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Electron diffraction by gases. The molecular structure of 1,3-cyclohexadiene
>oufnal of Molecular Structure
Elsevier Publishing Company, Printed in the Netherlands
Amsterdam
ELECTRON DIFFRACTION BY GASES. THE MOLECULAR TURE OF 1,3-CYCLOHEXADIENE
G. DALLINGA
AND
STRUC-
Miss L. H. TONEMAN
Koninkl~ke~Sl~ell-Lnboraioriun~, Amsterdam (SheIt Research N. V.)
(Received
1.
Aprii 21st 1967)
INTRODUCTION
The structure of 1,3_cyclohexadiene has several interesting features. The degree of non-planarity of the ethylene groups and the conjugated double-bond system depends on the relative importance of resonance and steric strain. The molecule contains carbon-carbon single bonds of three different types and their differences in length provide a test for theories of the chemical bond. From the gas-diffraction point of view it is a very suitable molecule for demonstrating what details this technique can bring out and what assumptions are necessary in the structure analysis. We have in mind here the small differences between some of the carbon-carbon bond lengths and the structure of the CHa groups. While we were studying this compound, other workers published the results of microwave investigations’* ‘. They suggest C, symmetry for the equilibrium conformation and offer a semi-quantitative estimate of the degree of non-planarity. This prompted us to restart our calculations, now combining our diffraction data with the moments of inertia obtained from the microwave work.
2. 2.1.
DETERMINATION
OF THE STRUCTURE
ExpeFinlental
1,3Cyclohexadiene was prepared by chlorination of cyclohexene with tertbutyl hypochlorite and subsequent abstraction of HCl. It was purified gaschromatographically to a purity of 98.5 o/o (1.2 o/o benzene, 0.3 o/o cyclohexene, no 1,44somer). Diffraction diagrams were made with the sector apparatus described before’, using a wavelength of about 0.06 A (&0.00003). Photometer records of the diagrams were obtained with the Joyce-Loebl double-beam automatic recording J. Mol.
Strucrure, 1 (1967-68)
11-23
12
G. DALLINGA,
microdensitometer, TABLE
The results are summarized
in Table
1.
1
DESCRIPTION
No.
Mark IIIB.
L. H. TONEMAN
OF DIFFRACTION
DIAGRAMS
Sample-film Secror ased
-lfiposure time
distance (mm)
Film density
(see)
Range of s-values used in the transfofmation (See 2.2)
for calculfzlions 1.5 1.5
1 2
306.9 306.9
r2(r,,
=
25 mm)
25 45
0.18-0.68 0.40-1.52
3 4 5 6
306.9 306.9 306.9 153.8
P(r,,
=
60 mm)
60 120 120 120
0.14-0.36 0.26-O-62 0.30-0.82 0.56-I -68
rejected because of asymmetry 4 -I 9.75 4.875-19.625 3.375-19.625 ‘4.875-19.625 6.75 -37.75 9 -36
120 120 120 120 240
0.52-1.56 0.44-l -44 0.44-1.40 0.34-I .2 0.48-l .6
7 -37.5 8.75 -44.75 11 45 1 I .625-50.875 14.25 -51
7 8 9 10 11
153.8 123.8 123.8 102.1 102.1 *Only
2.2.
left part of photogram
6 5
-
8.5 8.5
9 11.75 11.75 14.25 14.25
-8 -8
-36 44.75 -44.75 -51 -51
used.
Calculations
readings were rtransformed to s-scale and, by a linear The photometer transformation, scaled to the theoretic& background. For calculating the background we used Berghuis’s4 atomic scattering functions and Keating’s’ incoherent scattering functions. Subtraction of the background from the scaied experimental intensities yielded the moiecular scattering functions, s41,(s). Details of an automatic procedure for the whole process are given in Appendix II. The procedure was applied to diagrams 9, 10 and I 1; diagrams 3-8 were treated by the same procedure, but only semi-automatically. The procedure did not work for diagrams I (s < 6 A-‘) and 2 (s c 5A-‘); here a suitable background had to be drawn. As discussed before6, this failure must be attributed to the theoretical background being incorrect at low s-values. Minor corrections were necessary at high s-values (s > 38 A-‘), due to some extraneous scattering. These corrections were made in the final s41,,,(s) curve obtained after averaging, successively, left and right parts of each diagram, a11 diagrams at each sample-fiIm distance and the diagrams at all distances. The final result is shown in Fig. la. The intensity data, l:Js), were used to calculate a radial distribution func. tion, p(r), according to p(r) = (2/~~~~s~I~(s)(p(s) J. Mol.
Structure,
1 (1967-68)
11-23
sin srAs,
STRUCTURE
OF l&CYCLOHEXADIENE
where q(s) = exp(--a(s*). in Fig. 2.
BY ELECTRON
The curve obtained with As = l/8 and a = 0 is shown _.__.
------
-
_jyjI :-,~d__..__~-~
Ij_jI
Fig. 2. Radial
for 1,3-cyclohexadiene:
distribution
function
(a) experimental;
(a
I
I
1
Fig. 1. In(s)
13
DIFFRACTION
i
I
i
i
..-----,c__..,.--‘;-______---._-_ ‘-_A
I I
-------
/
(b) calculated;
ia -b
I =
(c) difference
b-a.
= 0).
We then applied our computer program for least-squares calculations to the intensity data6. There proved to be a systematic difference between the experimental intensity data and those calculated for various models, the latter being invariably higher ?t s > 23 A- ’ and practically always lower at s c 23 A-‘. Analysis showed .that diagrams taken at 307 mm were not responsible for this 1.
Mol.
Sfrucrure,
1 (196748)
II-23
14
G. DALLINGA,
L. H. TONEMAN
effect: It might arise from extraneous s_dttering in the outer regions of the diagrams taken at small camera-film distances. We corrected the intensity data for this effect and did a number of least-squares calculations, both with the corrected and tne uncorrected s”l,(s). The effect of this particular error on the value of the parameters proved to be small but of course, it did affect the standard deviations. Table 2 shows the results of calculations of the carbon-carbon distances with the two sets of intensity data. TABLE THE
2
EFFECX
OF CORRECTIONS
OF THE INTENSITY
DATA ON PARAMETER
VALUES
AND THEIR
STANDARD
DEVIATIONS
Carbon-
uncorrecred I=(s)
carbon bonds*
w0
RI-2
1.339
h-3 Rl-6 &-I3
1.467 1.492 1.517
* For numbering
corrected I,(s)
St. dec. G-f)
w0
1.340 1.473 1.493 1.512
0.0021 0.0070 0.0017 0.0080
St. deu. (A) c.0013 0.0050 0.0010 0.0060
see Figure 3.
As mentioned in the Introduction, during the course of this work additional data were published in the literature, viz. moments of inertia from microwave measurements’*2. This led us to consider the possibility of using combined diffraction and microwave data in one and the same calculation. Our least-squares program provides for the introduction of additional structure information in the form of constraints, which are linear relations between two or three interatomic distances. It was easy to adjust the program so that moments of inertia could also be introduced as constraints. In general, such a procedure gives a very large weight to the additional information, but in our case this was no drawback, because microwave data are much more accurate than diffraction data. The procedure is described in some detail in Appendix I. The first calculations of this type were not successful. Both diffraction and microwave data yield an absolute scale for the dimensions of t’ne molecule. These scales may differ for the two methods as a consequence of experimental errors or the inherent differences between R-values obtained. Therefo::e, we introduced a scale factor, a, between microwave and diffraction da:a. The calculations then converged more rapidly. Theoretically, this combination of diffraction and microwave results would provide a possibility of determining the absolute scale of the molecular structure as obtained from electron diffraction far more accurately than usual. In practice, J. Mol. Structure, 1 (1967-68)
11-23
STRUCTURE
OF I,~-CYCLOHEXADIENE
BY ELECTRON
DIFFRACTION
15
however, difficulties arise. The differences mentioned above between R-values obtained by the two methods depend in a very complex way on the structure of the molecule. By introducing a scale I’actor one approximates the situation only roughly. 2.3.
The radial distribution function
The radial distribution function (Fig. 2) was used to find a starting model for the least-squares calculations on the intensity data. Tt shows a C-C bond length of 1.335 A and an average C-C bond length of nearly 1.50 %i. The other peaks do not contradict the assumption, made for the model, that the equilibrium conformation of the molecule has C2 symmetryl. 2.4.
Refinement cf the structure by least-squares calculations
From experience we know it is usually not possible to refine all structural parameters of a complex molecule by least-squares calculations. For that reasdn calculations are started with a reduced number of parameters. The reduction is obtained by employing some or all of the following methods: a. Keeping the amplitudes of vibration, I~ij, constant; b. Introducing constraints, by which interatomic distances can be kept constant or linearly related to each other; c. Keeping the scale factor constant. Ideally, the calculation converges rapidly and is repeated with an increased number of parameters by dropping one or more of the above restrictions. The choice of the additional parameters is arbitrary; whether the addition is meaningful can be decided by a statistical test6. In practice, many,complications may arise. The first calculation may go wrong, in which case the number of parameters has to be further decreased or different parameters have to be chosen. The problem may be ill-conditioned, i.e. contain parameters which show correlation or are poorly determined by the experimental data. The intensity data may turn out to contain a systematic error. Preliminary calculations are sometimes necessary to determine the nature of the complications. In our case correlations could be expected between the lengths of the three slightly
different
single
C-C
bonds
and #C-c.
We
therefore
kept the uij constant
initially, using the uij values obtained in calculations on l&cyclohexadiene’. Intensity data in the range 0.25 s s 6 46.0 A were employed with As = 0.25 A- ‘. Fig. 3 shows the numbering of atoms in the molecule. In the first calculation the only constraints used were those on the carbonhydrogen bonds (a, b, c, Table 3). We found: a. large differences between experimental and theoretical intensities below J. Mol.
Structure,
1 (196748)
11-23
16
G. DALLINGA,:L.
H. TONEMAN
s = 4.75 A-’ (S ee Fig. 1.) In this region the intensity is very insensitive to variations in the structure. On the other hand, the experimental data are rather inaccurate for s < 5 A- *. We decided, therefore, to use the intensity data in the range 4.75 $ s s 46.0 A- ’ in all following calculations; b. R2m3 > RI_, > R5-6, contrary to what one would expect’on the basis of a large body of experience; c. considerable inaccuracy for various coordinates of hydrogen atoms, especially the z-coordinate of HT. Their values oscillated during the iterative calculations; this prevented the calculation from converging. TABLE
3
CONSTRAINTS
USED
IN
LEAST-SQUARES
Type of distance
carbbn-hydrogen
carbon-hydrogen, carbon-carbon
bond
non-bonding
bond
CALCULATIONS
No.
Constraint
a b
R:_7 R,_,
= =
R=-a R5_-l,-0.01
C
&r-l,
=
RI-I=
d e
&--rz = &-IJ Rs-13 = &-~a
f
R2--3 f R,_, = R,_, = R,_, =
0
h i
RL-,, R,_, R,_. R,_,
= 2RI_6 -0.03 -0 03
-0.02
Fig. 3. Numbering of atoms in 1,3syclohexadiene.
In subsequent calculations we added the constraints d through i in various combinations and kept H, from oscillating by relating y7 and z7 to yl and zl. Constraint f, or g+ h or g+i, kept the C-C bonds in the “right” order (Rs_s > Rr _6 > RZ_-3); the use of d and e clearly increased the chance that the calculation converged. During the calculations it proved necessary to correct the I&r) for asystematic error (Section 2.2). J. Mol. Structure,
1 (1967-68)
11-23
STRUCTURE
OF l,%CYCLOHEXADIENE
BY ELECTRON
17
DIFFRACTION
Apparently, the structure cannot be determined unambiguously from diffraction data alone. Constraints are necessary to reasonably fix the hydrogen positions and to arrive at theoretically acceptable values of the C-C single-bond distances. In this stage of the investigation the microwave result- became availabler. The introduction of moments of inertia into the calculation (see App&dix I) made it necessary to choose a new coordinate systein, x’u’z’ (Fig. 3). This was obtained from the original system, xyt, by shifting the origin along the +y-axis and rotating the xz-plane about the y-axis so that in the x’y’z’ system &Z~JJ: = 0 and ClniXlZi’ = 0, mi representing
the mass of atom i (i = 1,2, . . ., iQi
The two
i
equations balance the two extra coordinates (JJ; and zi) necessary in this coordinate system. In the following we have omitted the accents in t’he x’y’z’ system. TABLE BOND
No.
4
LENGTHS
(s-i)
OBTAINED
constraints and relations between coordinates
FROM
LEAST-SQUARES
CALCULATIONS
C-C bonds RI-Z
C-H bonds -
&-3
RI-6
abc
1.339
1.471
I so4
ac
1.340 1.340 1.339 1.339 1.339 1.338 1.339 1.339
1.467 1.462 1.468 1.471 1.472 1.464 I.453 1.465
1.495 1.497 1.494 1.489 1.488 1.503 1.511 1.492
C at.!
actav acfvw actvw actvw acj’mv
1.488 1.509 1.508 1.510 1.518 1.518 1.493 1.494 1.519
RI-7
&-8
1.12 1.08 1.06 1.07 1.07 1.07 1.09 1.08 1.07
(1.12) (l.OSj 1.09 (1.07) (1.07) (1.07) (1.09) (1.08) (1.07)
(1-i 1) 1.14 1.15 1.14 1.14 1.14 1.12 1.13 1.14
As discussed before, a scale factor was introduced between diffraction and microwave data. The calculations converged better than before and it proved possible to decrease the number of constraints (Table 4, calculations 1, 2 and 3). The convergence was not complete however, owing to uncertainties in y,, z7 and zs, whose values kept oscillhLmg in consecutive iterations. These uncertainties can be understood from Fig. 3: changes in the parameters concerned have only a small effect on the distances between the hydrogen atoms involved and the other atoms. Consequently, the determination of these parameters from the diffraction data is inaccurate. For this reason and in order to speed up calculations we decided to do some more calculations in which the above-mentioned hydrogen coordinates were related to the coordinates of the adjacent carbon atoms by some of the following relations: YH-I= 0.25 y,, (t); z&f,= Zc, (U); z& = 4zc, (0); zH, = 22,, (w)_ The relations (u) snd (w) were used alternately in order to test the influence of the value of y in zu, =yz,-,. J. Mol. Srrucrure, 1 (1967-68)
11-23
18
G. DALLINGA,
L. H. TONEMAN
There was another uncertainty, viz. in the assignment of I,, and lYy, whose YXperimental values were very close (99.6313 and 99.8607; 1, = 187.1159 amu x A2 with 1 amu = 1.6598 x 10VzJ g). In calculations 1, 2 and 3 we had chosen IXX> Zw; calculation 2 was repeated with I,, c I,, (see 4, Table 4). The effect of the interchange on bond lengths w3s small. Since the difference between Icolo and &,(.s) became a little smaller, we used IX, -K I,,,, in all following calculations. This is probably in agreement with the order used in ref. 1. Calculations 5 and 6 show that the system is not very sensitive to the restrictions imposed upon some hydrogen coordinates. It is seen, however, that the olefinic and parafhnic C-H bonds are rather different in all calculations where constraint b is omitted. We repeated 6, using a larger value for UC-_H(0.079’ instead of 0.071 A). Indeed the differences between the Rc_n became smaller (7), which shows again the correlation between changes in u and R. In the same calculation, the order of R1_-6 and R,_, changed as well. This also happened when we used slightly different (smaller) values for some of the z.+_~ (8). Therefore, in the next calculation (9) we again introduced constraintf to keep the Rc_= in the “right” order. In this calculation we also allowed uc =c and uc_= to vary. uczc changed from the constant value chosen in previous calculatiOOS 0.0375 A, to 0.0415 A; uc_c (0.0540 A) did not vary at all; Uc_u was again kept constant at 0.071 ip. In a final calculation, not reported here, uczc and three independent uc_c were allowed to vary. This calculation converged, but impossible ~c-= values were obtained. TABLE
5
AMPLITUDES
OF VIBRATION
Distance
u,,(h
c=c c-c
0.0375 0.054 0.071 0.065 0.085 0.120 0.100 0.120
C-H C C (non-bonding) C I’-’ --’ H 8 C Ho H:;: : . H,2 H...H
Table 4 suggests that some parameters can be determined accurately (RI _*) or reasonably well (R,_,). For other parameters (R, _-6 and R,_,) even their relative order cannot be obtained with certainty. There is no clear-cut method of determining the best result. For further discussion we chose the result of calculation 4 to represent the structure of 1,3_cyclohexadiene. It has the merit of (a) containing only two, most reasonable, assumptions, viz. that all olefinic C-H bonds are equal and all aliphatic C-H bonds are equal, and jb) being a reasonable J. Mol. Strucrure, 1 (1967-68) 1l-23
STRUCTURE
OF l,%CYCLOHEXADIENE
BY ELECTRON
DIFFRACTION
19
average of all results. The structure is represented in Fig. 4; the set of amplitudes of vibration used in the calculation is given in Table 5. The scale factor between diffraction and microwave data was found to be 1.0067, which indicates that distances in the microwave structure are larger than those found from diffraction. When all calculations had been performed, the microwave data of L.ass and Harmony2 came to our attention. Since these data were only slightly .iifferent from those of Butcher’, we did not repeat the calculations.
Fig. 4. Structure
3.
of 1,3-cyclohexadiene
(standard
deviations
between brackets)
DISCUSSION
Table 6 gives some figures for the length of carbon-carbon double bonds and sp2-sp2 single bonds in various compounds, as found in the literature. The average length of a double bond is 1.340f0.004 A and of an sp*-sp* single bond 1.470+_0.006 A. Deviations from the average values are found for the single bonds in 1,3,5-lrans-hexatriene and, to a smaller extent, in 1,3-butadiene and 1,3,5,7~yclooctatetraene as obtained from the earlier measurements. There is, of course, no reason to expect the length of a certain type of bond to be exactly the same in all molecules, but Table 6 shows that for some bonds this is approximately true. The differences found do not warrant further discussion, since they are of the same order of magnitude as those found in the careful investigations of butadiene and cyclooctatetraene and their duplications. The amplitudes of vibration for carbon-carbon double and single bonds show good agreement and they appear to be reproducible, in different structures and obtained by different workers, with an accuracy of perhaps 5 Ok., The C, symmetry of the equilibrium conformation shown by the microwave investibations is confirmed. The ethylene groups (C1C2C3C6) are planar; in J. Mol. S~rucrure, 1 (1967-68)
I l-23
20
G.
TABLE
DXLIIN.GA.
L. H. TONEMAN
6
CARBON-CARBON
BOND
LENGTHS
AND
AhiPLITiJDES
Bond length
(A)
OF VIBRATION
Amplitudes
of
ReJ
uibrution (/I) RC=C
RC-C
1,3-Butadiene
1.344 1.337
1,3,5,7-Cyclooctatetraene
1.340
1.467 1.483 1.476 1.467 1.462 1.450
1,3,5-rruns-Hexatriene Cyclopentadiene 1,3Kyclohexadiene* Ethylene (average) Perylene * This work:
1.334 1.345
1.342 1.339 1.336 -
rr-values of calculation
1.469 I.468 I.473
IlC,C
w-c
0.044
0.051
0.046
0.054
0.043
0.053
0.042
0.054
9 10 11 12 13 9 14 15-17 18
9 were used.
various calculations the carbon atoms are between a few thousandths and a few hundredths A out of the best plane through them. The two ethylene planes make an al ,& of 17”. Bond angles, within the accuracy of the measurement, do not deviate from predictions based on existing experimental evidence, except perhaps the HCH angle in the methylene groups, which proved to be unexpectedly small. We found the same for a few other cyclic molecules6, but lack of accuracy invariably prevented any definite conclusions. The necessity of a scale factor between diffraction and microwave structures is to be expected. It reflects the different way in which interatomic distances are averaged over molecular vibrations in the two cases. It is very difficult even to estimate the magnitude of this effect and there is no sense in discussing the value found (1 XJO7). APPENDIX
I.
INTRODUCTION
DETERMINATION
OF
BY GAS ELECTRON
MICROWAVE
INTO
MOLECULAR
STRUCTURE
DIFFRACTION
From microwave measurements moments of inertia defined by
4, = C mdyf+.&
DATA
one
can
calculate
the
three
principal
(1)
i
and analogous expressions for I,,, and 1,. The moments can be used as a check on a structure determination by diffraction methods. In such a comparison, one has to take into account that the interatomic distances, Rij, obtained frrJm diffraction are in principle not the same as the distances, (ro)ij, which reproduce the observed values of I,,, Iru, I,“. Differences may be of the order of 0.01 A. J.
1 (1967-68)
11-23
STRUCTURE
OF I,%CYCLOHEXADIENE
BY ELECTRON
21
DIFFRACTION
It occurred to us that one might also try to use microwave data directly in the determination of a structure from diffraction data. A possible approach would seem to be the simple addition of equations (1) to the set of normal equations of the least-squares problem. There are, however, two complications. Firstly, equations (1) hold only if the coordinate system is formed by the three principal axes of inertia. This means that the quantities I,, = Cmixlyi, I,, i
and I,, must be zero. Part of these requirements may be met by imposing a certain symmetry upon the system, others must be fulfilled by adding the expressions as constraints. In the case reported here (C, symmetry with the y-axis as twofold axis) I,, = 0 and IX,, = 0. To keep the origin in the centre of mass we add as constraint C miyi = 0; further C nzixizi = 0. The &ond complication rilates to the fact that both diffraction and spectroscopic data yield an absolute scale for the dimensions of the molecule. These scales are different ior the reason mentioned above and because of experimental errors. When this happens, the calculation may go wrong. It is then necessary to intro d uce a scale fr.ctor - which is close to 1 -between diffraction and microwave data. It should be realized that such a factor provides only a rough compensation for the differences between Rrj and (r,)ij values. However, at the present state 0’: the art of diffraction and spectroscopy it would seem to be adequate (see for example ref. 8), at least in the case of complex molecules. Our method of least-squares on intensity data, with constraints, has been described elsewhere3 and we give here only the additional equations necessary to introduce the microwave data. To equation (6) of ref. 3 (p. 797) one has to add the matrix equation M(X,
a) = j,
-
where M is a vector-valued function representing the moments of inertia as a function of the independent coordinates, X,, anci the scale factor, a and j represents the experimental values of the moments of inertia. As in the case of the other equations we assume that, at the beginning of the calculations, the choice of the parameter values is such that the M-equations are not obeyed exactly, but that M(X, a) = j+w,
(7)
The process of linearization and finding the best fit to the experimental data proceeds as usual3 and finally we obtain the following equations for the corrections Jj (coordinates, amplitudes of vibration, scale factor for intensities) and a (scale factor between diffraction and spectroscopic data): AA’ 0 0 0 K 0 BCO
-K’ 0 0
-B’ -C’ 0, 0 J. Mol. Structure,
1 (1967-68)
11-23
22
G. DALLINGA,
L. IL TONEMAN
where
p is a Lagrange multiplier. The other quantities have been defined previously3.
APPENDIX OPTICAL
II. PROCEDURE
FOR AUTOMATIC
CALCULATION
OF &(S)
FROM T)IE OBSERVED
DENSITIES BY AN ELECTRONIC COMPUTER
We assume that we have available, in numerical form, the optical densities of an electron diffraction diagram as a function of the distance r from the eentre of the diagram. We then apply the following procedure for obtaining i,,,(s). a. The r-scale of the diagram is transformed to s-scale: s = 4n sin e/A. The diffraction angle, 28, is calculated from tan 28 = r/L, where I. is the distance from the sample to the centre of the diagram on the film. The wavelength, A, is where E is the electronobtained from f, = (150.332/(E+0.9788 x 10-6E2))*, accelerating potential in volts. b. The theoretical background is calculated: Ia = C((Zi-~i)2 + ZJj) i c. I&Y) is transformed to experimental scale: 28; (rcn,c/rm;lx)x is the sector function; rmaxis the GM = -WI (~~,,,/r,,,Y se4 cos3 radius of the sector, rcnlccorresponds to the set of s-values used and x depends on the sector opening (we use r2 and r3 sectors). The factor cos328 corrects for the non-spherical form of the photographic film. d. In order to obtain the best possible transformation a least-squares calculation is performed to find p and q in I&s) = p&,(s) +q, where I&,(.s) are the experimental intensities corrected for non-linearity of the optical wedge. A somewhat arbitrary, but sometimes useful weight factor, s’, with k = 0, I, . . . can be introduced. At the inner and outer limits the diagram may by less accurate; the range of s-values to be used is determined after careful inspection of the photoSgram. e. Calculation of IO,(s) = pI&(s) + q. f. Corrections for sector, etc., as in c: I&) = ~,,,(s)(r,,i,/r,,,)-x~4cos-32e. g. Substraction of background: I,(S) = l,(s) - I&). The seeming detour c, d, e (instead of cakzulating p and q from I&) = pl&+q is necessary to counteract base-line errors in the photogram. In our experiments the photogram of a diffraction diagram consisted of curves on 2 to 8 sheets of paper. The computer program provides means to anneal these curves to one I,(S) curve. Not all diagrams can be treated successfuily with this program. At low diffraction angles it fails because of the very inaccurate theoretical background in that region (s < 5 A-‘), at high angles sometimes some extraneous scattering occurs. This effect can be partly eliminated by introducing into the least-squares
STRUCTURE
OF l,%CYCLOHEXADIENE
BY ELECTRON
DIFFRACTION
23
calculation a contribution e.g. of the form ms or ns2, which represents the extraneous scattering. The p-.-ogram has been written in Fortran-IV for an IBM 7094. ACKNOWLEDGEMENTS
The authors thank Miss C. J. Becker and Mr. M. J. van den Brink for synthesizing the 1,3-cyclohexadiene, Mr. W. Boog, who wrote the original version of the computer program mentioned in Appendix II, and Dr. H. Bolder for a fruitful discussion on moments of inertia. SUMMARY
The molecular structure of 1,3_cyclohexadiene has been determined from electron-diffraction data combined with the results of microwave investigations. The ethylene groups are planar, their planes making an angle of about !7”.
REFERENCES 1 S. S. BUTCHER, J. Chem. Phys., 42 (1965) 1830. 2 G. Luss AND M. D. HARMONY, J. Chem. Phys., 43 (1965) 3768. 3 H. C. CORBET, G. DALLING~, F. OLTMANS AND L. H. TONEMAN, Rec. Truu. Chim., 83 (1964) 789. 4 J. BERGHUIS.~. M. HAANAPPELAND M. POT-I-ERS, Acra Crysr.. 8 (19X)478. 5 D.T. KEATING AND G.H. VINEYARD, Acta Cryst.,9 (1956)895. 6 G. DALLINGA AND L. H. TONEMAN, Rec. Trau. Chim., 86 (1967) 171. 7 G. DALLINGA AND L. H. TONEMAN, J. Mol. Structure, 1 (1967-1968) in press. 8 D. R. LIDE JR.. Tefruhedron, 17 (1962) 125. 9 W. HAUGEN AND M. TRAETTEBERG, Acta Clrenl.Scat& 20 (1966) 1726. 10 A. ALMENNINGEN,~. BASIANSEN AND M.TRAE-I-I-EBERG, Acta Chem.Scand.,12 (1958)1221. 11 D.J. MARAIS,N.SHEPPARDAND B.P.STOICHEFF, Tetrahedron.17 (1962)163. 12 M. TRAETTEBERC, Acra Chent. Stand., 20 (1966) 1724. 13 0. BASTIANSEN.L.HEDBERG AND K.HEDBERG, J. ChemPhys.,27 (1957) 1311. 14 L. H. SCHARPEN AND V. W. LAURIE, J. Chem. Phys., 43 (1965) 2765. 15 H. C. ALLEN AND E. K. PLYLER, J. Am. Chem. Sac., 80 (1958) 2673. 16 L. S. BARTELL AND R. A. BONHAM, J. Chern. Phys., 31 (1959) 400. 17 J. M. DOWLING AND B. P. STOICHEFF, Can. J. Phys., 37 (1959) 703 18 G. DALLINGA, L. H.TONEMAN AND M. TRAEI-I-EBERG,Rec. Trav. Chim., in press.
J. k4of. Structure, 1 (1967-68) 11-23
Elsevier Publishing Company, Printed in the Netherlands
Amsterdam
ELECTRON DIFFRACTION BY GASES. THE MOLECULAR TURE OF 1,3-CYCLOHEXADIENE
G. DALLINGA
AND
STRUC-
Miss L. H. TONEMAN
Koninkl~ke~Sl~ell-Lnboraioriun~, Amsterdam (SheIt Research N. V.)
(Received
1.
Aprii 21st 1967)
INTRODUCTION
The structure of 1,3_cyclohexadiene has several interesting features. The degree of non-planarity of the ethylene groups and the conjugated double-bond system depends on the relative importance of resonance and steric strain. The molecule contains carbon-carbon single bonds of three different types and their differences in length provide a test for theories of the chemical bond. From the gas-diffraction point of view it is a very suitable molecule for demonstrating what details this technique can bring out and what assumptions are necessary in the structure analysis. We have in mind here the small differences between some of the carbon-carbon bond lengths and the structure of the CHa groups. While we were studying this compound, other workers published the results of microwave investigations’* ‘. They suggest C, symmetry for the equilibrium conformation and offer a semi-quantitative estimate of the degree of non-planarity. This prompted us to restart our calculations, now combining our diffraction data with the moments of inertia obtained from the microwave work.
2. 2.1.
DETERMINATION
OF THE STRUCTURE
ExpeFinlental
1,3Cyclohexadiene was prepared by chlorination of cyclohexene with tertbutyl hypochlorite and subsequent abstraction of HCl. It was purified gaschromatographically to a purity of 98.5 o/o (1.2 o/o benzene, 0.3 o/o cyclohexene, no 1,44somer). Diffraction diagrams were made with the sector apparatus described before’, using a wavelength of about 0.06 A (&0.00003). Photometer records of the diagrams were obtained with the Joyce-Loebl double-beam automatic recording J. Mol.
Strucrure, 1 (1967-68)
11-23
12
G. DALLINGA,
microdensitometer, TABLE
The results are summarized
in Table
1.
1
DESCRIPTION
No.
Mark IIIB.
L. H. TONEMAN
OF DIFFRACTION
DIAGRAMS
Sample-film Secror ased
-lfiposure time
distance (mm)
Film density
(see)
Range of s-values used in the transfofmation (See 2.2)
for calculfzlions 1.5 1.5
1 2
306.9 306.9
r2(r,,
=
25 mm)
25 45
0.18-0.68 0.40-1.52
3 4 5 6
306.9 306.9 306.9 153.8
P(r,,
=
60 mm)
60 120 120 120
0.14-0.36 0.26-O-62 0.30-0.82 0.56-I -68
rejected because of asymmetry 4 -I 9.75 4.875-19.625 3.375-19.625 ‘4.875-19.625 6.75 -37.75 9 -36
120 120 120 120 240
0.52-1.56 0.44-l -44 0.44-1.40 0.34-I .2 0.48-l .6
7 -37.5 8.75 -44.75 11 45 1 I .625-50.875 14.25 -51
7 8 9 10 11
153.8 123.8 123.8 102.1 102.1 *Only
2.2.
left part of photogram
6 5
-
8.5 8.5
9 11.75 11.75 14.25 14.25
-8 -8
-36 44.75 -44.75 -51 -51
used.
Calculations
readings were rtransformed to s-scale and, by a linear The photometer transformation, scaled to the theoretic& background. For calculating the background we used Berghuis’s4 atomic scattering functions and Keating’s’ incoherent scattering functions. Subtraction of the background from the scaied experimental intensities yielded the moiecular scattering functions, s41,(s). Details of an automatic procedure for the whole process are given in Appendix II. The procedure was applied to diagrams 9, 10 and I 1; diagrams 3-8 were treated by the same procedure, but only semi-automatically. The procedure did not work for diagrams I (s < 6 A-‘) and 2 (s c 5A-‘); here a suitable background had to be drawn. As discussed before6, this failure must be attributed to the theoretical background being incorrect at low s-values. Minor corrections were necessary at high s-values (s > 38 A-‘), due to some extraneous scattering. These corrections were made in the final s41,,,(s) curve obtained after averaging, successively, left and right parts of each diagram, a11 diagrams at each sample-fiIm distance and the diagrams at all distances. The final result is shown in Fig. la. The intensity data, l:Js), were used to calculate a radial distribution func. tion, p(r), according to p(r) = (2/~~~~s~I~(s)(p(s) J. Mol.
Structure,
1 (1967-68)
11-23
sin srAs,
STRUCTURE
OF l&CYCLOHEXADIENE
where q(s) = exp(--a(s*). in Fig. 2.
BY ELECTRON
The curve obtained with As = l/8 and a = 0 is shown _.__.
------
-
_jyjI :-,~d__..__~-~
Ij_jI
Fig. 2. Radial
for 1,3-cyclohexadiene:
distribution
function
(a) experimental;
(a
I
I
1
Fig. 1. In(s)
13
DIFFRACTION
i
I
i
i
..-----,c__..,.--‘;-______---._-_ ‘-_A
I I
-------
/
(b) calculated;
ia -b
I =
(c) difference
b-a.
= 0).
We then applied our computer program for least-squares calculations to the intensity data6. There proved to be a systematic difference between the experimental intensity data and those calculated for various models, the latter being invariably higher ?t s > 23 A- ’ and practically always lower at s c 23 A-‘. Analysis showed .that diagrams taken at 307 mm were not responsible for this 1.
Mol.
Sfrucrure,
1 (196748)
II-23
14
G. DALLINGA,
L. H. TONEMAN
effect: It might arise from extraneous s_dttering in the outer regions of the diagrams taken at small camera-film distances. We corrected the intensity data for this effect and did a number of least-squares calculations, both with the corrected and tne uncorrected s”l,(s). The effect of this particular error on the value of the parameters proved to be small but of course, it did affect the standard deviations. Table 2 shows the results of calculations of the carbon-carbon distances with the two sets of intensity data. TABLE THE
2
EFFECX
OF CORRECTIONS
OF THE INTENSITY
DATA ON PARAMETER
VALUES
AND THEIR
STANDARD
DEVIATIONS
Carbon-
uncorrecred I=(s)
carbon bonds*
w0
RI-2
1.339
h-3 Rl-6 &-I3
1.467 1.492 1.517
* For numbering
corrected I,(s)
St. dec. G-f)
w0
1.340 1.473 1.493 1.512
0.0021 0.0070 0.0017 0.0080
St. deu. (A) c.0013 0.0050 0.0010 0.0060
see Figure 3.
As mentioned in the Introduction, during the course of this work additional data were published in the literature, viz. moments of inertia from microwave measurements’*2. This led us to consider the possibility of using combined diffraction and microwave data in one and the same calculation. Our least-squares program provides for the introduction of additional structure information in the form of constraints, which are linear relations between two or three interatomic distances. It was easy to adjust the program so that moments of inertia could also be introduced as constraints. In general, such a procedure gives a very large weight to the additional information, but in our case this was no drawback, because microwave data are much more accurate than diffraction data. The procedure is described in some detail in Appendix I. The first calculations of this type were not successful. Both diffraction and microwave data yield an absolute scale for the dimensions of t’ne molecule. These scales may differ for the two methods as a consequence of experimental errors or the inherent differences between R-values obtained. Therefo::e, we introduced a scale factor, a, between microwave and diffraction da:a. The calculations then converged more rapidly. Theoretically, this combination of diffraction and microwave results would provide a possibility of determining the absolute scale of the molecular structure as obtained from electron diffraction far more accurately than usual. In practice, J. Mol. Structure, 1 (1967-68)
11-23
STRUCTURE
OF I,~-CYCLOHEXADIENE
BY ELECTRON
DIFFRACTION
15
however, difficulties arise. The differences mentioned above between R-values obtained by the two methods depend in a very complex way on the structure of the molecule. By introducing a scale I’actor one approximates the situation only roughly. 2.3.
The radial distribution function
The radial distribution function (Fig. 2) was used to find a starting model for the least-squares calculations on the intensity data. Tt shows a C-C bond length of 1.335 A and an average C-C bond length of nearly 1.50 %i. The other peaks do not contradict the assumption, made for the model, that the equilibrium conformation of the molecule has C2 symmetryl. 2.4.
Refinement cf the structure by least-squares calculations
From experience we know it is usually not possible to refine all structural parameters of a complex molecule by least-squares calculations. For that reasdn calculations are started with a reduced number of parameters. The reduction is obtained by employing some or all of the following methods: a. Keeping the amplitudes of vibration, I~ij, constant; b. Introducing constraints, by which interatomic distances can be kept constant or linearly related to each other; c. Keeping the scale factor constant. Ideally, the calculation converges rapidly and is repeated with an increased number of parameters by dropping one or more of the above restrictions. The choice of the additional parameters is arbitrary; whether the addition is meaningful can be decided by a statistical test6. In practice, many,complications may arise. The first calculation may go wrong, in which case the number of parameters has to be further decreased or different parameters have to be chosen. The problem may be ill-conditioned, i.e. contain parameters which show correlation or are poorly determined by the experimental data. The intensity data may turn out to contain a systematic error. Preliminary calculations are sometimes necessary to determine the nature of the complications. In our case correlations could be expected between the lengths of the three slightly
different
single
C-C
bonds
and #C-c.
We
therefore
kept the uij constant
initially, using the uij values obtained in calculations on l&cyclohexadiene’. Intensity data in the range 0.25 s s 6 46.0 A were employed with As = 0.25 A- ‘. Fig. 3 shows the numbering of atoms in the molecule. In the first calculation the only constraints used were those on the carbonhydrogen bonds (a, b, c, Table 3). We found: a. large differences between experimental and theoretical intensities below J. Mol.
Structure,
1 (196748)
11-23
16
G. DALLINGA,:L.
H. TONEMAN
s = 4.75 A-’ (S ee Fig. 1.) In this region the intensity is very insensitive to variations in the structure. On the other hand, the experimental data are rather inaccurate for s < 5 A- *. We decided, therefore, to use the intensity data in the range 4.75 $ s s 46.0 A- ’ in all following calculations; b. R2m3 > RI_, > R5-6, contrary to what one would expect’on the basis of a large body of experience; c. considerable inaccuracy for various coordinates of hydrogen atoms, especially the z-coordinate of HT. Their values oscillated during the iterative calculations; this prevented the calculation from converging. TABLE
3
CONSTRAINTS
USED
IN
LEAST-SQUARES
Type of distance
carbbn-hydrogen
carbon-hydrogen, carbon-carbon
bond
non-bonding
bond
CALCULATIONS
No.
Constraint
a b
R:_7 R,_,
= =
R=-a R5_-l,-0.01
C
&r-l,
=
RI-I=
d e
&--rz = &-IJ Rs-13 = &-~a
f
R2--3 f R,_, = R,_, = R,_, =
0
h i
RL-,, R,_, R,_. R,_,
= 2RI_6 -0.03 -0 03
-0.02
Fig. 3. Numbering of atoms in 1,3syclohexadiene.
In subsequent calculations we added the constraints d through i in various combinations and kept H, from oscillating by relating y7 and z7 to yl and zl. Constraint f, or g+ h or g+i, kept the C-C bonds in the “right” order (Rs_s > Rr _6 > RZ_-3); the use of d and e clearly increased the chance that the calculation converged. During the calculations it proved necessary to correct the I&r) for asystematic error (Section 2.2). J. Mol. Structure,
1 (1967-68)
11-23
STRUCTURE
OF l,%CYCLOHEXADIENE
BY ELECTRON
17
DIFFRACTION
Apparently, the structure cannot be determined unambiguously from diffraction data alone. Constraints are necessary to reasonably fix the hydrogen positions and to arrive at theoretically acceptable values of the C-C single-bond distances. In this stage of the investigation the microwave result- became availabler. The introduction of moments of inertia into the calculation (see App&dix I) made it necessary to choose a new coordinate systein, x’u’z’ (Fig. 3). This was obtained from the original system, xyt, by shifting the origin along the +y-axis and rotating the xz-plane about the y-axis so that in the x’y’z’ system &Z~JJ: = 0 and ClniXlZi’ = 0, mi representing
the mass of atom i (i = 1,2, . . ., iQi
The two
i
equations balance the two extra coordinates (JJ; and zi) necessary in this coordinate system. In the following we have omitted the accents in t’he x’y’z’ system. TABLE BOND
No.
4
LENGTHS
(s-i)
OBTAINED
constraints and relations between coordinates
FROM
LEAST-SQUARES
CALCULATIONS
C-C bonds RI-Z
C-H bonds -
&-3
RI-6
abc
1.339
1.471
I so4
ac
1.340 1.340 1.339 1.339 1.339 1.338 1.339 1.339
1.467 1.462 1.468 1.471 1.472 1.464 I.453 1.465
1.495 1.497 1.494 1.489 1.488 1.503 1.511 1.492
C at.!
actav acfvw actvw actvw acj’mv
1.488 1.509 1.508 1.510 1.518 1.518 1.493 1.494 1.519
RI-7
&-8
1.12 1.08 1.06 1.07 1.07 1.07 1.09 1.08 1.07
(1.12) (l.OSj 1.09 (1.07) (1.07) (1.07) (1.09) (1.08) (1.07)
(1-i 1) 1.14 1.15 1.14 1.14 1.14 1.12 1.13 1.14
As discussed before, a scale factor was introduced between diffraction and microwave data. The calculations converged better than before and it proved possible to decrease the number of constraints (Table 4, calculations 1, 2 and 3). The convergence was not complete however, owing to uncertainties in y,, z7 and zs, whose values kept oscillhLmg in consecutive iterations. These uncertainties can be understood from Fig. 3: changes in the parameters concerned have only a small effect on the distances between the hydrogen atoms involved and the other atoms. Consequently, the determination of these parameters from the diffraction data is inaccurate. For this reason and in order to speed up calculations we decided to do some more calculations in which the above-mentioned hydrogen coordinates were related to the coordinates of the adjacent carbon atoms by some of the following relations: YH-I= 0.25 y,, (t); z&f,= Zc, (U); z& = 4zc, (0); zH, = 22,, (w)_ The relations (u) snd (w) were used alternately in order to test the influence of the value of y in zu, =yz,-,. J. Mol. Srrucrure, 1 (1967-68)
11-23
18
G. DALLINGA,
L. H. TONEMAN
There was another uncertainty, viz. in the assignment of I,, and lYy, whose YXperimental values were very close (99.6313 and 99.8607; 1, = 187.1159 amu x A2 with 1 amu = 1.6598 x 10VzJ g). In calculations 1, 2 and 3 we had chosen IXX> Zw; calculation 2 was repeated with I,, c I,, (see 4, Table 4). The effect of the interchange on bond lengths w3s small. Since the difference between Icolo and &,(.s) became a little smaller, we used IX, -K I,,,, in all following calculations. This is probably in agreement with the order used in ref. 1. Calculations 5 and 6 show that the system is not very sensitive to the restrictions imposed upon some hydrogen coordinates. It is seen, however, that the olefinic and parafhnic C-H bonds are rather different in all calculations where constraint b is omitted. We repeated 6, using a larger value for UC-_H(0.079’ instead of 0.071 A). Indeed the differences between the Rc_n became smaller (7), which shows again the correlation between changes in u and R. In the same calculation, the order of R1_-6 and R,_, changed as well. This also happened when we used slightly different (smaller) values for some of the z.+_~ (8). Therefore, in the next calculation (9) we again introduced constraintf to keep the Rc_= in the “right” order. In this calculation we also allowed uc =c and uc_= to vary. uczc changed from the constant value chosen in previous calculatiOOS 0.0375 A, to 0.0415 A; uc_c (0.0540 A) did not vary at all; Uc_u was again kept constant at 0.071 ip. In a final calculation, not reported here, uczc and three independent uc_c were allowed to vary. This calculation converged, but impossible ~c-= values were obtained. TABLE
5
AMPLITUDES
OF VIBRATION
Distance
u,,(h
c=c c-c
0.0375 0.054 0.071 0.065 0.085 0.120 0.100 0.120
C-H C C (non-bonding) C I’-’ --’ H 8 C Ho H:;: : . H,2 H...H
Table 4 suggests that some parameters can be determined accurately (RI _*) or reasonably well (R,_,). For other parameters (R, _-6 and R,_,) even their relative order cannot be obtained with certainty. There is no clear-cut method of determining the best result. For further discussion we chose the result of calculation 4 to represent the structure of 1,3_cyclohexadiene. It has the merit of (a) containing only two, most reasonable, assumptions, viz. that all olefinic C-H bonds are equal and all aliphatic C-H bonds are equal, and jb) being a reasonable J. Mol. Strucrure, 1 (1967-68) 1l-23
STRUCTURE
OF l,%CYCLOHEXADIENE
BY ELECTRON
DIFFRACTION
19
average of all results. The structure is represented in Fig. 4; the set of amplitudes of vibration used in the calculation is given in Table 5. The scale factor between diffraction and microwave data was found to be 1.0067, which indicates that distances in the microwave structure are larger than those found from diffraction. When all calculations had been performed, the microwave data of L.ass and Harmony2 came to our attention. Since these data were only slightly .iifferent from those of Butcher’, we did not repeat the calculations.
Fig. 4. Structure
3.
of 1,3-cyclohexadiene
(standard
deviations
between brackets)
DISCUSSION
Table 6 gives some figures for the length of carbon-carbon double bonds and sp2-sp2 single bonds in various compounds, as found in the literature. The average length of a double bond is 1.340f0.004 A and of an sp*-sp* single bond 1.470+_0.006 A. Deviations from the average values are found for the single bonds in 1,3,5-lrans-hexatriene and, to a smaller extent, in 1,3-butadiene and 1,3,5,7~yclooctatetraene as obtained from the earlier measurements. There is, of course, no reason to expect the length of a certain type of bond to be exactly the same in all molecules, but Table 6 shows that for some bonds this is approximately true. The differences found do not warrant further discussion, since they are of the same order of magnitude as those found in the careful investigations of butadiene and cyclooctatetraene and their duplications. The amplitudes of vibration for carbon-carbon double and single bonds show good agreement and they appear to be reproducible, in different structures and obtained by different workers, with an accuracy of perhaps 5 Ok., The C, symmetry of the equilibrium conformation shown by the microwave investibations is confirmed. The ethylene groups (C1C2C3C6) are planar; in J. Mol. S~rucrure, 1 (1967-68)
I l-23
20
G.
TABLE
DXLIIN.GA.
L. H. TONEMAN
6
CARBON-CARBON
BOND
LENGTHS
AND
AhiPLITiJDES
Bond length
(A)
OF VIBRATION
Amplitudes
of
ReJ
uibrution (/I) RC=C
RC-C
1,3-Butadiene
1.344 1.337
1,3,5,7-Cyclooctatetraene
1.340
1.467 1.483 1.476 1.467 1.462 1.450
1,3,5-rruns-Hexatriene Cyclopentadiene 1,3Kyclohexadiene* Ethylene (average) Perylene * This work:
1.334 1.345
1.342 1.339 1.336 -
rr-values of calculation
1.469 I.468 I.473
IlC,C
w-c
0.044
0.051
0.046
0.054
0.043
0.053
0.042
0.054
9 10 11 12 13 9 14 15-17 18
9 were used.
various calculations the carbon atoms are between a few thousandths and a few hundredths A out of the best plane through them. The two ethylene planes make an al ,& of 17”. Bond angles, within the accuracy of the measurement, do not deviate from predictions based on existing experimental evidence, except perhaps the HCH angle in the methylene groups, which proved to be unexpectedly small. We found the same for a few other cyclic molecules6, but lack of accuracy invariably prevented any definite conclusions. The necessity of a scale factor between diffraction and microwave structures is to be expected. It reflects the different way in which interatomic distances are averaged over molecular vibrations in the two cases. It is very difficult even to estimate the magnitude of this effect and there is no sense in discussing the value found (1 XJO7). APPENDIX
I.
INTRODUCTION
DETERMINATION
OF
BY GAS ELECTRON
MICROWAVE
INTO
MOLECULAR
STRUCTURE
DIFFRACTION
From microwave measurements moments of inertia defined by
4, = C mdyf+.&
DATA
one
can
calculate
the
three
principal
(1)
i
and analogous expressions for I,,, and 1,. The moments can be used as a check on a structure determination by diffraction methods. In such a comparison, one has to take into account that the interatomic distances, Rij, obtained frrJm diffraction are in principle not the same as the distances, (ro)ij, which reproduce the observed values of I,,, Iru, I,“. Differences may be of the order of 0.01 A. J.
1 (1967-68)
11-23
STRUCTURE
OF I,%CYCLOHEXADIENE
BY ELECTRON
21
DIFFRACTION
It occurred to us that one might also try to use microwave data directly in the determination of a structure from diffraction data. A possible approach would seem to be the simple addition of equations (1) to the set of normal equations of the least-squares problem. There are, however, two complications. Firstly, equations (1) hold only if the coordinate system is formed by the three principal axes of inertia. This means that the quantities I,, = Cmixlyi, I,, i
and I,, must be zero. Part of these requirements may be met by imposing a certain symmetry upon the system, others must be fulfilled by adding the expressions as constraints. In the case reported here (C, symmetry with the y-axis as twofold axis) I,, = 0 and IX,, = 0. To keep the origin in the centre of mass we add as constraint C miyi = 0; further C nzixizi = 0. The &ond complication rilates to the fact that both diffraction and spectroscopic data yield an absolute scale for the dimensions of the molecule. These scales are different ior the reason mentioned above and because of experimental errors. When this happens, the calculation may go wrong. It is then necessary to intro d uce a scale fr.ctor - which is close to 1 -between diffraction and microwave data. It should be realized that such a factor provides only a rough compensation for the differences between Rrj and (r,)ij values. However, at the present state 0’: the art of diffraction and spectroscopy it would seem to be adequate (see for example ref. 8), at least in the case of complex molecules. Our method of least-squares on intensity data, with constraints, has been described elsewhere3 and we give here only the additional equations necessary to introduce the microwave data. To equation (6) of ref. 3 (p. 797) one has to add the matrix equation M(X,
a) = j,
-
where M is a vector-valued function representing the moments of inertia as a function of the independent coordinates, X,, anci the scale factor, a and j represents the experimental values of the moments of inertia. As in the case of the other equations we assume that, at the beginning of the calculations, the choice of the parameter values is such that the M-equations are not obeyed exactly, but that M(X, a) = j+w,
(7)
The process of linearization and finding the best fit to the experimental data proceeds as usual3 and finally we obtain the following equations for the corrections Jj (coordinates, amplitudes of vibration, scale factor for intensities) and a (scale factor between diffraction and spectroscopic data): AA’ 0 0 0 K 0 BCO
-K’ 0 0
-B’ -C’ 0, 0 J. Mol. Structure,
1 (1967-68)
11-23
22
G. DALLINGA,
L. IL TONEMAN
where
p is a Lagrange multiplier. The other quantities have been defined previously3.
APPENDIX OPTICAL
II. PROCEDURE
FOR AUTOMATIC
CALCULATION
OF &(S)
FROM T)IE OBSERVED
DENSITIES BY AN ELECTRONIC COMPUTER
We assume that we have available, in numerical form, the optical densities of an electron diffraction diagram as a function of the distance r from the eentre of the diagram. We then apply the following procedure for obtaining i,,,(s). a. The r-scale of the diagram is transformed to s-scale: s = 4n sin e/A. The diffraction angle, 28, is calculated from tan 28 = r/L, where I. is the distance from the sample to the centre of the diagram on the film. The wavelength, A, is where E is the electronobtained from f, = (150.332/(E+0.9788 x 10-6E2))*, accelerating potential in volts. b. The theoretical background is calculated: Ia = C((Zi-~i)2 + ZJj) i c. I&Y) is transformed to experimental scale: 28; (rcn,c/rm;lx)x is the sector function; rmaxis the GM = -WI (~~,,,/r,,,Y se4 cos3 radius of the sector, rcnlccorresponds to the set of s-values used and x depends on the sector opening (we use r2 and r3 sectors). The factor cos328 corrects for the non-spherical form of the photographic film. d. In order to obtain the best possible transformation a least-squares calculation is performed to find p and q in I&s) = p&,(s) +q, where I&,(.s) are the experimental intensities corrected for non-linearity of the optical wedge. A somewhat arbitrary, but sometimes useful weight factor, s’, with k = 0, I, . . . can be introduced. At the inner and outer limits the diagram may by less accurate; the range of s-values to be used is determined after careful inspection of the photoSgram. e. Calculation of IO,(s) = pI&(s) + q. f. Corrections for sector, etc., as in c: I&) = ~,,,(s)(r,,i,/r,,,)-x~4cos-32e. g. Substraction of background: I,(S) = l,(s) - I&). The seeming detour c, d, e (instead of cakzulating p and q from I&) = pl&+q is necessary to counteract base-line errors in the photogram. In our experiments the photogram of a diffraction diagram consisted of curves on 2 to 8 sheets of paper. The computer program provides means to anneal these curves to one I,(S) curve. Not all diagrams can be treated successfuily with this program. At low diffraction angles it fails because of the very inaccurate theoretical background in that region (s < 5 A-‘), at high angles sometimes some extraneous scattering occurs. This effect can be partly eliminated by introducing into the least-squares
STRUCTURE
OF l,%CYCLOHEXADIENE
BY ELECTRON
DIFFRACTION
23
calculation a contribution e.g. of the form ms or ns2, which represents the extraneous scattering. The p-.-ogram has been written in Fortran-IV for an IBM 7094. ACKNOWLEDGEMENTS
The authors thank Miss C. J. Becker and Mr. M. J. van den Brink for synthesizing the 1,3-cyclohexadiene, Mr. W. Boog, who wrote the original version of the computer program mentioned in Appendix II, and Dr. H. Bolder for a fruitful discussion on moments of inertia. SUMMARY
The molecular structure of 1,3_cyclohexadiene has been determined from electron-diffraction data combined with the results of microwave investigations. The ethylene groups are planar, their planes making an angle of about !7”.
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J. k4of. Structure, 1 (1967-68) 11-23