of these C A~/T) - IT-‘I. t%) + ix/71 . A(T)}. (14) atoms restores at least part of the overlap lost. This latter argument is very effective in predicting the signs of oop bending deformations provided the torsion angle which is Therefore, errors of a given magnitude in x and T produce under a constraint is properly recognized. Many of the above ideas have been anticipated by tit shouldbe notedthat the oop force ccdanCs of MH, seem several authors. Dun@ has discussed AgNO,-adducts of to be systematicalty smaller than those of C_CH-C: H, cyclodecene derivatives showing nonplanar double (ethylene)” is less than If, (berrzene),’and for butadiene,H, (C, bonds. On the basis of thermochemical arguments he terminal) is than
H. B. BCnol and E.
2980
investigated by ab initio calculations? Oop deformations calculated for constrained cis- and trans-torsion angles agree quantitatively with structural data. In a study of the photoelectron spectra of polycyclic[l4]annulenes Heilbronner et al. have suggested that the use of skeletal torsion angles leads to an overestimate of the correction for the nonplanarity of the v-electron system.m We find the twist angles to be 75-100% of the skeletal torsion angles. As pointed out in the introduction oop vibrations of polyenic and aromatic compounds have been badly neglected, The results presented in this paper should provide most of the qualitative information needed to set up raw force fields which may then be retined on the basis of the experimentally observed frequencies. A large and consistent set of such force constants together with structural data would open the door to a description of the energy surfaces of polyenes based more directly on experimental data and therefore, less dependent on the approximations inherent in semiempirical molecular
!%lEFTEX
shown that 2d’(l-cosa)=2(d2-Ay(l-cos
Acknowledgement-The
A=(i!?k&~n~d For the less symmetrical cases of chemical interest the following approximation may be used a(l2R2)- a(Rz23)Y. WW+ WI+ 17.25 3
A = (360”- a(lU)-
A relationship between bond angles and oop coordinate 6 is obtained from (Al)
,J(2&).
d*(2R,)
= 1=
d(2R,).
d(12)d(nph a(123).
3-dimensionalcoordinate system. It is useful to define a reciprocal coordinate system d*(iix d*(jk), d*(s) first system by d(ij).d*(jk)=O
which is related to the
I
i.e. tbe vector d*(ik) is perpendicular to the vectors d(ij) and d%). The angle between d*(i)
(0
which leads to (AS)
There is a simpler relationship between bond-angles and oop coordinate x cos x(12) = cos a(123). cosWRJ . cosaW3) sina(123)cos(12R2) ’
w
The transformation from 6- to x- coordinates is achieved by sinx(ij) = sin G(k)/sin a(ijk).
(Alo)
j#k (Al)
d(ij).d*(ii)=
-8)
sin 6(R2) = D/sin a(l23).
This appendix gives some formulae relating bondlengths and angles with various oop bending coordinates. One may consider the three bonds radiating from a tri--coordinated atom as vectors (d(ij), d(jk), d(j&)) defining a
4%)) (Aa)
cos(W authors thank Dr. 0. Ermer and Prof. J.
(A4)
A4 is expanded to linear terms about a, = 120”to give
orbital methods.3’32 D. Dunitz for discussion and critisism.
120”)
and d*(ik) is equal to x(jl). The
two types of coordinate systems are well known in crystallography as the direct (d) and reciprocal(d*) coordinate systems. Detailed derivations of most of the following formulae are found in books on X-ray crystallography. The volume of the parallelepiped spanned by d(lz), d(23) and d(2R,) is
V= d(12). d(23). d(2RJ. R R’= I-cos’a(123)-cos*a(12Rz)-cos’a(Rz23) +2cosa(l23)cosa(l2R~)cosa(R223) (A2) The distance A of atom 2 from the plane of atoms 1.3and R, is A = {sin*a(R33)/d’(12) + sin’ a(12R,)/d2(23) + sin’ a(123)/d’(2R2) +(cos a(lZR,)cos a(l23)-cos a(R&U))/d(23). d(2R,) +(cosa(R,23)cos a(l2R2)-cos a(l23))/6(12). d(23) (A3) +(cos a(l23) cos a(R,23)-cos a(12Rz))/d(2R,) . d(l2)}-‘“. R. If the local environment of atom 2 shows C,. -symmetry, it can be
‘W. Klyne and V. Prelog, Experienlia 16, 521 (1960). ‘F. K. Winkler and J. D. Dunitz. I. MO/. BioL 59. 169 (1971). ‘A. Warshel, hf. Levitt and S. Lifson, 1. Mol. Spectrosc. 33; 84 (1970). ‘0. Ermer and S. Lifson, J. Am. Chem. Sot. 95,412l (1973).
‘For transformations between 6’s and Cartesian displacement coordinates see: D. Whiffen, Phil. Trans. Roy. Sec. A248, 131 (1955). “r. R. Scherer, Spectrochim. Acta 23A, 1489(1967). ‘C. La Lau and R. G. Snyder, Ibid. 27A, 2073 (1971). ‘R. A. Kydd, Ibid. 27A, 2067 (1971). ‘P. K. Gantzel and K. N. Trueblood, Acta Crystollogr. 18,958 (l%S). “C. L. Coulter and K. N. Trueblood, Ibid 16, 667 (1%3). “A. W. Hanson, Ibid B27, 197 (1971). “R. Bianchi, G. Morosi A. Mugnoli and M. Simooetta, Ibid B29, 11% (1973). “R. L. Beddoes and 0. S. Mills, Israel J. Chem. 10, 485 (1972). “C. M. Gramaccioli, A. S. Mimun, A. Mugnoli and M. Simonetta, I. Am. C&m. Sot. 95, 3149(1973). “C. M. Gramaccioli, A. Mugnoti, T. Pilati, M. Raimondi and M. Simonetta, Acto Crystal/ogr. BM, 2365 (1972). “0. S. Mills, private-comm-&cation. “hf. J. Barrow and 0. S. Mills. Ckem. Comm. 220 (1971). “0. Ermer, Angew. Chem. 86,672 (1974);Ibid. Internat. Edit. 13, 604 (1974). “R. L. Amett and B. L. Crawford, 1. Chem.Phys. 18,118(1950). “N. Neto, C. di Laura. E. Castelluci und S. Califano, Spectrochim. Acta Au. 1763(1967). “E. M. Popov and G. A. Kogan, 1. Opt. Speclr. 17, 362 (1964). =E. Heilbronner and H. Bock, DOS HMO-&dell und seine Anwendung, Vol. 3. Verlag Chemie, Weinheim (1970). =W. L. Mock, Tetmhedron Letters 475 (1972);L. Radon. J. A. Pople and W. L. Mock, Ibid 479 (1972).
Out-of-plane deformations of cyclic polyenes “J. D. Dunitz, Pcrspccf. Szruct. Gun. (Edited by J. D. Dunitz and J. A. lbers), Vol. 2, p. 57. Wiley, New York (1968). =R. McClure, G. &I, P. Cqgon and A. T. McPbail, Chm. Comm. 128(1970). “0. Kenoard, D. L. Wampkr, J. C. Coppola, W. D. S. Motberwell and D. G. Watson, Acto Ctystollogr. B27. 1116(1971). =H. fmgartinger, Chem. Bn. 105,2068 (W72). “D. N. J. White, Helu. Chim. Acta 56, 1347(1973).
Ti?TRAVOL.31NO.Z3-I
2931
“F. K. Winkkr and J. D. Dun&z, Acta Cryddlogr. B31, 251 (1975). -C. Baticb, E. Heilbronoa and E. Vogel, Heh. Chim. Acto 57, 2288 (1974). “N. L. Allinger aad J. T. Sprague, 1. Am. Chem. Sot. 95,3893 (1973). “A. Warsbel and hf. Karplus, Ibid 94, 5612 (1972).