An Electron Diffraction Investigation of the Molecular Structure of 1,1,1-Trichloroethane, 2,2-Dichloropropane and 2-Chloro-2-Methylpropane

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P U R D U E U N IV E R S IT Y

THIS IS TO CERTIFY THAT THE THESIS PREPARED UNDER MY SUPERVISION

51------- John Wallace C o m t t s

e n t it l e d

---------------- -----

AM ELECTRON DIFFRACTION INVESTIGATION OF THE

MOLECULAR STRUCTURE OF 1,1,1-TRICHL0R0ETHANE, 2 r2-UICHL0R0PR0PANE A M D 2-CHLQR0-2-METHYLPE0PAME

COMPLIES WITH THE UNIVERSITY REGULATIONS ON GRADUATION THESES

AND IS APPROVED BY ME AS FULFILLING THIS PART

OF

THE REQUIREMENTS

FOR THE DEGREE OF

Doctor of Philosophy

Professor m

H eap

«June 14

i9

of

C h a r o e o f T h k s is

Sc h o o l

bO

TO THE LIBRARIAN:----IS THIS THESIS IS NOT TO BE REGARDED AS CONFIDENTIAL

G R A D , S C H O O L FO R M

9 —3 - 4 9 —XM

or

D epar tm en t

AN ELECTRON DIFFRACTION INVESTIGATION OF THE MOLECULAR STRUCTURE OF l,l,l»TRICHLOROETHME, 2,2-BICHLOROPROPANE AND 2wCHL0R0~2«*METHYLPR0PANE

A Thesis

Submitted to the Faculty

of

Purdue university

by

John Wallace Coutts

In partial Fulfillment of the

Requirements for the Degree

of

Doctor of philosophy

June, 1950

ProQuest Number: 27712261

All rights reserved INFORMATION TO ALL USERS The q u a lity of this re p ro d u c tio n is d e p e n d e n t u p o n the q u a lity of the co p y su b m itte d . In the unlikely e v e n t that the a u th o r did not send a c o m p le te m a n u scrip t and there are missing p a g e s, these will be n o te d . Also, if m a te ria l had to be re m o v e d , a n o te will in d ic a te the d e le tio n .

uest P roQ uest 27712261 Published by ProQuest LLC (2019). C o p y rig h t of the Dissertation is held by the A uthor. All rights reserved. This work is p ro te cte d a g a in s t u n a u th o rize d co p yin g under Title 17, United States C o d e M icroform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346

ACKNOWLEDGMENTS

I wish to express my sincere thanks to Dr. R. L# Livingston for his guidance and constant interest in this problem. I am also very grateful to Dr* H. J . y earlan for his willing advice in many phases of the work. My colleagues, T. T. Broun and H. P. Lemaire have been very helpful with much of the experimental work* Thanks are due Dr* S* js* Wirt and Miss B. Bradley of the Purdue Tabulating Division for their wholehearted co­ operation in making the I. B. M. tabulations. I am also grateful to the Research Corporation for their financial support of the largest share of this work.

TABLE OP CONTENTS

Page AB S T R A C T .................

1

I - INTRODUCTION.......................................

!

II - THE ELECTRON DIFFRACTION METHOD..................

3

A - History......

3

B - Theories of Scattering as Applied to Electron Diffraction. .. .....................

4

1.

Introduction...............................

4

2.

Scattering by Rigid Molecules..............

5

3.

Scattering by Non-Rigid Molecules

....

7

4.

Simplifications of the Scattering Expression....................

C - Recording the Diffraction pattern............

11 13

D - Visual Observations on the Diffraction Pattern.........................

17

E - Determination of Molecular Structure from the Visual pattern...........................

21

Correlation Method.........................

21

1. a.

principle of the Method.

........

b.

Calculation of Intensity Curves for Rigid Molecules...........

c.

21

24

Calculation of Intensity Curves for Non-Rigid Molecules......................

26

2*

The Radial Distribution Method.............

30

3.

Accuracy of the Method.......

32

Page III - THE MOLECULARSTRUCTURE OF METHYL CHLOROFORM I* Experimental.................. . ..............

34 34

2* Correlation of Theoretical and Visual Intensity Curves.............................

38

IV - THE MOLECULAR STRUCTURE OF 2,2-DICHLOROPROPANE....

50

1.

Experimental..................................

50

2. Correlation of Theoretical and Visual Intensity Curves......... »..................

54

V - THE MOLECULAR STRUCTURE OF TERTIARY BUTYL CHLORIDE..........................................

79

!• Experimental..................................

79

2. Correlation of Theoretical and Visual Intensity Curves............................. VI - DISCUSSION OFRESULTS........... BIBLIOGRAPHY.......................................... VITA..................................................

83 99 104

LIST OF TABLES Table 1*

PaSe Exposure Conditions for Measured patterns of Methyl Chloroform.........

36

2.

Data for Visual Curve for Methyl Chloroform.......

37

3.

Quantitative Correlation for Methyl Chloroform.•.•

45

4.

Principal Distances in Acceptable Models for Methyl Chloroform*

5*

••

48

Exposure Conditions for Measured patterns of 2, 2-Dichloropropane...................... ......*•

53

6.

Data for Visual Curve for 2,2-Dichloropropane

7*

Quantitative Correlation for 2,2-Dichloropropane.. 71

8.

principal Distances in Acceptable Models for 2, 2-Dichloropropane..............................

9.

92

Quantitative Effects of Temperature Factors for Tertiary Butyl Chloride..........................

13.

82

Quantitative Correlation for Tertiary Butyl Chloride.............................

12.

81

Data for the Visual Curve for Tertiary Butyl Chloride...................................

11.

74

Exposure Conditions for Measured Patterns of Tertiary Butyl Chloride....................

10.

55

94

principal Distances in Acceptable Models for Tertiary Butyl Chloride..................

97

LIST OF FIGURES Figure

Page

1

Parameter Chart for Methyl Chloroform*...........

2

Visual and Theoretical Intensity Curves for Methyl Chloroform................................

3

Radial Distribution Curves.......................

40

42 49

R.D. 1 - Methyl Chloroform R.D* 2 - 2,2-Dichloropropane R.D. 3 - Tertiary Butyl Chloride 4

Parameter Chart for 2,2-Dichloropropane for ^

5

- 1094°

...........

64

Visual and Theoretical Intensity Curves for 2, 2-Dichloropropane. .....

66

6

Parameter Chart for Tertiary Butyl Chloride......

85

7

Visual and Theoretical intensity Curves for Tertiary Butyl Chloride.................... ......

86

i

Department of Chemistry and furdue Research Foundation furdue university, Lafayette, Indiana

ùïï ELECTRON DIFFRACTION INVESTIGATION OF THE MOLECULAR

STRUCTURE OF 1,1,1-TRICHLOROETHANE, 2,2-DICHLOROPROPANE AND 2-CHL0R0-8-METHÏLPR0PANE * ^Contains material from the doctoral thesis of J. W» Coutts. By J» W» Coutts^and R. Le Livingston tPresent address;

Dept, of Chemistry, Mount union College,

Alliance, Ohio.

-AN ABSTRACT

Introduction. - One of the important applications of electron diffraction has been the study of the effects pro­ duced in the configuration of a molecule when certain groups or atoms therein are replaced by other groups or atoms.

The

purpose of the present investigation was to determine whether any structural differences result from the substitu­ tion of methyl groups for chlorine atoms in the series; 1,1,1-trichloroethane (methyl chloroform), 2,2-dichloropropane and 2-chloro**2-methylpropane (tertiary butyl chloride). It was also hoped that accurate values could be obtained for the bond distances and angles in these compounds so that comparisons with distances and angles in other compounds

ii.

would be possible*

Of these compounds, the first and third

have been previously investigated by J. Y* Beach and D* if* Stevenson^-* (1)

The diffraction patterns obtained by Beach

Beach, J. Y. and Stevenson, D, t *, J . Am. Chem. Soc., 60, 475 (1958).

(2)

Beach, J. Y. and Stevenson, D. if., ibid., 61, 2645 (1959).

and Stevenson did not extend to as large a scattering angle as is generally obtained with the apparatus available in this laboratory.

Since the outer part of the diffraction

pattern is often the most valuable in structure determina­ tions, repetition of their work was considered worthwhile in the hope that a more precise determination would result. Furthermore, their determinations were made without allow­ ance for the displacements of atoms caused by intramolecular vibration.

In the case of a molecule such as tertiary butyl

chloride with a large number of hydrogen atoms vibration factors assume considerable importance and the study of this compound can not be considered complete without an investi­ gation of their effects. When this investigation was undertaken there was no report in the literature of a structural investigation of 2,2**dichloropropane.

When the work was near its completion,

however, mention was made in a review article5 of unpublished (5)

Allen, r • W. and Button, L. is., Acta Crystallograph!ca, 5 (jrart 1), 46 (1950).

ill. work on this compound, by J * 0 1Gorman and V. s chomaker *

The

authors have been informed by Dr. Schomaker that mention of this work in the review article was premature as the work had never been completed. .experimental. - Commercial samples of methyl chloro­ form and tertiary butyl chloride were purified by washing with appropriate reagents followed by rectification in all­ glass columns.

The sample of 2, 2-dichloropropane was pre­

pared from acetone and phosphorus pentachloride following in most essentials the method outlined by omyth and Turkevich^. (4)

The refractive indices of the samples used were as

Smyth, C. f . and Turkevich, A., J. Am. Chem. Boc., 62, 2468 (1940).

follows:

methyl chloroform, n ^

- 1*4380; 2,2«dichloropro-

pane, n^° - 1.4148; tertiary butyl chloride, n£0 - 1*5862. The electron diffraction photographs were prepared us­ ing an apparatus built by professor H. J . Year!an of the Department of Hiysics of fur due University.

The wavelength

of the electrons was determined from the transmission pat­ terns of zinc oxide and was about 0.055 &• tance was about 11 cm.

The camera dis­

The diffraction patterns extend to

about q - 90 for each compound and are represented by curves Vis in Figures 2, 5 and 7.

The q0 values are given in

Tables 2, 6 and 10. Interpretation. - In interpreting the photographs both

iv.

the visual correlation method^ and the radial distribution (5)

Brockway, L. 0-, Revs. Mod. Fhys., 8, 231 (1936;.

method6 were employed. (6)

The intensity curves used in the

Shaffer, *. A., Schomaker, V. and Pauling, L., J. Chem. Phys., 14, 659 (1946).

visual correlation were celculated from the equation I(q) z

Zj.2. exp

sin (IQqr^ .)

rij 7r The summations were performed from punched cards6*^ by an (7)

Shaffer, Jr. A., Schomaker, V, and Pauling, L . , J. Chem. Phys., 14, 648 (1946;.

I.B.M. tabulating machine.

In the calculation of all inten**

sity curves the bonded C-H distance (1.09 A) and the short non-bonded C-H distance (2.16 a ) were damped by the factors b - 0.00018 and b s 0.00035, respectively.

These values

have been found by schomaker^*^ to account adequately for (8)

Schomaker, V. and Shaffer, p., J. Am. Chem. soc., 69, 1555 (1947).

(9)

Lipscomb, W. N- and Schomaker, V., J. Chem. phys., 14, 475 (1946).

the distribution of these distances due to vibration in several similar compounds.

The values of b^j assigned to

other distances will be discussed separately for each compound.

V

The radial distribution functions were calculated from the equation^

The values of I(q) were read from the visual curve at in­ tegral values of q.

Intensity values for the inner portions

of the patterns which do not appear on the visual curve were read from theoretical intensity curves.

The factor b was

chosen so that exp (»»bq*j*fty) - 0*1. Methyl chloroform. - The intensity curves were not af­ fected appreciably either quantitatively or qualitatively by the omission of the terms involving non-bonded H atoms. Therefore these terms were omitted from the summation.

Ap­

plication of a small damping factor to the non-bonded C-Cl distances, which was found necessary for the other two com­ pounds caused no appreciable change in the appearance of the methyl chloroform curves.

These distances were there­

fore included without damping, i.e. b - o.

The range over

which the shape parameters ( 0,0003. For a preliminary investigation of the effects of these vibration factors it was found convenient to omit completely the rotation—dependent distances, i,e. b was made infinitely great.

Several models over a wide parameter range were

calculated in this manner but it was found impossible to ob­ tain correspondence between the calculated and visual curves. The curves which appeared closest to fitting the visual curve were then altered by reducing the damping factor to its minimum value of 0.0003.

No curves with this value of

b were compatible with the visual curve.

It was evident

that correlation was not possible between the two assumed extremes of b. Next, with the rotation-dependent distances still omitted, a damping factor was applied to the non-bonded C-Cl distance to allow for vibrations of the carbon atoms and chlorine atoms attached to the central carbon atom. Such a damping factor should necessarily be small since the amplitude of vibration of the carbon atoms will be less than that of the hydrogen atoms.

The chlorine atoms will be sub­

ject to even smaller amplitudes of vibration and hence the Cl-Cl distance should not require a damping factor.

Further­

more, it turned out that the Cl-Cl distance in most of the acceptable models was almost the same as the heavily damped Cl-H distance so that the vibration factor applied to the latter will effectively include any slight factor which

should possibly be applied to the Cl-Cl distance*

Ho factor

was applied to the non-bonded C-C distance since its effec­ tive weight in the summation is very small*

It was found

that the use of a factor b of the order of 0*0001 in damp­ ing the non-bonded C-Cl distance was very effective in bring­ ing the theoretical curves into agreement with the visual curve. Several curves were then calculated with various com­ binations of the factors b^ (non-bonded C-Cl) and bg (nonbonded Cl-H).

Because of the complexity of the problem it

was assumed that the same temperature factor applied to each of the non-bonded Cl-H distances.

The relatively unimport­

ant rotation-dependent C-H distances were omitted completely* The factors b^ and bg fortunately affected the curves in such a way that it was possible to find optimum combina­ tions of the two*

A further assumption made in choosing

b^ and bg was that a satisfactory combination should give good results for both 2#2®dichloropropane and tertiary butyl chloride*

Although this assumption may not be strictly cor­

rect it does not seem to be an unreasonable approximation in view of the similarity of the two molecules.

The factors

which give the best results were b-L

b

0*00006

*>2

st

0.0008.

It is not claimed that these factors necessarily represent the correct values; the problem was already one of four parameters without the evaluation of temperature factors.

It

xii.

can only be asserted that they are probably of the correct order of magnitude and that they were chosen after a carefnl consideration of their effects on several curves.

Further,

in the selection of acceptable models allowances were made in doubtful cases for the effects which would be produced by a reasonable variation in b^ and bg. The features of the visual curve (see Figure 5) which are most sensitive to changes in parameters and temperature factors are the sixth and seventh maxima.

These broad peaks

are not symmetrical but show a measurable maximum of inten­ sity on their inner side and gradually taper off in intensity toward the out side. sixth.

The seventh maximum is broader than the

Neither of these peaks showed any doublet character.

The outer region showed a small maximum (q - 77) partially resolved from a larger maximum at q - 84 with a measurable minimum between the two* The calculated curves were very sensitive to changes in the angle &

(Cl-C-Cl), with the great majority of ac­

ceptable models being found at

- 1094°.

Although oc­

casional acceptable models were found with /4 - 110-J0 and with these values*

/!& greater or less than

It is believed that enough models of this

type have been calculated to Insure that qualitative agreement is impossible for values of more than 14°. studied for $ of

differing from 109-i° by

Figure 4 represents the parameter range - 109^°.

The range covered for other values

^4 was by no means as complete.

xiii.

The curves of Figure 5 will illustrate the principal criteria used in the selection of acceptable models*

First­

ly, curves showing partial resolution of the seventh maximum into doublets have been rejected unless the degree of resolu­ tion is such that a reasonable variation of temperature fac­ tors may be expected to bring the curve into agreement with the visual curve*

dee, for example, curves f and V which

were considered acceptable on this basis*

Curve AA was re­

jected since any combination of b^ and bg which tends to im­ prove the appearance of the seventh maximum confers pro­ nounced asymmetry of the wrong type on the sixth maximum (i.e., it causes a more gradual slope toward the inside than the outside).

Similarly, curves showing pronounced resolu­

tion of the fifth maximum (e.g. X and E) were rejected, while curves similar to s were accepted as borderline cases in view of the changes which might result from changes in temp­ erature factors.

Models giving rise to curves similar to B,

in which the seventh maximum is insufficiently asymmetric and the sixth broader than the seventh, have also been eliminated.

Curves showing asymmetry of the wrong type in the

seventh maximum (see curves H and AD) were rejected since it may be safely predicted that the changes in temperature fac­ tors required to reverse the asymmetry of the seventh maxi­ mum would bring about the appearance of a shoulder on the side of the sixth maximum. Curves Ui and Ug illustrate the effects of changing the angle $

from 107-g° to 111%° (the other parameters correspond

xiv.

to those of model U)«

These effects were found to be of the

same general nature for all values of

In the parameter

range studied* The range of acceptable models Is Indicated In Figure 4. Selecting the middle of this range gives the best value of oC

as llOjg-0*

value of

It has already been pointed out that the best

is 109^°*

The q/qo ratios for the acceptable

models are listed in Table 7»

Calculation of the C-Cl dis­

tance from the q/qQ ratios for models near the middle of the range, viz. M and U, gives 1*775 or 1*776 &•

The average

value for all the acceptable models is 1*776* The final values for the parameters and other important distances and angles are listed below*

In imposing the

ranges of uncertainty a somewhat greater latitude has been placed on

and

ceptable models.

than is indicated by the range of ac­ This arises from making an additional al­

lowance for possible uncertainties caused by temperature factors.

In summary, the accepted values are; C-Cl (bonded)

1.776

+

C-C (bonded)

1.54

*#

C-Cl (non-bonded)

2.71

+ 0.04 A + 0.05

Cl-Cl (non-bonded)

2.90

-w

0.05 A

JL C-C-C

110*0

Is ee

5°

Z.C1-C-C1

109*0 t 2°

Z. Cl-C-C

109.20+

0*02

A

3°

The values for the bonded and non-bonded C-Cl distances are in excellent agreement with those taken from the main

XV,

peaks of the radial distribution curve (RD-2 in Figure 3), viz. 1*78 A and 2.71 A. Tertiary Butyl Chloride. - Tertiary butyl chloride was assumed to have the symmetry of the C^v point group, with the methyl groups oriented so that the hydrogen atoms were as far as possible from the chlorine atom*

The vibration

factors were the same as those used for 2,2-dichloropropane; in addition, the non-bonded C-C distances were modified by a vibration factor equal to that used for the C-Cl distances, viz. 0.00006.

The longest non-bonded Cl-B distance and the

longest non-bonded C-H distance were omitted.

Many addition­

al models were calculated with temperature factors different from those mentioned above in order to note the effects of variations in temperature factors.

The range of parameters

studied is shown in Figure 6. The features of the pattern which proved to be of par­ ticular value in choosing among models may be seen from curve Vis in Figure 7.

The fifth maximum appeared more in­

tense than those on either side of it, with the sixth mini­ mum being more shallow than the fifth or seventh.

The

seventh maximum is particularly sensitive to parameter changes.

It appears on the plate as a very broad peak,

falling off in intensity toward its outer edge.

It shows

little, if any, detectable doublet character but has a measurable maximum of intensity on its inner side.

The ninth

maximum is also asymmetric. Increasing in intensity toward the outside.

This peak has a point of inflection or weak

shoulder on Its inner side* Several curves representative of the different areas of the studied parameter range are shown in Figure 7.

Curves

B, L and TJ and J, L and 0 illustrate the general changes oc­ curring upon crossing the parameter field in two directions. It is evident from curves B, L and V that increasing the angle

causes a reversal in the asymmetry of the seventh

and ninth maxima, with the curve at 107-^° being in best agree­ ment with the visual curve*

Models B, xu and 17 were rejected.

L i s the best model and R was accepted as a borderline case after making due allowances for possible effects resulting from alterations in temperature factors. In going from curve J through 0 the fifth and sixth maxima approach each other In intensity, the seventh maxi­ mum reverses the nature of its asymmetry and the shoulder moves down the side of the ninth maximum and becomes resolv­ ed into a small but definite maximum.

These variations, like

those observed in changing ©< , combine to give the best agreement with the visual curve in the vicinity of model L. Model J was rejected on account of the pronounced resolu­ tion of the seventh maximum, while N and 0 were rejected be­ cause of the inversion in the relative intensities of the fifth and sixth maxima and the reversed asymmetry of the seventh maximum.

Models K and M were accepted.

Models 0 and p lie at the extreme ends of thediagonal field of acceptable models.

Although curves showing resolu­

tion of the seventh maximum of the degree occurring in curve

xvii

P were not accepted in the case of the curves for 2,2-dichloropropane they are more difficult to eliminate for the present compound for two reasons ;

the appearance of the

seventh maximum is more shelf—like than in the dichloropro— pane pattern, and the effects of temperature factors in this compound are in general greater than for dichloropropane. Models D and S have been shown in Figure 7 in addition to those already mentioned so that the appearance of any curve of the parameter field may be readily judged by interpo* lation*

The range of acceptable models is indicated in

Figure 6 and the q/q0 ratios for the accepted curves are listed in Table 11* The imposition of temperature factors has a quantita­ tive as well as qualitative effect on the curves.

In the

case of tertiary butyl chloride this effect Is considerable because of the large number of distances affected by vibra­ tion factors.

In order to test the effects of variations of

these factors a number of different curves were calculated for model L®

It was found that the use of either extreme

value of bg, i.e. bg = o or bg = ©o , causes an appreciable Mdrift” in q/q0 1s with increasing q.

In the former case

the q/qQ ratios increase with increasing q, while in the latter case the drift is in the opposite direction.

It was

also found that the largest changes In q/qQ ratio occur up­ on the initial introduction of temperature factors of the order of 0*0001 or 0.0010 and In going from a factor of

xvlii such size to an infinite factor.

Therefore, provided

reasonable temperature factors are used at all and provided important distances are not omitted completely the quanti­ tative results are not greatly affected by changes in temperature factors over the usual range.

Another point

which might be noted is that the curves calculated with the values of b^ and bg which were considered to give the best qualitative agreement also gave the least mean deviation of those studied. From the parameter chart it is evident that the best model is very closely represented by model L*

The best

values of the structural parameters have been derived, there­ fore, from model L and the limits of uncertainty have been imposed after a consideration of the extreme values of the various parameters at which acceptable models have been found, with the usual allowances for the probable error in exp eriment al me asur ernents. The final values with their estimated limits of un­ certainty are : C-Cl (bonded)

1*80 Î

C-C (bonded)

1*54 ^ 0.03 a

^Cl-C-C

107*

0.04 a

t 1*°

Other values of interest are : .

The factors used in curve L (06-60) are

the values which were selected as the best values, as has already been discussed, viz, b^s 0.00006 and b^- 0.00080. It will be seen that application of the factor for the C-Cl and C-C distances has reduced the mean value of q/qD to ap­ proximately the same values as for L (30) and L (80).

The

96*

slight difference between the mean of the first seven and the last seven readings is not large enough in L (06~Q0) to be of significance in view of the experimental error in­ volved.

Several significant points regarding the data of

Table 12 should be noted.

Firstly it is evident that there

is a considerable difference between the mean q/qo ratios calculated for the two extreme cases, L (o) and L (T), and that there is bad drift for both.

Secondly, it will be

noted that introduction of a temperature factor considered to be of minimum magnitude, i.e. 0*0003, also produces an appreciable increase in the q/q0 ratio*

A comparison of

curves L (30) and L (80) shows, however, that variation of the factor over a considerable range has not changed the ratio appreciably*

The largest changes, then, occur upon

the initial introduction of temperature factors of the order of magnitude of

0*0001

or

0*0010

and in going from a factor

of such size to an infinite factor.

Therefore, provided

reasonable temperature factors are used at all, and provided important distances are not omitted completely, the quanti­ tative results are not greatly affected by changes in tempera­ ture factors over the usual range.

Another point which might

be noted is that curve L (06-80), calculated with the factors which were considered to give the best qualitative agreement, also shows the least mean deviation of those studied* The q/q0 ratios from Table 12 were used in order to cal­ culate the values of the most important distances In the ac­ ceptable models.

These are recorded in Table 13.

From the

Table 13 principal Distances in Acceptable Models

Parameters

Model F

Model Gr

Model K

Model L

Z.ClCC(z>U

106*0

106*o

107*°

107*0

OC1

1*82

1*84

1*78

1*80

C-C

1*54

1.54

1.54

1.54

C-H

1*09

1.09

1.09

1.09

obtained

0*999

0*995

1*008

1.000

C-C1(bonded)

1*818

1.831

1.794

1*800

C-C(bonded)

1.538

1*532

1.552

1.540

C-Cl(non-bonded)2* 697

2.696

2*701

2.700

C-C(non-bonded) 2.557

2*547

2.560

2.540

principal Distances

(Continued)

Parameters

Model M

Model t

Model Q

Model R