Application of Quantum Mechanics to Molecular Structure. Electron Diffraction Investigation of the Structure of Gas Molecules, and the Dependence of Interatomic Distance on Single Bond-Double Bound Resonance Citation Beach, John Youngs (1936) Application of Quantum Mechanics to Molecular Structure. Electron Diffraction Investigation of the Structure of Gas Molecules, and the Dependence of Interatomic Distance on Single Bond-Double Bound Resonance. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/nyk7-yk34. https://resolver.caltech.edu/CaltechTHESIS:11022024-090738724 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): Pauling, Linus Thesis Committee: Unknown, Unknown Defense Date: 1 January 1936 Record Number: CaltechTHESIS:11022024-090738724 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11022024-090738724 DOI: 10.7907/nyk7-yk34 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16830 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 05 Nov 2024 20:09 Last Modified: 05 Nov 2024 20:23 Thesis Files PDF - Final Version See Usage Policy. 18MB Repository Staff Only: item control page

Applica tion of Quantum i\<Lchauics to Molecular Structure Electron Diffr action Iuvest.igation of -the Structure of Gas f.[ol ecules and The Dependence of I nt er atomic Distance on Single Bond- Doubl e Bond Resonance Thesi s 'fr.J John Y. Beach In Partial Bhlfil.lment of the Requirements f or the Degree of Doctor of Philosophy Calif ornia Insti t ute of Technology Pasadena , California 1936 Contents Page Propositions /. Application of Quantum Mechanic s to Moh::cula r Structure Introduction If Quant.um · echanicul Treatment of Hel ium Hydri de I/ Molecule I on Van der We..al s Forces bet\s1 een Hydrogen Atoms 'J. 6 El ectron Diffr action Imresti gation of t..he Structure of Gas 33 Molecules Dependence of Interatornic Dist ance on Singl e Bond- Doubl e Bond Resonance '-/6. /. PROPOSI TIONS 1. The electron diffraction method i s the most generally ap_pliea.ble met.hod f or deter mining the structure of gas mol ecule s . 2. The mole cular structures of the chloroethyleues afford evidence f'or the r esonance of molecules among differ ent el ectronic structures. 3. The v&.n der Vl aals forces between hydrogen a toms c an be conveniently calculated. by the use of 2. set of ortho-normal f im.ctions which ar e differ ent from t he hydrogen atom wave f unctions. 4. The molecule-ion" Hea+, is st abl e, t he stability being due to the partial f ormation of c:.. cov<.ilent bond nnd to the pol.e.I'iza.tion of the helium atom. 5. The molecule-ion HeH++ , i s unste.bl e . The mole cule-ion HeH2 ++ i s probabl y stable i f constrained t o the symmetrical linear con.fi gura tion but is pro ba bl y u.." lstabl e with r espect to di ssociation into HeH+ cand H+. 60 The group V pentahe.lide molecules are trigonal bipy r ami ds o 7. I n spite of the gen e r a l applicf.:.bili ty of Pauling ' s theory of bond directions it does not lead to unambiguous resul ts in the ca se of' the group VI t e trahalide inolecul es a.D.d ion s i soelectronic with them. 8. The treatment of double bonds by double bond orbitals is simple analytically but faces serious theoretical difficulties and, in the case of POC1 3 , is in disagreement with experimento 9. The existence of covJ.ent compounds and i ons containing the inert pai r is due to the utili zation of excited orbitals for bond f'ormation 6 10 . I n the Al( OHh sol a.nd the C:r(OHh polymeriz,ation products the oxygen bonc:s shoul d be equiv&lent, contr· ry to the picture given by Thomas1 A 11. It is time to stop writing nitrogen with f i "ITe covalent bonds o 1 2. The explanation of .Lessheim and Samuel 2 for the incr ease in dissociat,ioo energy of NO upon exciw.tion is incorrect o The "lambda pointn of liqu.i< helium w.1d other simila r phenomena are nei ther second order nor third order phase equilibr i a but, exhibit some of the characteristics of both~ 1 4~ Chemistry l at C.I. T. wou.l d be improved if the lec·tures and teJ<,.,-t books placed more emphe.sis on writing che!nictl equations in the ionic form . 1 5. The r ailroad compani es should speed up t.he service between Los Angeles and Sc.n :F'rancisco by operatine; buses between Los Angeles and Bakersfield which woul d meet the trains at Baker sfiel d . Application of QuantU.!ll Mechanics to Molecular Structure Introduct;ion The f i r~;t of the quantum mechanice l cal culati ons of the stability o:f molecules 1u1s the t r eatment by Burrau1 ) in 1927 of the hyarogen molecule ion, H2 +. PrevioU f'. classical treatment of this •probl e:m by Pauli 2 ) and Niessen 3 ) had f ailed to give re : i+i.l lts competibl e with experiment. Burr e.u.' s cc:.l culation showed t hat the iJOtenti r.J. ener gy of' the syst em for .il l finite i nternuclear di stance s gre,. ter t han <.ibout 0 06 A was less t han the ener gy of the hydrogen atom. The potential ener gy curvepossessed a minimum of . 20•:3 Rhc a t an inter - nucleRr distance of 1.06 A. This r esult was obt&ined by a numerical integrat i on of the we.ve equa•tion after separation of ~r~.riables in confocal elliptic coordinates. Hy 1 ....1.eraas tl) ...J .?...nu Other treatme.ts 'i?er e given by J Q1,...c•.l. e 5) 'mo ., , tmi -, •1 e using • d J•. .1."'f·eren t met hods , arrived a t t he same results. The value o bt aineo. for the dissociation energy after taking into account t he zero point vibrational ener gy i s 2 . 639 ± . 00 2 v . c. whi ch i s i n complete aireemen t ·- i th and probabl y 1 mor e accura t e than the exi sting exper ir:1ental va.lu.e . This calculation was t he f irst theoretical justification of the conce t, of the electron bond, in this Ca.Se a one el e ctron bond . A l ess precise but mu ch si :apl er di scussion of this molecul e was gi vcn by Pa.u1L-ig 6 ) who treated t he probl em by the Vf,r:i.ation me t hod using a Vb.ri ation function of the f oll1Y11ing f orm.: 'f' = a + b. ,,. denotes ~,n el ectron occupying a hydrogen ground st-a.t,e wave f11nction on proton ~ and b •enotes fill electron oecupying a hydrogen ground state -wave function on proton B. Thi s calculation gives a. dissocie.t.ion energy of a bout t wo thirds the observed ~ralue , agreeLr1g quali t· ti vely with experiment~ An obvious interpreta tion of this r esult is that to .:.. f'air ap:proximt'ttion t he stability of the mol ecule i s due t.o t h e r esona.ri.ce or exchange , of the electron between t he -two po sitions given in the va ria tion function. The variot,ion methoct j) due to P..itz 7 ) beforEl the ({dvent of quantum mechanics, will be described in some detail in the section on the helium hydride molecule-ion; its <lis-tinct;ive fee.ture., however 11 i s t .ha.t it gives an upper limit for t,he ener gy of t he system.. The only restrictions on the va ri e.tion func tion a1.·e t hat it obey the bowi ·a.ry conditions and tha t it perm.it the eva lua tion of' the necessary integr al s . Oth.r trea t ments of intermedi a t e precision have been given thi s mol ecule b<J Finkelstein and Horo ·l itz 8 ), Guille1llin &nd Zener 9 ) a nd Dickinsou1 0). This triumph o.f quantum mechanics i1:dthL11 a year of it.s iriception foreshadowed t he subsequent success it, was t-0 meet in trea ting other more complica.t eo mol ecules where the cl ass.ical t.heory invariably failed. I n the s ame yeo.r of the v.ppearance of Burru:J.1 s paper on the hydrogen molecule-ion t he f irst trem:t:.ment of the hydrogen molecule was started by Heitler and London11 ). This ctlcul atian was compl et ed, also during 1927, by Sugiural 2 ) who evaluated an integr.-1.l only estL ated 6 by Heitler ,:,. nd London. The Ifoitl er London Sugiura treai.~raent of the hydrogen mole cule .ras .much le ss a ccura te than the Burra« trea t ment f or t he hydrogen mol e cule -ion, being a va riation trea:t,ment using a varia tion function silliilt..r to the one mentioned above, used in 1928 by Pauling in discussing t he hydrogen molecule -ion. T.b..is c alcula tion sho·,Bd t hat the pot enti• 1 ener gy of the syst em f or all finit e iuternuclee.r di s t ances gr e"'ter than about . 5 A w:a s less than the energy of t \ o i solated hydrogen atoms . The minimum in the potential energy curve was found to be 5 .14 e.v. a t an inte rnuclear distance of . 80 A. This derivc:,tiou of t b.e exi stence of t he zj_on pol ar bond, never reali zed by cl -- ssi ca l mechanic s , .ua.de the electron pair bond , which up to th&t time h ad mer el y been ari assertion made reasonabl e by ch endcticl evidence and bearing no r e l a tion to physica l l aws , a na tural conseqlle?lCe of a generally ap;)licabl e mechanics . This dis so ciation energy , 3 .14 e.v., i s only about t wo t.hirds of the observed ve.lue , 4 . '1 2 e . v., and t he i nternuclear di s t ance , BO A, i s somewhat. gr eat,er t hnn t he observed distance , . 74 A. function USt:;;d was 'f = a (l) b( 2) + b (l) a ( 2) ; a an si ~nificunce a., above. The v,.,,riation b have t.he san1e 1 and 2 refer to the t wo elect,rons. Thi s r esult admi ts a qual itative inte r pr e t a tion aw ogous to t he one made in the ca se of the hydro .~.en . oleeu.le -ion; t hHt i s , that to c: fai r degr ee of approxime.tion the st.a bility of the ;aolecule is clue to t.he r esonance of the mole cule between the t o e4 ival ent conf i 5t1ra:tions 7 given in the variation .fu..ri.c'Gion. ~fore d.ett:.il ed t r eutt.1ents approach- i ng more clo sel y the correct value f or tb.e dissoci ation ener gy of t he w.ole cu l e have been given by Wmig13 ) , Hun ct1 4) , Rosenl 5 ) , Mu.lliken 1 6 ) and W,~i nbaum17 ). The desired a c cur a cy i n t he hydrogen molecule Cfalculci.tion was obt a i ned by J ames and Coolidgel8 ) . Intro- ducing t he interel e ctronic dist.;nce e:xpl icitl ;,r i nto the varir ·t.ion f llllct i on, not done D'J en y of: the other i nvestigators , they obta.ined a val ue f o r the di s·ocia.ti ou ener gy coin cident wi th t he experi ment.al vul ue . The interpretation mttd- above t hat t he bond f o rmation in t he hydrogen mol e ettle end the hycl:r-ogen molecu.le- ion is due t o the reson- &nee of t,b.e molecul e betwE:en t wo 0l ectronic structures is of no theoreti <~a.l si gnific0.nce . In the detailed ci:lcula.tion for these t wo mole cules no thi ng ap~::;ea.rs th&t sug•;es~s ,t~iat it i s ·true. 'l'he int,er- pr etation was based on the v·a r iation t r eatmentt::-1 of the hydrogen mole cule by H.ei tler , London "nd Su gi ur a ~md the hydrogen molecule- i on by Pauling t o whi ch no ux1ique theoretica l signi f :!. cance can be ascribed. I n spi t e of i ts quail t c>.t i ve natu.re , however , i t pr ovi des ru1 extreme l y val uabl e and pr acti cally i ndi ~per sabl e l anguage f or discussing the l a r ge number of mole cul es wher{o: r i gor ous trea t ment i s impo ssi bl e . Ac curacy compar abl e t o t ha.t in ti:rn two CHl cu l ati on s j u st descr i bed has not been r eali zed f or any other molecule. Sev er ,il other mol ecul es have , however , been treat ed by the v-aria tion aethod u sing 8 V,: riation f un.ctions of vnrying com: _) lexi ty and tipper limit s f or their ene r gy have been obtained . The mol ecuL., s for which these ca lculations 23 ) L_-i11 23), have been :aade are: n,.3 +19), HJ 23), He~ ++ 21), He H+ 22), nu+ ~ 9 C"Jd) ?5) , 2o'~) . +~--•• - -I.J.2 , Li~ , He..:· . Li4 is the .mo s t complicnted mol ecule f or which a VHri a tion treatr.-:ient has been made . Many other ,nolecules have been treatBd by more approximd;e methods but 1.vill not be mentioned here~ In the following section t he variation treatment .L or HeR+ will be giYen . l. Oyvind Bur r aa, Det. K._-:1. Danske Vid. Sel skab. 1 , l (1927) 2. H. Pauli, Ann . d. . Phys . 68 ., 1'17 (1922) 3. K. F. Niessen, Dissertation Utrecht (1922) 4. E. A. Hylleraas , Z. f. Phys. 71 , 739 (1931 ) 5. G. ,Jai'fe , Z. f . Phys. 87, 535 (1934) 6. .L. Pauli ng, Chem. Rev . f, 173 (192B) 7. 1-<.i tz, J. f. reine und angewa Mat h . 135, 1 ( 1909) 8. B. N. Finkel stein and G. E. Horowitz~ Z. f. ?hys . 48 , 118 (1928) 9. V. Guillemin; J r. and C. Zener, Proc. Na.t . Acad . Sci. 1 5, ;51 ,'.!: (1929 ) 10. B. N. Dic.kim,on, J. Chem. Phys . !, 317 (19.33) 11. W. Beitler and F . Londoi1, Z. f. Phys . 1:l:4, 455 (19 27) 12. Y. Sugiura , Z. f. Phys . 1§., 484 (1927) 1 3. S. C. Wang, Phys . Rev . 31, 579 (1928) 1 4. F . Hund, Z. f . P 1ys • .§1., 759 (1928 ) ; Th, l (1931) 15. f,. Rosen, Phys. Rev. 38, 2099 (1931) 16. R. S. Mulliken, Phys. Rev. 32 , 18G, 761 , (19t~8) 18 . H. M. J ames and Ao S . Coolidge , J. Chem. Phys . 1, 825 (1933) 19. J . Hirschf elder, H. :E:yring and 1:f. Ro sen, J. Cheiil . Phys. _±, 1:50 (1956) 20. J. Hi rschfelc.er, H. Eyriug u.d N. Rosen, J·. Chem. Phys . 4, 121 (1936) JO Pauling,. J. Chem . Phys. 1, 56 (1953) 21* T .u. 22 0 J. V ~-- 0 E,et:..cl'!, J,. Che. . Pl1ys . June {1936) -i 23 0 Je K. Knipp, J. Cl'1eruo Pb.ya . _, :soo (1936) 3 9 (1935) Jam.es, J ~ Chem. Pl\y-s. _, 24 . H. 25. 1-I. !t . Ju.rue~ , J. Gliem .. PhJs . £:..., " 794 ( 1939 ) 26 . ..,C' . L ei11 br:uru, J. Chem. Ph;;,rs. ~,, , 5<k7 (19 35) '{,J i l:rl ~ QUANTIJ.M MECHANICAL TREATi\1ENT OF HEI,IUM HYDRIDE MOLECULE-ION ., HeH+, // QUANTUM I,lf:CHA.rICAL TR.EA'I':.mHT OF HELIUM HYDRIDE MOUXfJLE-ION hel.P- Singl y char ged helium ns-dride mol ecule- ion has been. known1 ) f or many yea.rs from mf...SS spectroscopic investi gations of helium hydrogen mixt.urea~ I ts ,.,osit,ion wi th respect to t hat of n2H?.Hl+ in the mass spectrogr·aph has been used to determine accur ~.-tely the mass of the deuteron 2). Spectroscopic work has not sho ·n the existence of thi s ion. The onl y _pr eviou s qu 1:;.ntu.m u1och8.llici,. l discussion of BeW i s a variation treatment made by Glockl er and li'uller 3). t wo probl ems: They consider (1) the intera ction of a proton with a n excited helium a tom, t he wave fun ct i ons f or the el e ctrons being r, = e.e - ..... ../1' 1:-.nd \fa = ( b + e ./1~ )e _A5 ../J;i. , and ( 2) the interaction. of l>. hydrogen atom wi t h a singl y ionized. heliu.ru ator.., t he wave functions f or the el ectrons bein ;::,'.l' f irst case o<. and /3 ae ltre - c,1-... ..//a I ., an o. When i n t he put equal to 2 ai:--id Oe 7 r espe ctively the i nteraction i s repulsi ve a t all distcinces o The .function s '-f. and Cf'J.. 2 a r e the 1-2. end 2s hydrogenli.ke vmve functions, the helium a.tom being appr oximately in the first excited s t a t e . I n case t 'RO o<. and /3 a r e put equal to 2 and 1 , t he r e~,u.lting potenti·,1 ener gy cur·ve has a minimum of 8o l e .v. ~t 1.3 "'O • In this calculation the unsym:aetriettl Sugiura type integrul was llet:lected. Ls pointed out by Glockler and Fuller, incluaing t hiB integr'<!:,l will r t..i se the energy b y an undet er- mined amount. The above calculation is not evidence for the existence of Hen+. The dissociation energy 9 8.1 e .v., obtained by making the above approximation , is rti th respect to the dissoci ation products hydrogen atom &nd helium ion, 'Nhile the ciiffer ence in the ioniza tion potentials of hydrogen an..: helium is 10 . 9 t1.v. Under these conditions the molecule would be unstable with respect to dissocint,ion into heli-wn atom anr a hydr-ogen ion. To prove the stability of thf; mole- cule it is necessr.ry t.o shm'! t .. a t f or some inter-medi&te internuclee.r distct..'1Ce the energy of the molecule is l ess than the energy of the helium a:tom c 'fhe variat.iona l integral w =Jr tfd1/f 'f~'f a:n upper limit. for the energy of' a system.. wave .function. .7'--r gives ~f-J '{ is en approximate li is the Ha·niltonian operator and in this case i s !ll When a variation f'uncT,ion of the form f =2_ e <fn 0 is used, n:::l the condition that! be a minimum with res ect to the constants c11 is that the determinantal equation ·/ H1• J( - Ll /\ 1.J ·.; / be sat:isf'ied4 ), where Hij = j lfl H 'rJ· d Y e nd =0 ( eq . 1) =f Lf/'{;- The f,.pproxi, ate E:mergy for the ground state will be the lo·:1est root 13 of this equation. The constB ts, en, are obteined by solving the set o.f linear equations ~ CJ (Hij - D ijW), i = lj2 ..• m. i.iost ,/ :::. 1 of the integrals required in evP , uating Hij mid 6 ij were use Rosen 5) and Weinbau.m6) in discussing the hydrogen mol e cule. by Dr . Weinbaum has not publi shed his integrals but kindly pl e.ced them at my di sposal. The integrc.ls used in this calculation, not ti:ibuls.ted by Rosen 5 ) , are given in the appendix . The varie.tion functions used her e I ing functions: I z3 )~ - Z..//Q., a ( 1) == ( a,/ e. 71 t:l o _ Z../Ja..1 a 1 (1) cos GG:t, e for electron ( 2). .::{,o cu1d the sAme f unctions Each fun ction i s nor'Iae.lizeci to uni t y . E;:.(1) is the ground state wave funct.ion for e_ hy-drogen.ic c1_tom of nuclee..r charge !.• and atisfie6 the equation : Ze.:. ) o.(l} = ../Ja.. I 2 wher e WH = _JL_ • a' (1) 28.o is a first excit ed vrave functi on for a hydro- genie atom of nucleRr ch.t:.trge 2Z a.!ld satisfi ~s the equation Tr...is form of ~ , (1) wa.s u sed by Rosen 5 ) in intro-ducine; ;Jol · riu,.tion into t he hydrogen molecule calcuh.t.ion. It is ver:1 conv"'ni G:i:tt ns i t let:ve~~ t he exl-)011ents in Q.(l) and .§1' (1) i entical makin.i the ctlculation of i nt,egrals much simpler. In the pre::w·nt calculu'tion the s r.,.me ef.f'ecti Ye nuclear charge, 1, is used in each term of the va ria tion functionso This i s done to eliminate '~he necessity of cal.cul·· ting t he unsymmetrical Sugiura integral a..>1d othe:r. integr als. Thi s procedure W"S adopte by Pe.uling 7 ) in treat,ing the helium molecu.le- ion He:.:+ .. It is evident that the stable state of' the molecule is c1. singlet stat.e so we sh?..J.l require t he vari ation function to be symmetric in t.he spatial coordinates of the t wo electrons (and 1:-.nt.isymmetric in the spin coordinates). In the following par agr aphs we shall calcu- l ate t he en0,r 8y of the sy stems for sever,:,..J. S llCh ve.ria tion functions .. It l)!Jil+' be found ;X>ssible, by taking a linear combina tion of the lowest l ying st~tes f or simpl e wave functions, to obtain & calculated vnlu~ of t hr~ energy of the ,uole cule t.ha.t is lower t han t he experimental va lue f or the helium a tom. 1. Normal helium at.om and proton. The mo st stable state .for the S-Jstem f or n single rm.ve f unction of the type we a.re considering , eq . ( 2.), is the state helium a tom unct proton . It will t her efore be necessar-y, if t he lowest vr ue of the energy is desir ed., to include this stHt.e in the c ·lculations. as the electronic wave f unction of the· system If = a (l) a ( 2), equation ( 1) for t he ener gy bHcomes Taking This i n tere.ction i... r epulsive a t all distances and is shorn.1 by curve .Q.., Fi g . 1 f or the c a se t.,.11a t Z == i~. When , e qual s i n.fini t.y this expre ssion beco.m.es the fnmiliar expression f or the ener ts of a helium , tom for hydrogenlike wave functions "Ni th variable effective nu.clear char ge, W = ( 2~ - ~7 Z) 1P.H . minimum ,1hen Z == 27 the 1 W = - 5 o695 WI{ o 16 Thi s energy- is a The e:x,--perimental value ! t1d the value calcula ted by Hylleraas 8 ) f or the helium atom i::1 - 5 . 807 WH• 2. Hydrogen a.tom e.n d hel ium ion. The sy,mnetric vari ation fun ction, 'f = a ( l} b ( 2) + a ( 2) b(l ) , corresponds to an el ectron i;air bond bet.ween t he t wo atoms. energy is shown by curve B in Fig . l f or the ce.se tha t The f == 27/16 = 1. 69. This energy can be mini mized with respect to z_ f or each value of \ When t hi r:; i s done a value of !1_ only 0 . 0L1% lower ·than that f or f = 27/16 i s o bt,dned , the effective charge then being equa.l t o L 72 . When equals infinity the ener gy becomes W = ( 2.Z"i - 6Z) WH which i s a minimum when f ::.: 5/ 2 and !: ;:: - 4 . 5Wa . Th correct value for the sum / of t he ener gies of a hydrogen atom and a helium ion i s - 5 Wr-r . This discrepancy i s due t o the f act that we have t aken the same effective char ge on the t v10 atoms . Thi s calG-ulation i s t..~e same as that made by Glockl er and Full er except that her e t he s ame effective nuclear charge i s used on both atoms and consequently all t he integrals can be ealculated. Thi s /6 curve i ndi ca t e s t h e existence of 0. met6.s t ablc attr a ct.i ve s t a t e for He 3. Exei ted hel ium at om and pr oton . The V"'ria t.ion funct ion t/ = a (l}a ( 2) 4 a '( l ) a ( 2) i n conju...7lc1 tion wi t h a (l) a ( 2) , wi ll be u sed to r e pr esent a £-'O lar i zed heliu.111 8.tom. The pot ential ener gy cor-.ce spondi ng t o t he variation i"u.fJ.ction a t( l ) a ( 2) + a ( l ) a ' ( 2) alone i s shmm by curve fl i n Fig . l .for the ca se t.riat E. = 27/16. This curve is hi gher t han t hat f or the corre s- pondi ng tri pl e Ht a t e , t.i 1 (l) a ( 2) - a ( l) a. ' ( 2) 9 but the t r i pl et; s t,ate does not combi ne wi th t he mor e i .mpor t w1t singl et stE<.tes, non-di agonal mat rix el ements bei rtg equrd to Zl:rro, so c am o t, be used e Vfb.en t he ener g-J f or t he singl et sta te i s minimized with r espect to t he ef fect i ve nucl ear char ge , the ene:fgy curve i s lower ed by about 0 . 55 WH, the corresponding ve.lue of the ef f ective nuclear char ge a t the minimum lYJing l . 26. 4. Combination of 1 and 3. Using tJ! = ~( l) u ( 2) + k ( a '( l) u ( 2) + a ( l ) a 1 ( 2)) &s a vax i a tion f unc'tion, t he ener gy appear s as a r oot of a qm~.d:re.tic secul a.r equat ion , equa tion ( 1). The r esul t, f sol ving t he equa t i on i s shown by curve 11 i n Fig . 2 for t h€: ca se t hat t he effective nucl ear char ge i s 2'7/16. I t i s not po s sibl e to minimi :l.e ead.l y '1'ith r espect t,o t he effec t i ve micl ear ciw.r ge so the quadr-;.tic is me r ely solved f or sever al val ues 17 of the ef'fective charge. I t is f ound that the ene1~gy is worse for any value of the effective char ge mu.ch different th£ n fZ 7/16. The minim.um energy f or thi s curve is VJ == -5 .. 750 .,,t R = L 60 a 0 • The normHli zed function i s a. (l) a ' ( 2)) 9 '/ = 0 . 99 a ( l ) a ( 2) + 0,.l3(a( l ) a ( 2) + The best, va lue o f the energy of t ht;; system. at, ~ e'-l_uals infini t y for t his t:§rpe of v,.;.ria tion f unction i s - 5 . 695 WH, giving f or the dis ~oci ~-tion energy of the molecule O. Q,.)5 'fil'H or O.. 7 4 v . e . The stabili zation re sulting from the u se of t his vnria tion function ce.n be att ributed to a polarization of the helium atom9 5. Combina tion of l and 2o The energy corre sponding to t he vari a tion f u11ction f = a (l) a ( 2) + k ( t=:.(l) b ( 2) + b( l) ::t ( 2) i s s hown in curve g_ in Fi ,.; . 2 f or t he case t ha.t, I = 27/16. The normtlizGd £'unction is 0.97a (l) a. ( 2) + 0 . 24 (a (l) b( 2) + h (l) o.( 2) ) .. f= I t is not. easy to minimi ze with respect to the effective cha.rge so agai n the quadr a tic secalar equation is sobred for sev-e r ul value s of the effecti Ve chl!.rge, t he calculated ener oY being much hi gher when t,i'le effective c 1a rge i s much different than 2'?/1 6 . R =1 .. 60 a 0 , The lowest value obta.ined is W = - 5 .80 5 fft which iYhen subtrac ted f rom t he value -5 . 695 WH f or the helium a tom, give s a dissocia tion energ,1 of 0. 110 Wu or 1. 49 v.e. Tbe form of t he variation function suggests that this i s due to the partial f ormv.tion of a covZcl ent bond o {g 6. Combin:tio11 of 1, 2 and 5 . The best variation functi on used is of the form 'f :::. u( l) a ( 2) + k ( a 1 ( l) a ( 2) + a. ( l)· 1 ( 2)) + ~( a (l) t ( 2) + b( l) a ( 2)) . On the basis o f' c alculi:-.-.tions 5 and 6, in which it yms found that the best value of the effective clw.rbe Wf"'S not far from 27/16, caleulations were 1,1ade o n l y for the, cnse tha t ~ = 2 7 /1 6 . The energy t1.ppears as a root of a cubic secular e .;_uation and is shor:n by cu.rYe D, Fi g . 2 and. D'J curve Q, Fig . L The norm· lized f-w.1ction is '.l = 0 . 97a(l ) a ( 2) + 0 . 2 0 ( a t ( l ) a ( 2 ) + a ( 1 ) ~ ' ( 2) ) + 0 . 10 ( a ( l ) b ( 2 ) lowest energy obtained. is - 5 . 84 ! Wu a.:b 1 + b (l) a ( 2 ))c R = L 57 H-0 • The T_lis , when b'Ub- tracted from the helium atom vr.lue of - 5 . 695 Vi , gives a value f or the dissociation energy of 0.149 WH or 2. 02 v.e . This dissoci a tion ener gy is oht,c,ined by u;:,ing the ener gy of the helium ~,tom calcula ted from a variation functi on of' t,he same type as t,he one used f or the m1dis8ocia.ted molecule . This procedure asswnes that the error in t h e energy is the se.me at the equilibrium dist,ance and at infinity. A lower limit for the dissociation energy is obtained by subtracting t he minimwn energy for the molecule from the experimental value, - 5.807 Wn , fo r the helium atom. 'rhe re sult is 0 . 037 WH or ~50 v . e . This positive resul t proves conclusi vely t.he.t the molecule is stabl e . The vtlu.e 2. 02 e .v . ho\';ever, is probably much clo ser t o t he a.ctu&.l value of the di fJsoc i a - t ion energy .. Fitting a. parabola. tot.he bottom part of the curve leads to ;q a v.:1lue fo r the force constant of 0 ,. 43 megadynes/ cn . a.8 a f undamental vi br ati on i're,.iuency of 2800 cnc 1 . The Molecule-Ion Helf+ Doubl y char ged helium hydride mol e cule- ion was treated by the s ame -type of va riati on fu.:.'1cti on , 'f ;; a (l) + k1b(l) + lrz a '(l) . The r eslll t i ng potential energy cur ve shows no mi.11.im.um and di ffers ~ only slightl y f r om - 4WH + : o The one el ec t ron bond betwe en the t wo a toms i s not Sl.tffic i ent l y strori_<s to 0vercome t he coulomb r epul sion .. Another mol ecul e i u which t her e i s coulomb r epulsion bet ween the at..oms, the heliWll mol ecul e ion, He.a++, h ~ s been trea t ed by P"".uling i ncluding i onic t erm,~ in t he variation f unction. In. thi s case t he el ectron pair bond formed i s s t,rong enough t o ov ercome t he coulomb r epul sion of t he t wo atoms , there bei ng a mi nimUJ in the poten tial ener gy curve . The instability of HeH++ i s due to t he lack of degener c:.cy of t he two s t at e s }Ii 1-r+ an d Ht + H. Di scu ssion From a compari son of curves ~ E:.nd Q., Fi g . 2, i t i s seen t h.at t he introduction of polari zation into the variation f unction i s about hal f as effective i n s t abi li zing t he mole cule as t he introduction o.f the cova l ent bond . It is t her efore qualit,ati v-el y corre et to say t hat t wo t hirds of t he sta.bilizution i s due t o t he f ormation oi' a cov1.:, l en-t. bond and one third i s c:ue t o pol fari za-ti on of t he helium cd:,mn.v The 8--0 strength of the bond, 2 o0 2 e .v. , i s SLlrprisingly grea t in view of the small coeffi cien-ts of t he covalr:mt bond ter::ns rmd t he 1:;olt-;rization t erm, 0. 20 and 0.10, in the v,n•i tion function gi ven in section 6. It is r oughly h,1.lf as stronr; as the bond in the li;/drogen molecule, 4 ~10 e . v-.:, c a lcula ted 6 ) from the same t ype of v .:; ri a tion fu..YJ.ction. I ·,Yi oh to t;ha.nk Pro:fe s~,or Linus Pauling f or mt=:.kin ; many valuabl e s uggestions dLu~ing the prepa r a tion of t h i s paper. References 1. Fo W. Aston, "I sotopes 11 , Both editions, T. R. Hogness and E. C. Linn, Phys. Rev., 26, 44 (1925) 2. Ke T. Bai nbrid.ge j Phy s. Rev., 44, 57 (1933) 5. G. Glockler &nd D. L. Fuller, J . Chem . Phys., !, 886 (1953) tl. See for example Pauling and Wilson 11 Introduction to Qu al'ltl.L"!! I,1echanics 11 • 5. , r.i '7.e ~· ,~a ~ ~g (1°~ ~) N . R,osen, Pnys. nev., -~' .., ,'.l.L 6. S . Wienb~ um, .J. Chem . Phys ., 1, 593 (1933) 7. Linus P1:i.uling , J. Chem. Phy::: ., 1., 56 (1933 ) 8. E. A. Hyller aas, Z. f . Phys., 65, 209 (1930) Appendix The simpler integr,ds a.re orui tted . The integrals, M001 , 0 and P.io10,o, are evaluated by the use of the Neumann expansion in confocal elliptic coordinates . The f unctions H(m,n, ) end S(m~n) are defined by Rosen5) . · -f illoo1 o ' 6 / (1l1 a • (• 2) b(•?.) av, , , _ ( .. + z C ,., H( 5, 4, • f) + av_z - - EIIQOO O - - --, [ A [ - 2r-:5 JJ,'J.. 2 ' ao 480 + 135 H( 5, 2, ( ) + 195 H(5,4, f ) - 117 H( 3,2, e) + 225 S ( 3,5, e) - 60 :3(1,5, (_"} + 225 S( t1,4, e) - 1 20 S( 2.,4, e} - 195 S ( 3,:S, e) + n[ 375 H( 5,4 , e ) - + 1 2 s ( 1,3, e ) - 1 3s s ( 4 ~~, d + ? 2 s ( 2 , 2, d ] + - 225 H(S,2, e) - 195 H(l ,4!> e) + 117 ll (l ,2, cl - 375 S(3,5, e) + + 100 S(l,S, e) - 37$ S(1,4, ( ) - 125 S( 2J4, e) + 120 S(0,4, c) + + 225 S(4,2, r) + 75 8 ( 2, 2 , e) - 7r?.8 ( 0 , 2, e) + 195 S( 3 .l , t ) - 72 8 ( 1 ., 1, e) j +C f - 375 H( 3,4, {) + 225 H( l,4, e) + + 225 H( 5,2 , ('.') - 155 H(l,?., e) + 375 S(;-'~, ,l , e) - 100 S ( 0 , 4 , e) + + 375 S(3,3, e) - 225 S( 3,l, ~) - 100 S(l ,5, o - 225 S(2, 2, r) + + 60 S (0,2, {') + 60 S( l, l , (') 5J - 3 H( 5,0, t ) + 9 H( 3, 2, e) - 5 H(3,0 , c) - 9 8(4, 2, ~ + + 3 S( 4 , 0, ( ) - 9 S(l,S, (') + 3 S(;~,0, f ) 3 B[ - 27 H( 5 , 2, ('.') + + + 9 H(s,o, c) - e H(1? 2 1 {') + ::; H(1,o, l') + 21 s ( 4 , 2, d - 9 8 ( 4 ,0, (') + 9 S( 2, 2, e) - 3 S( 2:,0, e) + 27 S(l ,5, (') - 5 S(l, l, e)} + cf 27 H( 3,2, e) - 9 H(5 , 0 , e ) - 9 H(l, ?. , ('.') + + 3 H(l,0, c ) - 27 S( 2,2 , ~ + 9 S( 2,0, ~ - 27 S(l, 3, e) + + 9 S{l,l , e) v1here 3J A -3 - 'I B -S C D - 6 ( Fig. 1. B, 'f J... Potential energy curves. A, tf= £,.(1) {; _ '( 2) + e. '( l ) a{2); = aa( l)b( 2) + b( l ) a ( 2) ; C, If :: e. (l) a ( 2) ; D, 'f :::: e.(l) a.( 2} + K ( 8. (l ) b( 2) + b(l~1a(2)) + K~ ( ct (l) e '( 2) + a 1_(l) a ( 2) ) -S. 6 A 3 R/ ao Fi g . 2. B, f Potenti al energy curve s . A, ':f z a (l) a ( 2) ; = a (l) a.( 2) + K n(l) a f( 2) +.u 2 (l) a ( 2) ; C, cf = a(l) a(2) + K(a(l) b(2) + b(l) &(2)) ; D, f = n (l) a ( 2) + K ( a (l) b ( 2) + b( 2) a ( l) ) + K~( a (l) a ~( 2) + a '(l} v.( 2)) VAN DER WAALS INTERACTI GN OF HYDROGE.'N A'l'OMS MAY 1, 1935 PHYSICAL REVIEW Printed in U. S. A. VOLUME 47 The van der Waals Interaction of Hydrogen Atoms LINUS PA ULING AND J. Y. BEACH, Gates Chemital Laboratory, California Institute of Technology (Received March 18, 1935) The van der vVaals interaction energy of two hydrogen atoms at large internuclear distances is discussed by the use of a linear variation function. By including in the variation function, in addition to the unperturbed wave function, 26 terms for the dipole-dipole interaction, 17 for the dipole-quadrupole interaction, and 26 for the quadrupole-quadrupole interaction, the interaction energy is evaluated as W"= 6.49903 e2 124.399 e2 1135.21 e2 aop." in which p = R/ao, with R the internuclear distance. Some properties of the functions F,xµ(t ,'J, ,p), which are orthogonal for the volume element ~d~ sin OdOd,p, are discussed, and their usefulness in atomic problems is pointed out. Wang claimed to have obtained an exact solution; it was, however, pointed out by Eisenschitz and N approximate second-order perturbation London that Wang's result is necessarily in error. treatment of the inverse sixth power interIt seemed to us possible that Wang's work might action energy of two hydrogen atoms a large distance apart (corresponding to the so-called have contained only a numerical error, and that dipole-dipole van der Waals attraction) was given the method might actually be capable of giving in 1930 by Eisenschitz and London. 1 This treat- an exact solution. Because of the usefulness ment led to the result W"= -e2 A / aop6, with which a method of exact solution of problems of this sort would have, we thought it worth while p=rAn/ a 0 (rAn being the internuclear distance for the two atoms), A being evaluated as 6.47. Ap- to study the problem thoroughly. We have found plications of the variation method by Hasse2 and that the method used by Wang does not give an by Slater and Kirkwood 3 verified this result es- exact solution, 6 but that it can be extended to sentially, the constant A being shown to be give as closely approximate a solution as is deequal to or greater than 6.4976. • sired. The results of the treatment are comAs early as 1927 this problem had been at- municated in this paper. . tacked by Wang, 4 using the method developed A rough treatment of the dipole-quadrupole by Epstein 5 for the treatment of the Stark effect. and quadrupole-quadrupole interactions of two hydrogen atoms has been published by Mar1 R. Eisenschitz and F. London, Zeits. f. Physik 60,491 INTRODUCTION A (1930). 2 H. R. Hasse, Proc. Camb. Phil. Soc. 27, 66 (1931) . 3 J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931). 4 S. C. Wang, Physik. Zeits. 28, 663 (1927). 6 P. S. Epstein, Phys. Rev. 28, 695 (1926) . 6 Wang does not present the final steps in his calculation in detail, but states that he set up a sixth degree secular equation from which he obtained an accurate value for t he energy. We believe that the error in his treatment occurs at this point. VAN DER WAALS 687 INTERACTION genau. 7 We have applied our method to obtain reasonably accurate expressions for these interactions also. terms represent the dipole-quadrupole and quadrupole-quadrupole interactions, respectively. It can be easily shown that in the calculation of the second-order perturbation energy the terms FORMULATION OF THE PROBLEM can be considered separately, their contributions In the wave equation for two hydrogen atoms being additive. In the solution of the problem we shall make Hi/;= Wi/; let us putH=H0 +H', H 0 being the part use of the functions F,x,,, (t iJ, cp) discussed in the of the Hamiltonian corresponding to two isolated appendix. Each of these functions can be made hydrogen atoms and H' representing the interacidentical with a hydrogen-like wave function tion of the two hydrogen atoms. In order to do Vlnlm (r, iJ, cp) by choosing a suitable linear relathis we neglect the resonance phenomenon (which tion between r and ~; the functions F,x,,, all conis unimportant at large distances), taking as the tain the same exponential function, in contraunperturbed wave function the product function distinction to the functions Vlnlm• We have defined V110o(Al) i/;1oo(B2); that is, we consider electron 1 h and 6 in such a way that Fioo (h, iJ-1, ,p1) and to be attached to nucleus A and electron 2 to Fioo (b, iJ-2, ,p2) are identical with i/;100 (r1, t? 1, ,p1) nucleus B. We also write W= W 0 + W" (W', the and i/;10 0 (r2, iJ-2, ,p2), respectively; that is, the unfirst-order perturbation energy, being equal to perturbed wave function can be written as zero). We shall consider only the interaction of Fioo (h) F100 (~2). We now apply the variation two normal hydrogen atoms, so that W 0 is equal to -e2/ ao. Making the substitutions h = 2rAi/a0 method in treating the perturbed wave equation, using as the variation function a linear combinaand ~2= 2rs2/ ao, the wave equation becomes tion of the product functions F,,x,,,,, (h, iJ- 1, ,p1) F,,x ,,,,, (~2, iJ-2, ,p2), with arbitrary coefficients. It can be seen that the second-order perturbation energy for the perturbation function +f3b~2 2 cos iJ-1(3 cos 2 iJ-2 -1 ) in which X= - W"a 0 / 2e2 • The interaction H' is equal to +Y~1 2~2 2(3 cos 2 iJ-1 -1) (3 cos 2 iJ- 2-1 ) -e 2/ rA2 -e 2 / rs1 +e 2/ rAs+e 2/ r12. If R(=rAs) is large, this can be expanded 7 in inverse powers of R to give the expression _+(2x1x2+2y1y2-3z1z2)(z1-z2)l + ¾(e 2/ R 0) Ir1 2r2 2-Sr2 2Z1 2- Sr1 2z2 2 (2) In this expression X1y 1z1 are Cartesian coordinates of electron 1 relative to nucleus A, and x 2y 2z 2 those of electron 2 relative to nucleus B, the z axis for each being directed towards the other nucleus. The first term represents the mutual energy of two dipoles; this term alone is important for large values of R. The second and third 7 H. Margenau, Phys. Rev. 38, 747 (1931 ). a= (6)½ao 3/ 8R 3 , (3) -y= (7O) la i>5J l28R 5 , (4) {3 = (3O) ½ a 0 4/ 32R4, and II'= (e 2/ R 3) (x1x2+Y1Y 2- 2z1z2) + (3 / 2) (e 2/ R 4) Ir/z2-rlz1 -15zi2z22+2(x1x2+Y1Y2+4z1z2)2 l + • • •. + · · · l, in which is identical with that for the function of Eq. (2), and, moreover, that to obtain the first-on'ler perturbed wave function and the second-order perturbation energy, the variation function used need contain, in addition to the unperturbed part F100 (~1) Fioo (~2), only the terms F, 1 10 (h, iJ-1) F,,10 (~2, iJ-2) (for the dipole-dipole term in a), F, 1 10 ( h, iJ-1) F,, 20 ( b t?2) (for the term in (3), and F, 20 (h, iJ-1). F,,20 (~2, iJ2) (for the term in -y). 8 1 On application of the ordinary methods of perturbation theory, it is seen that the first-order perturbed wave funct~on for a normal hydrogen atom with perturba tion funct10n f (r)T (t'J, rp ), where T is_a tesseral harmonic, has the form 'l.t1oo(r ) +<l> (r ) T (t'J, rp ), the perturbed part involving the same tesseral harmonic as the perturbation function. The statements in the text can be verified by an extensio,.., of this argument. 8 688 L. J. Y. BEACH PAULING AND For a linear variation function I: 1ai<I> 1 (where and 4> 1 represents the product functions F, 1 x1 µ 1 (ti, tJ1, \01) F, 2 x2 µ 2 (b, ?J2, \02)) the secular equa.tion corresponding to the wave equation (1) is the determinantal equation 9 (5) in which xik 0 = f.f cf>/(v'1 2+(1/t1)-¼ +v12 2+(1/t2)-¼)<I>kdT1dT2, being equal to h 2 sin 81dt1d?J1d\01 (with dT2 differing only in the subscripts) and the integrals extending over the configuration space of the system. It is to be noted that the volume element dT1dT2 is not such as to make the functions <I> mutually orthogonal. The integrals can be eva.l uated with the help of the relations given in the appendix. To obtain the second-order perturbation energy we need introduce the term in A in the row and column corresponding to the wave function for the unperturbed system only. dT1 THE DIPOLE-DIPOLE INTERACTION The secular equation for the dipole-dipole interaction is vi">.. 1µ,v;A;µ; 100100 210210 210310 310210 310310 210410 410210 310410 410310 410410 -4>.. 32a -16a -16a 8a 0 0 0 0 0 32a -8 2 2 0 0 0 0 0 0 -16a 2 -14 0 4 ✓ (10) 0 0 0 0 -16a 2 0 -14 4 0 ✓ (10) 0 0 0 8a 0 4 4 -24 0 0 2 ✓ (10 ) 2 ✓ (10) 0 0 0 0 0 0 ✓ (10) 0 0 -20 0 6 0 0 ✓ (10) 0 0 -20 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 ✓ (10) 2 ✓ (10) 6 0 -34 0 0 6 0 -34 =0, (6) 3 ✓ (10) 3 ✓ (10) 3 ✓ (10) 3 ✓ (10) -48 the rows and columns corresponding to the values of v1 • • • µ 2 shown at the left. We obtain successive approximations to the solution of this equation by neglecting rows and columns beyond the nth. This process has been carried out for n= 2, 5, 10, 17 and 26. Some simplification is achieved by combining rows (and columns) with v1 and v2 interchanged, the corresponding functions having the same coefficient. The results of the calculation are given in Table I, in terms of the constant A in the exTABLE I. The dipole-dipole interaction constant A. TABLE I I. The dipole-quadrupole interaction constant B. TABLE III. The quadrupole-quadrupole interaction constant C. Degree of approx. Terms included A Degree of approx. Terms included B Degree of approx. Terms included C 2 5 10 17 26 v1:C::2, v,::::2 3 3 4 4 5 5 6 6 6 6.4822 6.4984 6.49899 6.49903 2 5 10 17 v1:C::2, , 2 ::::3 3 4 4 5 6 5 115 .7 124.10 124.386 124.399 2 5 10 17 26 ,,,::::3, .,::::3 4 4 5 5 6 6 7 7 1063.1 1132.6 1134.35 1135.12 11:35.21 pression W"= -Ae2 / a0 p6 , with p=R/ a0 (A being equal to 3}-/ 16a2). It is seen that the convergence is rapid, the fin~! value of A, 6.49903, being trustworthy to within one unit in the last decimal place. 9 See, for example, L. Pauling and E. B. Wilson, Jr., I ntroduction to Quantum Mechanics, with Applications to Chemistry, McGraw-Hill Book Co., 1935, Chap. VII. VAN DER WAALS 689 INTERACTION THE DIPOLE-QUADRUPOLE INTERACTION The secular equation for the dipole-quadrupole interaction is v1X1µ1vi>..2µ, 100100 210320 310320 210420 310420 410320 210520 410420 310520 410520 -4>. -192 ✓ (3)/5 96 ✓ (3)(3 -192 ✓ (3)(3 -14 4 96 ✓ (3)(3 96v1(3 -48v1(3 0 0 0 0 0 4 -24 0 2 ✓ (6) ✓ (6) 0 0 0 0 0 0 2 ✓ (10) 0 0 0 0 96v1(3 -48v1(3 0 ✓ (6) 0 0 2 ✓ (6) 2 ✓ (10) 0 -20 6 0 -34 6 0 -34 0 0 ✓ (14) 0 0 0 3 ✓ (10) 3 ✓ (6) 0 2 ✓ (14) 0 0 0 0 0 0 0 0 0 0 0 0 0 3 ✓ (10) 0 3 ✓ (6) -26 0 -48 0 8 0 0 3 ✓ (14) 2 ✓ (14) 0 0 0 ✓ (14) 0 8 0 -44 4 ✓ (10) 0 0 0 0 0 0 0 =0. (7) 3 ✓ (14) 4 ✓ (10) -62 We have solved this equation approximately, the results being given in Table II in terms of the constant Bin the energy expression -Be2 / a0 p 8 . The error in the final value of B we estimate to be less than one unit in the last figure quoted. THE QUADRUPOLE-QUADRUPOLE INTERACTION The secular equation for the quadrupole-quadrupole interaction is v,X,µ,v2X,µ, 100100 320320 320420 420320 420420 320520 520320 420520 520420 520520 -4>. 3456-y 3456-y -24 -576 ✓ (6)-y 2 ✓ (6) -576 ✓ (6)-y 2 ✓ (6) 576-y 0 0 0 0 0 0 0 0 0 0 0 -576 ✓ (6)-y -576 ✓ (6)-y 2 ✓ (6) 2 ✓ (6) 0 -34 0 0 0 3 ✓ (6) 2 ✓ (14) 3 ✓ (6) 3 ✓ (6) -48 0 2 ✓ (14) 0 0 3 ✓ (14) 3 ✓ (14) -44 0 4 ✓ (6) 0 0 4 ✓ (6) a -44 -62 4 ✓ (6) 0 0 4 ✓ (14) -62 4 ✓ (14) 0 4 ✓ (6) 0 -80 0 0 4 ✓ (14) 4 ✓ (14) -34 0 3 ✓ (6) 2 ✓ (14) 0 0 0 0 0 2 ✓ (14) 0 0 0 576-y 0 0 3 ✓ (14) 3 ✓ (14) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 =0. (8) The results of the approximate solution of this equation, in terms of the constant C in the energy expression - Ce2 /a0 p10 , are given in Table III. The final value, C= 1135.21, is reliable except for the last figure. DISCUSSION OF RESULTS We have thus found for the interaction energy of two normal hydrogen atoms at the large distance R = pa 0 the expression 6.49903 e2 W"= 124.399 e2 1135.21 e2 (9) It is interesting to note that the value A= 6.4976 found by Hasse for the dipole-dipole coefficient by the use of a variation function ,j,100 (r1) ,j,100 (r2) (l+H'(A+Br 1r2+Cr/r 22+Dr 13ri3)l is very close to our value for v1 ~4, v2 ~4, which is based on a variation function involving all terms (unsymmetric as well as symmetric) out to r13 ri. This indicates that the unsymmetric terrris are of minor importance. 30 690 L . PAULING AND J. Y. BEACH The approximate second-order perturbation energy W 0"= (H00 ')2/ W 0 °, with W 0°d -e2 / a0 , leads to the values A= 6, B= 135, and C= 1417.5, the last two values being given by Margenau. 7 It is seen that the value of A is too low, and those of Band Care too high. This means that the dipole-dipole interaction is due more to excited states with negative energy (less than e2 / a0 above the normal state of the system of two hydrogen atoms) than to excited states with positive energy, whereas the dipolequadrupole and quadrupole-quadrupole interactions are due more to the latter than to the former states. As has been pointed out by earlier authors, the van der Waals forces are more important than exchange forces for values of R greater than about 7a 0 . At this distance the dipole-quadrupole force is about one-half as large as the dipole-dipole force, and the quadrupole-quadrupole force is about one-eighth as large, the dipole-dipole attracaon ·becoming relatively still more important at larger distances. This van der Waals calculation, based on product wave functions, is not significant for values of R much less than 7a 0 because of neglect of the resonance phenomenon and because of failure of the expansion given in Eq. (2). While our treatment has not led to an exact soJution of our problem, the use of the functions F,xµ. (t tJ, ip) has permitted the reasonably accurate approximate solution to be made with considerable ease, and we feel that these functions may be found useful in the treatment of other problems of atomic and molecular structure. 11 APPENDIX The functions F,xµ. (t, {}, ip) We define F,xit -(), ip) =A,x(t)0xµ(-())<l>µ('P), with A,xW = l (v-X-1) !/[(v+X) !]3) le~V2tXL,+x2Hl(t). L being an associated Laguerre polynomial, P being an associated Legendre function, and The functions are orthogonal and normalized for the weight function t sin tJ, satisfying the equation F,xµ.(t -(), ip) satisfies the differential equation . The following relations involving the A's can be easily derived from the properties of the associated Laguerre polynomials: tA,x = -1 (v-X)(v+X+ 1) j½A,+1, x+2vA,x -1 (v+X)(v-X-1) )!A,-1, X· fA,x = - l (v-X)(v+X+ l)(v+X+2)(v+X+3)) ½A,+2, x+1 +2(2v-X) l (v+X+ l)(v+X+2)) lA,+1, x+1-6vl (v+X+l)(v-X-1)) ½A,, x+1 • 1 ' Added in proof: Professor J. H. Van Vleck has pointed out to us that the functions were used in the' treatment of dispersion by. hydrogen-like atoms by B. Podolsky, Proc. Nat. Acad. Sci. 14, 253 (1928). 3/ VAN DER WAALS INTERACTION 691 eA,x = - {(11-i\)(11+x+ l)(v+x+2)(11+X+3)(,+i\+4)(v+x+s) l ½A,+a, x+2 +2(3v-2i\) I (v+x+ 1) (v+X+2)(v+x+3)(v+X+4) l ½A,+2, X+2 -5(3v-i\) I (11-i\- l)(v+x+ 1) (v+i\ +2) (v+i\+3) }½A,+1, x+2 +2Ov( (v+i\,+ l)(v+X+2)(v-i\- l)(v-i\-2) j½A,, x+2 -5(3v+i\) I (v+x+ l)(v-i\-1) (v-i\-2)(11-i\-3) }½A,-1, >-+2 + 2(3v+2i\) I (v-i\-1) (11-i\- 2) (11-i\-3)(v-i\-4) }½A,- 2, x+2 - {(v+i\)(v-i\- l)(v-i\-2)(lJ-i\-3)(v-i\-4)(11-i\-S)} ½A,-a, x+2From these we obtain similar relations in the F's. The following special cases are needed in evaluating the matrix elements in Eqs. (6), (7) and (8): tF100= -v2F200+2F100, f cos iJF100= -2-../2F310+4v2F210, e(3 cos 2iJ-l)F1oo= -24F420+24✓ (6)Fa20. The evaluation of the matrix elements In order to illustrate the method of construction of the matrices in Eqs. (6), (7) an.cl (8), we shall evaluate some of the integrals. Let us first consider the integral1° By using the differential equation for F,xµ (t iJ, cp), the equation becomes JC 0 ,1'•2'•1• 2 = .r... JF*,,, F*,,'[(v2- l) {(v1 - l)(v1+2) l ½F,,+tF,,+ (v1 -1) I(112- l) (112+2) )½F,,F,,+1 . - 2(2v1112- v1 - v2)F,,F,,+ (v2 -1) {(v1 + 1) (v1 -2) )½F,,-1F,, + (v1 -1) {(112+ 1) (v2- 2) j½F,,F,,-1]ht2 sin iJ1 sin i'J2dhdil1dcp1dt2dif2dcp2, and hence we obtain JC 0 ,,+t, , 2 , ,,. , 2 =(v,-1){(v1-l)(v1+2)j½, :ic 0 ,,. ,,+1. , 1 , ,,=(v 1-1){(112-l)(v2+2))½, JC 0,,_ 1, ,,, ,,. ,,= (v2-l) {(v1+l)(v1-2) }½, JC 0 , 1 , ,,-1. , 1 , ,,=(v 1 -1){(v2+l)(v2-2)j½, and all others being equal to zero. • 3J.. 692 L. PAULING AND J . Y . BEACH In Eq. (7) we have and hence JC' 23, 11 = 192 ✓ (3)/3, JC'33 , 11= -96✓ (3){3, JC' 24, 11 = - 96v2{3, ani JC':u, n = 48Y2{3, all others being equal to zero. To illustrate the evaluation of the integral Li let us consider Liu, 11 : J···J/F1oo(t1)F1oo(t2)) 2t12h 2 sin sin t'i2dt1dt'i1dcp1dt2dt'i2dcp2 = J···JF1oo(h)F1oo(t2) /2F2oo(h)F2oo(t2)-2Y2F2oo(h)F1oo(t2)-2Y2F1oo(h)F200(t2) Li11, 11 = t'i1 +4F1oo(h)F1oo(h) l bt2 sin t'i1 sin t'i2dt1dihdcp1dt2dt'i~cp2, or, making use of the orthogonality and normalization of the F's, Li 11 , 11 =4. El ectron Di ffr act ion Investi gation o.f the St ructure of Gas Mole cules 33 [Reprint from the Journal of the American Chemical Society, 57, 2693 (1935). ] [CONTRIBUTION FROM THE GATES CHEMICAL LABORATORY, CALIFORNIA INSTITUTE OF TECHNOLOGY, No. 508] The Electron Diffraction Investigation of Phosgene, the Six Chloroethylenes, Thiophosgene, a-Methylhydroxylamine and Nitromethane 1 BY L. 0. BROCKWAY, J. Y. BEACH AND LINUS PAULING Introduction The importance of the use of interatomic distances as a test for resonance of molecules among several valence-bond structures has been mentioned in earlier papers, 2 in which it was pointed out that the interatornic distance for two bonded atoms in a resonating molecule is determined mainly by the strongest of the bonds between the two atoms provided by the resonating structures. We have now obtained evidence regarding the quantitative dependence of interatomic distance on bond type for resonance between a single bond and a double bond, and have made use of this relation in the discussion of the electronic structure of a number of molecules involving single bond-double bond resonance. The investigation is based largely on the determination of the atomic configuration of molecules by the diffraction of electrons; the description of this work is given in this paper, and the interpretation and discussion of results in the following one. • Electron diffraction photographs of the gas molecules investigated were prepared in the usual way, 3 with film distances of about 12, 20 or 30 cm., the electron wave lengths being about 0.06 A. The photographs were measured on a comparator and interpreted both by the radial distribution method 4 and the usual visual method. 5 The results are given below; in each case the interatomic distances and bond angles are provided with estimated probable errors, which indicate the extent to which we consider them to be reliable. We are indebted to Dr. S. Weinbaum and Dr. J. Sherman for aid in connection with the extensive calculations involved in the interpretation of the photographs. Phosgene.-The phosgene used was prepared by the action of fuming sulfuric acid on carbon (1) Some of the results communicated in this paper were presented at the meeting of the A. A. A. S. in Berkeley, June, 1934. (2) L . Pauling, Proc. Nat. il cad. Sci., 18, 293, 498 (1932); L. O. Brockway, ibid., 19, 860 (1930).; L . 0. Brockway and L. Pauling, ibilt., 19, 868 (1933); L . Pauling and M. L. Huggins, Z. Krist., 87, 205 (1934) . (3) R. Wier!, Ann. Physik, 8, 521 (1931); L. 0. Brockway and L. Pauling, Proc. Nat. Acad. Sci. , 19, 69 (1933). (4) L. Pauling and L. 0. Brockway, THIS JOURNAL, 57, 2684 (1935). (5) See L . Pauling and L. O . Brockway, J . Chem. Phys., 2, 867 (1934) . tetrachloride in the presence of a catalyst (infusorial earth), and was purified by distillation. The photographs show eight rings, with values of s = (4 1r sin 0/2)/).. (averages for eleven photographs measured by two observers) and estimated intensities given in Table I. The eight-term radial distribution function (Fig. 1) shows two peaks, the first, with maximum at 1.79 A., representing C-Cl, and the second, a broad peak with maximum at 2.74 A., representing Cl-Cl and Cl-0 (unresolved). In applying the usual visual method we calculated curves for sixteen plane symmetric models. Three parameters are involved, the Cl-C-Cl angle, the C-0 distance, and the C-Cl distance, the qualitative appearance of the curves being dependent on the angle and the ratio of the distances. The angle was varied from 110 to 125° and the ratio C-Cl/ C-0 from 1.23 to 1.60. Most of the models are eliminated at once by qualitative comparisons. The model corresponding to the Cl valence bond structure Cl "-/C=O, with C-Cl = i.76, C-0 = 1.28, and the angle Cl-C-Cl = 110°, leads to curve A of Fig. 2, which is unsatisfactory in regard to the fourth and sixth rings, each of which is observed to be close to the preceding one. Curve B of Fig. 2 represents the model (with C--0 = 1.12, C-Cl = 1.80, angle Cl-C-Cl = 110°) reported by Dornte 6 as the result of the study of electron diffraction photographs showing only four rings; it is seen that this curve is unsatisfactory, showing no maximum corresponding to our observed fourth ring. It was found that models with the Cl-C-Cl angle equal to about 117° and the ratio of distances C-0/ C-Cl equal to about 1.28/1.66 lead to curves in reasonably good qualitative agreement with experiment, all other models tried being unsatisfactory. Thus in Fig. 3 the curve for a = 117° (C) is reasonably satisfactory, the fourth and sixth rings being represent ed by humps rather than maxima; the curve for a = 115° (D) shows no sign of the sixth ring, and that for 120° (B) is unsatisfactory with regard to the clearly ob(6) R. W. I)ornte, Tms !0URNAL, 55 1 4 126 (1933). I 3'/ 2694 L. 0. BROCKWAY, J. Y. BEACH AND LINUS PAULING Vol. 57 TABLE I PH0SGENE, COCl2 Max. Min. 1 s C 20 3.00 3.95 5.05 6.33 7.44 8.53 9.34 10.33 11.51 12 .54 13.51° 14.52 15.70 16 .94 18.10 2.88 3.88 5.15 6.50 7.70 8.90 9.63 10 .76 12.25 10.25 11 .75 . 14 .50 16.00 17 .38 18.25 14.25 15.62 16 .88 17 .80 2 2 15 3 10 3 4 2 4 5 5 5 6 1 6 7 7 3 8 8 Values of" D I 1 F 2 .84 3.83 5.07 6.35 7.66 Values of C-Cl D C 14.40 1.638 15.76 1.671 17 .12 1 .683 18.00 1.654 Average 1. 687 A. 1.649 1.671 1.674 1.652 1.681 A. Model C: Cl-C-Cl = 117° C-O = 1.28 A. C-Cl 1.64 A D : Cl-C-Cl = 117° 1.28 1.68 F: Cl-C-Cl = 118° 1.28 1.66 1.68 :1: 0 . 02 A. Results: C-Cl Cl-Cl 2.87 ='= .02 A. C-O 1.28 ='= .03 A. 117 ± 2° Angle Cl-C-Cl Angle Cl-C-O 121 °30' ± 1 ° 0 This ring or shelf does not appear as a maximum on the simplified theoretical curves for these models. served fourth ring. The effect of changing the C-O/ C-Cl ratio is shown by the lower four curves in Fig. 2. Of these the curves for 117°, 1.28/ 1.64, and 117°, 1.28/1.68, agree very well with the photographs in qualitative appearance except that the sixth ring is not quite so well represented as expected, appearing only as a shelf on the curves. In the figures of this paper showing intensity curves the observed values of s for apparent maxima and minima are indicated by small vertical lines. In comparing these with the calculated _intensity curves it must be borne in mind that a linear change of scale may be made; the indicated s values are shown in each case for the x/s ratio determined by quantitative comparison for the model finally accepted. The results of the quantitative comparison of the photographs with the curves for three models are given in Table I. Bearing in mind the qualitative comparison, we write as the probable configuration of the phosgene molecule a = 117 ± 2° (angle Cl-C-CI), C-Cl = 1.68 :1: 0.02 A., C-O = 1.28 :1: 0.03 A., and Cl-Cl = 2.87 :1: 0.02 A. The values of Dornte (whose work has been referred to above), a = ll0 ± 5°, C-Cl = 1.80 ± 0.04 A., and C-O = 1.12 = 0.02 A., we believe F (1.574 A.) (1.590 A.) (1. 605 A.) (1.629) (1.631) (1. 610) 1.672 1.687 1.680 1.685 1.678 1.684 1.727 1.697 1. 730 1.726 1. 711 1.653 1.691 l.'Z08 1.667 1.687 1.723 1.745 1.715 2 . 90 3.88 5.11 6.40 7 . 7~ 8 .87 9 .30 10.50 11 .95 1.646 1.666 1 .678 1.651 1. 683 A. to be less accurate than the errors assigned to them indicate. Vinyl Chloride.-Photographs of vinyl chloride (from the Carbide and Carbon Chemical Company) were taken with a film distance of 12.19 cm. (the same distance being used also for the other chloroethylenesr The photographs show about six rings: the first very weak, the second strong, the third and fourth medium, the fifth weak and the sixth very weak. In addition there is apparent a very weak ring or shelf beTABLE II VINYL CHLORIDE Max. Min. 2 I s 30 5.19 6.45 7.61 9 .84 10.76 12.15 13 .51 15 .27 19.10 3 20 5 3 Shelf "for model D C-Cl 5.14 6.48 7.77 1.663 A. 1.687 1.714 1. 680 · 10.76 1.687 12.21 1.676 13.48 5 l.Q53 15.01 5 10 1.707 5 19.40 6 Average 1. Mf3 A. 1.38 A. (assumed) Results: C-C C1-Cl 1.69 :1: 0.02 A. 2.70 ='= .02 A. C2-Cl Qi = 122 ± 2° 4 4 15 • Dec., 1935 2695 AN ELECTRON DIFFRACTION INVESTIGATION tween the third a,nd fourth rings, somewhat closer mentally established value for the carbon-carbon to the third than to the fourth. • Measured values double bond, the non-resonating structure is unof s obtained by two observers from nine photo- satisfactory. graphs are given in the fourth column of Table II and estimated intensities for the rings in the third column. The six-term radial distribution curve is given in Fig. 1. It shows two pronounced peaks, with maxima at 1.695 and 2.69 A., with indication of a subsidiary peak at 1.35 or 1.40 A, The first two we associate with the two carbon-chlorine interactions, and the last with carboncarbon. If we accept the value 1.38 A. for C-C, the distances 1.695 and 2.69 A. lead to the value 121 ° for the C- C-Cl bond angle a. In applying the usual visual method, . we have found that the photographs do ·not provide enough information to permit us to evaluate simultaneously the two distances C-Cl and C-C and the angle Cl-C-C with much accuracy. Accordingly we have assumed the C-C distance to have the double bond value 1.38 A., and have calculated curves for C-Cl = 1.68, ~ ~ C-C = 1.38 and the angle a = 130, 125, ~ ~ 122.5 and 120°. These are shown as B, C, D and E in Fig. 4. Of these B and C are qualitatively unsatisfactory in that the hump corresponding to the faint ring or shelf observed between the third and fourth rings is too large, and E in that the hump is too small. Comparison of measured 2 4 1 3 values of s with the x values for model D A . (Table II) leads to the carbon-chlorine Fig. 1.-Radial distribution curves for phosgene and the six distances 1.683 and 2.70 A. chloroethylenes. Combining the results of the two methods, we write C-C = 1.38 A. (assumed), Ci-Cl= 1,1-Dichloroethylene.-The substance was 1.69 ± 0.02 A., Ci- Cl = 2.70 ± 0.02 A., a = prepared by treating 1,1,2-trichloroethane (made 122 :1: 2°. No earlier electron diffraction work by passing vinyl chloride into antimony pentaon this substance has been reported. chloride) with alcoholic potassium hydroxide, and It is of interest to consider also the model with was purified by fractional distillation. C1-Cl = 1.76, C-C = 1.38 and a = 125°, The photographs, showing seven well-defined corresponding to the non-resonating structure rings, have the following qualitative appearance: Cl H the first medium, the second strong, the third H)c=c(H . This gives the curve A of Fig. 4, medium, the fourth weak and the fifth medium which, is in satisfactory qualitative agreement weak, these five being about equally spaced; with the photographs. The quantitative com- then a wide minimum and a weak ring, and parison leads to the distances C-Cl = 1.68 A. and another wide minimum and weak ring. Measured C-C = 1.32 A., however, and since the latter dis- values of s (averages for four photographs) and tance should not fall below 1.38 A., the experi- estimates of I are given in Table III. .. . . . .. 36 2696 L. 0. BROCKWAY, J. Y. BEACH AND LINUS PAULING Vol. 57 TABLE III 1, 1-DICHLOROETHYLENE Max. Min. 1 I s Model C 15 2 .87 3.83 5 .02 6.28 7.33 8 .42 9 .38 10.52 11.76 15.55 2.75 3.73 4 .96 6 .23 7.40 8.33 9.24 10.50 11.78 15 .75 2 2 40 3 15 3 4 8 4 5 5 6 20 10 Model C: C-Cl/C- C = 1. 70/1.38, f3 Model D: C-Cl/ C- C = 1.70/ 1.38, f3 Results: C-C 1.38 .A. (assumed) C-Cl = 1.69 ± 0 .02 .A. Cl-Cl = 2 .86 ± 0.02 .A. Angle Cl-C-Cl = 116 ± 2° Angle Cl-C-C = 122 ± 1 ° D 2.74 3.67 4 .95 6 .20 7.34 8.22 9 . 15 10.33 11.73 15 .56 Average C-Cl Cl-Cl Model C C-Cl for (1.629 .A.) (1.656) 1.679 1.687 1 .717 1.681 1.674 1 .697 1.703 1 .722 1.695 .A. 2.858.A. D (1.623 .A.) (1.629) 1.676 1.679 1.702 1.660 1.659 1.670 1.896 1.701 1.680 .A. 2 .870.A. 115° 117 .5° The six-term radial distribution function (Fig. l) shows a C-Cl peak with maximum at 1.67 A., and a large peak due to both Cl-Cl and C-Cl interactions. The lack of resolution of this peak (maximum at 2.81 A.) makes its interpretation difficult. CO C l2 :x for 110 and 125° and the C-Cl/C-C ratio varied between 1.76/ 1.38 and 1.64/1.38. It was not found possible to evaluate the C-Cl/C-C ratio as well as the angle (3 with much accuracy; accordingly we have assumed the C-C distance to have the double bond value 1.38 A. Of the models tried, only those with (3 equal to about 115° agree qualitatively with the photographs. For example, the model with C-Cl = A 1. 76 A., C-C = 1.38 A., and /3 = 110°, corresponding to the vaCl H B lence-bond structure Cl ) c=c(H , is unsatisfactory in that the fourth maximum on the curve (curve A of Fig. 5) is higher than the third, whereas the fourth D ring is observed to be much weaker than the third and fifth. Quantitative comparison with E this and other curves shows the C-Cl distance to be about 1.70 A. Curves B, C, D and E of Fig. 5 are calculated for C-Cl = 1. 70, 5 15 20 C-C = 1.38, and the angle (3 = x-+ 112.5, 115, 117.5 and 120°, reFig. 2.-Simplified intensity curves for phosgene. A, a (angle Cl-C-Cl) = 110°, r (ratio C- O/ C-Cl) = 1.28/1.76; B, a = 110°, r = 1.12/1.80; C, a = spectively. Of these curve B •is 117°, r = 1.28/ 1.64; D, a = 117°, r = 1.28/1.68; E, a = 117°, r = unsatisfactory in that it shows 1.28/1.72. an additional maximum between We have calculated intensity curves for twelve those corresponding to the fifth and sixth rings, models, with the Cl-C-Cl angle (3 varied between and curve E in that the fourth maximum is too c· 37 r Dec., 1935 AN ELECTRON DIFFRACTION INVESTIGATK>N 2697 peaks, with maxima at 1.67 A. (C-CI) and 3.21 A. (CI- CI). These correspond to the value 123°15' for the angle Cl-C-C, and to another C-Cl distance of 2.69 A., some indication of which is visible in the curve. C OCl 2 On calculation of theoretical intensity curves it was found that A all models giving rough qualitative agreement with the photographs lead to a Cl-CI distance B close '[to 3.23 A. In order to determine the C-Cl distance, curves were calculated for a series C of models with C-C = 1.38 and Cl-Cl = 3.23, the value of C-Cl being varied. It was found that D the shelf beyond the second ring changes rapidly in this series; only for C-Cl = 1.68 (curve C in Fig. 6) does the shelf corre2(J 10 5 15 spond to its appearance on the photograph (about one-fourth as Fig. 3.-Intensity curves for phosgene, with r = 1.28/1.68, and ix =- 125, pronounced as the second ring). 120, 117 and 115° for A, B, C and D, respectively. Decrease by 3% wipes it out enThe quantitative comparison of the measured tirely. We accordingly accept model C. The values of s and the x values for models C and D, comparisou.._of observed s values and x values for given in Table III, leads to the interatomic distances C-Cl = 1.69 ± 0.02 A., CI-Cl = 2.86 ± 0.02 A., with the angle Cl-C-Cl = 116 ± 2° and C-C = 1.38 A. (assumed). A The only previous investigation of this substance, that of WierF by electron diffraction, gave the value Cl-Cl = 2.9 ± B o.3 A. cis-Dichloroethylene.-The sample of cis-dichloroethylene used was obtained C from a mixture of the cis and trans compounds by fractional distillation with a D 90-cm. column. The photographs show five well-defined rings, with the following qualitative apE pearance: the first ring medium; the second strong, with an outer shelf; the third medium; the fourth weak, and somewhat closer to the third than to the fifth; the 20 5 10 15 fifth medium weak. Observed values of s (averages for nine photographs) and estiFig. 4.-Intensity curves for five models of vinyl chloride. mated intensities are given in Table IV. The six-term radial distribution function is this model, given in Table IV, leads to C-Cl shown· in Fig. 1. It shows two well-defined 1.671 A., Cl-CI = 3.223 A. Curve A1 calculated for the non-resonatin~ (7J. R. Wier!, Ann. Ph:ysik, 13, 453 (1932). bigh. Curves C and D are satisfactory, and we accordingly accept for the CI- C-Cl angle the value fJ = 116 ± 2°. 2698 L. 0. BROCKWAY, J. Y. BEACH AND LINUS PAULING TABLE Vol. 57 IV cis-DICHLOROETHYLENE Max. Min. 1 I s Model A 10 2.480 3.489 4.652 6.00 6.958 8.095 9.22 10 . 10 11 . 16 12 . 17 2 2 Shelf 30 8 3 3 20 4 4 3 5 5 Model A: C-C Model C: C-C Results: C-C C-Cl Cl-Cl Angle Cl-C-C 10 1. 38, C-Cl 1. 76, 1. 38, C-Cl 1. 68, 1.38 A. (assumed) 1.67 ± o.o3 A. 3.22 ± 0 .02 A. 123 . 5 ± 1 ° fJ fJ x for C-Cl for C Model A 2.36 3.20 4 .33 2.46 3.35 4.62 (1.675 A.) (1. 614) (1. 639) (1.666 A.) (1. 613) (1. 668) 6.57 7 . 74 8.78 9.62 10 . 54 11.63 6.88 8.14 9.23 10 .01 10 .98 12 .15 1.662 1.682 1.675 1 . 677 1.664 1.682 Average 1.674 A. 1 . 661 1.689 1 . 681 1.665 1.653 1.677 1.671 A. C 125° 123.7° model with the Cl-C- C angle {3 = 125 °, C-C = 1.38, and C-Cl = 1. 76, is in satisfactory qualitative agreement with the photographs, quantitative comparison, however, giving C-Cl = 1.674 A. and Cl-Cl = 3.233 A. (Table IV), the only essential Combinj ng the results of the two methods, we accept for the structural constants the values C-C = 1.38 A. (assumed), C-Cl = 1.67 ± 0.03 A., Cl-Cl = 3.22 ± 0.02 A., {3 (angle Cl-C-C) = 123.5 ± 1°. Previous investigations have given the less accurate values Cl-Cl = 3.30 ± 0.1 A. (Wierl,7 electron diffraction) and Cl-Cl = 3.6 A. (Debye, 8 x-ray diffracA tion). trans-Dichloroethylene.-The sample of trans-dichloroethylene was separated B from a mixture with the cis compound by fractional distillation. The photographs show seven measurC able rings, with apparent intensities as given in Table V (the second ring showing an outer shelf). Values of s (averages D for ten photographs) are also given in the table for the features which could be measured with accumcy. E The eight-term radial distribution function, given in Fig. 1, shows three welldefined peaks, with maxima at 1.675, 2.70 and 4.27 A. These we correlate with the two C-Cl interactions and the Cl- Cl 5 10 15 20 interaction, the three interactions being x-+ of about equal importance. The disFig. 5.-Intensity curves for 1,1-dichloroethylene. tances correspond to the values C-C = difference from the results for model C being in the 1.38 A., {3 (angle Cl-C- C) = 123°. In applying the usual visual method we obC-C distance, for which the low value 1.30 A. is obtained. Curve D, calculated for {3 = 130°, served that the quantitative comparison with the C-Cl = 1.72, and C-C = 1.38, shows the extreme photographs of all the models tried gave values qualitative disagreement caused by a relatively close to 4.27 A. for the Cl-Cl distance. We then small change in model. (8) P . Dehye, Physik. Z., 31, 142 (1930). AN ELECTRON DIFFRACTION INVESTIGATION Dec., 1935 2699 TABLB V trans-DICHL0R0ETHYLENB C- Cl for x for Max. Min. 1 5 2 20 5 2 Shelf 3 3 4 5 6 7 10 1 1 5 2 s Model C D Model C D 3.11 3 .98 4.90 5.68 6.78 7.74 9.18 10.59 12.11 14.97 3 .25 3.95 4 .87 3 . 26 3 .95 4.87 (1. 777 A.) 1.687 1.690 (1803 A.) 1 .707 1.716 6 .71 7 .75 9.17 10 .63 12 . 10 15 .05 1.682 6 .77 7 .74 1.702 9.18 1.698 1.706 10 .62 1.699 12 . 11 1.709 15 .02 Average C-Cl 1. 697 A. Cl-Cl 4. 272 A. 1. 717 1. 720 1 .720 1.724 1.720 1.725 1. 719 A. 4.277 A. Results: C-C = 1. 38 A. (assumed) C-Cl = 1.69 :1: 0 .02 A. Cl-Cl = 4.27 :1: 0 .02 A. Angle CI-C-C = 122. 5 :1: 1 ° calculated curves for models with Cl-Cl = 4.28, (average for three photographs, two observers) C-C = 1.38, and C-Cl = 1.68, 1.70, 1.72 and 1.76 are given in Table VI. The five-term radial distribution function (curves B, C, D and E, respectively, of Fig. 7). All of these agree qualitatively with the photo- (Fig. 1) shows three peaks, with maxima at 1.73, graphs except in so far as the weak fourth and 2.85 and 4.37 A., the first corresponding to the fifth rings are concerned; the approximate small C-Cl distances, the second to the larger Cequality of these rings is best represented by curve D. The results of the quantitative comparison for C and D are given in Table V; it is seen that the value of the Cl-Cl distance is essentially independent of the model. A Averaging the results of the two methods, with about equal weights, we assign to the structural parameters the values B C-C = 1.38A. (assumed), C-Cl = 1.69 ± 0.02 A., Cl-Cl = 4.27 ± 0.02 A., {3 (angle Cl-C-C) = 122.5 ± 1°. Previous studies C gave the values Cl- Cl = 4.33 ± 0.1 A. (Wierl,7 electron diffraction) and Cl-Cl = 4.1 A. (Debye, 8 x-ray diffraction). In D addition a note has been published by de Laszlo 9 in which the C- Cl distance in this molecule is given as 1.74 .A. Trichloroethylene.- The photographs 10 15 20 5 of trichloroethylene (Eastman) show six x--+ rings, with intensities weak, strong, meFig. 6.- Intensity curves for cis-dichloroethylene. dium, weak, medium weak, weak. Characteristic features are that there is some indica- Cl distances and to two Cl-Cl distances, and tion of a small shelf between the second and third the third to the trans CI-CI distance. The lack rings (closer to the second than to the third) of resolution of the second peak makes it of little and that the weak fourth ring is closer to the value. In discussing the possible molecular models we third than to the fifth. The measured values of s have restricted ourselves mainly to those in which (9) H. de Laszlo, Nat_ure, 136, 474 (1935). , L. 0. BROCKWAY, J. Y. BEACH AND LINUS PAULING 2106 Vol. 57 the three Cl-C-C angles are equal. Curve A (Fig. 8) is calculated for C-C = 1.38, C-Cl = 1.76 and the angles Cl-C-C = 125° for the CCb group and 123° for the CHCI group. This curve agrees with the photographs qualitatively, and leads on quantitative comparison to the value 1.69 A. for C- Cl (and hence 1.32 A. for C-C). Cl-C-C = 125, 122.5, 121.5 and 120°, respectively. Of these curves C alone is in satisfactory qualitative agreement with the photographs. Quantitative comparison (Table VI) leads to the values C-Cl = 1.69 A., Cl-CI = 2.85, 3.19 and 4.27 A. Giving somewhat more weight to these than to the radial distribution values, we accept as representing the configuration of the molecule the values C-C = 1.38 A. (assumed), C-Cl = 1.71 ± O.Oi A., Cl-Cl= 2.72 ± 0.04 A. (in the CCb group), and Cl-Cl = 3.23 ± 0.05 A. and 4.33 ± A 0.05 A. (between CCh and CHCI), with the angles Cl-C-C = 123 ± 2°. Trichloroethylene previously has been B studied by this method by Domte, 10 who reported the values C-C = 1.32 ± 0.08 A., C-Cl = 1.82 ± 0.08 A., Cl-Cl = 3.41 ± C 0.08 A., and CI-CI = 4.52 == 0.08 A., in approximate agreement with our values. Tetrachloroethylene.-The rather weak D . photographs of tetrachloroethylene (Eastman) obtained at room temperature show six rings, the first medium, the second E strong, the third medium, the fourth weak, the fifth medium weak and the sixth weak. Averaged values of s • (for four photographs) and estimated intensities 10 20 5 15 are given in Table VII. X-+ The six-term radial distribution funcFig. 7.-Intensity curves for trans-dichloroethylene. tion (Fig. 9) shows three peaks, with Other models lead to about the same C-Cl value. maxima at 1.72, 2.86 and 4.37 A., the first repreCurves B, C, D and E of Fig. 8 are calculated senting a C-Cl distance, the third the trans CI-CI for C-C = 1.38, C-Cl = 1.70 and the angles distance, and the large second peak representing three distances. TABLE VI Curves A, B, C and D of Fig; 9 are calculated TRICHLOROETHYLENE Max. Min. I 1 2 2 30 3 3 15 s 2 .91 3.88 4 .86 6.43 7 .81 8.68 9.67 10.91 12.04 15.49 " 2 .80 3.73 4.79 Model C . C-Cl (1.639 A.) (1. 633) 1.675 7.73 1 .683 4 1.682 8.59 4 4 9.11 1.602 5 10.97 1.709 5 10 12.03 1.699 6 4 16.20 1.778 Average 1.690 A. Results: C-C = 1.38 A. (assumed) C-Cl = 1. n ± o . 03 A. Cl-Cl 2 . 72 ± o.04 A. 3.23 ± 0 .05 A. 4.33 ± 0.05 A. Angles Cl-C-C 123 ± 2° TABLE VII TETRACHLOROETHYLENE Max. Min. 1 I s x for model B C-Cl 15 2.78 3.81 4.76 7.46 9.00 11.84 15 . 17 2.70 3.68 4.63 7.52 9.12 11. 72 (1.69 A.) (1 . 68) 1.691 1.753 1. 761 . 1.722 2 2 3 4 5 6 Results: 30 10 3 8 3 C-C C-Cl Cl-Cl Angle Cl-C-C Average 1. 732 A. 1.3~ A. (assumed) 1. 73 ± 0.02 A. 2.87 ± 0.03 A. 3.30 ± o.o3 A. 4.28 ± o.o3 A. 123° 45' ± 1° (10) R . W. Dornte, J. Chem. Phys., 1, 566 (1933). Dec., l935 AN ELECTRON DIFFRACT:CON INVESTIGATION 2701 for models with C-C = 1.38, C-Cl = 1.74 and the angles CI-C-C = 125°, 123°45', 122°30' and 121 °15', respectively. It is seen that the curves peak is not sufficiently displaced toward the third. We accordingly accept for the CI-C-C angle the value 123°45' ± 1°. The quantitative comparison leads to C-Cl = 1.732 A, and Cl-CI (trans) = 4.38 A., in essential agreement with the results of the radial distribution treatment. We accordingly accept for the structural parameters of the moleA cule the values C-C = 1.38 A. (assumed), C-Cl = 1.73 ± 0.02 A., Cl-Cl = 2.87 ± 0.03 A. (in the same CCiz group), and CI- Cl= B 3.30 ± 0.03 A. and 4.28 ± 0.03 A. (between CClz groups), with the angle Cl-C-C = 123°45' ± 1°. C A previous electron diffraction investigation by Dornte 10 gave the values C-C = 1.32 ± D 0.08 A., C-Cl = 1.82 ± 0.08 A., and CI-CI = 3.41 ± 0.08 A. and 4.52 ± 0.08 A., in approximate agreement with our results. De Laszlo E has also reported the value C-Cl = 1. 74 A. in a preliminary note.9 Thiophosgene.-The thiophosgene used was 5 10 15 20 prepared by the chlorination of carbon disulfide and subsequent reduction, and purified by fracFig. 8.-Intensity curves for trichloroethylene. tional distillation. 11 The photographs show nine rings, for which change very rapidly with change in angle. The appearance of the photograph is closely repro- measured s values and estimated intensities are duced by curve B, and not so well by C or A; the given in Table VIII (averages for four photolatter is unsatisfactory only in that the fourth TABLE VIII Max. Min. I THIOPHOSGENE x for model B s C-Cl 2 . 67 (1.598 A.) 2 3.67 (1. 664) (1. 646) 2 25 4.84 6.04 1.689 3 12 7 . 13 1.693 3 8.06 1.692 4 12 1.685 4 9.09 10 .29 1 . 707 5 11.49 1 .713 5 6 12 . 51 6 1.711 1 .709 6 3 13.48 14 .56 1.706 7 7 15 .75 1.721 3 16 .88 1 .710 8 1. 711 8 1 17.90 18.87 1.697 9 19.97 1 . 706 1 9 Average 1 . 704 A. 1. 63 A. (assumed) Results: c-s Angle Cl- C-Cl 116 ± 2° (assumed) C- Cl 1.10 . ± 0.02 A. Cl-Cl 2.88 ± o.o4 A. Cl-S 2.90 ± 0 .04A. 1 10 2.84 3.75 5.00 6.08 7 . 16 8.10 9 . 17 10 .25 11.40 12.43 13.41 14.51 15.56 16.78 17.79 18.90 19 .90 A B C D E 5 10 15 X->- Fig. 9. -Intensity curves for tetrachloroethylene. (11) "Organic Syntheses," John Wiley and Sons , Inc., New York City Coll., Vol. I, p. 493. L. 0. BROCJ{WAY, J. Y. BEACH AND Lrnus PAULING 2702 graphs 12). The nine-term radial distribution curve (Fig. 10) shows two well-defined peaks, the first, with maximum at 1.59 A., corresponding to the C-S and C-Cl distances, and the second, with maximum at 2.87 A., to the Cl-Cl and Cl-S distances. The sharpness of the secm;i.d peak (which csci, 1 I 2 A. 3 Fig. 10.- Radial distribution curves for nitromethane, a-methylhydroxylamine and thiophosgene. • closely resembles the Cl- Cl peak for carbon tetrachloride) indicates strongly that the Cl-Cl and Cl-S distances are nearly equal. The position of the first maximum is unreliable, being strongly dependent on the estimated intensity values. On calculating. simplified intensity curves for e1ght models it was found that all of the parameters determining the structure of the molecule could not be evaluated. The curves out to the tenth ring are affected very little by small changes in the Cl-Cl/Cl-S ratio, no perceptible differences existing between those calculated for C-S = 1.63, C-Cl = l.70, and the angle Cl-C-Cl = 114, 116, _118 and 120°, respectively, the Cl-Cl/CI-S ratio changing from 0.97 to 1.02. All of these curves agree satisfactorily with the photographs in qualitative appearance; the 116° curve is shown as A in Fig. 11. The curves are also not very sensitive to small changes in the position of (12) Some measurements made on five rings of \,ery weak photographs, disagreeing with those in the table by .about 2%, were discarded. • Vol. 57 the light carbon atom; however, the curve for C-Cl = 1. 76, C-S = 1.44 and the angie Cl-C-Cl = 110° (B of Fig. 11) is qualitatively unsatisfactory in regard to the fourth ring, which is observed to be as strong as the third. We have assumed for the C-S distance the double bond value 1.63 A., as given by the table of covalent radii (and verified by the value 1:64 A. reported for crystals of thiourea), 13 and for the angle Cl-C-Cl the value 116 ± 2°, as in phosgene and 1,1-dichloroethylene. The observed size of the Cl-Cl-S triangle then requires that C-Cl be close to 1.70 A. The quantitative comparison with curve A leads to C-Cl = 1. 704 A.; taking some cognizance of the 2.87 A. peak on the radial distribution curve, we accept for the structural parameters the values C-S = 1.63 A. (assumed), angle Cl-C-Cl = 116 ± 2° (assumecl.), C-Cl = 1.70 ± 0.02 A., Cl-CI = 2.88 ± 0.04 A., and Cl-S = 2.90 :1: 0.04 A. a-Methylhydroxylamine.-The photographs of a-methylhydroxylamine (Eastman) show three well-defined rings, with s values (average for four films) and estimated intensities given in Table IX. The three-term radial distribution function (Fig. 10) shows peaks with maxima at 1.39 and 2.31 A. The first we interpret as showing the C-O and O-N distances, unresolved, the table of covalent radii giving for them the values 1.43 and 1.36 A., respectively. The C-N distance 2.3{ A. then leads to the value 112° for the C-O-N angle. Intensity curves were calculated for the following values of the angle: 114, 110 and 106°. In these the ratio of the O-N and C-O distances was A 8 10 15 5 Fig. 11.- Intensily curves for thiophosgene. 20 taken as l.36/ 1.43. The curves for the first two are shown in Fig. 12 and the x va1ues in Table IX. Model A (110° angle) is superior to Model B (114° angle) as indicated by the poor agreement between the x· ands values for the second maxi (13J R. W. G. Wyckoff and R . B. Corey, Z. Krist., 81, 386 (1932). 'i3 AN ELECTRON DtFFRACTION INVESTIGATION Dec., 1935 2703 TABLE IX Mio. Max. 1 I $ 5 6.00 7.95 9.49 12 . 13 13.94 2 2 2 3 1 3 a-METHYLHYDROXYLAMINE Model A "' 6.10 7 .93 9.93 12.20 14.30 N-0 1.383 A. 1.357 1.418 1.367 1.393 Average 1.384 A. Model B X N-0 6.04 7 .88 10.20 12.17 14.18 1.369 A. 1.349 (1.461) 1.362 1.381 1.365 A. Results: N-O 1.37 ± 0.02 A. 1.44 ± 0 .02 A. C - N = 2 .31 ± 0.03 A. Angle C-0-N = 111 ± 3° C-O mum in Model B. The 106° model was rejected because it leads to widely fluctuating values for the size of the molecule as calculated from the various maxima and minima. The quantitative comparison for Model A leads to N-O = 1.38 A., C-O = 1.45 A. and C-N = 2.31 A. Combining the results of _the two methods we obtain the values N-O = 1.37 ± 0.02 A., C-O = 1.44 ± 0.02 A., angle C-O-N = 111 ± 3°, C-N = 2.31 ± The photographs do not permit the evaluation of all the structural parameters. We expect, however, that the C-N distance has the single • 110° o.o3A. 114° Nitromethane.-The photographs of nitromethane (Eastman) show four welldefined rings followed by a very wide minimum and two more very faint maxima. The s values averaged from 5 10 15 20 six photographs and the estimated intensities are given in Table X. The Fig. 12.-Intensity curves for a-methylhydroxylamine. six-term radial distribution function (Fig. 10) shows two peaks with maxima at 1.19 and bond value 1.47 A. and the N-O distances ap2.18 A., the first corresponding to the N-O dis- proximately the double bond value 1.22 A. The tance with the N-C distance unresolved and the three curves in Fig. 13 correspond to models second fo the 0-0 and C-O distances. having the relative dimensions determined by the above distances and the three values of the TABLE X O-N-O angle, 120, 125 and 130°, respectively. NITR0METHANE N-Ou.-.o X s I Max. Min . The qualitative features of the photographs (1.09 A.) 3.27 3 .65 10 1 fix the angle at about 127°. Thus, the promi(1.07) 4.43 5 .04 2 nence of the third maximum relative to the 1 . 175 6.50 6.26 30 2 fourth and the· position and character of the 1.226 7 .96 8.00 3 fifth minimum as observed on the photograph 1.219 9 .29 9.28 12 3 1.178 10.17 10.54 4 eliminate the 120° model. In the 130° curve 1.228 11.83 11.75 12 4 the position of t_he seventh maximum is a little 1.231 13.47 13 .61 .5 better than in the one for 125° but the hump ·17 .21 1.181 17.77 3 5 following the fifth minimum is too pronounced. 1.227 19 .50 19.37 6 21.02 2 6 The quantitative comparison (Table X) leads Average 1 . 208 A. to the values C-N = 1.46 ± 0.02 A. and 1.21 ""0 .02A. Results: N - O N- O = 1.21 ± 0.02 A., with the angle O- N-O = 1.46 ± 0 .02 A. C- N 127 ± 3° . 127 ± 3° Augle O-N-O '-f 2704 L. 0. BROCKWAY, J. Y. BEACH AND LINUS PAULING Vol. 57 120° 125° 130° 5 20 15 X->- Fig. 13.-Intensity curves for nitromethane. Discussion of Results In phosgene, thiophosgene and the six chloroethylenes the carbon-chlorine distances vary between 1.67 and 1. 73 A., being between 5 and 2% less than the normal single bond value 1.76 A. This decrease is due to the partial double bond character caused by resonance resulting from the conjugation of an unshared pair of electrons on the chlorine atom with the adjacent double bond. The values found for the angle CI-C-X are somewhat smaller than the tetrahedral value 125°16' for the angle between a single bond and a double bond, as a result of the same phenomenon. A detailed discussion of these effects is ITTVen in the following paper. For the other interatomic distances the values found are in good agreement with the table of covalent radii. In phosgene the carbon-oxygen distance has the double bond value 1.28 :±: 0.02 tion of the photographs. The N-0 distance 1.21 :±: 0.02 A. for the nitro group in nitromethane is close to the double bond value 1.22 A. The distances N - 0 = 1.37 :±: 0.02 A. and C-0 = 1.44 :±: 0.02 A. in a-methylhydroxylamine and C- N = 1.46 :±: 0.02 A. in nitromethane agree well witq the single bond values 1.36, 1.43 and 1.47 A., respectively, given by the table of radii. Summary The arrangements of atoms in molecules of phosgene, the six chloroethylenes, thiophosgene, a-methylhydroxylamine and nitromethane have been determined by the use of electron diffraction photographs, interpreted both by the radial distribution method and the usual visual method, with the following results. (Values given without attached probable errors were assumed to be correct in the investigation.) Phosgene : C- Cl = 1.68 ± 0. 02 A.; C-O = 1 . 28 ± 0 . 02 A.; angle Cl-C-O = 121. 5 ± 1 °. C-C C-CI Angle CI-C-C Vinyl chloride 1.38 A. 1.69 ± 0 .02 A. 122 ± 2° l,l; Dichloroethylene 1.38 1.69 ± 0 .02 122 ± 1° cis-Dichloroethylene 1.38 1.67 ± 0 .03 123 . 5 ± 1° trans-Dichloroethylene 1.38 1.69 ± 0.02 122.5 ± 1° Trichloroethylene 1. 38 1. 71 ± 0. 03 123 ± 2° Tetrachloroethylene 1. 38 1. 73 ± 0. 02 123 . 75 ± 1 ° Thiophosgene: C-Cl = 1. 70 ± 0. 02 A.; C- S = 1. 63 A.; angle Cl-C-S = 122 °. a-Methylhydroxylamine: N-O = 1.37 ± 0 . 02 A.; · 0--C = 1.44 ± 0.02 A. ; angle C-O-N = 111 ± 3°. Nitromethane: N- O = 1.21 ± 0 .02 A.; C-N = 1.46 ± 0.02 A.; angle O-N-O = 127 ± 3°. A. (1.28 A. from the table), and for thiophosgene the double bond value 1.63 A. for the carbon- The discussion and interpretation of these results are given in the following paper. sulfur distance leads to a satisfactory interpreta- PASADENA, CALIFORNIA RECEIVED SEPTEMBER 24, 1935 'I 'fhe Depende.1ce of Interatomic Distance on Singl e Bond- Double Bond Resonance [Reprint from the Journal al the American Chemical Society, 57, 2705 (1935).1 [CONTRIBUTION FROM THE GATES CHEMICAL LABORATORY, CALIFORNIA INSTITUTE OF TECHNOLOGY, No. 509] The Dependence of lnteratomic Distance on Single Bond-Double Bond Resonance 1 BY LINUS PAULING, L. 0. BROCKWAY AND Introduction Three years ago it was pointed out 2 that observed values of interatomic distances provide useful information regarding the electronic structu_res of molecules, and especially regarding resonance between two or more valence bond structures. On the basis of the available information it was concluded that resonance between two or more structures leads to interatomic distances nearly as small as the smallest of those for the individual structures. 3 For example, in benzene each carbon-carbon bond resonates about equally between a single bond and a double bond (as given by the two Kekule structures); the observed carbon-carbon distance, 1.39 A., is much closer to the carbon-carbon double bond distance, 1.38 A., than to the single bond distance, 1.54 A. In benzene the two Kekule structures contribute equally. In general, however, the coefficients of the functions corresponding to different structures in the approximate wave function of a molecule may have arbitrary values, and a bond between two atoms may have any intermediate character between the extremes of a pure single bond and a pure double bond. For a series of bonds covering the range between a pure single bond and a pure double bond we expect the interatomic distance to change continuously from the single bond value to the double bond value. In this paper we present evidence regarding the nature of the function expressing the dependence of interatomic distance on single bond-double bond resonance, and then make use of the function in order to obtain information regarding the electronic structures of resonating molecules for which experimental interatomic distance values are available. The effect of resonance on bond angles is also discussed. (1) Part of the material in this paper was presented at the meeting of the A. A. A. S. in Berkeley, June, 1934. (2) L. Pauling, Proc. Nat. Acad. Sci., 18, 293 (1932). (3) The following argument, suggested by Professor P. M. Morse, shows that this conclusion is reasonable. Of two potential functions corresponding to two structures, the one with the smaller value of the equilibrium internuclear distance wlll have the greater curvature in the neighborhood of the minimum [see for example, R. M. Badger, J. Chem. Phys., 2, 128 (1934) ]. Rene~ the more stable of the two resultant potential functions corresponding to resonance between these two structures will tend to have its minimum in the position determined by the original curve of greater curvatu~e, that is, near the smaller value of the internuclear distance. J. Y. BEACH The Interatomic Distance Function for Single Bond-Double Bond Resonance.-The carboncarbon single bond distance is 1.54 A. (diamond, aliphatic compounds). For the carbon-carbon double bond distance we shall use the value 1.38 A. given by the table of covalent radii; 4 some support for this is given by Badger's value 6 1.37 A. for ethylene. These give the two extreme points of the interatomic distance fonction for single bond-double bond resonance. The midway point, for fifty per cent. double bond character, is provided by the value 1.39 A. for benzene; 6 the electronic structure of benzene is represented in the main by resonance between the two Kekule structures (the contribution of excited structures being small7), and this makes each bond resonate equally between a double and a single bond. Another point on the curve is provided by graphite (C-C = 1.42 A.), in which each bond has onethird double bond character, corresponding to resonance among many structures such as Through these four points we draw a smooth curve, as shown in Fig. 1, which we accept as representing the dependence of carbon-carbon interatomic distance on double-bond character for single bond-double bond resonance. We believe that by a suitable translation and a change of vertical scale (to give the correct end-points) the same function can be used for bonds between other atoms, and probably also for double bondtriple bond resonance. This use of the curve will be illustrated below. (4) L. Pauling, Proc. Nat. Acad. Sci., 18, 293 (1932); L. Pauling and M. L. Huggins, Z. Krist., 87, 205 (1934). (5) R. M. Badger, Phys. Rev., 46, 648 (1934). (6) L. Pauling and L. 0. Brockway, J. Chem. Phys., 2,867 (1934). (7) A possibly more accurate value for the double bond character of the bonds in benzene (0.46) is obtained by considering all five canonical structures with weights equal to the squares of their coefficients in the wave function. There is some uncertainty as to the significance of this, however, because of the non-orthogonality of the wave functions for the canonicaf structures. and for chemical purposes it is sufficiently accurate to follow the simple procedure adopted above. 2706 LINUS PAULING, L. 0. BROCKWAY AND It is seen that a small amount of double bond character causes a large decrease in interatomic distance below the single bond value, whereas only a small change from the double bond value is caused by even as much as fifty per cent. single bond character. 8 In consequence, the interatomic distance criterion for resonance provides quantitative information only through about onehalf of the bond character region. C-C 0 ~ 1.45 GRAPHITEQ 1.40 f---- - -C=C ~ - - -- / BENZENE~ I Double Bond Single Bond Fig. 1.-The empirical function expressing the dependence of carbon-carbon interatomic distance on bond character for single bond-double bond resonance. The applicability of the curve to bonds other than carbon-carbon can be tested with the data for carbon-oxygen and nitrogen-oxygen bonds given in Table I, the predicted distances being obtained from the table of covalent radii for pure single and double bonds, and from the curve (with end-points determined by the table) for bonds of intermediate type. The carbonate and nitrate ions resemble graphite in that the double bond resonates among three positions, whereas in the carboxyl and nitro groups, as in benzene, the double bond resonates between two positions. It is seen that there is approximate agreement ·between predicted and observed values. The Use of the Interatomic Distance Function in Discussing the Electronic Structure of Molecules.-The observed interatomic distances listed in Table II are interpreted with the aid of the curve of Fig. 1 to lead to the assignment of fractional double bond character as given in the last column. It is seen that a single bond between conjugated double bonds (cyclopentadiene) or (8) Verification of the shape of the curve has bet,n obtained with the collaboratiou of Dr. J. Sherman by the theoretical treatment of a somewhat simila r problem (the effect or s- P hy bridization of bonrl orhitals on interatomic distance). Vol. 57 TABLE I COMPARISON OF PREDICTED AND OBSERVED lNTERATOMIC DISTANCES Carbon-Oxygen Bonds Amount of double Predicted Observed bond character distance 0 1.43 A. 1 /s 1.32 1.29 1.28 ¼ 1 0 1.501---- - - -- - - - - - - - - - -1 J. Y. BEACH distance 1.42 A. in (CH1)2O" 1. 44 in CH3ONH," 1.31 (1.26) in (CO 3)-b 1.29 in (HCOOH)2c Nitrogen-Oxygen Bonds 1.36 A. 1.37 A. in CHaONH2" 1.26 1.22 (1.26) in (NO3)-b 1.23 1.21 in CH 3NO2" 1.22 • Preceding papers. b Preliminary values obtained through redetermination of parameters in crystals (calcite and sodium nitrate) by Mr. Norman Elliot. The values in parentheses are based on older parameter determinations. c L . Pauling and L. 0. Brockway, Proc. Nat . Acad. Sci., 20, 336 (1934). The value 1.25 A. reported in crystals of oxalic acids and oxalates is probably less reliable. conjugated benzene rings (biphenyl) p-diphenylbenzene, has about 15 or 20% double bond character, 9 whereas a single bond between conjugated TABLE II INTERATOMIC DISTANCES AND BOND TYPE Amount of Observed double bond Molecule Bond distance character Cyclopentadiene C- C 1.46 A.• 0.20 1.48b p-Diphenylbenzene C-C . 14 1.48c Biphenyl C-C .14 1.43d Cyanogen C-C .29 1.43d Diacetylene C-C . 29 Urea C-N 1.37' .28 C-N 1.371 Thiourea .28 Cyanuric triazide C-N 1.38" .25 1.30h Carbon suboxide .20' C=C 1.20h .20' C=O • Preliminary result of electron diffraction study in this Laboratory. b L. W. Pickett, Proc. Roy. Soc. (London), Al42, 333 (1933). c J. Dhar, Indian J. Phys., 7, 43 (1932). d L. 0. Brockway, Proc. Nat. Acad. Sci., 19, 868 (1933). • R. W. G. Wyckoff and R. B. Corey, Z. Krist., 89, 462 (193.4). 1 R. W. G . Wyckoff and R. B . Corey, ibid., 81, 386 (1932). • I. E. Knaggs, Proc. Roy. Soc. (London), AlSO, 576 (1935). h L. 0. Brockway and L . Pauling, Proc. Nat. Acad. Sci., 19, 860 (1933). • Triplebond character. (9) The comparison of these results with the simple theory of conjugated systems [Pauling and Sherman, J. Chem. Phys. , 1, 67\J (1933)] is not straightforward because of non-orthogonality of the canonical structures'. If we assume that the double bond character r--7 is given by the square of the coefficient of the structure - = - in the normalized wave function for a system of two conjugated double bonds the simple theory leads to the value 0.25 . in approximat,• agreement with the ex perime ntal result Dec., 1935 SINGLE BOND- DOUBLE BOND RESONANCE AND INTERATOMIC DISTANCES triple bonds (cyanogen, diacetylene) has about 30% double bond character. In cyanuric triazide the reported distance for the bond linking an azide group to the cyanuric ring corresponds to_about 25% double bond character. In urea and thiourea the ionic structures such as H2N+ NH2 ~/ are of sufficient importance to give C I ;Q;_ each C-N bond about 28% double bond character. In case these structures were equivalent to the H2N NH2 "-._ / struc1!'1re giving complete resonance, C II each bond would have one-third double bond character, as in the carbonate, nitrate, and guanidonium ions. The experimental result shows that resonance is nearly complete. In carbon suboxide, for which the structure : O=C=C=C=O: is most important, structures such as -: 0-C=C-~O :+ contribute about 20% triple bond character to each bond. Resonance of a Carbon-Chlorine Bond and an Adjacent Double Bond.-We expect the phosgene molecule to resonate among the structures ·er :ci• . "-._ / • ·d~ /"·c1· • c I C II and :o:- 0: of which the first is the most important, the second and third making only small contributions to the normal state of the molecule. The value 1.68 ± 0.Q2 A. for the C-Cl distance, reported in the preceding paper, 10 corresponds to 17% double bond character for the carbon-chlorine bonds, calculated with the curve of Fig. 1, with end points C-Cl = 1. 76 A. and C=Cl = 1.58 A. The value C-Cl = 1. 70 ± 0.02 A. in thiophosgene leads to 12% double bond character, indicating that the ionic structures make a smaller contribution for this molecule than for phosgene. We might predict that of the six chloroethylenes the C-Cl distance would be smallest in vinyl chloride, which resonates between the structures H) H •. /~I : C=C, 'H and .. + H) •• _ ;,Cl: C -C, H 'H (10) Brockway, Beach and Pauling, TIID }ovltNAL, 117, 2693 111311) . . and largest in tetrachloroethylene, in which the effect of the double bond is divided among four carbon-chlorine bonds. This is found experimentally (Table III), except that the distance reported for cis-dichloroethylene is about 0.02 A. smaller and that for vinyl chloride somewhat larger than expected. The amounts of double bond character shown in the table are reasonableabout 15% for mono- and dichloroethylenes, 10% for trichloroethylene, and 6% for tetrachloroethylene. TABLE III lNTERATOMIC DISTANCES AND BOND TYPE FOR CARBON-CHLORINE BONDS Molecule :0 2707 Phosgene Thiophosgene CH2CHCI CH2CCl2 cis-CHCICHCI trans-CH Cl CH CI CHCICCl 2 -C2Cl, C- Cl distance Double bond character 1.68 ""0.02 A. 1.70 ='= .02 1.69 ='= .02 '1 .69 ='= .02 1.67 ± .03 1.69 ='= .02 1.71 ± .03 1.73 ='= .02 0.17 . 12 . 14 .14 .20 . 14 . 10 .06 The effeot of this partial double bond character on 1.4he chemical properties of chlorine atoms cqnjugated to double bonds is well known; it corresponds in the main to a diminution in reactivity. The correlation with bond angles is discussed in a later section of this paper. Inasmuch as the conjugation properties of a double bond and a benzene ring are closely similar, 11 we expect for the halogen substituted benzenes interatomic distances similar to those for the halogen ethylenes. De Laszlo 12 has reported the values C-Cl = l.69 A., C-Br = 1.88 A., and C-I = 2.05 A., corresponding to 14_, 6, and 10% double bond character, respectively. 13 The Dependence of Bond Angles on Single Bond-Double Bond Resonance.-In a molecule such as phosgene or 1,1-dichloroethylene the value 125°16' for the angle Cl-C-O (/3) is predicted by the theory of the tetrahedral carbon atom in case that the C-Cl bonds have no double bond character. If the double bond resonates equally among all three positions, giving the Cl-C bond one-third double bond character, we expect from symmetry (11) L. Pauling and J. Sherman, J . Chem. Ph ys., 1, 679 (1933). (12) H . de Laszlo, Proc. Roy. Soc. (London), AU.6, 690 (1934). (13) The interatomic distance values given are stated to hold for hexahalogen, sym-trihalogen and P-dihalogen benzenes. We expect the distances in these different compounds to be different, however; and investigation of these and other halogen compounds is under way in this Laboratory . De Laszlo has al; o reported values of C-Br and C-I distances in ethylenic and acetylenic compounds in a letter to Nature, 1311, 474 (19311). .. 2708 LINUS PAULING, L. 0 . BROCKWAY AND the value 120° for fJ. Between these limits the dependence of fJ on the amount of double bond character of the C-Cl bonds may be postulated to be given by a curve similar in shape to the interatomic distance curve of Fig. 1. If this is so, there would be a linear relation connecting the angle fJ and the C-Cl distance, between the points fJ = 125°16', C-Cl = 1.76 A. and fJ = 120°, CCI = 1.64 A. (as given by the curve for one-third double bond character). From ·a similar discussion, the other chloroethylenes (except vinyl chloride) would be expected to fall roughly on the same line. A test of this relation is shown in Fig. 2, the seven points being those determined experimentally for phosgene and the six chloroethylenes. • 124° • • • 122° • 120° ~-....___ _.__ __.__........._ _.______, 1.64 1.68 1.72 1.76 C-Cl. Fig. 2. -The relation between bond angle and carbon-chlorine distance for phosgene and the chloroethylenes. It is seen that, with one exception, the points lie very close to the straight line drawn according to our assumptions, the displacements being much less than the probable errors of the experimental values. The exceptional point is that for cisdichloroethylene, the discrepancy confirms the suggestion made in the preceding section that our results for this substance are slightly in error. 14 It is surprising that vinyl chloride obeys the same relation, as there is no apparent reason for the angle to be much different from 125°16'. Interatomic Distances in Polynuclear Aromatic Hydrocarbons.-As a further example of the use of the interatomic distance function we shall discuss the polynuclear aromatic hydrocarbons. For naphthalene there are three unexcited (14) The discrepa ncies are removed by increasing the C-Cl distance by 0.02 A., to 1.69 A., and decreasing the angle by 1 °, to 122°30'. These changes are within the estimated probable errors of our determination, *0.03 A. and =1°, re•pectively. • ,. structures J. Y. B E ACH Vol. 57 (X) ()/) and ('\-0 '\f' ~ '\/ "-..f' Giving these equal weights, we obtain the result that the 1-2 bonds have 2/ 3 double bond character and all others 1/ 3 . These correspond to the values 1.39 A. for the 1-2 interatomic distances (including 3--4, 5-6 and 7- 8), and 1.42 A. for all others, 7 giving the configuration 0 1.42 1 ) J -42 ~✓ A possibly more reliable prediction can be made on the basis of Sherman's wave function for naphthalene, 16 by considering all 42 canonical structures. The fractional double bond character of a bond can be considered to be given approximately (neglecting non-orthogonality of the canonical wave functions) by the expression in which a; represents the coefficient for the ith structure; the unprimed slim includes all structures of the set of 42, and the primed sum those for which the bond in question is a double bond. The results of this calculation are the following: 16 Bond 1-2 2-3 9-1 9-10 Double bond character 0 . 60 0 .33 0 .26 0.43 C-C distance 1.39A. 1.42A. 1.44A. 1.40A. corresponding to the configuration ( 8)/1~ 9 1.40 21 1-42 ~ 10~✓ - The four unexcited structures for anthracene, given equal weights, lead to 3/ 4 double bond character for the 1-2 bonds, ¼ for the 9-11 bonds and ¼ for others, the predicted configuration . being (15) J. Sherman, J . Chem . Phys., 2, 488 (1934). (16) It may be pointed out that the . changes from the simple calculation given abover (sult e x~inly from the fact that the coeffi- I cient for the structure • unexcited structures . ~ I is larger than that for the other ,I' Dec., 1935 SINGLE BOND-DOUBLE BOND RESONANCE AND lNTERATOMIC DIST~CES For phenanthrene the five unexcited structures lead to 4/s double bond character for the 9-10 bonds, 3/ dor the 1-2 bonds, 2/ dor the 1-11, 11- 12 and 4-12 bonds, and 1/ 5 for the 10- 11 and 12- 13 bonds, and the predicted configuration is 1-38 / ~6 4 8--'{_ / ~ ~ " '~ l - 4 1 / ~ 146 "-.1.39 :--. "-.. 1/ 1.41"-...4_ _3/ 1.41 1-39 Similar prediction can be made for larger molecules. It must be pointed out that the contributions of excited structures become important for bonds with small double bond character, inasmuch as in conjugated systems excited structures alone may lead to as much as 20% double bond character; it is probable that the maximum carboncarbon bond distance in aromatic hydrocarbons is about 1.46 A.-, the minimum being the double l:1ond distance 1.38 A. The predicted average interatomic distance is 1.41 A. in naphthalene, anthracene and phenanthrene, this value being somewhat larger than •the benzene value (1.39 A.). With increase in size of the hydrocarbon the carbon-carbon distances should all approach the graphite value 1.42 A. The only experimental values of sufficient accuracy to permfr a test of the predicted values are those obtained by Robertson in his careful and thorough x-ray investigations of the structure of • 2709 crystals of naphthalene 17 and anthracene. 18 In each of these molecules Robertson reports the value 1.41 A. for the average carbon-carbon bond distance, in complete agreement with the predicted value. 19 He does not discuss individual variations from the average; however, measurements made on his reproduced electron distribution projections (Fig. 2 for naphthalene, Fig. 2 for anthracene) show differences of 2 or 3% in the predicted directions. 20 Summary Using experimental values for carbon-carbon bonds, a function is plotted showing the dependence of interatomic distance on bond character for single bond-double bond resonance. This function is tested with data for other bonds, and used in the discussion of the electronic structure of molecules containing conjugated double or triple bonds or aromatic nuclei and of molecules containing carbon-chlorine bonds adjacent to double bonds. The dependence of bond angfes on single bond-double bond resonance is discussed. Values of carbon-carbon bond distances in polynuclear aromatic hydrocarbons are predicted and compared with the existent experimental data. (17) J.M. Robertson, Proc. Roy . Soc. (London), A142, 674 (1933) . (18) J. M. Robertson, ibid., A140, 79 (1933) . (19) The data for chrysene [J. I ball, ibi d., A146, 140 (1934)) are also compatible with this value. (20) The value 1.41 A. has also been reported for the carboncarbon distance in benzene derivatives. We think it probable that this is 0.02 A. too large. PASADENA, CALIFORNIA RECEIVED SEPTEMBER 24, 1935