I. The Vibrational Spectrum of Water Vapor. II. An Analysis of the Vibrational Spectrum of Ethylene Citation Bonner, Lyman Gaylord (1935) I. The Vibrational Spectrum of Water Vapor. II. An Analysis of the Vibrational Spectrum of Ethylene. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/d65m-js48. https://resolver.caltech.edu/CaltechTHESIS:10042024-224650227 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemistry) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Badger, Richard McLean Thesis Committee: Unknown, Unknown Defense Date: 1 January 1935 Record Number: CaltechTHESIS:10042024-224650227 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10042024-224650227 DOI: 10.7907/d65m-js48 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16774 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 07 Oct 2024 17:36 Last Modified: 07 Oct 2024 17:47 Thesis Files PDF - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

I THE VIBRATIONAL SPECTRUM OF WATER VAPOR II AN ANALYSIS OF THE VIBRATIONAL SPECTRUM OF ETHYLENE Thesis by Lyman ~aylord Bonner In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1935 S E P T E l\I B E R 15 , I934 PIIYSICAL REVIEW VOLUME 46 Printed in U. S. A. The Vibrational Spectrum of Water Vapor LYMAN G. BONNER, Gates Chemical Laboratory, California Institute of Technology (Received June 27, 1934) The nonlinear triatomic molecule of the type X Y 2 has been treated and the expression for the energy of vibration has been obtained when second, third and fourth powers of the coordinates are considered in the potential function. The higher powers of the coordinates have been introduced by the use of first and second order perturbation theory. A method has a lso been outlined for treating the more com- plicated types of kinetic energy expressions by a perturbation method. Using the results of this treatment there have been evaluated for the water molecule, from the known spectrum, the primary binding constants and vibration frequencies for infinitesimal amplitudes. Finally, a prediction of the infrared vibration spectrum of the symmetrically substituted heavy water has been made. I NTRODUCTION be more comp letely presented and further results will be discussed. The present case is somewhat complicated by the fact that the kinetic energy expression is of such a form that it cannot be treated rigorously. However, it is hoped that this difficulty has been successfully surmounted by using a perturbation method. It is desired in this article to find the form of the energy expression when the deviation of the potential field from that of a harmonic oscillator is considered. Third and fourth powers of the coordinates will be included in the potential ECENTLY Adel and Dennison 1 have published a quite thorough treatment of the linear triatomic molecule of the CO2 type, and have obtained the energy expression when higher powers of the coordinates are considered in the potential function. In this paper it is proposed to discuss the nonlinear molecule of the H20 type from a similar standpoint. Some of the results of this treatment have already been published. 2 In the present paper the method will R 1 2 l A. Adel and D. M. Dennison, Phys. Rev. 43, 716 (1933) . L. Bonner, Phys. Rev. 45, 496 (1934) . 2. VIBRATIONAL SPECTRUM function. In addition, expressions will be found for tne true mechanical frequencies of the molecule for infinitesimal amplitudes of vibration. A brief outline of the method employed may now be given. First, the classical normal coordinates will be found, which, for infinitesimal amplitudes of vibration, reduce the kinetic and potential energies to sums of squares. The wave equation will then be written in terms of three coordinates - having a simple significance iu the molecule. The potential energy must, however, be expressed in terms of coordinates giving displacements from the equilibrium configura- OF WATER VAPOR 459 tion, and a transformation to these displacement coordinates must be made. The normal coordinates will next be introduced and the Hamiltonian separated into three parts. The first of these will be solved rigorously, and the other two applied as perturbations. This leads to the desired energy expression and to expressions for the mechanical frequencies of the molecule. The known spectrum of water vapor will then be used to evaluate certain of the constants of this molecule. Finally, a prediction will be made of the spectrum of the symmetrically substituted heavy water. CLASSICAL TREATMENT The kinetic energy expression for the nonlinear triatomic molecule has been given by Cross and Van Vleck; 3 and is, for the present case of a symmetric molecule, T = ½(µ1 -Aµlr2 2 sin 2 a)r12+½(µ1 -Aµlr1 2 sin 2 a)i'2 2+½Ar12rl(µ1 2- µ2 2 cos2 a)a 2 + ( - µ2 cos a+Aµ2 2r1r2 sin 2 a)r1r2+Aµ2r1r2 sin a(µ1r1 - µ2r2 cos a)i'ia Aµ2r1r2 sin a(µ1r2- µ2r1 cos a)r2a, + µ1 =m(M +m) / (M +2m), Here r1 and r2 represent the 0- H distances, a is the angle between these two bonds, and m and M are, respectively, the masses of H and of 0. For the present the potential energy expression may be taken to include all terms quadratic in the coordinates, and may be written V 0 = ½a ( (t.r1)2+ (M2) 2) +½b(t.a)2+cM1Lir2+dt.a(Lir1 +M2). Displacement coordinates may now be substituted for the above r1, r2 and a, as follows r1=R+p, where R and {3 are the equilibrium values. 3 P. C. Cross and J. H. Van Vleck, J. Chem. Phys. 1, 350 (1933). 3 If, in the coefficients of the dotted terms in T, the displacements in the coordinates may be considered neglibible compared with the equilibrium values, T takes the following form: The coefficients B, C, D, and E are given by the following expressions: B = /' ~ J, fl /1i R ~ 13 ' 1 - l. C ~ 'll R~ (?, 1/'1. ~ /3) /) :; fl /I/ R. i j:y(fa } -/ u ~fa £ - Yi R?'l~ P 1- In these displacement coordinates, the potential energy takes the fonn The customary norme.l coordinate derivation of the .fumdamenta.l frequencies of a. vibrating system may now be carried out. The procedure is first to apply the Lagrangian differerential equation of moti on, cd cit to the above expressions for T and V. This gives three equations containing the coordinates and their second derivatives with respect to the time. Next there is assumed a periodic solution in which the dependence of each coordi•4te upon the time is given by an expression of the forrn iAt q = Ae • When this substitution is made the following set of q 4 equations is obtained: Rf(a- )1B)+fl~(<--).'-O) + fl<f(d-)'E) -= O Af (c- ).,.D) -f A(j"Co.-).'8) + R<P(cl-)'-£):: o A,P (d-) E) +f!~(cJ- )'E)-1-fl<f)(iJ-AlC) == O 1 The condition that this set of linear homogene ous equations in the amplitudes, Aq , shall have a solution is , of course, that the deterrrd.nant • of the coefficients shall vanish . The solution of this third order l determinant gives th.re.ft values of .A. . The ).. 's are related to the fundamental vibration frequencies of the system by the expression ll::L17T',.. 1~ A -r '-'V i • The /\. 'sin terms of the fundamental constants of the molecule are given be l ow. A12 = }.. 2 2 , 3 41r 2wi2 = (µ1 - µ2 cos (3) (a -c) / (µi2- µ2 2), =471' 2w 22 , 3 =R- 2 (µ 12 - µ l ) - 1 {b(µ 1 - µ 2 cos f3)+½R 2(µ1+µ2 cos f3)(a+c)-2dRµ2 sin (3} ±R- 2 (µi2-µ 22) - 1[ { b(µ 1- µ2cos (3) +½R2 (µ 1 +µ 2 cos f3)(a+c) -2dRµ2 sin (3} 2 - 2R 2(µi2-µ 22) {b(a+c) - 2d 2) ] 4. 2 }..2 is taken with the positive sign. The general solution for each coordinate of the above set of differential equations will be a linear combination of the three solutions , i . e. of the form separate 5 The amplitude ratios /lp, , etc. can be calculated ffom the initial fief. set of linear equations by substituting the values for ).. , , ).i ., )__ • 1 Using these ratios we geh P = fl, e, l ),t + R"l. ei)..1.t rs- = -R, e.i. A.l + fl2. e. i.)....,t We have here three equations involving the three f requencies., and AJ .A, , )"l. so we may eliminate them, two at a time., and find those combinations of the original c oordiaa.tes which involve just one frequency. These combinations a.re., by definition, the normal oo ordinates of the system and are those substitution that reduce T and V to sums of squares. f , o, and <[> in terms of the normal coordinates., Yp y 2 ., y 3 are as follows: P = ~(+di-. +<J3 1.. ' _ - J d +/\,_ /t.,JV:r¼ j3 ~ a-=--~, +ta+ ~3 ;:;_ ' + -Jot + ) 1 Rf!"l/),(Mp f /4 - Y2 A~ R. \v, 1/1'- l<Jof) a J- y.,;i_; R.."-(!, 1/4,_ /fl13 ) V3 6. 460 LYMAN G. BONNER vv AVE MEcHAN1cAL The assumption made above of constant coefficients in T introduces a certain amount of error, and it is desired to show here how this uncertainty may be considerably reduced, and a closer approximation obtained. We may write the wave equation for the system under consideration in terms of the original coordinates, r 1 , r2 and a, following the method of Podolsky. 4 This gives: A af A af + -(µ1 2rl- 2µ1µ2r1r2 cos a+ µ2 2r 12) - + - (µ1 2r12 - 2µ1µ2r1r2 cos a+ µlrl) r1 ar1 r2 ar2 87!'2(µ12 _ µ22) + - - - - ( W - V)if;=0. h2 If, as before, we substitute in this expression the equilibrium positions plus the displacement coordinates, we obtain expressions involving (R+ p)- 1, sin (~+ ip), etc. Although a rigorous treatment of this wave equation is impossible, a sufficiently close approximation to the true solution may probably be obtained by expanding these expressions in powers of the displacement coordinates, and neglecting all powers higher than the second. The justification of this procedure is that, at least for moderately small values of the vibrational quantum nurp.bers, these coordinates are small in comparison with the equilibrium distances. The expressions for the normal coordinates obtained above may now be substituted, and another transformation made to remove multiplying constants, after which the wave equation takes the form: af aif; aif; 871' 2 +c1x1-+c2x2-+c3X3-+-(W- V)if; = 0, ax1 ax2 ax 3 h2 where the a's, b's, and e's are constants involving the equilibrium positions, the binding constants, and thew's. Analytical expressions for these constants will not be given, since they are quite complicated and their contribution to the energy in the present case, is small. All other powers of the x's are rejected in the coefficients, and those given are the only ones which have nonvanishing diagonals in the Hermitian matrix. The advantage of the normal coordinate substitution is that this reduces to second order magnitude the contributions of terms such as a2y;/ ax1ax2 which cannot be treated rigorously and must be applied as pertu.rbations. We will now assume that the Hamiltonian, I-I, may be expanded in a power series in 'Y, a parameter of smallness. I-I =I-I°+-yH' +-y2H" corresponding to the expansion W = W"+-y W' +-y2W''. In J-J 0 we will include V 0 and those terms from the wave equation which have constant coefficients. I-I' contains all possible terms of V cubic in the coordinates, and I-I" contains, in addition to the • B. Podolsky, Phys. Rev. 32, 812 (1928). I, VIBRATIONAL SPECTRUM 461 OF WATER VAPOR quartic terms of V, the terms from the wave equation containing the coordinates explicitly. If we assume the potential function to possess the same symmetry as the molecule it follows that X1, being an odd function of the r's, may appear in V only to even powers. The terms in the expansion of H may now be written: The zeroth order equation separates, and may be solved at once, leading to the energy expression in which the V's are the vibration quantum numbers. The y;'s are then the regular Hermite functions of argument x;. If W' is the first order perturbation energy, we have the expression Since the method of obtaini'ng the necessary matrix elements of the Hermite functions is well known, it need not be gone into here and only the results will be given. For the total energy of the system we obtain an expression of the form where the coefficients X;; are given by the following equations: h { w1 a12 w1 a13 w2 a21 w2 a23 w3 a31 w3 a32 a22 a33 • - - + - -+ - - + - -+ - - +- - - - - l67r 2 w2 2 W3 2 w1 2 W 3 2 w1 2 w2 2 4 4 Xo=½(w1+w2+w 3)+- - . llf2+i2+J2+6Ji+6jj+2ij g2 i2 j2 k2 8w3 4(2w1+w2) 4(2w1+w3) 4(2w2+w3) 4(2w3+w2)' 462 LYMA G . BONNER T Jg2 g2 Ji2 i2 8(2wt+w2) 8(2wt-w2) 8(2wt+wa) 8(2wt-wa)' h {wt w2 w2 wa } 15e2 +6eg+6ek j2+3fj+ij X2=w2+-- -a12+-a2t+-a2a + -aa2 +a22 +½(3m+p+s)-- - - - - 16-1r2 W2 Wt W3 W2 4w2 2w3 g2 g2 3j2 j2 k2 k2 4(2wt+w2) 4(2wt-w2) 8(2w2+ws) 8(2w2-wa) 4(2wa+w2) 4(2wa-w2)' h { Wt w2 wa w3 } 15f2+6Ji+6Jj X a= wa+-- -ata +-a23+-aat +-aa2+aaa +½(3n+q+s) - - - - - - 2 1671" W3 W3 Wt W2 4wa i2 2w2 i2 j2 j2 3k2 k2 4(2wt - wa) 4(2w2+wa) 4(2w2-wa) 8(2wa+w2) 8(2wa-w2) g2 3 g2 ~ t Xn=-l---------+ 2 2w2 2wa 8(2wt+w2) 8(2wt - w2) h 3 15e2 j2 X22 =--a22+---m----1671" 2 2 4w2 2wa k 2 +3ek+gk i2 i2 8(2wt +wa) j2 , + 8(2wt - wa) ' j2 - - - - + - - - -, 8(2w2+wa) 8(2w2-w3) h 3 15j2 k 2 k2 k2 Xaa=--aaa+-n----------+----, 1671"2 2 4wa 2w2 8(2wa+w2) 8(2wa-w2) g2 g2 i2 i2 2(2wt - wa) j2 k2 k2 2(2w2-wa) 2(2wa+w2) 2(2wa-w2) In spite of the app1.rent complexity of these expressions, they are readily solved for the w's, leading to the result These equations for the w's in terms of experimentally determinable quantities are perhaps the most important result of this analysis, since it is from these that the zeroth order binding constants in the potential energy expression are evaluated, and it is about these constants that the greatest interest centers. APPLICATION TO THE SPECTRUM OF WATER VAPOR We have now obtained an expression which should fit the known vibrational energy levels of triatomic molecules to a fair degree of accu- racy. We have also obtained expressions giving the true mechanical frequencies of the molecule in terms of the coefficients in this energy expression. It is desirable to check the validity of the energy formula over as wide a range as VIBRATIO AL SPECTRUM possible in order to determine its general usefulness and the degree of reliability to be attached to the results. For this purpose the spectrum of water vapor is at present most favorable. In the case of water seventeen infrared and visible vibration-rotation bands are known, and have been measured with great care. In addition the recent work of R. Mecke, 5 Baumann and Mecke, 6 and Freudenberg and Mecke 7 on the rotational analysis of these bands should give the positions of the band centers with considerable accuracy. The figures for the band centers have been taken from Mecke's paper, and nine of the best known bands, distributed throughout the spectrum, have been selected to determine the coefficients X;;. Xo cannot, of course, be determined in this way, since the given band positions are differences between the given energy level and the ground level. In Table I are given the positions of the bands, TABLE I. Positions of vibrational bands. D esigPosition (crn- 1 ) nation calc. V, V2 Va obs. 0 0 0 1 1 1 1 1 3 0 0 1 0 0 1 1 2 0 1595.5* 1 1595.5 3152.0* 2 3152.0 3604. 6 0 (3600) 3756.5 * 0 3756.5 1 5332 .3 5331.0 7255.0 0 7253.0 1 8807.05 8810.6 0 10613.12 10613.1* 0 11032.36 11032.5* Designation Vi V2 V , Position (c rn - 1 ) calc. obs. 1 3 1 3 1 3 1 1 3 12151.22 12149.8 12565.01 12565.0* 13830.92 13830.8* 14318.77 14318.8* 15347.91 15348.6 15832.47 15832.4* 16821.61 16827.4 16899.01 16908.1 17495.48 17464.7 2 0 3 1 3 1 3 4 2 1 1 0 0 1 1 2 0 0 * Indicates the band cente rs used in evaluating the constants. observed and calculated, together with the designation of each in terms of the vibrational quantum numbers. The fundamental (0, 1, 0) is not observed due to heavy overlying of that region by neighboring stronger bands, but is predicted at about 3600 cm- 1 from combination relations. The value given above of 3604.6 cm- 1 confirms this, and gives a more reliable figure for the actual frequency. The check here is satisfactory, within experimental error for all cases but the last two. There is considerable doubt as to the complete • R . Mecke, Zeits. f. Physik 81, 313 (1933). 6 Baurnann and Mecke, Zeits. f. Physik 81, 445 (1933). 7 Freudenberg and Mecke, Zeits. f. Physik 81,465 (1933). OF W ATER VAPOR 463 correctness of the rotational analysis of these two bands, which appear very weakly in the solar spectrum. This agreement between the observed and calculated values of the band centers justifies the use of the coefficients X;; in further calculation. The values of the X's which have been calculated are: X1=3796.0 cm-1, X 11 = -39.S cm-1, 1 X2=3674.8 cm- , X22= -70.2 cm- 1, Xa=1615.0 cm- 1, Xa3 = -19.S cm-1, X12= -106.1 cm- 1, Xrn= -21.0 cm-1, X2a = -18.9 cm-1. If these quantities are substituted in the formulae of the previous section, we obtain for thew's w 1 =3899.0 cm- 1, w2=3807.S cm-1, wa = 1654.S cm- 1. We are now in a position to calculate the zeroth order force constants in the potential function, from the normal coordinate expressions given earlier. However, we have only three relations from which to calculate the four constants therein assumed. The reasonable assumption may therefore be made that the constant d is small in comparison with a, b and c, and has, in this case, been taken equal to zero. The values for the equilibrium positions of the molecule have been obtained from Freudenberg and Mecke's 7 extrapolation to the vibrationless state, and were /3=104° 36' and R=0.9558A. The equations may now be solved and we obtain a=8.233Xl0 5 dynes/ cm, b/ 2R2 = 0.376 X 10 5 dynes/ cm, c= -0.0757 X l0 5 dynes/ cm. bis divided by 2R 2 to give dimensional similarity. The above values of the force constants are in fair agreement with those obtained by Van Vleck and Cross 8 by an entirely different, and somewhat less accurate method. It is also of interest to note that the constants a and b may be obtained with fair accuracy from the relation 8 ] . H. Van Vleck and P. C. Cross, J. Chem. Phys, 1,357 (1933 ). /0. 464 LYMAN G. proposed by Badger 9 between force constant and internuclear distance. Three of the nine relations given by the X's have now been used, and it should be possible to calculate six higher order constants in the potential expression. However, it seems scarcely worth while to do this, since the additional information concerning the nature of the function would be small. It seems desirable to leave this until the nature of the interaction between rotation and vibration has been studied. PREDICTION OF THE SPECTRUM OF HEAVY WATER It may be assumed for the present that the equilibrium positions and force constants of the water molecule are unchanged by the introduction of the hydrogen · isotope of mass two. If this assumption be made we may calculate the mechanical frequencies and convergence factors of the symmetrical heavy water. Putting the changed masses in the normal coordinate expressions for the w's given above, we find the following values w1=2865.4 cm-1, w2=2764.7 cm- 1, ws = 1209. 7 cm- 1. The ratios of these three frequencies to the corresponding frequencies for ordinary water are rather close to 1.37. Making use of this fact, it is possible to calculate approximate values for the Xii. It will be assumed that the force constants l-s appearing in H" are small compared with the higher order constants, and that the contributions of the terms from the kinetic energy perturbation are negligible. Further, it should 9 R. M. Badger, J. Chem. Phys. 2, 128 (1934). BONNER be noticed that due to the substitutions previously made the coefficients e-k of H' must be multiplied by terms of the order of wl before the real force constants are obtained. These assumptions, then, make the factors X11,.X22, etc., but not Xi, X2, Xs, homogeneous functions of the order w2. Using the average value of the ratio of w's given above, we see that we may expect these X's for ordinary water to be approximately (1.37) 2 or 1. 78 times the same constants for heavy water. From these six values, and the values of the w's, X1, X2 and Xs may be readily calculated. This gives X1 = 2810 cm- 1, Xu= - 21.2 cm-1, X2=2694 "cm-1, X22= -37.7 cm-1, Xs= 1188 cm-1, X 3s= -10.5 cm- 1, X12= -56.9 cm-1 , Xis= -11.3 cm- 1, X2a= -10.1 cm- 1. The positions of the three fundamentals so calculated are 2790, 2655 and 1180 cm-1, respectively. These figures fit the band observed by Casselman 10 at 4.2µ very closely as being the first overtone of va. / 0. RL. ~ ' P~ R.w-. ~, ;;.;;.1 (lr11) II. Bartholom; and Clusius 11 have investigated the infrared spectrum of D2 0 vapor in the region 2 - 10 ).A and found the two. fundamentals ~ and ~ to lie at 2775 and 1185 cm- 1 with a probable error of 5 cm-l Rank, Larsen, and Bordner 12 found one line in the Raman spectrum of D2 0 vapor at 2666 which is almost undoubtedly~ . Considering the crudeness of the above prediction the a greement betvreen t he theoretical and observed fre quencies is quite satisfactory . In conclusion a few remar ks should be made concerning the Raman spectrum of H2 0. This spectrum has been carefully studied by Bender 13 , by Rank., Larsen, and Bordner 12 , and by others. It seems quite certain that H2 0 vapor possesses just one line at about 3655 cm -1 , although all three frequencies are permitted in the Raman effect. It vrould seem most likely that this were the fre quency ~ , but it has been shovm above from the combination relations of i nfrared bands that 1 lies e.t 1 Furthermore, a frequency of 3655 cm-1 cannot be made to 3604.6 cm-. fit in with the infrared vibrat ion, analysis. As yet no satisfactory explanation of this state of a.ffairs has been found although the fact that the fundamental fre quency of the OH molecule lies a.t 3652 cm-l may be significant. Some support of the 3605 cm-l interpretation is given by the good a greement between the value of ~ for D2 0 calculated from this fre quency and the observed Ra.man lines of this molecule. 11. Bartholome and Clusius, Naturwiss. 22, 420 (19~4) 12. Rank, Larsen, and Bordner, J.Chem. Pli'ys. 2, 464 (19)4) 13. Bender, D., Phys. Rev. fl, 252 (19,5) • II .An .Analysis of the Vibrational Spectrum of Ethylene 13. The first investigation of the infrared absorption spectrum of ethylene gas was made by Coblentz 1 in 1905. Using a prism spec- trometer he was able to locate a number of absorption maxima but did not resolve any of the fine structure. In 1928, Levin and Meyer 2 reinves- tigated the spectrum of this molecule, using higher dispersion instruments and ta.lcing greater care to obtain pure gas and to exclude atmospheric effects. In the region 2 - 15 ~ they found a total of ten absorption bands, most of which were at least indicated on Coblentz 1 prism curve. The spectrum of ethylene in the photographic region of the 3 infrared has been investigated by Badger and Binder, who found only one band, at 8720 A. The most recent work on the Raman spectrum of this molecule 4 was carried out by Dickinson, Dillon, and Rassetti . They used the gas at high pressure and mercury resonance excitation and found a total of six lines. Any attempt to obtain a complete vibrational analysis of ethylene, for force conste.nt evaluation, showed, however, that there were a good many gaps and inconsistencies in the experimental data. There has been made, then, in the present work, a complete reinvestigation of the photographic infrared and Raman spectra. The results of this work lead to a complete and quite satisfactory determination of all the fundamente.l frequencies of vibration of the ethylene molecule. Ten new absorption bands have been found in the photographic infrared. Experiments using liquid ethylene have shown the presence of three new Raman lines., and have failed to show two of those reported by Dickinson, Dillon, and Rassetti 4 . EXPERIMENTAL All infrared absorption experiments were done with the stainless stee l absorption tube and twenty-one foot grating spectrograph recently constructed in this laboratory. The tube is approximately seveneighths inches in diameter and has a maximwn length of seventy feet. The full length was used in investigations of all regions of the spectrum except that beyond 10,000 A. For this region, due to the low sensitivity of a.11 available plates, it was necessary to reduce the pa.th length to twenty-eight feet. Pressures of gas used in the tube ranged from one to seven atmospheres and the entire spectral region investigated lay between 6,000 A and 12,000 A. Eastman spectroscopic plates were used for wave lengths below 10,300 A, but for longer wave lengths their sensitivity was too low, even with the reduced tube length. The Eastman plates were always hypersensitized before using by treating for one mi nute with a 4o anunonia. solution at 10 °C. One series of measurements was ma.de in the regi on 10,000 A - 12,000 A using special Agfa infra.red plates very kindly supplied by Dr. T.J. Dunham, Jr. of the Mt. Wilson Observatory. The ethylene used in these experiments was a commercial product, supplied for anaesthetic purposes , and guarante e d by the manufacturers to be better than 99% pure. Raman scattering experiments were made on gaseous ethylene using ). = 2537 A excitation, but under these conditions only the three strong lines reported by Dickinson, Dillon, and Rassetti were found. Experiments were then made with the liquid in the hope of obtaining a larger number of shifts. 'l'wo types of experiment were ma.de, one using glass apparatus and and ,A.. ).. = 4358 A excit ation, and the other with quartz = 2537A. In each case liquid ethylene wa.s confined in a tube about 15 mm. in diameter and 20 cm. long. '!'his was then immersed in a Dewar flask filled with a mixture of alcohol and solid carbon dioxide. At this temperature, -78°c, the vapor pressure of ethylene is a.bout 3. 5 atmospheres. The De,var was silvered on one side only, and the light from a mercury arc lamp was passed through the clear side. The scattered light was then focussed by means of a prism and lens on the slit of a prism spectrograph. For the glass apparatus, the dispersion of the spectrograph was about 50 A per mm., and for the qua rtz, 10 A per mm . Also, in the case of the visible excitation, an aqueous solution of quinine and CoS0 4 was placed between the lamp and the Dewar to serve as a filter, thereby restricting the exciting light as closely as possible to the line 4358 A. A total of seven lines was observed. Three quite strong ones were measured with frequency shifts of 3009, 1619, and 1341 cm- 1 . Dickinson, Dillon, and Rassetti measured these same lines in the gas at . . 3019 3 , 1623 3 , and 1342.4 cm- 1 , so it may be seen that. for the low ~ frequency lines particularly, the shift in passing from the gas to the liquid is not large. iiowever, gas values will be used where~er possible. In addition to the three strong lines there were four weak ones measured at 3069, 2880, 1654, and 950 cm- 1 . The last one in particular was extremely weak and diffuse, and in consequence the measurement may be in error by as much as 10 cm- 1 • The line at 2880 cm- 1 was observed by the previous investigators and was given by them as somewhat weaker than a pair, also observed by them, at 3240 and 3272 cm- 1 . Since in these experiments the line at 2880 showed up quite definitely but no trace was obtained of the other two, they may probably be disregarded. ASSIGNMENT OF FREQUENCJES In Fig. I are shovm diagrams of the twelve fundamental modes of vibration of ethylene as given by R. Mecke 5 • The fir3t pair of letters following each mode gives its symmetry to rotations of 1i about the axes of greatest and least moment of inertia respectively. Thus mode 4 is antisymmetrica.l to a rotation of 180° about the axis of greatest moment of inertia (the axis perpendicular to the plane of the figure), but symmetrical to the same rotation about the axis of lea.st moment of' inertia (the figure axis). Of the letters in brackets, R indicates that the fundamental is active in the Raman spectrum, I that it is active in the infrared, and In that it is completely inactive. Since the molecule possesses a center of symmetry, no modes may be active in both Raman and infrared spectra. Linear combinations must be made of the bracketed modes to obtain the true vibration forms. From the figure it may be seen that, on the basis of symmetry, there are three types of fundamental active in the infrared, and each of these is typified by its distinctive band envelope. The type AS is 17 Figure I The Funde.mental Modes of Vibration of Ethylene Y, SS (R) SA (R) SS (R) SA (R) SS (R) AA (I) AA (I) + AS (I) + + AA (R) SS (In) recognized by its sharp central maximum, or Q branch. The type AA. has no central maximum and appears., under low r esolution., as a widely spaced doublet. The Type SA can have no real Q branch., but may appear to have a small one due to piling up of lines near the center of the band. These two last types are distinguished., under low resolution, by a series of widely spaced maxima.. The three modes ~ ,~ , ~ a.re of the completely symmetrical type., and should appear very strongly in the Raman effect, so to these may be assigned at once the three strong observed Raman lines. The assi gnment of frequencies to the remainder of the modes is not at all such a clear cut case. Table I shows what seems a.t pre sent to be the most reasonable assignment of frequencies to the fundamental modes and to the combinations and overtones appearing in the infrared a.nd Rarnan spectra. In assigning the infrared frequencies the following principles have been used. First, if two or more modes are excited simultaneously the syrnmetry of the combination level is the product of the symmetries of the individual modes. The multiplication of symmetries obeys the s~~e law as the multiplication of signs: S x S s x A = = S., Ax A = S., A. The type of the band resulting from a transition to this combination level is that associated with the symmetry of the level. Thus., if a mode of symmetry M combines with one of symmetry SA the resultant level is of symmetry AS and the band resulting from a combination of this with the ground level is of the Q branch type. It follows from this that any combination leading to the symmetry SS cannot appear in I 'i. the infrared. The second principle is that no two inactive fundamentals can combine to give an infrared active band. In Table I the columns headed v , V2 , etc. give the quantum 1 numbers of the upper level. The lower level is in all cases the ground level. 1'he column headed "Type" tells whether the band possesses a Q branch (Q) or has a doublet structure (D). The infra.red band at 949.7 cm- 1 is almost without a doubt not a single band but a superposition of two fundamentals. In fact, from the ,- 2 absorption curve published by Levin and meyer, the maxima belonging to -1 each may be picked out, and from this the spacing seems to be abcmt. 10 cm It is interesting that three fundamentals seem to fall in the re gion of 950 cm- 1 , and this fact leads to some ambiguity in the assignments. It is in some cases impossible to tell whether a given band is an harmonic containing ~ , ){ or ~ 0 • The frequency )1r has not yet appeared in the spectrum although it should be active in the Ra.man effect. However, the ini'rared band at 2047 cm- 1 is abnost without a doubt a combination of this with one of the 950 cm- 1 frequencies, thus fixing ~ -1 at about 1100 cm ))j'~ cannot appear alone in either the infrared . = 6 or Raman spectra, but ~ucken and Parts, from specific heat data, have estimated it to lie in the re gi on 750 - 800 cm- 1 . If the weak absorption reported by Coblentz 1 at 1724 cm-l is real, it could be a combination of ~r r,~ vrith one of the 950 cm-l fundamentals, thereby 1ocat·ing at about 775 cm- 1 , in good a greement with Eucken and Parts. other possibilities are, however, that the weak Raman line observed in the present work at 1654 cm-l is either the first overtone of this frequency, which . rJO. I TABLE Assignment of Transition Frequencies Frequency( cm- 1 ) Type Vl v2 v3 v4 v5 V v7 v8 v9 vlO vll vl2 D 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3 1 0 2 1 1 3 3 3 5 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 2 3 1 2 2 0 0 0 0 0 1 0 0 0 1 0 0 0 3 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 6 Infra.red: 950 940 1443 .9 1724 1889.7 2047.0 2988. 2 3107.4 4207.9 4324.3 4515.5 4729.0 8758 * 8982 * 9134 * 9776 * 10119 * 10286 * 10366 * 11465 11724 * 11780 * 14100 * D Q ? Q Q Q D D D D D Q Q, D Q, D Q Q D Q, D ? 3 Ra.man: 950 * 1342.4 1623.3 1654 * 2880;1 3019.3 3069 * Frequencies marked* are re ported for the first time in this pa.per. oU. would be permitted, or a combination of it with a 950 cm- 1 band. This would give to Y,~ the values 825 or 700 cm- 1 • Considering the nature of the specific heat estimation either one of these would be in about as good agreement with experiment as the above value of 775 cm- 1 . It is at present impossible to make a unique choice and to state definitely the value of the ~ fundamental. However, the Ra.man line is certainly real, and there is considerable doubt as to the infrared band. Also the overtone would, in this case, be more likely to appear than the combination, thus giving some wei ght to the 825 cm-l value. In concluding this section, it might be to the point to give a table showing the frequencies at present assigned to all the fundrunental vibrational modes of the ethylene molecule. Table II vibration frequency vibration frequency Y, 1623.3 Yi 950 Yi 3019.3 ~ 3107.4 Yj 1342.4 ~ 950 --;-., 2988 . 2 "'Vi'o 940 Y's- 1443.9 "Yi, HOO ~ 3069 °Yicl 825 EVALUATION OF THE FORCE CONSTANTS The most important results to be obtained from such an evaluation of the fundamental vibration frequencies of a molecule as has been made in the previous section, are the values of the force constants of the bonds holding the atoms together. These force constants, together with such constants for the interaction forces between bonds as may be obtained are of interest for the deeper insight they give in to the nature of intra-molecular forces. For a really accurate evaluation of bond force constants the mechanical frequencies for infinitessimal amplitudes of vibration should be used, but for most molecules, particularly the more complicated ones, the data are insufficient to bbtain these, so the observed frequencies must be used. It is probable, though, that for most types of vibration the deviations will not be so large as to affect the calculated force constants greatly. The expressions relating the fundamental frequencies with the bond force conta.nts and other constants of the etpylene molecule have been derived by Dr. E.B. Wilson, Jr. Figure II shows the assumed structure of the ethylene molecule. 1 • = carbon • = hydrogen C _J_ __ _ -----d--- A somewhat simplified potential function containing eight ccmstants may now be written down as follows: l V~ K (D. R)"°+ q/Uio. sif+Ha._ 2. (c,,oc,f + J, a~ 2-?/+ J_ L [ 6S, .LIS~+ 6S, LIS~1+ )__ ect-[ L,oc, LloC.,_ +f'ioc; bx:~ l Here + J.11 ad. [6 a,£le(3 +.6oc~_LlCC't} + l n 0-d-{f\µ3 +)Ai/'If] b. R refers to stretch of the C-C bond, 6Si to stretch of the .6 oCi to bending of the C-Hi bond in the plane, and ~ L C-H. bond, l. to bending of the C-Hi bond out of the plane. The secular determinant in terms of the constants and of the frequency parameter A factors into sub-determinants corresponding t o the bracketed modes of Fig. I. These factors may now be written down for each of the sets of modes. The following abbreviations have been adopted: m = mass of hydrogen M - (:s = cos _oc m ~ N = 1 + 2 My Y = sin OC. F=I + E mass of carbon = 2 !!!,M a +b b(? y I i*(a.~111f- Q:: I + d---~ f3~ Y,J~J~ K -t-f '\<{ +L) + Y-__ (H+0-t/}.. )- ½fl\)_ j (i+L) Y( H-1 0 -ti\) -p(1+L) (q+L)-n,1,A_ 0 0 ( H-10-1-/1 )-Wt), y{H48-1f\) =0 ~ ){ I = ~~ [ Qq/ +N /1'-r KQl+!V!I')~ ~fll (NQ-p•t] 1 I Cf; == <iJ + L I H-= H1 0 -ll ~,}? 11 f ==-:}_; [ Nf+Ff/''+ ~ri11+Fll"t-11 fl {NF-["l] 11 ~II~ 1;-L 11 f-/ = H-0-11 ~ Yq I )__ = -/m [ N({ + QH'"± q/':: 1J- L 'Yto gN{'i Qll"f- tt q//l'"(NQ-P)t] H H- (J +A Ill::: If we assume the dimensions of the ethylene molecule obtained by Badger 7 to be correct, viz: C-H = 1.04 A, C-C = 1.37 A, and c;H \H angle - 126°, we a.re in a position to compute values for these constants. It is to be noted that the five constants q, L, H, 0 and 11 occur in all the expressions for the first nine frequencies so there will be ample opportunity for checks on their values. Due to approximations in choosing the potential function, the cubic factor Yi,){,~ has no real solutions for the force constants. However, by a. variation method, it has been possible to find the values of the force constants which give frequency values fitting most closely to the observed values. These a.re probably quite close to the true values. In Table III there a.re given the values of the force constants or force constant combinations occurring in the expressions for each frequency set. TABLE III ,,,,#.., 1:,3 K = 8. 2 x 10 ,~ --1- H+0+L1 = 5 1.18 X 10 dynes/cm 5 ii,/; q + L ){,/; q - L = 5.08 X 10 5 5 = 4.88 X 10 Yi, Yq q - L = 5.02 X 10 fl. 0 45 X 10 Yio J..-f I,; )i-il = 0 37 X 10 11 11 5 11 5 11 5 5 dynes/cm 11 0 0 q + L = 5.09 x 10 11 H+0-11 = 1.09 X 10 H-&-A 0 34 X 10 H-0tA = 0.52 X 10 5 5 11 11 0 5 11 It can be seen that the values of q +Land of q - L calculated from two different frequency sets check satisfactorily. Also, if the values of fl, 0, a~d i\. sets, the value of be determined from the second., third., and fourth H+0+i\. so obtained is 1.26 x 10 6 as compared with 1.18 obtained above., which is satisfactory when the nature of the solution obtained for the cubic factor is considered. It is now possible to give a table of the values of all the individual force constants used in this work. TABLE IV K = 8.2 X 10 e= 0.38 5 q 5.02 A = 0.09 L 0.01 h H = o.so X 10 5 0.41 .n = 0.04 The interaction constants are all quite small except for the constant 0 . The size of this, comparable with the primary constant H., is a result of the definition in the potential function. The half angle in the CH2 bond was used as the coordinate., and hence interactions between the two half angles of the same bond appear quite large. A different choice of coordinates., one in which the entire angle was used., would lead to a much smaller value of the apparent interaction. ACKNOWLEDGEMENTS I wish to express my appreciation to Professor R. M. Badger for the valuable advice and assistance he has given throughout the course of this investigation. I wish also to express · my · gratitude to Dr. i. B. Wilson, Jr. for his kindness in permitting me to use the ethylene frequency forrnulae in advance of publication. BIBLIOGRAPHY 1. Coblentz, W.W., Pub lications of the Carnegie_ Institution of Washington, 35 (].905). 2. Levin, A. and Meyer, c.F., J.o.s. A. and R.S.I., 16, 137 (1928). 3. Badger, R. M. a.nd Binder, J.L., Phys. Rev., 38, 1442 (1931). 4. Dickinson, Dillon, and Rassetti, Phys. Rev., 34, 582 (1929). 5. Mecke, R., Zeit. physik. Chem., Bl 7, 1 (1932). 6. Eucken a.nd Parts, Zeit. physik. Chem., B20, 184 (1933). 7. Badger, R. lVI., Phys. Rev., 45, 648 (1934).