The molecular structure of 1,3,5,7‐cyclooctatetraene has been studied by a sector‐microphotometer technique using data extending to very much larger scattering angles than were obtained in earlier investigations. An application of the method of least squares to sector‐microphotometer data in electron diffraction worked out by one of us (KH) has led in three refinement stages to unusually precise values for the parameters. The following are the more interesting parameter values with standard errors. It should be noted these results do not include a possible error of up to 0.2% in the scale of the molecule because of uncertainties in the electron wavelength, nor do they include the effect of correlations among the observations on the standard errors, which we estimate might increase the standard errors by as much as the factor 2½. Molecular symmetry D2d,C=C=1.334±0.001 A,C–C=1.462±0.001 A,C–H=1.090±0.005 A,∠C=C–C=126.46∘±0.23∘,∠C=C–H=118.3∘±5.9∘,aC=C(=〈δl2〉C=C/2)=(103±7)×10−5,aC–C=(147±10)×10−5,aC1···C3=(256±13)×10−5, aC1···C4=(530±100)×10−5, aC–H=(327±36)×10−5, aC2···H=(520±110)×10−5,aC3···H=(640±180)×10−5⁠.

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4.
K. Hedberg and V. Schomaker, American Chemical Society Meeting, San Francisco, 1949.
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7.
The necessity for such long exposure times was at first somewhat surprising when compared to times of the order of seconds required in other apparatuses. This was presumably due partly to a smaller rate of gas flow in our experiments (probably occasioned by the hydrodynamic characteristics of the nozzle), partly to a slower photographic emulsion than is used in visual work and partly to our larger camera dimensions. With the newly constructed California Institute of Technology apparatus, exposure times of about 2–4 minutes with 0.2 μa beams and Kodak process plates have been found necessary to obtain sufficient scattering in the s = 40–50 region.
8.
We believe that these larger scattering angles are due to a low rate of gas flow (sample bulb vapor pressures of 5–20 mm for cyclooctatetraene, for example) combined with efficient trapping of the emergent gases by the nozzle cold trap, which together act to prevent gas spreading about the nozzle tip and consequent elongation of the scattering region. Experiments with the new California Institute of Technology apparatus by K. H. have since shown that if the rate of flow is too high, even efficiently trapped gases spread markedly about the nozzle tip.
9.
A mechanism was subsequently built and is now in use which oscillates the plates during microphotometry in order to reduce the effect of graininess in the emulsion.
10.
Our plates had no direct evidence of the center of the diffraction pattern, such as would be provided by an image of the undiffracted beam.
11.
Density = ln(I0/I), where I/I0 is percent transmission. The initial arbitrary scale of photometer readings was placed on a 0–100 scale by using the approximately 20 readings made on both scales to evaluate an average I0′ from the relationship I/I0 = I′/I0′.
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13.
We assumed, as is usual, that the photographic density is proportional to exposure over a certain density range. A check on this assumption was always provided by the ratios of plate densities obtained during the nonlinearity correction procedure (reference 12): these ratios were essentially constant for densities less than about 1.5.
14.
The proper weighting of the data from each plate is a difficult matter, such factors as plate quality being of great importance and yet difficult to estimate except by examination, possibly biased, of the microphotometer traces. In current practice the corrected densities are not weighted.
15.
Current practice in the reduction of the photographic data is simpler than that outlined here and gives better results. The plates are oscillated about their centers during tracing, which is done with a continuously recording microphotometer, and the amplitudes of the curves are read off directly with a logarithmic scale.
16.
As a matter of interest we also calculated BT and attempted to fit it to the observed curves by adjusting the arbitrary constant k. In general the calculated function could be made to fit the middle parts of the curves fairly well, but began to deviate in the direction of greater density over the inner and lesser density over the outer parts of the curves. The f curves of
H.
Viervoll
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Ögrim
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17.
A bothersome situation was discovered here. The 19‐cm camera distance curve was found to be displaced slightly with respect to the other curves, the displacement being a maximum of about 18s unit at s = 35. Nothing was done to bring the curves into closer agreement.
18.
This function yields a Fourier transform, the peaks of which are essentially Gaussian. See
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19.
Amble
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20.
This equation usually includes a factor exp(−as2) to minimize diffraction effects in the Fourier transform resulting from series termination. However, our intensity data, extending to such high angles, were sufficiently attenuated by the real molecular vibrations as to make this unnecessary.
21.
Since the observed intensity was divided by (Z−f)C2 to give IE, the calculated curves and IE are not strictly comparable, all terms involving hydrogen being in error by an amount related to deviation of the ratio (Z−f)C/(Z−f)H from constancy. This error is very small because the large vibration factors associated with hydrogen terms reduce their contributions rapidly with increasing angle. The intensity curves were calculated at intervals Δs = 12 using r values accurate to 0.001 A. The calculations were carried out with a desk calculating machine: due to the range of our data the punch‐card methods were inapplicable.
22.
In this method theoretical intensity curves are calculated for ranges of values of the parameters. Comparison of these curves with observation leads to a region of acceptability in parameter hyperspace from which estimates of the best parameter values and their limits of error can be made. The number of curves required is roughly 3n, where n is the number of parameters being adjusted; hence structures having more than four parameters (the D2d model of cyclooctatetraene has a dozen or more important geometrical and vibration parameters) are seldom thoroughly investigated.
23.
See, for example, N. Arley and K. R. Buch Introduction to the Theory of Probability and Statistics (John Wiley and Sons, Inc., New York, 1950), Chap. 12.
24.
W. C. Hamilton and V. Schomaker have discussed the least squares method in detail in connection with analysis of visual electron diffraction data (see W. C. Hamilton, thesis, California Institute of Technology, 1954).
For an application to visual data see
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25.
This interval is somewhat arbitrary and it is possible that considerably more data could have been properly included by choosing a smaller one; however, a smaller interval, besides greatly extending our computational labor, surely would have tended toward increasingly correlated observations. The observations at Δs = 12 are not uncorrelated in fact, since the very process of drawing and adjusting the background establishes a relationship among ranges of points.
26.
The independent distance parameters (and assumption of bond coplanarity about each carbon atom) define the geometry of the D2d model completely. The choice of but seven vibration parameters from the fourteen corresponding to all important interatomic distances includes the arbitrary simplifying assumptions aC1⋯C4 = aC1⋯C5 = aC1⋯C6, aC2⋯H = aC8⋯H, and aC3⋯H = aC4⋯H = aC5⋯H = aC6⋯H = aC7⋯H.
27.
The uncertainties quoted from the earlier investigations are limits of error, which are naturally larger than standard errors, perhaps about equal to 2σ.
28.
The approximate s value reached in Hedberg and Schomaker’s experiment was 33, in Karle’s 22.
29.
The explanation of this discrepancy most likely lies in a greater extent of gas spreading about the nozzle in Karle’s experiment, with resultant more rapid apparent damping (see reference 8). Vibration factors were not evaluated in the other earlier electron diffraction investigations.
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1957
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