Some Electron Diffraction Studies: I. Electron Diffraction Studies of Some Heavy Metal Hexafluorides. II. The Molecular Structure of Gaseous N₂O₄. Ill. Applications of the Method of Least Squares in Electron Diffraction Investigations Citation Smith, Darwin Waldron (1959) Some Electron Diffraction Studies: I. Electron Diffraction Studies of Some Heavy Metal Hexafluorides. II. The Molecular Structure of Gaseous N₂O₄. Ill. Applications of the Method of Least Squares in Electron Diffraction Investigations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/A9G6-3H34. https://resolver.caltech.edu/CaltechETD:etd-02132006-132126 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Part I. An electron diffraction investigation has been made by the sector-microphotometer method of WF6, OsF6, IrF6, UF6, NpF6 and PuF6. The photographs of all these compounds exhibit a phase shift anomaly which gives rise to apparently assymetric structures, the anomaly setting in at smaller angles [...] of scattering for the heavier molecules and at smaller values of q = (40/[...])sin [...]/2 the greater the electron wavelength. There is good evidence for the symmetrical octahedral structure of all the compounds. The metal-fluorine distances were found to be (in Angstrom units) l.833(W-F), l.831(Os-F), l.830(Ir-F), 1.996(U-F), l.981(Np-F), and l.971(Pu-F). Part II. An electron diffraction molecular structure determination of gaseous N2O4 has shown that the N-O distance, the N-N distance, and the ONO bond angle are 1.177 A, 1.752 A, and 134.8° respectively. The molecule is planar, which is surprising in view of the long N-N bond. Part III. A structure refinement procedure, which was applied to the above compounds, has been programmed for the Burroughs 205 computer. It involves the fitting of a theoretical intensity curve to the experimental one. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemistry and Physics) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Schomaker, Verner F. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1959 Record Number: CaltechETD:etd-02132006-132126 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-02132006-132126 DOI: 10.7907/A9G6-3H34 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 623 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 14 Feb 2006 Last Modified: 19 Oct 2023 20:47 Thesis Files Preview PDF (Smith_dw_1959.pdf) - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

SOME ELECTRON DIFFRACTION STU DIES Il ELECTRON DIFFRACTION STUDIES OF SOME HEAVY METAL HEXMAPLUORIDOES ll THE MOLECULAR STRUCTURE OF GASEOUS N,O 4 lll APPLICATIONS OF THE METHOD Of LEAST SQUARES IN ELECTRON DIFFRACTION INVESTIGATIONS Thesis by Parwin Waldron Smith In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1959 ACKNOWLEDGEMENTS It is with great pleasure that I acknowledge the friendly, inspiring guidance and valuable criticism of my research supervisor, Professor Verner Schomaker. I wish to thank Dr. Kenneth Hedberg for suggesting that I work on his N2O4 data which he gathered in Norway, and for several dis- cussions concerning the least squares refinement of sector data. To both cand. real. Lise Hedberg and Dr. Kenneth Hedberg go my thanks for their guidance in the reduction of the N 2° 4 data. lam very glad to acknowledge collaboration with Dr. Masao Kimura in the refinement of the hexafluoride compounds, and I wish to thank him for our many friendly discussions. To Professor Jurg Waser and Dr. Richard E. Marsh I express my thanks for reading and criticising several sections of this thesis. My appreciation also goes to Professors Linus Pauling, Harden McConnell, Richard M. Radger and others for helpful discussions on several phases of my work. I wich to thank Dr. Robert Nathan, Dr. Joel N. franklin, and Mr. Kindrick J. Hebert for discussions relating to the programming of the Burroughs 205 computer. To the California Institute, the Allied Chemical and Dye Co., the Eastrmman Kodak Co., andthe E. I. du Pont du Nernours Co. I express my sincere appreciation for the scholarships and fellowships which have made graduate school possible. My thanks also go to the many staff members and graduate students of the Institute who have made my years here so pleasant and rewarding. Finally I wish to dedicate this thesis to my wife Jo Ann, whose love and encouragement mean 80 much, to our two daughters Shelly and Lorraine, and to our wonderful parents. ABSTRACT I, An electron diffraction investigation has been made by the sector-microphotometer method of WE es O8F gs InP ys UP 6s Nek, and Pur,. The photographs of all these compounds exhibit a phase shift anomaly which gives rise to apparently assymetric structures, the anomaly setting in at smaller angles @ of scattering for the heavier molecules and at smaller values of q = (40/\)sin 6/2 the greater the electron wavelength. There is good evidence for the symmetrical octahedral structure of all the compounds, The metal-fluorine dis- tances were found to be (in Angstrom units) 1.833(W-F), 1,831(Os8-P), 1,.830(Ir-F), 1,.996(U-F), 1.981(Np-©), and 1.971(Pu-f). Il, An electron diffraction molecular structure determination of gaseous N34 has shown that the N-O distances, the N-N distance, and the ONO bond angle are 1,177 A, 1.752 A, and 134, g° respectively. The molecule ie planar, which is surprising in view of the long N-N bond, I, A structure refinement procedure, which was applied to the above compounds, has been programmed for the Burroughs 205 computer, It involves the fitting of a theoretical intensity curve to the experimental one. TABLE OF CONTENTS PART TITLE PAGE I. ELECTRON DIFFRACTION STUDIES OF SOME HEAVY METAL HEXAFLUORIDES Introduction l Theory 2 Experimental Intensity Curves 5 Calculation of Theoretical Curves 14 Least Squares Refinement 19 Results of Least Squares Refinement 22 Temperature Parameters 38 Systematic Errors 4) Plutonium Hexafluoride 44 Evidence for Octahedral Structure 51 Conclusions 52 il. THE MOLECULAR STRUCTURE OF GASEOUS N,O 4 introduction and Theery 56 Reduction of Microphotemeter Traces 57 Analysis of Data 62 Least Squares Refinement 64 (Publication: The Molecular Structure of Gaseous Dinitrogen Tetroxide) 65 Internal Rotation in N,O 4 72 The N-N Bonding in N,0%, 75 Til. APPLICATIONS OF THE METHOD OF LEAST SQUARES Introduction © 83 Description of the Program 86 Interpretation of the Standard Deviations 88 Appendices Appendix I: Theoretical Intensity Curves Program 90 Appendix II: Radial Distribution Function Program 94 Appendix II: A Least Squares Program 98 «lj. Il ELECTRON DIFPRACTION STUDIES OF SOME HEAVY METAL HEXAPLUORIDES An electron diffraction investigation has been made of six gaseous heavy metal hexafluorides: those of tungsten, osmium’, iridium, uranium, neptunium, and plutonium. Early electron diffraction work of UF, and WF 6 had resulted in anomalous unsymmetrical structures (2), (3), while spectroscopic cbservations (4) - (7) had favored the symmetrical octahedral structure. However, in 1952 Schomaker and Glauber (8) showed that failure of the first Born approximation, used in cornputing scattering amplitudes, had led to erroneous electron diffraction results for these and certain other compounds containing both heavy and light atoms. Subsequently UF 6 (9) and WE 6 (10) have been successfully interpreted in terms of a symmetrical model by using angle-dependent phase shifts in the scattering amplitudes. The present investigation is of interest in confirming the sym- continuing the study of the phase shift by providing move extensive data at various electron wavelengths for all six compounds. This investiga-~- tion was instigated by Dr. Bernard Weinstock of the Argonne National Laboratory who supplied all of the compounds except W Fy The WF, was kindly supplied by Dr. John M. Teem of this Institute. The sam- ples were considered to be sufficiently pure for structure determination. * The volatile yellow solid OsF, has been incorrectly called OsF, for many years. However Weinstock and Malm (1) have shown by molecular weight determination and chemical analysis that it is the hexafluoride and that the compound previously called Os¥ 6 was also identified incor- rectly. ~2- Table I lists the electron accelerating potentials, wave lengths, and sample temperatures for each compound photographed. The photo- graphs were prepared by the sector method by Professor Schomaker, Dr. Weinstock and Dr. Kimura. Dr. Kimura prepared microphoto- meter tracings and reduced them to experimental intensity curves. The experimental curves were interpreted by the radial distribution method and by a least squares refinement procedure in which theo- . . : : . te retical intensity curves are adjusted to the experimental curves . ‘Theory The scattering of electrons by gas molecules containing both heavy and light atoms has been discussed by Schomaker and Glauber (8), and Hoerni and Ibers (9). For electrons of wave length \ scattered at angle 6, the expression for the totai scattered intensity is eo ~o767/2 sin r,.s E(s) = B(s) + KP, if, it2) jl tg, js)! cos Qi, jee 4 = (2) ij . where s = (47/)) sin 6/2; B is the background or structure independent in,(s) scattering; K is a constant; f, = lal e ? is the complex scattering amplitude (for electrons) of atom j; ani =, - 943 re is the root mean square variation of Tay the distance between atoms i and j. The second term (the molecular part) in the right member of (1) may be called I sothat we have I1=B+tl.. m t m For precise sector work, especially where a determination of vibrational parameters is desired, theoretical intensity curves should * I gratefully acknowledge collaboration with Dr. Kimura in interpreting the data and refining the results. “5 +" ca] woe sy Se op ra 3 a FT ¥ T ELECTRON ACCELERATION POTENTIALS USED IN MAKING ELECTRON DIPPRACTION PICTURES Sample sa ge Compound temp. Potential (kev) and wavelength (A) oa) wf a) ony ut? oF uty fan NO ana > on on com © oo tn re 2 Qo Qi & SS mt ND mt co ao Ko 2 oO nO tt pet SO et m3 a ao aa) ao i O in bi oy ie) “a oa oat won i tt ni non HoH n tt hit > > > > pA > > Pae) WE, #55 x x x x x x o Os, «39 x x o InP “42 x x o UF, cas) % x x o NpF, “23 x x x ake be calculated from an expression of the form a(i, - B) _ If, il; i ae? 5 In this expression f(s) and £,(8) may be scattering factors for a particular pair of atorns in the molecule or may be some sort of "average" scattering factors for the molecule (averaged in some way with respect to the atoms). They are introduced in order to make the coefficients in the Fourier sum nearly constant. Such a theoretical intensity curve has the familiar appearance of a damped sine series. In the present case as explained later, it is impossible to determine absolute vibrational parameters; therefore it was considered adequate to calculate theoretical curves from the expression 22 si(s) = / = cos An, (s)e sin T55 (3) where Ci is independent of s. In the case of a symmetrical octahedral metal hexafluoride (MF >), we have 2 2 6C -o, 6 /2 el(s) = M:? cos An sin rse ME 120... ao”, s“/2 4 ——————= sin /2 ree 2 Yr 3C. -c% _8*/2 + EF gin 2rse * °° * (4) : or where r= Ter and 4m = SnyeE This expression leads to an «5 apparent split M-F distance, for if we make the approximation An =6s, 6C = cos 4m, sin rs 3¢ 3c, . == sin (x + 6)e + =~ sin (r «- &)}s fl (5) 3C 2S, sin (r + 8)s + x. sin (r - s)s + O(8/r*). Hy The apparent splitis Ar = 26, and early anomalous structures were reported with just such splits. The critical region in the pattern, known as the cut-off point, occurs at that value of s for which 4n = 7/2. Inthe split model the two sine terms beat out at 6. nf 28, and again at ao = 37/26. The split model can be used for approximate theoretical curves out to about its second cut-off point, but since the data extend beyond s = 37/28, (and do not show a second cut-off point in this range), a more accurate treatment than the split model was needed. This will be discussed under "Calculation of Theoretical Gurves. " Experimental Intensity Curves The electron diffraction photographs were made on Kodak process plates, developed in Microdol. The microphotometer used to obtain density data is the Chemistry Division's Sinclair Smith model, made by the Fred C. Henson Company, Pasadena, in which the radiometer detecting unit has been replaced by an RCA 1P39 phototube. An R-C electronic filter circuit serves to damp out part of the noise due to photographic imperfections. The signal is amplified by an Allied * Physics Corporation vibrating reed electrometer , and traced by a : ; We wish to thank Professor Thomas Lauritsen for loaning us the electrometer. ~ B= Brown potentiometer. The illumination is by parallel light, the lamp being powered by storage batteries. The photometer has high stability; the drift is negligible under the conditions of illumination and amplification present in making a trace providing the lamp is warmed up for about two hours and the phototube is warmed up by being illuminated with a low light level for 30 minutes to an hour. Exact tests of phototube linearity were not necessary because small deviations are absorbed in the unknown non-linearity of the photographic emulsion, discussed below. That the system is approximately linear is shown by the fact that doubling the slit width showed approximately double the phototube current. Indeed, a plate traced twice at slit widths differing by a factor of two yielded nearly identical density curves. The slit width was 0.2mm as measured on the plate except for low potential (long wavelength) pictures, where, because of the broader rings, 0.4 mm was used. Such a width and a slit length of 2 mm intro- duced only a small broadening of features inthe pattern. That this broadening is indeed small may be seen by considering the highest frequency to be expected in the pattern: that due to the long F--.F bond of about 4 A. This bond contributes to the pattern a term of period about 1.4 mm on the plate when the wavelength is 0. 056 A, the shortest wavelength used. The plate speed was the slowest availa- ble, 1.4 mm/min., and the recorder chart speed was 10 ft/hr. Pip marks were superimposed over the full length of the trace by impressing the recorder pen with a voltage pulse every 0.5 mm translation of the plate. This pulse came from a microswitch driven oJ by an arm on the microphotometer screw. These pip marks enabled distances to be found without the possibility of errors caused by the paper shrinking or stretching. The plates were spun on a rotating stage to average out graini- ness and defects in the emulsion and to reduce the effect of possible unsymmetrical spurious background caused by scattering from the “apparatus. The centering of a plate on the rotating stage was done with the aid of the beam stop shadow. The error in centering the diffraction pattern includes two effects. First, the error in centering the bear stop shadow is about 0.l mm. The electron beam can be centered on the axis of the sector with an error of about 0.l mm. The beam stop rotates with the sector, therefore its shadow is concentric with the axis of the sector. Thus the error in centering the pattern on the rotating stage is probably less than 0. 2 mm. Two types of microphotometer traces were obtained from each plate. In one, the range from zero to 100% transmission was recorded, In the second, a greater amplification and a shift of the zero line re- sulted in an expanded scale, making readings more accurate. The first type of trace was used merely to locate the zero and 100%. lines on the expanded record. The plates were traced along the full clameter of the pattern, giving two identical haives. The center of the trace could be located to well within 0.1 mm either by measuring the positions of maxima and minima on the trace or by making a mark in the clear glass near the center of the beam stop shadow before tracing the plate. A distance scale was marked off on the trace and readings were taken at every ae 0. 2 mm (as measured on the plate) except for the low potential pictures where 0.4 mm sufficed. This interval is of the same order of magni- tude as the slit width and as the uncertainty in centering. Microphoto- meter transmission veadings were converted to densities. in sector work it is desirable to convert densities into relative electron intensities as accurately as possible. At low exposures the density D andinmtensity I are proportional; a plot of D vs. I is a straight line. At higher exposures, the D vs. I curve falls below the straight line: 7 ] and a positive correction must be applied to the density. Other cor- rections which might be applied include a factor of 1/ cos” 6 to correct for the increased distance from the scattering center to the outer parts of the plate, and, assuming a thick emulsion, a factor of i/cos i] because of its angle of tilt with respect to a diffracted ray. In order to correct for the non-linearity of the photographic emulsion, one needs two plates at different exposures of the same exact diffraction pattern, developed together (11). Unfortunately in the present work the electron accelerating voltage was changed with every plate in a set because of limited amounts of some of the samples, because of the primary purpose of determining the effect of the voltage on the cut-off point, and because of the necessary overall effective haste in which the work was done. -9- Therefore the above described correction for the non-linearity of the photographic emulsion could not be made and the amplitudes of the rings in the dense inner part of a picture are too small compared with the amplitudes of the Ught outer rings (before dividing by the background). Thus the experimental intensity curves (strictly density curves} are in error by being multiplied by an unknown but smooth modification function. This is not really serious however, because although it will render impossible the determination of absclute vibra- tional parameters, it will not affect bond distances or the position of the cut-off point. Our results should certainly be more accurate than visual results because we still have quantitative measurements of shapes of diffraction rings and of relative amplitudes of a number of neighboring maxima and minima. However, another method for at least partially correcting for the non-linearity of the ermulsion as well as the I/cos” 6 effect is to divide by EG. The procedure therefore was as follows: densities obtained from raicrophotometer traces were corrected for the presence of the sector. The total intensity curve i, thus obtained as a function of r was then plotted and a scale of q = ~ sin ; = ae was marked off on it. The intensities were then read off at integral values of q. A smooth experimental background B was drawn in by hand and sub- tracted from i. The result t - B was divided by the experimental : tn t ae background as is the general practice in sector work . When photo- graphic density corrections are possible it seems better to divide by ee See, for example, reference (12). the theoretical background if it is known, or by the product of scattering powers for a particular pair of atoms as suggested previously under "Theory". An experimental background line is certain to contain errors of judgement in drawing it as well as errors caused by spurious background scattering. Since the theoretical background is not known exactly, especially for our samples, it is difficult to say what the best procedure is, Dividing by the experimental background is perhaps justified in view of the possibility that under ideal experimental condi- tions, the background obtained may be more nearly correct than the theoretical background, and may approach the shape of the true scat- tering factors closer than theoretical ones do. In the present experi- ments, the background was not ideal, but the background used in dividing was monotonic and was adjusted, where necessary, to conform to the general shape of the theoretical background. Let this adjusted divisor be B'. Any detailed deviation of B' from a more proper divisor is of the same general nature as the non-linearity of the emul- sionand dividing by B or 5B! compensates somewhat for this non- linearity and also for the 1/ cos’ factor. The above considerations and the decision to approximate Cy; by a constant in the expression (equation 3) for the theoretical curves probably justify the foregoing procedure, but we must keep in mind the possibility that i, bas been multiplied by a non-constant modification function. The final experimental curves were obtained by multiplying the result of the above procedure by s: (I, ~ B) si{s) = (6) RB! ~ lie to conform with equation 2. Radial distribution (RD) curves were ob- tained for all of the compounds at several voltages. These RD curves support the octahedral model (see "Evidence for Octahedral Structure") and show two M-F peaks separated by the apparent split distance. From the M-F distance, the apparent split, and the approximate temperature factors ae read from the RD curves, preliminary theoretical curves were obtained. They were used as a guide in re-adjusting the back- ground where necessary. The sector deviates markedly from its intended shape at L1 + L2@em and at 13-15 cm. It was designed to have the angular opening a proportional to R for R greater thani.27¢cm. For R less than this value the opening is greater in order to give a flat background with carbon compounds. The turnover point on the sector corresponds to 1. 35 cm on the plate, one of the peints of difficulty. The sector cali- bration curve, (a plot of ai/ a}, was obtained from readings made with a travelling microscope. For small a large errors are introduced in n/a and it is difficult to draw a curve through the widely scattered calibration points. Visual comparison of non-sector plates with sector plates or with curves obtained from sector plates shows a definite discrepancy near l.4cm. Theoretical intensity curves also show the same dis- crepancy. Therefore, in the process of refining models, the region of the sector calibration curve near 1.4 cm was redrawn a number of times. The attempt was made to obtain a calibration curve that would fit the calibration points fairly well and yet would lead to experimental intensity curves having reasonable agreement with theoretical curves. alZa Somewhat of a compromise had to be made in the atternpt to satisfy these two requirements. The final sector calibration curve that was used for all of the pictures is shown in Fig. l along with the measured calibration points and the original curve through the points. The original (broken line) and final (solid line) sector calibration curves near l.4dcm. The radius R corresponds to distances on the photographic plate, and a is the angular sector opening in radians. -|3- 19-0 -- 18.8 -- 18-6 18.4 }— 18.2 — 18.0 }- 17.8 | SECTOR CALIBRATION R(ON THE PLATE) Fig. t Calculation of Theoretical Curves A program for calculating theoretical intensity curves on the . Burroughs 205 digital computer is described in the appendix. This is abasic program, written primarily for the calculation of intensity curves I, of the form sl(s) = 1, = > Ae sin 1,8 | (7) i where A; does not depend on s, and the double summation over atoms of equation 2 has been replaced by a single summation over pairs of atoms, i being the jth pair of atoms or ry the i th interatomic distance. This simplified expression, {equation 7), is satisfactory for most of the compounds studied by electron diffraction, but could not be ueed without modification in the present investigation. The reason is that the anomalous phase shift requires the use of the factor cos 4n inthe M-F term as in equation 4. Ibers and Hoerni (13) have calculated for 39.47 kev electrons the atomic scattering amplitudes including the angle-dependent phase shifts »(@). These calculations were based on the partial-waves scattering theory and were carried out for selected atoms. In Fig. 2 are plotted values of Ay = Tage Vp as a function of q = = = = = ein § for the metals with lowest and highest atomic numbers of interest here, tungsten and plutonium. Curves for the other metals fall between these two curves. All these curves have approximately the same shape. 15 Fig. 2 Graph of the dependence of the phase shift on scattering angle: 4nfO} vs. qs < sin § for 40 kev electrons. PueF and W-¥F, (solid lines}; from calculations of TIbers and Hoerni (13), A, (broken line): Ony F approximated by equation 9. 0 20 40 60 80 Toye) 120 i40 Av Fig. 2 olF= Their calculations applying directly only to V = Vo = 39.47 kev, Ibers and Hoerni proposed the following approximation ior V not too different from Vv, Z,6,V) = o(Z565 Vi) if Z! = Z(v,/v); ein(O'/2) = (2/zP Fay sin (9/2) . (8) 10 om /sec, Here, v is the velocity of the electrons, vo * 1.048 x 10 and he ® 0.06056 A. It is thus seen that curves of 4n vs. q for various voltages also have the same general shape. An analytical expression was used to approximate Arn in the calculation of intensity curves. The cut-off point a 108 / # could then be used ag an adjustable parameter to be determined from the data and compared with the theoretical predictions. It was convenient to use the expression 44 Ana) = 3+ 2elatq- a.) + bla- a) (9) with three adjustable parameters, Qo Be and b. (By An(q) is meant 47(@) expressed as a function of gq.) The data, however, do not justify more than a determination of de and a very rough deter- mination of a (which fixes the gradient rh =z 27a at cut-off), and it was decided to fix b at b = +6.0000129 for every compound at every voltage. This value had been chosen in a preliminary refinement of NpF ,-47 kev to make the second cut-off approximately correct for that particular set of data. It gives roughly the correct curvature to ar. Any error in b cannot affect r or q. significantly, and, al- 7 c -18- though it does affect the absolute temperature factors, it should have little effect on their ratios. In Fig. 2 is included a plot of An as calculated from equation 9, with a and a. taken from the final least squares refinement of WE g~ 40 kev. The deviation from the theoretical Ayn curve for W-F is less than 10% over the whole range of gq, with very good agreement near q? qu. The theoretical intensity curves prograra was modified to caleu- late cos 4 by equation 9 andto form I c from the expression ~ora"/2 A,e cos 4y sin r,8 (10) I= i c oa el The amplitude coefficient A, (or z FARA) ; 2 as a function of gq. Although it in equation 3) was determined bd fle? by plotting the ratio Theoretical was not constant, the value of this ratio near q = 60 was used to ob- tain Ay Any error introduced by assuming that A, is constant will again affect mainly the absolute values of the temperature factors. To examine the effect of varying the Ay, WE ,-40 kev was refined by least squares using three choices of the ratio ApAgiAs. The re- sults are listed in the following table, the relative value of A, being 1000. A, As oy v2 3 Ie a 300-455 0. 0379 0. 0944 0. 0654 43.9 0.00503 285 50 0. 0386 0. 0936 0. 0650 43.9 0, 00496 174 30 0. 0395 0. 0665 0. 0356 43.7 0. 00414 #19 « The subscripts 1, 2, and 3 refeyto Wehr, Fo, and F+-s respec- tively; oy and a are in Angstrom unite and a, is in reciprocal Angstrom units. In the first row the amplitude ratio corresponds to A, obtained from the scattering amplitudes of Ibers and Hoerni at q = 50; the second row foxy gq = 65; and in the last row the ratio was estimated from atomic numbers. As expected, the results are insen« sitive to small changes in the ratio AyAgiAg. There was no change in r which was 1.530 Ain every case, and the agreement in Ge is essentially perfect. In the first two rows the differences in the tern- perature factors are considerably less than their standard deviations. The use of atomic numbers to obtain the scattering amplitudes is seen to cause larger differences, however. The amplitude ratios used in the final least squares refinements were 1000:285:50 for W Fe, Ost, and InP; and 1000: 278:50 for UF, Npl'p, and Duk ¢. Least Squares Refinement In Part Ul a basic least squares program is described for the refinement of electron diffraction structures. This program mini- mizes the weighted sum of squares of the differences I evi ¢ between experimental and caiculated intensity curves by adjusting the distances, temperature parameters and amplitude scale factor in I = Overlays were written to modify this program for the special case of heavy atom hexafluorides. Since it was desired to adjust Qh and a by least aquares, the computer was programmed to caiculate the necessary partial derivatives. Ii Le? ig the M«F contribution to Io then «-2£0= oa “7 _8°/2 Lapep * = — cos An sinrse “~~ (11) 22 2 . mw -¢ g /20 : 1 =z A cos An sin +4 e* & /2 (using yoy * ay) gl 22 c on rat eo q /20; 5, . ars | Ta = A sin Ay sin =~ e (29a + 49b(q-q.)] (12) 81 22 wai = -2nlq- a.) sin An sin EG oe 7/20 (13) Another modification that was made was provision for semiauto-« matic recycling of the iterative refinernent procedure. At the end of every refinement cycle the computer stops on a breakpoint and the operator inspects the results which have just been printed out. If another cycle is desired, the operator so indicates and the computer automatically applies the least squares adjustment to each independent variable and calculates a new model. Thus the new value of Ye? is multiplied by ai/ 2 and by 2 to obtain the two new dependent bond distances r,. .. and r a? mee Then 0.005 A is added to Tay and a second model is calculated as required by the program for approxi- fet mating the partial derivative of the intensity with respect to Ofek by aN taking differences: ol, 7 Lt aor + 0. 005) » Lira.) 04) OF ee ~ 0. 005 “ The scale factor on the calculated intensity 1, is adjusted by multiplying every amplitude A, by the quantity (1+ 6 ) where scale «2Zl- § is the least squares correction to the scale factor. The other scale independent variables, (s;, de and a), are adjusted by simple addi- tion of the least squares corrections. A diagonal weight matrix P is used in the least squares pro- gram, with Pag z=(q- je” ¥9 . The same weighting parameter, y = 0. 0012 a 4°/100, was used for every compound at every voltage. Our weighting function is very similar to that of Bastiansen, Hedberg and Hedberg (14), who used P= ee” 0012s” it had originally been planned to adjust the background by least squares, but this was not attempted. (It was tried unsuccessfully in the refinement of N39, as described in Part II of this thesis.) Although preliminary theoretical curves were used as a guide in drawing the background, it became evident from ive i curves obtained during least squares refinement that the background sometimes needed further adjustment. Therefore, where necessary, a smooth background correction line was drawn by eye, using as a guide the i “ i curves, This adjustment to the background was not allowed to follow the fluctue ations in L, “ I), of course, but only its general trend over regions of 30-50 qeunits or so. After this final background adjustment, several further least squares refinement cycles were carried out until no significant change in the parameters occurred. Results of Least Squares Refinement Figures 3-7 show the final i Ls and Bel curves, labeled E (experimental), C (calculated) and E-C. The calculated curves are based on the parameters of Tables II-VI. These tables and figures are the results of the least squares refinement of the structures of WE, Oss, IrFy, UF ss and NpF ¢. (The structure of Buk’, is dis- cussed on page 44.) Our experimental data do not extend to small scattering angles due to the large beam stops which were used. The curves in Figs. 3-7 are drawn to cover that range in q which was used in the least Squares adjustment, except for NpF,, where the calculated curves are extended downto qg=0. The extra range for small q is shown for only one compound as an exarnple of the general effect on the diffraction patterns of ail MF, compounds of varying the accelerating voltage of the electrons. In Tables [I-VI are listed the best values and the standard deviations of six of the seven parameters adjusted by least squares. The seventh is the scale factor on I. The standard deviation is the external estimate, based on the assumption that the observations are independent. However, the observations are not independent; they are the amplitudes at integral q-values of a continuous experimental intensity curve. a aae Figs. 3= 9 Experimental (E), calculated (C}, and difference (E-C) curves for WEF 6 Osk 6 IrF 6 Ur 6 and NpF ¢- (The exact values of the accelerating potentials are listed in Table IX.) The arrows indicate the position of the cut-off point on each calculated curve. WF, 47 kev E C E-C WF, 40kev Fig. 3 C WF, |2 kev WF, 6kev 20 40 60 80 Fig.4 rele I20q -26- ON oo E-G IrF, 40 kev VAAVAVAVAVAS IrFe II kev 20 40 60 80 Tele) 120 40 q Fig.5 ~27- E C UFg 47 kev E | C E-C UF¢ 40 kev N nN Nn | CG E-C UF, Il kev | | | | | | | | ] | 20 40 60 80 100 I20 40 q Fig. 6 -2 8- Np Fe 47 kev E C E~-C Np Fe 40 kev \ Nn G E-C NpFg II kev | | a | | l | | | | 0 20 40 60 80 100 120 140 q Fig. 7 Tables ll - Vi The least squares values of the M-+-F distance r, the temperature parameters o, (i=12,32M-F, F-F, and F:-F, respectively}, the cut-off point Ges and the parameter a (defined in equation 9). In these tables, Fy Oss and a are in Angstrom units and de is in reciprocal Angstrom units. Note that the individual temperature parameters , are physically meaningless (see page 38). Fora summary of these tables, with limits of error, see Table XIII. ad 30~ TABLE II LEAST SQUARES REFINEMENT OF WE ¢ a “ @ 4 4 Ww & @ ® 4 49 G4 G& 8 ©! FO 8 g 4 2 ww = G p WE 6-47 kev WE p40 kev Best value std. dev. Best value std. dev. 1.836 6. 0007 1.825 0. 0006 44, 3 0.6 43.9 0. 4 6. 635 0. G04 G. G39 0. 003 0. 090 0. 008 0. 094 0. 0066 0. 043 0. 012 0. 065 0. 014 0. 0050 0. 0004 0. 0050 6. 0004 WE p- 34 kev WE ¢- 24 kev 1.836 0. 0007 1. 828 0. 6609 40. 3 0. 6 36. 5 0.5 6. 040 6. 004 6. O51 , 0. 004 6.100 G. 009 0.106 0. 010 0, 067 6. 018 0. 056 0. O14, 0. 0046 0. 0005 0. 0052 0. 0006 WE pak 2 kev WE 6-6 kev 1. 828 0. 0016 1.838 0. 0019 27.0 9. 6 19.3 6.7 0. 070 0. G04 0. 052 0. 009 0.129 6. COT 9.109 0. 009 0. 090 & O16 0. 085 6. 017 0. O03) 6. 0007 0. 60031 6. 6006 ig AST SQUARES REI ae a Ae 43 & woh & g Pa @& & ww s -3i- TABLE Il *j ge OsF .-40 kev Beat value 1.832 43.2 0. 044 0. 087 6. 070 ©. 0045 OsF p= 2 kev 1.830 26.8 0. 063 0.112 0. O73 0. 0042 INEMENT OF std. dev. 0. 0005 0. 4 0. 002 0. 006 0. O13 0. 0004 0. 0013 0. 6 0. 003 0. 008 0. 018 0. 0005 CaF, LEAST SQUARES REFINEMENT OF Irf, irPy- 40 kev Best value 1.832 42.8 0. 044 0. 093 0. 080 0. 0043 Irfy-)l key 1.829 26.7 0. 064 0. 093 0. 042 0. 0074 std. dev. 0. 0006 0. 0. 002 0. O07 0. 6. 0004 eo oo © 4 OL? . 6014 - G05 a 009 « 024 0. 0007 LEAST SQUARES REFINEMENT OF UF ein a a ay Q 82 8 a G & w mf & @ 49 4 6 &# wom eo & «330 TABLE V UP ,<47 kev Best value 1.999 35.9 0. 049 0.126 0. 073 0. 0048 UF 67 40 kev 1.995 30. 6 0. 649 0.133 0. 073 0. 0048 UF ¢~ ll kev 1.994 22.90 0. O71 0.118 0. O74 0. 0078 6 std. dev. 0. 0005 0.8 0. 006 0. O11 G. 015 0. 0008 - 0007 - 006 - O13 - O18 - 0007 ao Oo Oo 89 82 28 - 0016 » 009 - O13 . 026 - 0008 oo eo 92 2 8 = 3he TABLE VI LEAST SQUARES REFINEMENT OF Np, NpF gra? kev Best value std. dev. x 1.983 0. 0008 a. 35.9 1.1 oy G. O54 0. 005 oe O.L17 0. 015 . 0. O77 &. 026 a 0. 0042 9. 6010 NpF ,-40 kev r 1.982 0. 0006 qo 32.8 0.6 , 0. 653 , 0. 005 v3 6.128 0. 019 en G. 099 9. 020 a 0. 0051 0. 0006 NpP -11 kev r 1.977 9. GGLS Go Zo. & 0. 6 o; G, 069 0. 008 T> 0.109 0. 012 v3 0. 077 0. 027 a 0. 0078 0. 0009 In order to correct for this lack of independence in the observations, the estimate is made in Part Il] that the true standard deviations are larger than the ones obtained by least squares by a factor of roughly (20/ 3r)'/ 2 where r may be taken as the M-¥F distance in the present compounds. Since r is about 2A we would estimate that the cor+ rection factor for the standard deviations is roughly 18 or 2. Of course systematic errors will also increase our estimates of the standard deviations. The conclusions about the error estimates are discussed later. Table VIL (page 37) gives for each set of data the value of a= [ LIPL/(n-m)}/2, the mean error of an observation of weight unity, (not to be confused with s = (40/d) sin (0/2) ) where L'PL is the sum of weighted squares of residuals, m is the number of obser- vational equations and m is the number of adjustable parameters. Thus s is a measure of the goodness of fit of the observed and calcu- lated curves. However, there is an arbitrary but different amplitude scale for each set of data, and s inthe table is based on this arbitrary scale (which corresponds to the scale in Figs. 3-7). Therefore s was put on a standardized scale for all sets of data, andis tabulated s' in Table VIL, corresponding to Aypr 1000 in equation 10. A rough estimate based on s' can be given of the error in i expressed as a fraction of the amplitude of a typical ring. Im the case of WE 6740 kev, for example, the maximum of highest amplitude in the photometered pattern is near q 280. If Ayer 2 1000, this maximum has an amplitude of about 520. A measure of the expected error in the observations is atp7i/ 2 where p is the weighting function (page 21). = 360 In the region q = 40 to 100, p takes on values between 30 and 40, so the expected error in the observations is about 7% to 9% of the height of the maximum at q= 80. Of course such estimates of error are very sensitive to any background corrections, (because of the depen- dence of L'PI. on changes inthe background). Although s' has the same order of magnitude for each curve (the range in s' covers barely more than a two-fold spread), the agreement among the curves would probably be improved if further background corrections were attempted. a3 fe TABLE VU MEAN ERROR OF AN OBSERVATION OF WEIGHT UNITY s s! {arbitrary scale) (standardized scale) WE ¢-47 kev 16} 320 40 135 240 34 136 280 24 117 279 12 163 147 6 117 : 143 OsF ,-40 109 192 12 i186 218 IrF ,-40 136 241 ll 176 321 UF ,-47 122 163 40 155 232 11 185 306 NpF ,~47 39 193 40 167 195 ll 217 322 Temperature Parameters it has been mentioned several times that we cannot determine the absolute temperature factors. Thie is the case in visual work alao, where in drawing a visual curve an effort is made not to show the de- creasing amplitudes of the diffraction rings with increasing angle of scattering. The visual method is inherently unable to reveal absolute temperature factors because comparisons of ring amplitudes can be made only among two or three neighboring rings. Although this limi- tation is no longer inherent in the sector-microphotometer method -- amplitudes can be compared for all the rings in a pattern -- this limi- tation still exists in the present investigation primarily because the correction for the non-linearity of the photographic ermnulsion could not be made. Other steps in our procedure which obscure the absolute temperature factors (but could be corrected were it not for the basic factor -- the inability to correct for the photographic emulsion) include division by a somewhat uncertain background and the use of constant coefficients Ay in equation 10. | However, ag inthe case of visual work, we can still say some- thing about relative temperature factors. Equation 4 can be rewritten as follows: 2 2 -o s“/2 6C,, .. sl(s) =e M-F [ at cos Ay sin rs , ra a 2 12 Co an =O, are ja" /2 + = sin -/2 rs @ Bee M-F Ve ¥ . 2 2 3Cp. Ep (on, Oh ple /2 i. sin drse “ } (15) 2 2 and one would anticipate better results for (o2, p°o pp) and for (o, , er pF) than for the individual temperature parameters Tree Cope OF Gop Therefore in table VIII are listed both the individual temperature parameters o, and the relative temperature parameters (deed, ah where a, = 3 o {often called ay in the temperature factor expres- ~as:8% sion e” “4j§ ). The agreement among relative temperature parameters of the same compound at different voltages is reasonable when the standard deviations of the individual temperature parameters are considered, In Ir grt kev, d.-d, is negative (indicated by * in Table VIII) and therefore evidently in error. -~40- TABLE VII TEMPERATURE PARAMETERS The subscripts 1, 2, 3 referto M-F, 47 0. 035 9. 090 0. 043 G. 0034 0. 0003 0. 049 0.126 0. 073 0. 0067 0. 0015 0. 054 0.117 0. O77 0. 0055 0. 0016 40 0. 039 0. 094 0. 065 0. 0036 0. 0013 0. 044 0. 087 0. 070 6. 0028 0. 0015 0. 044 0. 093 0. 080 9. 0034 0. 0022 0. 049 9.133 0. 973 0. 0067 0. 0015 - 053 -128 090 - 0067 - 0026 oo 82 9 8 34 0. 040 8. 100 0. 067 0. 0042 0. 0014 EF, FF 24 0. O51 9.106 0. 056 0. 0043 9. 0029 » respectively 12, li 0. 6. - 090 - 0034 - 0016 oo oc © oo 3 Oo 2 ©. © 0. 0. 070 129 063 ~112 073 . 0043 - 0007 . 064 . 093 042 . 0023 x - O71 118 - O74 - 0045 - 0002 - 069 - 109 O77 0035 0006 6 kev 0. 052 6.109 0. 085 0. 0046 0. 0022 Systematic Errors The standard deviations reported in tables I]-VI make no allowe- ance for systematic errors. By systematic errors is meant those which have a non-random effect on the observation equations. Probably the most important systematic error is that in the wavelength deter- mination, because for the present least squares procedure a simule- taneous measurement must be made of the q-value and amplitude of every point in the diffraction pattern. It is assumed that the gq-values are known exactly and that errors exist only in the amplitudes. This is a fairly good assumption provided a separate allowance is made for the uncertainty in the wavelength (and the resulting error in the q-scale). The coordinate q depends on \ through q = = sin ae and is determined from the accelerating voltage of the electron beam. The voltage is monitored by a potentiometer connected across a voltage divider. The potentiometer was calibrated with ZnO pictures made at 47, 40 and 34 kev. Zinc oxide pictures made at 24 kev were not used for calibration because their rings were broadened to the extent that measurements were very uncertain, At 12 and 6 kev, the rings were so broad that measurements could not be made at all. The potentiometer reading £ and the quantity 7 determined _ from the zinc oxide pictures at camera distance L are given in table IXa. TABLE IX WAVELENGTH CALIBRATION a. Determination of K = V/f from ZnO data (volts) Wid V paler) K opal’) 1. 4000 1. 8832 47, 250 33. 750 0. 05516 1.1837 1. 7210 39. 740 33.573 0. 06036 1. 0000 1.5742 33. 488 33. 488 0. 06599 E (best value) = 33. 600 b. Determination of } from Ef = V x {(voits) V (kev) V appr ox Reale) discrepancy 1.4000 47.040 47 0. 05529 0. 230%, 1.1837 39.772 40 0.06033 —0. OB. 1. 0000 33, 600 34 0. 06583 0. 249, 0. 7000 23, 520 24 0. 07906 0. 3500 11. 769 12 0.11252 0. 3413 11. 468 il 0. 11389 0.1750 5.880 6 0.15947 te V approx in the text and figures. is the value of the voltage rounded off for quotation . : 1 : ; The "observed" potential V obs was found from TR using Ls 9. 627 cm as measured with a cathetometer, and \ = h/ [2m eVieev/ 2m 5o)) , the relativistic expression for the electron wave length. Thus the proportionality constant K = could be determined from Zn0O data. its average value, EK = 33, 600, was used to determine both the best values of the three highest potentials and alse the values of the lower potentials for which ZnO data were not obtained. Intable IXb, ik was determined from V = Rf and this value of \ was used to fix the absolute scale in all of the molecular structure determinations. The discrepancy between the observed and calculated values of X for the three highest potentials is less than 0. 24% which is slightly higher than one might expect, since measurements on two different ZnO plates at the same voltage agree to within 0.1%. However, our value of \ differs irom that of several previous workers in this laboratory, who found \ = 0.6056A for the potentiometer reading £21.1837v. This discrepancy of 0. 36% in two independent determina- tions of A may be largely due to uncertainty in the camera distance. This writer estimates that 0.4% is a reasonable value for the systematic error inthe scale of the molecules due to uncertainties in A. It ie interesting to note that the M-& distance as determined at several different wave lengths shows good internal consistency with an average deviation of 0.2% for WE 6 and 0. 1%. or leas for the other compounds. Certain types of background errors might be called systematic and others random. However, itis very difficult to analyze the effect of background errors. Several generalizations are discussed in Part UI. whim Other errors which may be called systematic are those due ta uncertainty in the sector calibration, lack of correction for the non~ linearity of the photographic ermulsion, and the various approximations used in the calculation -- the approximation that the coefficients Ay of equation 10 are constant and the use of equation 9 to calculate Ay. jowever these errors will have little effect on the scale of the mole- cules. The main source of systematic error inthe M-F distance is probably the uncertainty in A. Therefore 0. 4% will be used in esti- mating limits of error (see "Conclusions"). Plutonium Hexafluoride Because of the poor quality of the Puk ¢ pictures, least squares refinement was impossible on the 47 and 40 kev data, and only partially successful on the ll kev data. The 47 kev experimental curve is shownin Fig. 8 From the re- sults on UF 6 and Np 6 at the game voltage, one would expect the cut- off point to be in the neighborhood of 30-36 q-units. Thus we may refer to calculated curves in which q, = 30. 6 (UF ,-40 kev), and qo * 36.1 (UP .-47 kev) in Fig. 6 In the experimental PuF ¢ curve, the nearly symmetrical shape of minimum 6 and the relatively large excursion from maximum 6 to minimum 7 (which is possibly in error) favors the higher value of q.: In order to find q, by the correlation method, comparisons should be made among the amplitude of the 4th, 5th, aad 6th rings (see Fig. 7 where the theoretical curves extend to smaller scattering angles}. Thus the experimental Pur a7 4! kev curve is come patible with qd. 3 30 to 36, but the lack of data for the 4th ring prevents a correlation treatment. The excessive amplitudes of maximum 6 and #450 minimum 7 and the peculiar appearance of the 14th ring seem to indi- cate large errors in the experimental curve in these regions. The absolute amplitudes of "maxima" and "minima" on the photometer trace are small compared to those of the other compounds, and it is impossible to get a good photometer trace. The experimental curve for PuF ,-40 kev (Fig. 8) is also poor. Because there are no data below gq = 40, about all that can be said is that Ge is less than 40. The unexpected shelf at q=52 is not due to the troublesome 1.4 cm region of the sector because this shelf corres- ponds to Rels4 cm. Maximum 6 and minimum 7 have excessive amplitudes in this case also. The Pu-F bond distance was found by comparing the positions of maxima and minima of the 47 and 40 kev experimental curves with those of the corresponding calculated curves for NpF ¢. These results are given in Tables X and Xi. | . The results of the least squares refinement of Pury-l.kev are shown in Fig. & and Table XIi. The best theoretical curve, ¢, is in satisfactory agreement with the experimental curve, E, from the standpoint of the visual method: the height of maximum n can in general only be compared with the average of the heights of maxima (n+l) and (n-1). However, the difference curve, E-C, leaves much to be desired. The resulting parameters are much more uncertain than those of the other compounds. The temperature parameters are espe- cially unreliable -- the least squares adjustment to Ty, makes the F temperature factor for the long F-> F distance an increasing rather than a decreasing function of q. However, in the calculated curve, ‘ or, FP * zero was used. ~45— Fig. 8 Experimental curves for Pur, at 47 and 40 kev; and experimental (5), calculated {C) and difference (E-c) curves at 11 kev. -47- PuF, 47 kev E PuFg 40 kev VM" I\A : V Cc I E-C PuFe It kev | | | | | | | Lt | | | | | | 20 40 60 80 100 120 40 q Fig. 8 TABLE X ELECTRON DIFFRACTION DATA FOR PuF .-47 kev Min. No. “obs V4555 5 35.18 (0.971) 6 42, 76 0.995 7 53. 84 0. 998 8 63, 44 0.998 9 73. 77 6.996 10 83. 72 0.997 il 93.13 1.004 12 103. 62 1. 000 13 113. 75 1.000 14 124, 62 0. 999 15 134, 54 0.995 16 145.83 0. 987 Average 23 features 0.997, Average deviation 0. 004 Poy. * 1.983 Ax0.997 = LOTTA & “obs 37. 83 48.29 58.18 68. 64 78. 21 88.15 98. 05 168. 56 11S. 44 128. 64 139. 38 150.14 Max. Wone 0. 994 0. 986 G. 999 0. 992 1. 601 1. 602 1. 005 1. 001 1. 064 1. 000 0.997 0.992 Oo a GE wi 9 16 il 12 13 14 Average, 16 features Average deviation r Soba he. 54 54. 06 63, 63 73. 67 $3. 01 93.17 262, 61 113. 44 b 24. 38 1. G07 0. 986 0. 992 0.992 1. G63 1. 000 1. 007 8.999 @.993 0. 998, 0. 006 Dyes 7982 Ax 0.998 = LOTBA 98. O1 168. 47 116. 29 128. 96 Max. Wong (0. 960) 1.003 8.990 0. 989 0. 996 1. 002 1. 002 0. 999 1. 009 0.996 LEAST 8Q TABLE XI ARES REFINEMENT OF Pulg-il kev Best vaiue std. dev. 1.9568 A G6. 0026 21.6A7: 1.0 0.031 A 6. 026 6. 043 A 0. G15 G6. OL11 A 9. 0004 ~ 5 |e Evidence for Octahedral Structure Radial distribution (RD) curves were prepared from diffraction data at 40 and 11 or 12 kev for all compounds. Preliminary theoretical curves, and later final ones, were used to supplement the experimen- tal data for the inner part of the pattern. These RD curves appear to furnish good evidence for the octa- hedral symmetry because the doubled M-F peak is explained by the phase shift, whereas the two F«F peaks have the expected appearance and position for a regular octahedron. This evidence is somewhat weakened however by the fact that the inner parts of the experimental intensity curves do not extend to as low a value of q as would be desired, and therefore the RD curves are heavily influenced by the inner rings of the intensity curve calculated assuming an octahedral model. Thus although it is common practice to supplement the first one or two rings with a calculated curve, it proved dangerous in the present case. It was necessary to supply from calculated curves for the 40 kev data three rings and sometimes part of a fourth. Perhaps more convincing results could be obtained by using experimental in- tensities alone, but multiplied by a modification function to smooth out the ripples in the RD caused by omitting the inner rings. Both RD and experimental intensity curves clearly show that the cut-off occurs at a lower value of q; the lower the accelerating voltage, that is, the longer the electron wavelength. Such a dependence of the cut-off on the wavelength (which was observed for UF, by Hoerni and Ibers (9)) eliminates any possibility that the anomalous diffraction pattern is due to an asymmetric structure. The fact that we now have ~52~ intensity data extending to 34, without showing a second cut-off is further evidence against the structural asymmetry. It is also evidence that Arn = és can hold only as a rough approximation, and that the conclusions of Ibers and Hoerni are correct as to the general shape of the Ay curves (see Fig. 2). The experimental curves seem to be completely compatible with a symmetrical octahedral model for all the compounds (with the reser- vations noted above for Pur ¢). There rernains little doubt that all six M-F bonds are of equal length, but it is hard te determine the F-F distances because of the relatively low scattering power of F. How- ever, to the degree that the F> F and F--F distances do not fit the data these terms will be effectively eliminated through being assigned large temperature parameters in the least squares refinement. That the temperature parameters are reasonable and do not approach in-~ finity is evidence for at least an approximately octahedral structure. Conclusions The results of this investigation are summarized in Table XII. For each compound, the average of the values of the M-F distance r obtained by least squares from data at different voltages is given as the best value. Presumably a weighted average would be better, but the large systematic error mentioned below would make the weights nearly equal. To find limits of error one must consider both the least squares standard deviations and the systematic errors. The least squares standard deviation of r, (multiplied by a factor of 2 to allow for lack aw 3a of independence in the observations), ranges from 0. 0010 to 0. 0052 A for different compounds at different voltages. On the other hand, the systematic errors are essentially 0. 4% due to uncertainty in 4. The total standard deviation s is given by | Pa = a” + Pt total least squares systematic assuming that the covariance of the two errors is zero. Thus ® otal is leas than 0. 0090A for ail compounds. | It has been estimated (16) that a reasonable limit of error is twice the standard deviation. This criterion would give 0.018 A, as the Umit of error for r, and this has been rounded to 0.02 Ain Table XIIIa. In Table XIIlb the experimental and theoretical cut-off pointa Io are compared. The systematic errors in Ye {estimated to be 0. 4°. here also) are very small compared to the standard deviations obtained by least squares. Therefore the least squares standard deviation was multiplied by a factor of 2 to correct for lack of independence in the observations, and by another factor of 2 to convert standard deviations to limits of error. Thus 4(s } is reported in Table XIIIb least squares as the limit of error in Ge The theoretical Gq, were obtained by graphical interpolation from the table of 4n given by Ibers and Hoerni (13) and the approximation formula (equation 8). Their table does not extend high enough in atomic number to calculate de for the lower voltages by the approximation formula. For UF, kev the theoretical q. was obtained from an earlier paper by Hoerni and Ibers (9). THE STRt UCTURES 0 540 TABLE XII OF SIX HEAVY METAL HEXAPLUORE DES a. M+ bond distances (r), and temperature parameter (a, Ey of) Compound r WE, 1.833 A OBk ¢ 1. 831 ir, 1.830 UE, 1.996 NpF , 1. 98] PuF , 1.971 WE 6 Os k 6 Fuk 6 limit of error 0.02 A 0. 02 0. 02 0. 02 0. 02 0. 02 d5- a, 9. 96039 0. 90035 8. 0029 5. 0063 0. 0052 od ave. dev. daed, aye. dev. 0. 0005 0. 0008 0. 0006 6. 0912 0. 9012 Cd So oo SG 8 ©& - 0012 - 0011 2 001) - 0011 . 0016 Se ow. b. Sxperimental and theoretical cut-off points 47 kev 40 34 24 12 6 40 12 40 il 47 40 11 47 40 ll 47 40 il Exp. 44.3 A 43.9 40. 3 36. 5 27.90 19. 3 43.2 26. 8 42.8 27. 6 35.9 30. 6 22. 0 35.9 32. 8 22. 2 on tt we 21.6 mit of error + ood 2.4 a7) bat oe NNN? BO Ob Om ° PPS NET Pe VS & ry ft pL bh AAG OM ibe ot 0. 0006 - 0004 901) . 0006 - 0007 So 2° 3 & > o Theor. 49.5 An 43.7 40.9 37.06 +55 The present investigation of UP, completes the preliminary work of Hoerni and Ibers (9) on this compound. These authors found fairly satisfactory agreement with visual curves obtained from 40 kev and Li kev photographs by using a symmetrical octahedral model with following parameters: ferig 2 O49 A, dy, gr “ad co 2. 2x107° A’, U+F dee e wrdyy an ® Oe qsx107> Ae. They also used their complex scattering | amplitudes corresponding to the theoretical cut-off points in Table 211. Chr vegulte seam to agree well with these earlier ones at 1] kev, but our calculated curve for 40 kev is in better agreement with both our sector data and the visual curve of Dr. Otto Bastiansen which Hoerni and Ibers tried to fit. This indicates that the theoretical cut-off point ig too high in the case ok Ur,-40 kev. | Our results on WE ,-40 kev are in satisfactory agreement with those of Nazarian (19), who found Tay op F 1.824 A, +3 ae é d : * 3 «i q, * 42,9 A°° by the correlation method with visual data. The limits of error on x and gq. in Table ¢ HI are believed by this writer to be reagonable estimates, and the Hmits of error on the temperature paramcters are probably somewhat higher than the average deviations in Table XII, il, THE MOLECULAR STRUCTURE OF GASEOUS N,O introduction and Theory The expression that describes the scattering of fast electrons ae by gas molecules is K; an . ea..g’ 31D r,.8 Wes) = Qi B+ ) (Z-£),(Z-£),e°°j : 2 ), (1) 8 “we ij ij where i'(s) = intensity of electrons scattered at angle @ KK = constant = >! (Z-8)? + w Pacenst 1 li incoherent scattering function (15) is] ul atomic number f = atornic scattering factor for M-rays s = (497 sin SVs, X is the deBroglie wave length ij i) Ty = distance between atoms i and j The term EB represents the purely atornic contribution to the dif- fraction pattern, while the surmmation {in which the prime indicates that terms with i= j are not to be included) represents the molecular contribution. In the intensity expression for the compound N29 » (2-f), is very nearly (2-52 /Zy Therefore, we may write * 7 os . The factor cos Sr discussed in Part I need not be included here --+ nitrogen and oxygen are both light atorns. ~ os Pe Ke ~~. Paf . Is) = Sib + loMell, (2) S with 2 wi Zi, 7a, .6 I{s} = 3) =~ e sin ¥ 453 (3) ij 4 The structure sensitive expression si{s) is used in the approximation to the radial distribution integral (RDI): Smmax rD(r) =) ei{s) sin rs. (4) s=0 Solving (2) for si({s) we obtain e's) = (s°1(6) - Bis/(Z-£)4 G) in which we have neglected any constant which affects only the scale intensity. Reduction of Microphotometer Traces The data consisted of microphotometer traces of electron dif- fraction photographs obtained by the sector method. There were three sets of photographic plates corresponding to three camera distances of approximately 49, 19 andlZcm. The plates were oscii- lated about the center of diffraction pattern during the process of being traced by the microphotometer in order to smooth out granularity and imperfections of the plate. In spite of the oscillation, the traces still showed high frequency fluctuations, se a smooth curve was drawn by hand through each trace. It was desired to obtain intensity data at integral gq-values (q = 10s/7), s0a g-scale was marked directly om each photometer trace. Densities were obtained from the microphotometer traces by the relation -log T =f, which defines the photographic density, D, in terms of the transmission, T, which is proportional to the de- flection of the recording needle. it was desired to convert photographic densities into relative electron intensities. At low exposures, say upto E', the relation between the exposure, E, andthe density, D, is linear for practi- cal purposes. At exposures above E' a plot of D vs. E deviates frorm linearity as is shown schematically inthe diagram. The reci- procity law may be assumed valid: The exposure is the product of intensity andtime. Therefore two plates differing only in time of exposure will have an approximately constant ratio of densities measured at the same q's, provided both density ranges fall on the linear part of the D-E curve. The ratio will begin to deviate, how- ever, as the exposures onthe heavy plate exceed E'. Typical plot of density vs. exposure ~59- Karle and Karle (11) have discussed the calibration of photo- gxaphic plates in electron diffraction and the present method is based on theirs. For each set of plates (a set being those made at one camera distance) a table was made of the ratio R= D,/D, of the density D, of the heaviest plate to that (2)) of the lightest plate for each q. Then. R was plotted against Dy and Do as shown in the diagram: Rt R, — oN / “ Ul _» D D D Da: . / 7 D D D7 Plot of vatios of densities vs. densities The point Eo Dt where ER begins to deviate from a constant value Ry represents the upper limit of the linear portion of the DeE curve. Since we are interested only in relative, not absciute intensities, we may let D= iy the electron intensity at the plate, for values of D upto D'. Inthe range D,=D' to Ds DM", the values of R may be used to correct Dy, toan electron intensity scale because Dy is stil on the linear part of the D-E curve. Im the above range, letting i 1 ~60« be the value of 1, on plate 1, we have the relation I , ft Bf eR I = RI = mee [> (6) p, 1 a Thus. R/R, was plotted against Dos giving a calibration curve good upto De DD", Then a was corrected over the range D, = >" to BY", The whole process was repeated as many times as was necessary to convert D into I, for both plates; once or twice was sufficient. The calibration curve was then used to obtain I, for the other plates in the set. 1, was then averaged point by point for all plates of a set to obtain the final I, curve for each camera distance. The above procedure certainly does not taxe into account all of the factors involved in the conversion of density into intensity. Among other things a complete treatment would consider the following vari- ations from plate to plate: the time of development, the amount of gas in the nozzle region of the camera, the strength of the electron beam, instrumental background, and chemical fog. Inside the electron diffraction camera a rotating sector modified the exposure in such a way as to partially compensate for the steeply falling background. Accordingly, 1 p must be divided by a, the angu- lar sector opening. Since the plate is flat, not spherical, a correction must be made for the increased distance of the outer parts of the plate as well as its angle of tilt with respect to a diffracted ray. Thus, assuming a thick emulsion, 1, must be divided by a factor of cos §@ because of the angle of the plate, and a factor of cos” @ because of the increased distance. The above corrections lead to the expression i (s) rts), 3 3 E = 2 (=-) =e (7) 1s) = : al(s} cos” @ 8 als)’ xr cos z where the factor r°/als) appears because, due to the shape of the sector, alr)/r° or its reciprocal was a convenient representation of the experimental calibration of the sector. Here, r is a radius of the sector. Ignoring a constant factor, one obtains the relation 3 =~) 3 (8) als)‘ cos”@/2 4a ol =e sia) = et (2)( Equation 5 may be written ali (s 3 8) = B a (cea) “| e-93 " al(s} = cos°6/2 The background term E was calculated by the theoretical expression, and, when multiplied by an empirical scale factor, could be used in subtracting the background for the 49 cm. plates. The backgrounds for the other two carnera distances, however, could not be fitted by a calculated B function, so a smooth background was drawn through a plot of 8“1'(s) in such a way as to divide the curve into approximately equal positive and negative areas over a range of s corresponding, say, to several major features. Then the background was subtracted out and the result was multiplied by s/ (2-098 . Presumably it would have been better to divide by (2-896 before making the above sube traction. - G2= Having obtained si(s) with a different arbitrary scale for each carnera distance, it was desired to join the curves together. There was considerable overlap between the three curves: The 49, 19 and l@écm. plates covered q ranges of 4-60, 25-150, and 73-195 respec- tively. Two of the curves were multiplied by an appropriate factor to put all three on the same amplitude scale by taking ratios of ampli- tudes near maxima and minima inthe overlap region. The curves were then averaged together in their overlap ranges to obtain the final experimental sl(s}) curve. This curve is analogous to a temperature factored visual curve. The first peak, from qg=0 to q=5 was drawn in by hand. Analysis of Data The RDI was obtained from sl(s) in the usual way. It has six maxima which can be interpreted in terms of a planar Do. structure: an —NeO 1.1794 0.002A LA77A NN 1. 755 + 0. 005 1.750 N N Oye+O, 2.173 + 0, 005 2.175 Ne+:O 2, 455 + 0. 005 2.454 Oy Oy, 2. 655 + 0, 005 2. 650 d 5 Oye e+ Oy, 3,430 + 0. 010 3. 428 & 2! The above set of parameters which were read from the RDI are cone sistent with a planar model to within 0.005 Ain all six bond lengths. A confirmation of the above assignrnent of peaks can be given in terms of peak areas in the RDI provided an allowance is made for a certain concentration of NO. Comparison of calculated (2,25/r,.) 636 and measured peak areas lead to the value 2Z mole %, NOs by taking the average of the relative excess areas of the measured N-O and Or Og peaks. The following table shows the agreernent between calcu- lated and measured areas, where 22% NOs has been included. Peak Calculated Measured %. Discrepancy Area Area {relative scale) N-O 217 227 $49) N-N 28 27 a4 4 eee C é ew Oo} Oz 67 64 5 Neer © 91 94 +3 Oy ve On 48 49 +2 Or ee Op 37 36 «3 The discrepancy can be accounted for as being due to the uncertainty mtg the wes in the base line o o Assuring gaussian peaks, the peak half width, wy /2 is related to the vibration parameter a by the expression w,, = 4{In 2) The half widths were measured and the following values of a were obtained: 0. 0008 (N-©), 09.0033 (N-N), 0.0011 (O)--+ 3), 6. 0025 (Nee O), 0. 0052 (Oy os Oras 0. 0025 (oO; ee S5,)- The value of aniN Was believed to be too large, so a smaller value was used in calculating preliminary theoretical curves. «~64= The intensity curve for the best preliminary (planar) model, the experimental curve, and the RDI curve are shown in the Letter to the Editor of the Journal of Chemical Physics on the next page. This roodel was refined by least squares as described in the next section. The refinement was restricted to conserve the syrmmetry Do),7mmma. Least Squares Refinement The least squares program for the Burroughs 205 cormputer de- scribed in Fart I] was modified so that the concentration of NO» in the sarnple of N3O,4 could be adjusted as a parameter. Thus the approxi- mate curve contained, in addition to the terms representing the struc- ture of No Oy the icllowing two terms” vl SETRE exp(-1. 0x10" *s“)ein 1. 187s + °F exp(-1, 3x107"s“)sin 2. 1825] with a composition parameter w corresponding to 22 mole percent NO>- During the refinement w was adjusted by least squares but the other NO, parameters were not varied. Another modification in the program was the provision for auto- matically repeating the least squares cycles in a manner analogous to that described on page 20 for ME compounds. A sub-routine was in- cluded in which the three long distances of NO, were calculated from the three short distances (N-©, N-N, and Oy ve 05). The original experimental intensity curve was used in the early cycles of least squares refinement in which the following parameters * These NO, parameters were supplied by Dr. Hedberg (17). Reprinted from Tue JourNaL or CuemicaL Puysics, Vol. 25, No. 6, 1282-1283, December, 1956 Printed in U. S. A. Molecular Structure of Gaseous Dinitrogen Tetroxide Darwin W. Situ, Gates and Crellin Laboratories on Chemistry,* Pasadena, California AND KENNETH HEDBERG, Depariment of Chemistry, Oregon State College, Corvallis, Oregon (Received October 5, 1956) HE N.0, molecule in the crystal has been shown! to be coplanar and to consist of —NOz groups joined by a very long (1.64 A) N—N bond. The preliminary results of our electron- diffraction investigation of N2O. reveal that the molecule is also coplanar in the gas phase but with an even longer (1.75 A) N—N bond. Excellent diffraction photographs were made in the new Nor- wegian apparatus? using.a rotating sector, 34-kv electron ac- celerating potential, and a special nozzle in which the gas sample was cooled to —20-+5°C. The emergent gas at the scattering point apparently contained less than about 25% NOk. Plates from three camera distances (12, 19, and 48 cm) were carefully selected and microphotometered while being oscillated to reduce the effect of graininess; the resulting traces led by the usual methods to an experimental intensity function (Fig. 1) for the large range g=5—195 (g=107—s=40"! sinv/2, where \ is the electron wavelength and » the scattering angle). The radial distribution curve (Fig. 1), obtained by Fourier inversion of the experimental intensity function without use of a modification function, has six well-resolved peaks which corre- spond to within a few thousandths of an angstrom to a coplanar model having the distance values shown in the diagram. The theoretical intensity curve (Fig. 1), calculated for pure N2O, using these distance values and the vibration parameters 10‘a = 104(51;;2)4y/2 = 9 (N—O), 22 (N—N), 11 (O1- + -O2), 25 (N---O), 36 (O1+- -Ov’), and 25 (O;-+-O2’), is in very good agreement with the experimental curve over the entire range. If allowance is made in the calculations for presence of NOz, the agreement is even better. The N—O distance and O—N—O bond angle in gaseous N20, are about 0.01 A longer and 8° larger than in the crystal; indeed, the dimensions of the —NO2 grouping are very close to those of NOs. itself: N—O=1.188+0.004 A, 2O—N—O=134°4'+15'3 The weakness of the N—N bond expected from the chemistry of N2O, is in accord with its extraordinary length (0.27 A longer than the Schomaker-Stevenson! covalent single-bond radius sum) and its large vibration parameter, which is in fact comparable to those found for nonbonded interactions through one bond angle. The coplanarity of the N20, molecule is remarkable in view of the long, weak N—N bond. An attractive possible explanation is that 6 20 40 60 60 10 120 40 160 180 200 q 6 i 2 3 4 - BA Fic. 1. Electron-diffraction curves for NzO4. Curve A, experimental intensity; curve B, theoretical intensity; curve C, radial distribution. The vertical bars beneath curve C correspond to the interatomic distances shown in the diagram and have heights proportional to the weights of the terms. The dotted part of curve C is part of a second calculation made necessary by our punch-card system in order to obtain points beyond about 3.5 A, the bond is largely of w- rather than o type; the length and weakness of the bond are reflections of the poor overlap. properties of the orbitals in the bond direction. We are currently refining our structural results; however, we do not expect the distances to change by more than a few thousandths of an angstrom. We are greatly indebted to cand. real. E. Risberg for design and construction of the low-temperature nozzle, to cand. real. A. Almenningen for preparation of the photographs, and to L. Hedberg for preparation of the figure. One of us (K.H.) wishes to express his gratitude to the John Simon Guggenheim Founda- tion and to the U. S, Educational Foundation in Norway (Ful- bright) for fellowships during which part of this work was carried out. He is especially grateful to Professor O. Hassel and Professor O. Bastiansen for many kindnesses during his stay in Norway. * Contribution number 2137, 1J. S. Broadley and J. M. Robertson, Nature 164, 914 (1949). 2 Bastiansen, Hassel, and Risberg, Acta Chem, Scand. 9, 232 (1955). 3G, E. Moore, J. Chem. Phys. 43, 1045 (1950). 4V. Schomaker and D. P. Stevenson, J. Am. Chem. Soc, 63, 37 (1941). ~66- were adjusted: the three independent distances, the six temperature parameters, the overall scale factor, and the compOsition parameter. Throughout the refinement, the weight function p = (g-lexpl-0. 0012 nq" /100) was used. The resulting difference curve EC (representing the difference between the experimental and adjusted theoretical curves) indicated that the original background was too high in the regions q = 10 to 40, and 65 to 100, and too low in the region q = 120-180. An attempt was therefore made to find a suitable background correction by least squares. It is to be expected that the background curve which was drawn visually during the data reduction may contain errors. However the background errors should be “smooth” since the background curve was drawn with much less curvature than the fluctue ations due to molecular scattering. One hopes that there will not be any Fourler components of significant weight in the background curve which correspond to molecular distances. Therefore a plausible type of background correction AB might be a cosine series having only very low frequency terms. The form chosen was AB « S b cosa (niq/200), fu be n=l 4, the coefficients b, to be adjusted by least squares to fit (E ~ AB) and C. Only nine parameters could be adjusted, since the program provides for a maximum of 20 adjustable parameters and ll molecular parameters were already being used. Such a form for AB has the essential property of not changing those details of the experimental curve which are important in structure determinaticns. Thus the lowest frequency component in the experimental curve expected from molecular scattering corresponds tothe L18A N-O bond. This come- ponent has a period of about 17 geunits. The highest frequency in the ~67« above expression for 4B has a period equal to 45 q-units and there- fore such a background correction will have a negligible effect both on the shapes of maxima and minima, and on the amplitudes of rings as measured from a maximum to neighboring minima. Although there does seem to be hope for finding a suitable method of adjusting the background by least squares, the SB which was ob- tained by the method described above was felt to be unsatisfactory. It did fit the E-C curve fairly well in the region where the latter was especially poor, but ripples were introduced making the background worse in other regions. The writer felt that a better background correction curve could be drawn by hand. Therefore, a smooth AB curve was drawn in the manner described on page 21 for the MP, compounds. This AB curve was subtracted from the original experimental curve and the result is shown in Fig. 9, curve E. The experimental intensities of the inner- most part of the pattern, q = 5 and 6, are not shownin Fig. 9 and wer not used in the succeeding least squares refinement because of their large residuals and the high uncertainty of measurements at such very small scattering angles. The observations covered the wide range q = 7 to 195. Curve Cy in Fig. 9 represents the final calculated curve, ad- justed by least squares. It corresponds to 7 mole percent NOs where- as curve Co corresponds to pure N3° A with the Same parameters as those for Cy As one would expect, the 7%. NOs has little effect, and both Cy and Co fit the experimental curve very nicely. The para- meters of N204 are listedin Table XIV. Aliso included are the least -58~ squares estimates of the standard deviations, which are probably teo small (because of lack of independence in the observations; see Part Wil) by a factor of about (20/3 x 1.2) )'/* = 2.4, In addition the wave- length uncertainty for the Norwegian apparatus was estimated to be 0. 2% (14) introducing a possible error of this magnitude in the scale of the molecule. Combining these two effects (see page 53) and multi- plying by a factor of two gave the limits of error in Table XIV. Calculated, including 7°/o NO, Calculated, pure Nj,O, Internal rotation, r.m.s. amplitude 20° Internal rotation, r.m.8. amplitude an? sm Experimental Cy C2 EC, Difference Rag Rag internal rotation, free Critical marks Curve Mark Ro 3 Dot Det circle Rag Cross Meaning Creation operator; 6 max. too low; 10 min. too lew, etc. Destruction operator; ll max, teo high on right. Completely unacceptable features, -70- rATa | fe LD et \ PIV fe) 20 40 60 80 100 120 140 160 180 200 q Fig. 9 N-9 N-N Oyoee 92 N. eo Opeee Oy a) © ao 2! wfle TABLE XIV LEAST SQUARES REFINEMENT OF N94 Parameters for Curves Cc, and Cos Fig. 9 ria std. dev. 0, 001 0, 005 0, 002 lim of error 0,003 2 O13 o 007 o 06,0418 2 9750 - 0493 9778 - 0958 2 0737 std, dev, 0, 0006 0064 0018 - 0029 - 9063 2 0047 Although the theoretical curve based on a planar NoOq model is in excellent agreement withthe experimental curve, it was desirable to deterroime whether introduction of internal rotation about the N-N bond would destroy this excellent agreement. Curves were therefore a a calculated with varying amplitudes of internal rotation and three such rniodels are listedin Table XV. Assuming a quadratic potential func- tion, the dihedral angle & between the -N ae) groups has a gaussian distribution in the ground state of this hindered rotation. Models Rao and © 40 correspond to root mean square angular displacements es of 20° and 49°, while free rotation was assumed in Model Rye In these three models, the N-<0, N-N, Oy se Oz and Ne... © distances and all six temperature parameters are those of the planar model Co Since there pote th negligible difference between C, and Gg, no Ng was included in the FR models. All of the internal rotation models are distinctly less satisfactory than the planar model, the free rotation model being completely unsatis- factory. In curves Roo and PF 40” relative amplitudes of the main features are satisfactory, but the shapes of a number of these main features are in disagreement as indicated by the critical marks (18) in Fig. 9. A model with ¢ o* 10° is expected to fit the experimental curve in an acceptable manner because during such a smal) oscillation the most probable values of the long O-++O distances do not change by raore than 0.003 A from their planar values. At an = 20° however, the shift is about 0. 007 A and the corresponding theoretical curve begins to disagree with the experimental one. However it is possible that a readjustment of temperature parameters and distances might bring 3 o, so . . curves with oy = 26° or 40° into agreement. From the above consider~ ations this writer estirnates that os is less than 40°, ? ~ 740 TABLE XV INTERNAL ROTATION IN Ng, < G -0fs"/2 ai(s) = Z, = e sin r,6 Models Rag and & 40 Model Rie Roo Rag r r Cc e r Cc eg Liv77A LTA 5326 0. 0418 A LAi77A 5326 0. 0418 A 1. 752 1. 752 1164 0. 0750 1.1752 1164 0. 0750 2.173 2.173 3041 0. 0493 2.173 3041 0. 0493 2. 458 2. 458 5326 0. O778 2. 458 5326 6. O778 3. 433 3. 433 622 0. 0737 3. 433 304 0. 0737 2. 657 2. 657 622 0. 0958 2. 657 " @. 0958 3.427 3. 412 1094 0. 0737 3. 427 " 0. 0737 2. 663 2. 684 1094 0. 0958 2. 664 " 0. 0958 3.412 3. 351 752 0. 0737 3. 4124 " 0. O737 2. 684 2. 759 752 0. 0958 2. 684 " 6. 0958 3. 386 3. 257 404 0. O737 3. 386 " 0. O737 2. 716 &. &7) 404 0. 0958 2. 716 " G. 0958 3. 35) 3.135 168 0. O737 3. 351 " 9. O737 2 759 3. 002 168 0. 0958 2B 759 " 0. 0958 3. 307 " 0. O737 z. 811 " 0. 0958 3. 257 " 0. 0737 2. 871 " 0. 0958 3.196 " 0. O737 Be 935 " 0. 0958 3. 135 " 0. O737 3. 002 " ©. 0958 3. 069 " 0. O737 3. 969 " §. 0958 The N-N Bonding in N,O, The least squares refinement confirms the conclusions made earlier” about the structure of Nag: (£ special interest is the co- planarity of the molecule in spite of the long (1.75 A) N-N bond. If the length can be explained in terms of weak bonding due to adjacent positive charges on the nitrogen atoms, (19) then the planarity of the molecule needs explanation: One would expect either free rotation or perhaps hindered internal rotation about an equilibrium dihedral angle of 90° because of repulsions of oxygen atoms or of #-bond pairs. On the other hand if the planarity can be explained in terms of # bonding, then one would expect a shorter N-N bond. Smith and Hedberg” suggested that the N-N bond might be largely of w- rather than o-type: the w-character would keep the molecule planar whereas the lack of g-character would mean a long bond. Coulson and Duchesne (20) have even proposed a new type of chemical bond: a "“q-only" bond. The N-N bond in N9O% is said to have partial #-character but not amy ¢-character atall. Let us first consider the molecular orbital (MO) description of NOs given by Coulson and Duchesne: There are 13 valence electrons, omitting the K shells and the oxygen 2s electrons. Ten of these occupy five o orbitals: the two non-bonding oxygen po orbitals, the N-O o-bond orbitals, and the lone-pair ¢ orbital of nitrogen. This last orbital is isolated on the nitrogen atom and directed away from the oxygen atoms: % - Page 65. on lone | pur (\ o The three rernaining electrons are placed two in the bonding molecular # orbital (inthe diagram below) and one in the non- bonding molecular # orbital ©: C+) OQ © QO@ Ss "6 666 Here, the Co axis is perpendicular to the paper. In the above described "lone-pair" m model of } VOos the bonding in dimerization is due to interactions of the # electrons alone because there are no co orbitals available for bond formation. The expected MO energy levels in the dimer are shown in the diagrarm below: R-R —__.___—___R energy —________. R+R —_______-Q-Q zero ~—_—e———-P -P P NO, ———ee—__ P+P N04 ~t7- The interactions between R+R and +P and between R-FR and P-P due to the identical syroametry of each pair serve to stabilize PsP and P-P and to raise the levela R+R and R-R. The binding energy of N30, is the difference between the energy of the molecular configura- tion (PsP*P-2)*(0.0)* and twice the energy (P)*(O). The N-N bond is a fractional # bond, but without any cecharacter. There would be no advantage in promoting one electron from each ge lone-pair orbital of nitrogen in order to get a g-type N-N bond because this would place two electrons in @-2 which is antibonding. Coulson and Duchesne point out that such a change in the electronic structure of each -NG, group would lead to distances and force constants ap- preciably different from what they are in the monomer, whereas they are nearly identical. Also, a o-type N-N bond would have a force constant considerably higher than the value 1. 29 x 10° dynes/cm calcu- lated by Coulson and Duchesne. The gv-only bond is an attractive explanation of the planarity of N30, im spite of its long weak N-N bond. It is based on a "“lone-pair"” model for NO>. However there is an alternative assignment of electrons in NOs which would lead to ¢-character in the N-N bond of The electronic structure of polyatomic molecules has been dis- cussed by Mulliken (21) in terms of delocalized molecular orbitals. The nuclei of NO, are in a bent symmetrical configuration, point group Cay The approximate MO's are therefore either unchanged or change sign only under the symmetry operations of Coy if the molecule is oriented inthe yz plane with the C, axis coinciding with the z-axis, - (3 ~ the MO's will have the following symmetry properties: Cole) oye) aN By * o Pe fe) 0 ap + - ; wae by - * Py Bo - - Py x Thus, for exarnple, the MD by is antisyrametric with respect to rotation about the two-fold axis and svimmetric with respect to reflece y p tion inthe xz plane as is the p_ atornic orbital on nitrogen, Mulliken gave the following MC assignment to NO, in 1942 (21a): 2 2 Z fran ye 2 Z 2 (las) (ib 3) (2a) "(ab 2) (by "(3a, (a) "(3b 2) "(4a,) Walsh (22) also discussed the electronic structure of NGo and his assignrment (in 1953) was very similar to Mulliken's: (1a,)7(1b 9)7(2a,)7(4b,)7(3a,)74 2) tia,)°( 3b 9)"(4a,) Although they order the orbitals differently, Mulliken and Walsh agres largely from qualitative arguments, that the odd electron is in orbital 4ay which corresponds to the lone-pair ¢ orbital on nitrogen, The following methods are useful in determining the unpaired electron distribution in a molecule: (1) Quantitative numerical calculations of the energy levels, such as a seli-consistent field, linear combination of atomic orbitals (SFC), (LCAC) MOtreatment; (2) Analysis of the électronic spectrum. In addition, the symmetry of the total electronic te oy, aa i ie wave function (and thus the syrnmetry of the half-filled orbital) may be revealed by rotational spectroscopy; (3) Determination of the unpaired electron spin density at certain nuclei by electron pararnagnetic reso- nance spectroscopy. These three considerations will be discussed in that order. To this writer's knowledge, no SFC, LCA treatments have been made on NO>. However, such calculations have been made for COs by Mulligan (23) and for Os by FPischer-Hjalmers (24). The MO energy levels of CO, and O4 are shown in Fig. 10", and the interpolation has been made for NOs. The energy levels have been drawn with zero slope at 180 because, as Walsh (22) has pointed out,there must be either a maximurn or a minimum here by syrametry. Although in the case of Oy, the lone-pair ¢ orbital 4a, falls below lag {2 in the nomenclature of Coulson and Duchesne}, it seems unlikely that it would do so in NO, according to Fig. 10. The lonization energies of the various MO ‘9 which are indicated in Pig. 10 are the calculated values (23), (24), except for amt iv and iv. where the experimental values were used in drawing the levels. (The calculated value is indicated in parentheses in each of thege three cages, ) Although the interpolation for NG» Was mnade on the basis of the bond angles, the bond lengths are also included for eomparison. @ : 7 This figure was suggested by similar ones given by Mulliken (Zla)}, (25) and Walsh (22). -30- fo) fe) rs (2.5%) 4a, 137 ee. laa Q 199 9815.7 (150) 148 Ca 3ba _—_—_—_—___ ww [Pag ght 7.3 (18.8) b/ 79 Pp Zon, ib: p/9A 23 epi g 2a. zo, —*) aay 32a ¥ NJ mw > v y lo. 4l0pe— . x Guslars § $ N g ls _——— isang : 2 ftp 117° 134° /80° O; NO, CO, 1264 AIGA /./6A Fig. 10. Molecular Orbital Energy Levels ~81- Discussions of the electronic spectrum of NO, have been given by Mulliken (25); Harris, King, Benedict, and Pearse (26); Price and Simpson (27); Walsh (22); and Mori (28). Only Price and Simpson (in 1941) have placed the odd electron in a # orbital; and perhaps current opinion is best expressed by Mulliken (in 1958) who gives an interpre- tation of the electronic spectrum in terms of an assignment of the odd electron to 4a,- Infrared and microwave studies of the rotational levels of NO» can give information about the symmetry of the electronic wave function. However on this basia it is impossible to say whether 4a, or la, is the half-filled orbital because they are both symmetric with respect to interchanging the oxygen nuclei. * Bird, Baird and Williams (31) have studied the electron paramag- netic resonance (EPR) spectra of solutions of NO, in CCl 4 and So. They observed a triplet with a splitting of 107 + 3 gauss (300 + 9 mc) between successive components for NO, at 0. 3% by volume in both solvents. Because of rapid molecular tumbling in solution, the spin- spin interaction Hamiltonian reduces to the spherically symmetric Fermi interaction (32). Thus the interaction Hamiltonian is "The infrared spectrum of NOz has been studied by Moore (29), who found from the alternation of intensities of rotational lines that K = 0 is permitted only for even J in the ground vibrational state; therefore the rotational levels (and hence the electronic state) are syrametric with respect to interchanging the ol6 nuclei. This is in agreement with either 4a) or lag for the odd electron. The assignments of micro- wave rotational lines by Bird (30) confirm the symmetry of the rota- tional levels. Gee nt log ELD Sp Bat pe gt 2 oye: S where wRlo) igs the odd-electron wave function at the nitrogen nucleus. The value of 20) in NO, was thus reported to be 6.3 x 10% ¢ m”>, or 187. of the value (34 x 10°* em”? } estimated by Dousmanis (33) for a nitrogen ator 2s atate. © The large value of (0) in NG, indicates that the lone-pair ¢ orbital 4a, rather than the m orbital la, contains the unshared electron. We have thus attempted to show from an MO correlation diagram, from electronic spectroscopy, and from EPR measurements that the odd electron is ao rather than a v electron, and therefore that the N-N bond in Ngo jg ray be expected to have g-character. That itis a weak bond may be due to the delocalization of 4a, in NO». (In Ox, Fischer- Haljmars (24) found that there are large contributions to 4a, from the in-plane p orbitals on the end atoms.) Perhaps the planarity of Ng, can be explained in terms of the w-electron interactions dis- cussed by Coulson and Duchesne (20), and the length of the N-N bond in terms of the adjacent charge rule, and the de-localization of the da, MO in NCig. # “2 I find wo) = 31% i9%4 em” ” by extrapolating to r = 0 the nitrogen 2s atomic wave function given by Hartree (34). «83. II, APPLICATIONS OF THE METHOD OF LEAST SQUARES IN ELECTRON DIFFRACTION INVESTIGATIONS Introduction | In recent years, the method of least squarss has been applied to several electron diffraction structure investigations. The least squares method in the analysis of visual data was discussed in detail by Hamilton and Schomaker (16), and was applied by Jones, Hedberg and Schomaker (35) in the structure investigations of formyl fluoride. An application of the method of least squares to sector-microphoto- meter data was described by Bastiansen, Hedberg and Hedberg (14) and was used by them in the refinement of I, 3, 5, 7-cyclooctatetraene. This writer has written a least squares program for the Burroughs 205 digital computer and has used it in the refinement of the structures of several heavy metal hexafluorides and of N,° 4 As applied to sector data, the least squares method involves the This method has many advantages over both the correlation method commonly used in electron diffraction studies, and the method of least squares on visual data. In the correlation method, a number of calcu- lated curves are compared with the experimental one. It becomes pro- hibitively laborious when limits of error are to be found for a large num- ber n of parameters, since about 3" curves must be calculated, Im the method of least squares as applied to visual data in these laboratories, the observations are the measured positions of maxima and minima and ~B40 qualitative estimates of the xelative heights of two or three neighboring features. In the present application of least squares, the data representing a continuous intensity distribution are used, not just certain special features which happen to be easily compared by eye. Thatis, every point on an electron diffraction ring carries equal weight--the whole ving and ita shape are used, not just the position of special features and the relative heights of neighboring maxima and minima. In the correlation method, great weight is placed on those ‘portions of the pattern which are especially sensitive to parameter changes. However, such emphasis on a small part of the pattern can be very misleading, especially if that same sensitive region is gubject to large experimental error. The latter is just what happened in many of the heavy metal hexafluoride patterns discussed in Part I of this thesis: The part of the pattern which was most sensitive to the cut-off point unfortunately coincided in several instances with the bad place in the sector. If the correlation method had been used with these sector data, it is very likely that errors in q, would have been considerably larger. But ia the least squares procedure, such regions of high uncertainty can be given armmall welght if desired, + Es f f 2, In the ME, determinations this wae not done because data near the cut-off point were so valuable, and in reany cases we did not have enough data below q_ to be able to afford to lose the data near q.. Hf there had been more’ data below q, then it would probably have been better to use a special weighting flinction. However, a standardized weighting function was used for all sete of data. 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This writer estimates that a rough rule would be that a spacing 4q = 1/3 would give independent observations, where IL is.a dominant wavelength. It is not proposed that observations be made at this spacing; to do so would probably be to lose information about the longer distances. Rather it is suggested that observations be made at Aq =1 for convenience in molecules containing distances less than 10 A, and that the computed standard deviations be corrected on the basis that Aq = L/3 would give independent cbhservations. The computed standard deviation of x; is e(x,) = (viPV(B™ yA (nemy V2 (8) and if the intensity curve is completely defined by n equally spaced observations, taking @n cbservations at half the previous spacing will give a new estimate et) = (avipy(B)y, /2)/ (ane)? a(x, 27/2 (9) Thus if the number of observations is increased by a factor of g, the new computed standard deviation ig too small by a factor of gi/ 2. Fror the foregoing considerations it is estimated that the standard deviations in r; resulting from observations at Aq =1 are too small i bya factor of (20/3r, V2 where Y; corresponds to a dominant term. ~89a- a a a rore compl) 2 ee 2 a HOoto« & gre om Med ih ite Rite als ey ons) @ ES ducing he an a etecd to be of constant te ‘the % eh eres & 2 2 Waid nt Loney ee bg. oe t he e vror A gy Hes rt caf tie comotant backer AF lk ae ney gry adding & ic i Bn eet oO Whee ao Oh BLY | 6 POeVig Re thene (A'P) whose com- peRments wi i, cf of constant cor PAN of a vector ge in ‘t akan in other words no ¢! ee “5 ae YER idal ponents oscillate in a sinusa parameters can simulate a constant background increment to the pattern However, although rneters will not change rauch. EE, os # anc therefore the wh wound error, LIP T, 4 heuiy y affected by the bac! hi toh aa 1 6 ot ap i or de ey ora Pees Pou a = Se mates on the parar atch the error esti ee themegelves PTICAL DB a ees: OR THE BURR: This program will calculate the theoretical intensity curves which are used in the electron diffraction structure determinations of Introduction Tae theoretical intensity function i, amolecule containing t R distinct distances ray be written ib Lia) =) %, ~3 sin(z,qa/10)= 54) {1} th where C, igs a constant depending on r, but not on q, zs ig the i 42 distinct interatomic distance q = (40/.) sin 9/2, and the ternperature parameter o, is the root mean square variation of Eye itis desired to find I c for integral values of q. 9 Seheme of Calculation A sine table and an exponential table form part of the program tape and are stored in memory as follows: sin (27x/1000) is stored in cell (0000+ rod 1060), «= integer; boy : ; _— eye expl-y") is stored in cell (1000 + 1000y/3),0<y< 2.997. # % Prom @& report submitted to the Rlectre Data Division of the Burroughs Corporation. & e 6 2 & ua BE ne Fas 2 «8 ~ 3 - 2 es pap ~ 2 an oS Lic a Bs <S set 2 oa mt 8 8 OS ss) ree os 3 oe 28 _ go © S oa 8 S 5 8 he Bes 3 _& Oo -s ord a «rt oS <i & 3 s = a && i » 2 & oe - § - Sek * : 97 & é Hs . & & «a & 8 2 ad al Go) gs at fy ¢ & 5S a at end : “~ me a s = © ee ed a ne * r fet a 2 a eel . Pad et on : ¥e Py, cam Gag % aed ara as 7 red Pa a “a oe a sesh 8 3 wot Se + a ey Set & #3 Q 2 aes aes iy wy LA a tet A eek £3 f 3 rel geet . Ry od et ie : om _ a a 8 ae a g a 8 & i oe oF Fe o ext oe 3 23 3 ~ a es 5S & : ort GQ reat & ct ss Fae ope et Rng Bet be ‘ es “ Ga S 2 a mo FO 4 8 if wow gt a a i 3 G Lt Be ey bon be ey Soe $ re 2 3 ry a 3 cy met ¥ al St ed cS ey a % “a a wet it os os . seed u ats raed "a a Be & we ¢ et 3 & © ed Sak 7 nae es o r% oe. mR : § cf samy 3 ue B Gg gt aa ng a o 2 “, =y 5 a + # 3 * - ~ Hoos iy ae tO} aa ra Ho 8 at ae iff Ged ae ea - eet 6 g to s ae “4 ry et a i & res git, pry erty gti ; a of as cs e% ret ea op; 9 Bed Bp get a Pant oS eS oS @ ea es] fo e a wd ed mo & a Bs a 75 : S S ‘ a wy So & ed aka Es sot vt m9 cmt ZS eg 4 oe . * oe en oF Sy a ‘ ™ § 68 * A od " g = a es st o = Ke Bey ca gt aS ae me & Ee Ty Ge 5 boy aoe 5 ; : egy F a 6 oO € apd ry tet Bh gy f Dikhit ets g ww > o oa eee a te * ome intenss At Bam eeert Corer an et 2LG wat GIR foe 4 Serene Len Zo spr, A a ee ry OWS Oa Agter t he last set + fo ost ah an nha Ton, go wy om td ee eye ee, tle py my ., lata there must bo, im addition to the & CHT 3159 “ire oy 9 E and finally , a ° Ses Tee ea -_ cm) = UOSifed to omice at = Be gm, ono eh, say TESS 23h é - 3h PTT Pay atere im S290, 3202, efe., an place & OI 3150 age By oy © Cperation of the Program wn “had § aay PR onteol =m Se as & i OL Caey,ecine & t i. spdinegue 9 dee oe ot a be Ormat: Grouping and Gounters: OF Berg Suppress: ; cam sts? acl im tae p £ gram tape. Following a sum check and some other short operations, the computer will stop with OP 1254 in &. (HE the sum check unsuccessful, ar we a AM apes cry toa te wn sc ae ka Bio's, to AeA ke Riche taut i ae fr we g i o f, é Swe ROLL ERE pions allied in e en Sar be - ate “ GS oy iy % 3 Bey ieey Fat i dah wean 2 60 we eh “ ey od he acing ie ont AE By et ON, breal the e LEEe wed, my anal eee Fixe gs a Ft go oh Mee Kate Ne aL o t- ee & Raat EY te ae EEE) tease mates Se SO yea garter he . “ath Se Sige oa wy i s o a é re co al 4 g 46 Aa ahs A ct AL fagP oo td a mT es, eel oo Tye 3 2 3 5 3 3 4 Ul calculate the racial distribution function which “gy, ve Wh aS This progra : a A . sil Nand é absence hay AbneIninararendnsson CULE used in the electron diffraction structure determinations of gassous Es Es LOD 3 o oe ee int a6 & “lated % ae he he to be cal e1 eae LEE £ bey Bed ¥ = Eoin BR, Py i €3 ae teh is * 3a Wiad ey "al i ere Iiq) i verte Wek mB Re = 7 *i ig g bed g 4 Pat Soak Oy a oak at ty 3 ed 7 5 7 “ « ae polnta. 9 Bae) i Led oD 2 * a and ti (2) _ GQ 4 2 points in r yore es of os Seheme of Galeoulation 4 ke SES ee Fm lea when a= 1, water? wag” -2aq 9a g ane i € d 1 weet, ote Sy <= 2 oe, c " in ~ ain foe F ~ spacing. Then rD(r) is found for r= @Ar, and so onuntil ro . i SE Lehn s is reached, Provision is made for variable Ar spacing in different vegions of vr, enabling one to have a fine spacing im the regions where a resolution is bmportant, but saving calculation time by having a coarse A® spacing in regione where no peaks ere expected, a ia Preparation of Data Tape. I. Store the calculation number (up to four digits) in the icast significant portion of cell 3500 to identify the output data. 2 Store the intensity data Ua} im celle 2000 to {as a maxirnum) 2399. The data must be in fined point; a convenient range being 600005 00000 to 00960 01000. (Overflow will occur if the numbers are needlesaly large.) Store (1) im cell 2060, 2) in 2001, etc., Hq) im 2000 + {q~1). Valuce of ig} for q less than dnin 2xe usually guessed or supplied fzom theoretical curves. Hit is desired to use et iq) = 9 for q less than q_, if is not necessary to store zeros ey Fa] & rs) re) jou pat Pa ot @ } & a 3 Fa) e > t ge i wi ui eu rg ra) Et s o ee ?) fe = cae ea a pa . 6 5 I wf sy ) ke 3 2 &s 3 Ps] 2 ep Oat «a » #f& «& 7+ oO 9 8 S @ aiid oo aa “od a “a oat # tog epee ig Ha 3 * Se oO 4 f : S os @ com 8 - 9 a “ Oo ~ © 8 8B oe od a 8 a ont Go 2 +3 a, a * c . 2 ‘4 8 a o 8 s Bea eed 2 © QJ ° 3 gS @ 8, 4 oF wa 2 3 & : a o w J 2g @ Sf = bt &f © 8 a] . a ‘ 8 g =) rs 9 e é ee: Sn 2 8 h 9 S 2 @ gf o 3 of a enig : ‘Se € e 5 oy fs Fe 5 3 2 6 eo ‘4 S gt vm b#) £ 4 ° : a Sd we 2 Fee g Pe 2 a * . ts & Pa) chy " 3 3 Ea) és E re) re be} : a 3 Qa a 3 a a os 5 Pe eg © 6 B 8 BA fom 6 o os cs a « _ a «¢ 8 @ @ & Oo 8 @ a § Oo ; Boag aa . gr al - ols oot a g ried yet a Sy ay ed ns a © Big re! a 3 Fe) a EN Pi a ma . a G pe ) 4 6 er a ) o 9° 3 = g 8 fF & a s 3° sd &3 3 <3 a 3 wt Par ~ & Sed 5 : = & e pot ey = aa 3 = a. 2 S oy aa) é te ) 5 B Fg o & 2 & & & # - os 5 2 : & ord * . red Bl ca) 3 ‘. iE & s ® ae a 3 ae ” a qj 2 8 Hl ae a : ao & 2 6G ao os 8S ss 8 a) . «a o~ om F axeg ca a rt & = a a Ss 2 | a ‘ . aS a ret 3° Cod 3 atl ef Pe ty a * 2) as; ° ett 8 > co oO 8 " 4 ae : A ea oe ad a B 2 8 & o BE ‘a 8 go g, 5 “a - 3 @ a ae “of a ea 24 ms ped oy s et oF at SS og oO _ & ad wet aad cat sped cael ot et Pas 2) 2 4@ 2 8 8 ® 3 & & & 8 $2 @ Pt 3 os 2 S Be 8 qd os &§ 8 um ag “oon 43 3 a eo © “ 6 @ o 7 & « - & & 4 © yi si. . 2 p 28 3 a 2 q 4 & So @ os 8 BG 8. x a 2 wey yo *e3 g eS & Fs oe Fal ; 8 i eq ia pd fa] Pa 4 ng a] = . g “of Ss P By, 3 : 6 8 @ 4 s p pF 8 OF a wi OG @ & x & a 3 ou : . re é 2 3 ab 2 @ 3 & & a} QD = +3 2 >: @ 8 2 8 & ° Moa gf 8B og & o 8 S$ 4 3 SB go : 4 *@ 2 8 4 = ot & © & #8 & 3 42 age od ah a wy - « 2 a hg Sed : g 5 f * EA lors ¢ By = . 5 : oe ved bas o a oa g 4 9 9 3 8 ce a > a wed 3 Sq Ee 37 Ba Ps) a> eae &4 3 a a rat “ & 9 5 a i‘ 5 "4 4 3 a gt net oS Ra wet , 33 @ et & eo F a gi a b> a & ci ~ BR . be < Fe hy, i = & og > 2 @B a ¢ S pra eS & Fa i. i a € = i a og 8 > fag 2 9T- 1. Read in the program tape. Following a surn check and some other short operations, the computer will stop with STOP 1234 in C. (If the surn check is unsuccessful, STOP 0019 will stand in C. ) 2. Read in the data tape twice. A sum check on the data tape . will be made each time, and if they agree, the calculation will proceed. Tf not, then control is transferred to clear the data sterage region and the cornputer stops with STOP 1234 in &. Try again. 3e Aiter the print-out, the complete program and all storage domains will be returned to their original condition and a sum check will be made on the program, and the computer will again stop with STOP iz34in Cc. This signifies that the computer is ready to corapute another RD. The print-out includes: i} Calculation Number 2) Input data . | 3) Spacing code word ttt0 00 OOuu 4) vD(r) (answers) for ttt points with Ar = 0. wu 5) Second spacing code word (if used) vvvo oo ooww &) rD(r) for second region: vvv points with Ars O.ww etc., until. Tmax 18 Feached, . ¥ A LEAST SQUARES PROGRAM FOR THE BURROUGHS 205 COMPUTER® Surmmar y: The purpose of this program is to fit an analytical function to a set of observations. Specified coefficients in the analytical function are adjusted by the method of least squares: the weighted sum of aquares of deviations of the observations from their predicted values ig minimized. The computer will form the least squares normal equations, and solve ther by a complete matrix inversion. The print- out includes the least squares solutions, their standard deviations, and the estimated mean error of an observation of weight unity. BS ¥ System Make-up: Burroughs 205 with A. F.P. and paper tape input and output. i. Inteeduction: It is desired to solve n equations of condition, =p / a.m. = § Vp S21, 2 3, «cs % (i} je in order to express some observed quantities A, asa linear combina- tion of the known quantities ais The m <n unknown coefficients x, andthe n residuals Vv, are to be determined by the method of J : least squares in that the sum of weighted squares of residuala ; BV; is], From a report submitted to the ElectroData Division of the Burroughs Corporation. shall be a minimum. The weight B; is defined to be proportional to the reciprocal of variance of error of the observation ay. if a linear relation such as equation 1 does not exist, it is usually possible to linearize the problem. Thus if the equations of condition are £ (yx. oe xn) = ayt Vp iz-1,2,3,...n, (2) the f; may be expanded in a Taylor's series about an approximate set x= (&), Kos . x) of the unknowns. Then one may write _ © a6) £.(x) = £,¢x) + > As, & 8.4 Vig OF i i fs an i i o-, 8%, jal °°} (3) where higher order terms have been neglected and ae; aise £, (2). Equation 3 now has the same form as equation 1, and the OX» the unknown coefficients, are the corrections to the approximate % This program is most casily described in matrix language since it consists largely of matrix subroutines. The reader who is not familar with matrix language will still find this program easy to use. Let matrix A be the array of the a,,'s of equation 1. itis a rectangu- lar array of a rows and m columns and is loaded into the computer by columns as described later. Let i. be the column of i,ts and V the column of v,'S- A description of least squares in matrix language is given by Arley and Buch (36). In matrix language, the equations of condition become ce Lav (4) we 2s -100- where A is the nx m matrix (a,5) xX is the m x1 matrix or vector (x;), and soon, IL and V each having n components. The minimised sur of weighted equares is V'PV where V' is the trans- pose of V and P is a diagonal weight matrix with Pay = Pp The least squares normal equations are expressed by the matrix equation A'PAX = A'PL, or BM = A'PL, with solution (5) x= BwpL (6) where & is written for A'PA. The sum of weighted squares is VipVvV = LIPL - X'A'PL (7) The external estimate of the mean error of an chservation of weight unity is s = (V'PV/nem)!/2, (8) The estimate of the standard deviation of x, is 3 -1, 1/2 -} = B os Be f 8(x;) ({ 33) | (9) Il Limitations: The number n of equations of condition is unlimited but the number rm of coefficients x, must be < 20. Ali data must be in # LOATING POINT. Wi. Preparation of Data Tape: A. Initial (instruction) section: i, (mel) x 19716 is stored in celi 1521; for example, . 06000 00G17 if m = 18. ~idl- 2. n- mis stored in cell 1524 in floating point notation; for example, 53148 00000, if n=166 and m = 18. & o rex GU 1530 is then punched, followed by blank tape. 4 Optional features: a. if P = unity (all equations have equal weight), re- place 6 CU 1530 by 6 CU 1533, b. If it ia desired to suppress the sum check on input data, replace 6 CU 1530 by 6 CU 1534, (= may or may act be unity. ) the Data sections: If mn is greater than 20 there will be several sections of data, each section save the last containing data for 20 equa- tions. (Only for the special case of n = 0 (mod 20) will the last section also contain 20 equations.) The first 20 rows of A are loaded, by columns, starting at 3600. Thus (3600) = ay, IN _SLOATING POINT (3601) = (3662) ete, ay ie a fh ee) add " (3621) = = B55 ete., celle 3600 to (as a maximum) 3999 b used, the first 20 components cf L are stored in consecutive celis starting at 3000. Thus (3061) = Ly etc., cells 3000-3019 being used. The first 20 diagonal elements of F are stored from 1560- « iGB~ L579. (1560) : if (1561) =P # ete. Var the special but common case of all observations having = dt equal weight, it is not necessary to store anything in 1560- 1579, as these cells will automatically contain 51100 00000. ah Unless the sur check on the input data is to be sup- &B * " pressed, tf a e sux (mod 1) of all the elements in the first 20 rows of each matrix or vector must be obtained. For the purpese of the sur check, the floating point numbers are considered to have their decimal point immediately to the ‘n position.“ These sums are stored as follows: 1527} # surn (mod 1) of all elements in 20 rows of Ll, (1523) i gum {mod 1) of all elements in 20 rows of A, if ket © (1529) = sum (mod ]) of all elements in 20 rows of If all observations are of equal weight nothing is stored in cell 1529. At the end of this section of data for 20 equations, the command ~~ a {a ; CU 1 ep. appears, followed by blank tape. Exactly the same form is followed for each succeecling * The following examples wiil illustrate: 50 1234 5675 - 50 1234 5678 -59 VA11 11Li ~. 59 LILL ULE) . US 2345 OTE9 -. U8 9876 5435 We have dropped any integers and retained only the decimal fraction. set of 20 equations, 26 rows of A into 3600, 20 elements of i. into 3060, and 26 elements of PF into 1560, the sum k surna inte their appropriate cells and the command 6 CU 1531 followed by blank tape. In the last section, which will be the only section if n= 20, a special precaution is needed unless n = 0 (mod 20). The murnber of rows lees one for the last section must be stored in cell 1520. Thue if n= 80, nothing is storedin 1520. If n= 83, 00000 00002 is stored in 1520 in the last section. Following the last data section is again the command 6CU 1531 and the blank tape, and then a final command & CU 1532. IV. Operation of the Program: The control switches are set as follows: Breakpoint Switch: OFS Cratput: TAPE (or PAGE for a short problem) Skip Switch: ON for short print-out (ptional) OFE for long print-out Flexowriter: Grouping and Counters OFF The program tape is readin, and, following a sum check and some other short operations, the computer stops with STOP 1234in CC. The data tape, prepared as previously described, is now read in and a sum check is made onthe first 20 equations. If successful, the computer will pro- ceed with the calculations and call in the next set of 20 equations. On a sum check failure, the cormmputer will stop with STOP 5555 in C. The #104— print-out of part I occurs as described below. At the end of part I the computer stops with Sv: OP 2i24in €.. For part li (which is optional) the rewotnd data tape is placed in the optical reader and the cormmand 6 CUB 0174 is entered in the keyboard. The computer will stop with PTR 3600 in CG. The Clear Button is pressed and the data read in a second time. At the end of part Il the computer stops with STOP 0000 in C. A second problem may be run after either the short or long print-out of part I or at the end of part Il provided the same options are desired regarding the weighting and the sum check. To run another problem, the cornmand 6 CUB 0006 ig entered on the keyboard, and, after the computer stops with STOP 1234 in C, the new data tape is read in. VY. ‘Print-out, Part I: wt om A. The short print-out (Skip Switch ON): x“, designated M. 04 (Matrix 4), the unknown coefficients. L'PL, designated i'pl, the | sum of weighted squares of the observations. V'PV, designated v'pv, the sum of weighted squares of residuals. 8, the standard deviation of an observation of unit weight. 8(2,), designated sdx, the standard deviations of the x x; ‘6. * The quantity L'PL, is of interest when a non-linear problem is treated by least squares. Our x's become corrections to approximate values of the unknowns and the least squares solution is obtained by iteration. One may compare L'PL withthe V'PV of the previous cycle of itera- tion to see if much trouble is caused by nonlinearity. Also, a compari- son of L'PL with V'PV for a particular cycle will indicate how far one is from convergence. #ioSe B. The long print-out (Skip Switch OFF), in addition to all the above items: B, designated M. 03, the coefficients of the left-hand side of the normal equations. A'PL, designated M. 05, the right-hand side. BX«A'PL, designated M. 00, a measure of the errors in solving the normal equations, obtained by inserting the computed X into the normal equations, and finding : the difference between the right and left members. «1 B ”, designated M. 02, the inverse of B. VI. Print-out, Part IZ (Optional): Vis the residuals of each observation equation, printed 20 at atime, without designation but with an extra carriage return between each group of 20. V'PV, designated v'pv dir, the sum of weighted squares of residuals calculated directly from the v's obtained in Part 0. diff, the difference between the directly calculated. V'PV and that calculated in Part I from equation 5. This difference should be negligible; it is a check on the formation and solution of the normal equations. VIL Description of the Code: In order to handle an unlimited number m of observations, pro« vision has been made for entering the data into the computer for 20 condition equations at atime. After the first set of 20 equations has been read in, its contribution to the normal equations and to L'PL is found. Only 20 equations are read in at a time in order to avoid relying on magnetic tape storage. At this point, the first set is destroyed and the second set of 20 equations is readin. Its contribution to the normal equations and to L'PL is found and added to that of the first set, so that one now has the normal equations corresponding to the first 40 condition equations. Thies process is continued until all of the data are read in. A set may actually contain any number h of equations (provided h < 20 and (n-1) x 107) is stored in cell 1520) but it is most convenient to take 20 equations for each set save the last. The partitioning described above is possible since equation 5 (the normal equations) may be written vv y ALP ALS = >» ALP, Ly (10) kK where A,, Py, and L, are sub-matrices or sub-vectors obtained by partitioning by rows. The kth subematrix of A must of course have the same number of rowa as the kth sub-matrix of P andas the kth sub-vector of L. The final command 6 CU 1532 on the end of the data tape sig~ nifies to the computer that the read+in is cormplete. The matrix B is now inverted and its inverse refined. The inversion is obtained by a SAC™ matrix sub-routine using stright elimination with positioning in Foc, semi-automatic coding, is a syatem of coding which automati- eally compiles relatively coded sections into a program with absolute addresses. The sections may all be written by the coder or some of them may be sub-routines taken from the SAC sub-routine brary. The present program makes extensive use of SAC sub-routines. Perhaps the most useful feature of SAC is the simplicity which it lends to the writing of programs that have matrix operations, since a number of rnatrix sub-routines are available. For example, in the multiplica- tion of two 20 by 20 matrices the 400 elements to be found are each the sum of 20 products. These 8000 multiplications and $000 additions are programmed by the coder who uses SAC by writing a simulated matrix command containing only three words (after some definitions have been given to the computer): 6606 31 6501 Matrix 9906 00 8512 Multiply 0102 63 9900 Matrix l times Matrix 2. Store the product in the doma: ain of Matrix 3. For a description of SAC, see reference (37). «| . rows. To obtain a better approximation to B than the computed ine verse EH, consider the error matrix E: EekR- I (definition of BE) (14) where I is the identity matrix. We may then write «1 Bh e(rs B)UR = (FER, (23) where the approximation may be verified by substituting = - ER for po! in equation 12, and is valid provided E is small. Let Ry = KR be the first approximation and then F 3 may be de« fined as Ry ~ EAR 4 Let Eo = R,S ~ ij andsooen. Thenthe i + 15% approximation is Rey Ry > ER, (14) The number of refinements is thus i-1 andis specified by storing 10 & gn {i - 2)x107°™ in cell 1523. If no specification is made, cell 1523 will contain zero and one refinement is automatic. The refinement of pvt is often necessary because of round-off errors or ill-conditioning. An idea may be had of the success of the inversion and refinement by come- paring BA-A'PL with A'PL. If 3 was not accurately inverted, % will be inaccurate and BX-A'PL will not be small compared with A'PL. If & is singular or nearly so an overflow will occur within the SAC inversion gub-routine. For matrices encountered by the writer in refining electron dif- fraction structures of molecules by least squares, one refinement gave £6 more than sufficient accuracy inthe results. However in certain cases, awe. el Ke for example in the fitting of power series polynomials to experimental data, a program with double-precision matrix inversion, (that is, operations with 18 rather than eight decimal digits) may be needed. Perhaps orthogonal polynormials could be used, however. After Bp) is refined, X is computed and the print-out begins. Since the input data are destroyed during part I, the data tape muat be read in a second time if the residuals for each condition equation are desired. Vil. Program Execution Times: Ag an example of the time required for the corplete program, 134 equations in 19 unknowns required about 35 minutes for reading in, calculation, and punching out answers on the high speed punch. IX. Variations: The program may be modified to print A or IL or both in part I. To prim A, 0000 311020 is stored in cell 0187; to print L, 0000 03 0510 is stored in cell 0206 and 0000 G3 0810 is stered in cell 0209. For each set of 20 condition equations, 20 rows of A will be printed as rnatrix 7. Then, if L is called for, 20 lines will be printed, each line having an &£, followed by ite corresponding Vee To obtain an extra refinement of gh 00000 00001 is stored in 9. 10, ll, 12, 13, 14, 16, 17, 18. 19. -109- 093 REFERENCE: B, Weinstock and J. G. Malm, J. Am, Chem, Soc. 80, 4466- 4468 (1958). — 5S. H. Bauer, J. Chem. Phys. 18, 27-41 (1950), ©. Bastiansen, private cormnmunication to Professor V. Scho- maker, 1950, J. Bigeleisen, M. G. Mayer, FP. ©. Stevenson, and J. Turke- vich, J. Chem, Phys. 16, 442-445 (1948), K. N. Tanner and A. B. F,. Duncan, J. Am. Chem, Soc, 73, 1164-1167 (1951). ~— T. G Burke, D. F. Smith, and A. H. Nielsen, J. Chem. Phys. 20, 447-454 (1952), J. Gaunt, Trans, Faraday Soc, 49, 1122-1131 (1953), V. Schomaker and R, Glauber, Nature 120, 290-291 (1952); R. Glauber and V. Schomaker, Phys. Rev. 39, 667-671 (1952), J. Ae Hoerni and J, A. Ibers, Phys. Rev. 91, 1182-1185 (1953); J. A. Ibers, Ph.D. Thesis, Calif. Inst. Tech., 1954, &. Nazarian, Ph. D. Thesis, CalizZ, Inst. Tech, 1957, J, Karle andi, L. Karle, J, Chem, Phys. 18, 957-962 (1950), F. A. Keidel and S, A. Bauer, J. Chem. Phys. 25, 1218-1227 (1956). ~~ J. A. Ibers and J. A. Hoerni, Acta Cryst. 7, 405-408 (1954). ©. Bastiansen, L. Hedberg, and KF. Hedberg, J. Chem. Phys. 27, 1311-1317 (1957). ~— B. v. Borries, 7. Naturforsch, 4A, 51-70 (1949); H. Koppe, Z%. Physik 124, 658-664 (1948); BU "Lenz, Z. Naturforsch, 9A, 186-204 (1954). ~ W. ©. Hamilton, Ph.D. Thesis, Calif, Inst. Tech, 1954; W. C. Hamilton and V. Schomaker, unpublished work, K&. Hedberg, private communication. W. FF. Sheehan, Jr., and V. Schomaker, J. Am. Chem. Soc. 74, 4468-44869 (1952), — L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, 1945, p. 27T, 20. 21, Za. 22, 23. 24, 25. 26. 32, 33. 34. 35, 36. 37. -110- C. A. Coulson and J. Duchesne, Bull, classe sci,, Acad. roy. Belg, 43, 5222532 (1957). vie R. S. Mulliken, Fhys. Rev. 43, 279-302 (1933), R. S. Mulliken, Revs. Mod. Phys, 4, 204 (1942), Ae De Walsh, J. Chem. Soc.,, 1953, 2260-2331, . Je F. Mulligan, J. Chem, Phys, 19, 347-362, 1428-1429 (1951), i, Fischer-Hjalmars, Arkiv for Fysik 11, 529-565 (1958), RK. S. Mulliken, Can. J. Chem, 36, 10-23 (1958). L. Harris, G W. King, W. S. Benedict, and R. wW. B. Pearse, J, Chem. Phys. 3, 765-774 (1940); L. Harris and G, W. King, J, Chem, Phys. %, 775-784 (1940), wg —— We. Co Price and > M. Simpson, Trans, Far. Sec. 37, 106- 113 (1941). K, Mori, Sci, of Light 3, 62-69 (1954); 4, 130-147 (1955), G. E, Moore, J. Opt. Soc, Am, 43, 1045-1050 (1953). G. R. Bird, J, Chem, Phys, 25, 1040-1043 (1956). GR. Bird, J. ©. Baird, and R. B. Williams, J. Chem. Phys, 28, 738-739 (1958), 5S. I, Weissman, J. Chem, Phys. 22, 1378-1379 (1954). G. ©. Dousmanis, Phys. Rev. 9%, 967-970 (1955), D. R. Hartree, The Calculation of Atornic Structures, John Wiley and Sons, Inc., New York, 1U57, M. E. Jones, K. Hedberg, and Y. Schomaker, J. Am. Chem. Soc, 77, 5278-89 (1955), N. Arley and K. R. Buch, Introduction to the Theory of Proba- bility and Statistics, John Wiley and Sons, Inc-, New York, 1650, Chap. 12. Warga, Semi-automatic Coding for the Datatron, Report No. 120.020, ElectroData, Division of Burroughs; K. Hebert and H. Fox, Semi-automatic Coding for the Datatron with Simulated Matrix Commands, Report No. 2.40.010-2.47_ 190, EleéctroData, Division of Surroughs; K. Hebert, SAC for Magnetic Tape, Caltech Computing Center. ~lil- PROPOSITIONS 1. In the analysis of electron diffraction sector data, a background line must be drawn in a somewhat arbitrary manner, leading to cor- related observational equations in the least squares refinement. Two methods are proposed for obtaining observational equations which are independent of the background. a. The first and second derivatives of the experimental curves ray be taken as the observations. b. The radial distribution function of the observed and calculated intensity curves may be fitted by least squares, with large weight near the peaks and small weight in the other regione. 2. Dipole mament measurements of the NA° a No, system at vari- ous ternperatures seem to indicate that one of two alternatives exists (1) Either (a) the dipole moment of NO, has a large tem- perature dependence and the dipole moment of N,o 4 is zero; or (b) the dipole moment of NO, does not have a measurable dependence on temperature, but the morment of N,O, is non-zero. It is proposed that neither of these alternatives is true. <A study of the dipole mo-~ ment of NO, by means of molecular beam spectroscopy should show that {a} cannot hold, while the fact that N 2° 4 is gyrametric rules out (b). 3. Collin and Lossing (2) observed an ionization potential of NOs by clectron impact. They pointed out that their value, 13.98 ev, ~iiz- is not the lowest ionization potential. It is proposed that 13.98 cor- responds to the removal of an electron from the 3b 2 molecular orbit. 4, The N-N bond in N,O, has been discussed in this thesis. Pauling (3) proposed that an oxygenbridge structure should be more stable | than the symmetrical structure on the basis of the adjacent charge rule, {a} The stabilitics of these two structures are discussed. {b) It is proposed that a detailed molecular orbital treat- ment of NOs would help us to understand the structures of both NO, and NoO4 5. A modified free electron model for the 2s and 2p electrons of triatomic molecules of first row elements is proposed. It enables qualitative predictions to he made about the number of de-localized MO's of each symmetry species, their general shapes, and relative energies. 6& In many problems of mathematical physics, such as those in- volving partial differential equations, the solutions cannot be ex- pressed in terms of familiar or named functions. In this age of computers, however, the possibility of obtaining such solutions to any degree of accuracy becomes a reality. Several suggestions are offered for making the computational results and prograrms of one computing laboratory move available to other scientists. For example, computer programs could be written in a universal language and special translating programs could -113- translate from this universal language to the machine language of any particular computer. 7. Simple valence-bond theory predicts a resonance energy for cyclobutadiene comparable with that of benzene. Craig (4) has com- pared the valence-bond and free-electron treatments for benzene and cyclobutadiene and argues that the value of the exchange integral O@ appropriate to the farmer is not valid for the latier. He claims that a much smaller magnitude should be used. However, it is proposed that the sarne value of ec should be used for both compounds, and that the failure of valence-bond theory to explain the lack of stabil- ity of cyclobutadiene is due to another cause. 8. A problern in topology is proposed - that of finding the total num-~ ber of possible Kekulé-like structures for a condensed polynuclear hydrocarbon. A preliminary study of this problem has revealed the following partial result: ora Kel+c+(i/2ypr clc-i- lye... i=l 4 s Lead 7 3 where Mis the number of Kekule structures, and C and oy are de- & s Ly 2 ¢ 7 z termined from a particular Kekulé structure as follows: C is the total number of closed circuits of alternating single and double bonds; and C, is the nurnber of such circuits which have bonds in i common with 1 other circuits. No higher terms in the expression are needed if the starting structure has at most two incependent circuits. Tt ig usually easy to find a starting structure satisfying this restriction for molecules of up to about six rings. -1i4- 9. Cubic ice is stable ait very low temperatures and hexagonal ice at higher teraperatures. Frank (5) atternpts to explain these facts, and algo the variation with temperature of the axial ratio c/a in hexagonal ice, in terms of ionic and covalent character of the hydro- gen bond. His arguments are based on the work of Keffer and Portis (6) who showed that when nearest neighbors in a wurzite-type crystal of a binary compound are oppositely charged, the distortion of the lattice which lowers c/a also lowers the electrostatic energy (sta- bilizes the crystal). KK is proposed that Frank's explanation is un- sound because the results of Keffer and Portis are not applicable to ice. 10. Axguments have been presented that the origin of life did not require an extremely improbable or cataclyemic event; see, for example, ''Chemical Evolution and the Origin of Life,"' an address by Melvin Calvin (7). It is proposed that in spite of such arguments, God is neceasary in the explanation of the origin of life. REPSRENGCES FOR PROPOSITIONS i, & © Addison and J. pews Journal of the Chemical Society, (1953), pp. 1859-187- 2 J. Gollinand F. P. Lossing, Journal of Chemical Physics, 28, (1958), pp. 900-901. 3. La Pauling, The Nature of the Chemical Bond, Cornell Uni- versity Press, Ithaca, (1945), p. 271. 4, D. P. Craig, Journal of the Chemical Society, (1951), p. 3175- an 7. ~115- EL S, Frank, Proceedings of the Royal Society (London), A247, pp. 491-492, (1958). — and A. B4. Portis, Journal of Chemical Phyeies, 27, re tchomesa pp. 675-682. ne RAS “(1957), M. Calvin, American Scientist, 44, (1956), pp. 248-263.