by JOHN EDWitRD RitUOl A TIESIS submitted to in partial fulfillment of the requirements for the degree of MS1ER OF SCIEN June 1960 in charge of ?jor QiáÌrman of Department of Chemistry Date thesis is presented Typed by Mrs. V. L 1iite, Jr. This thesis is dI«t.d to Profossai J. C. Decius whose efforts and suggestions uade this thesis possible. chapter Page ..s..ess$.ee*ø i DfrROOUCUON. Ii cLASSIcAL TLEORY OF :îii QOANTUMTJEoRYO'TlERAEFFEcr Iv ØYPERPGIARIZABILITY V VE VI TiE eh Sh vn TIE ah VI II NUJSION B ZBUOGRAPRY wPEM:Ix i s 2 RAMAN EFFEC) , , . . . . . . ,.,..,, . ... . , , . $1JNTRY POINT GROUP , a $1TRT . . a a a ea a General a a O1$T GROUP . a s.seU S a Forlas a e a * * a a 6 9 .. * C 25 a a 32 s s e ,s**s 37 ci Tx*asZoritks 1FvFhs1o*ssssss0s Tables of characters (nz2, 6,o°). s, $urpi tk e for an Operation APPEMIX 3 Gharacter for R . APPENDIX 4 EulerIan Angles and the Averages of Needed Products of Direction Co.insø . APPEMflX 3 ......... f orthe APPErmu TE I a 41 . 0 s 51 , 5 = Sunuary of the Forni and Range of the Various n of Point Groups the Species of the . . . s gq'ENDIx 6 The iverage Intensity From Oscillating Dipole APPEMIX 7 The Dependence of the Intensity of a Hyper-Ranun Une Upon Ion Concentration s a Classical . , . . s 5? HYPERPO[ARJZABILITY AND FORBIDDEN LISES Ri'MAN QL4PThR I INTRODUCTION In the weak Raman spectrum of certain molecules and ions tzansitions occur which are forbidden to take place when one considers the symmetry of the free molecule. article by Gray and Waddinyton ( 9O1O8 6, p. ) An presents a sunmiry of the articles which have been written on the azide ion which is usually a linear ny investigators syustric triatomic ion. ion have observed its symmetry, i. e., assund to have 2' and 3) bas of this in the Raman spectrum. In lead azide a partial covalent bond has been postulated to exist between the lead and the azide which destroys the symnietry for the azide ion. the alkali stals Ïiowever, for the azide salts of this could not be postulated. It ts stated that the normal selection rules are relaxed under the influence of the perturbing It electric fields of neighboring ions. has also been noted by Welsh, that carbon dioxide at high density the P and V sg., ( 11, p. 99-110 ) the existence of both shows bands in the Raman spectrum. They suggested that the molecule was distorted under an influence of the field of neighboring molecules in such a way that type symitry, i. e., the molecule the molecule took on remained linear but one carbon oxygen bond became longer than the other. conclusion was drawn from the fact the fainter nzin*zrn while the J band had a double 14. band was narrow, of low strong Q branch with rotational wings This indicating a intensity. They nade no depolarization measurennts. Evans and Bernstein ( 3, p. 11274133 ) found that in concentrated solution of carbon disuluide using cyclopentane as a solvent that the existing 7 polarizsd ( and 0.63 for both bands 1. ) and that the of carbon disulfide showed no indication triatomic molecule with be two polarized Iace ( J' V,3 ) C- bands were both J) bands of a double peak. A syraietry requires that there bands and one depolarized band ( it was concluded that distoxtion in a carbon disulfide molecule under the influence of a perturbing neighboring field is not only along this axis. the ScS axis but also at a right angle to They gave the distorted carbon disulfids molecule a symmetry ofe of the band to the 2-3 . They also found the ratio of the intensity 3' band of carbon disulfide to be proportional to the volume-fraction of carbon disulftde cyclopentane to the four-thirds power. This paper will be an attempt to discuss the appearance of such forbidden bands, retaining the symmetry of the free molecule. ). 3 EFFECT tIASSICAL flEORY OF ThE The introduction of an atom or molecule Into an electric field E induces an electric dipole moment P in the system. a component ai the electric dipole is If expanded by a Taylor's series expansion in terms of the components of the electric field, (1) - + F" Thus, the induced dipole, neglecting quadradic and higher terms of the components of the (2) PF the components of the polarizability. Let It electric field, is can be shown ( 12, p. 44 ) that a set of axes in a molecule exists such that (3) r' if light = c(, E p of frequency falls P} c on an ion or molecule, the varying electric field produced by the light (4) ThIs =- field produces t. E} ny be represented a0cos(ri-) a varying dipole moment which itself causes 4 emission of light of the same frequency as the incident light. For a classical oscillating dipole the intensity of the emitted radiation, is given by 1, ( ______ ± 5) see appendix 6) a This accounts for the Rayleigh scattering which is responsible for the phenomena of dispersion and the Tyndall effect. The polarizability must change if internuclear distances change ( vibrations ) or il the molecule undergoes rotation, since the polarizability depends on the orientation of the molecule. ability, o( (6) By expanding the components of the pclariz in terms of the c - cKy nornl coordinates - /i)0Qk* .: is a normal coordinate of ths of the molecule . . aolcule and changos with a frequency characteristic of the molecule. (7) Now, = Qfr coiuing (8) Q; Ccos(zi equations 3, 4. 6, and 7 7-.I (ck) Cos(t1t) (9) -F (L)0 ¼ Q; Cos2(idi)t CûS?Yj Thus, Whfl one considers stnall changas in o( with vibration, the induced dipole moment changes not only with the frequency z of the incident light but also with with frequencies Z2zfor rotations ). frequenc1esZ( The Rariìn or lines dis- placed toward the longer wave lengths are called Stokes lines (Z- ) and those displaced ) are tord calld anti-Stokes the shorter wave lengths lines. Thus, one should expect displacements on either side of the excitation line equal toz4 4 for vibrations and2i4 for is fixed while can take on any rotations. Cls*sicafly value. Hence, one should expect a continuous spectrum about the exciting Rayleigh line due to rotation and that displacent should have length the long wave length cononent of the the sane intensity as the short wave displacement. These two expectations are not observed. Thus, although the classical treatnnt gives a qualitative picture of the picture. C Rann effect it does not give a quantitative 12, p. 43-48 ) [i QUAPIflIM TIEORY OF TIE RAMI4N EFFF.( molecules in the kth state which are If there are of B louer energy and in the asrgy, the average intensity tr*nsitiin k-'n is th state which is cf a higher of the line arisIng proportional t, Nk frm )" while that arising Erom the transition n-.k is proportional to N IA (-i)V V0-V) N i since Nfl/Nk Is the 8eltzn the (V0k). -(v0v4 factor betmean the two states, This simple derivation explains why the intensity of the Stokes lines are generaIly mere intense than those of the antl-'Stokes lines. The mean rate of total radiation of an induced dipole is givan classically by equation (5). ( Dirac radiation theory ) The hypothesis is nade that the classical forzadas can be converted into quantum formulas by replacing P0 in which [F°] (11) = 5O by 4 d thus, (12) 1 d] pOJ nn I Now, (13) -y* 6fo(>< LPTTh + Ef&xy VCcfl* Ef°<2 V' ations hold for Similar and and ire the components light ve, the integrals and 04) xxJ [Ponm. the amplitudes of of \4ÇT /- Ex° the Ey0 incident f°(xx are the matrix elements of the six components of the polar- izability tensor. In effect the electric vector of the Raman the incident illumination, acting on the charges of the molecule, perturbs its wave function. using perturbed is found as the t'ave functions that for n=rn there itegra1s If the C are calculated insisdiug time factors J, it are terms o! the san* frequency incident light, but for n'm frequency4J±zj. P0m there are terms Z with the terms with unshifted frequency are responsible for Rayleigh scattering while those of frequency z4/give rise to the Rauan effect. allowed if at least one of the six the approximation that the product immn transition is quantitiesßF,/JmO. ve function can be written as a of a rotational function, and a vibrational wave function, with quantum numbers R \/f, the integrai or f,jr LJ (f5) since mo1ecuIarfid axu in coordinates of rotation ( ) 12, p. 52 ). f'dî does not involve the The orthoonality of the rotational wave £unctions, therefore, leads to the result I -SR/f]V E'F (16) ConsequtiJflm same. zero unless the rotational states a i These integrals can be treated exactly the sar 7 thà U the integrals of the electric moment; that Is to say, they vanish i unless the synuitry of the set of functions levels *iìd.r consideration species the f norl % etc. , e1lf y'ìfor the two some species In common with the For fundamentals this requires that vibrations for a given frequency fall into one of the species associated with the in the t,- 'rb, Rann effect. ( 's, if this frequency is to occur 12, p. 48uu53 J cnwTER IV HWERPOMRIZILITY in the case of the p.1ui*eb*Ift, a component of the As electric dipole of the tnon*nt ney be expanded in terus of the components ctric field utin9 upon the molecule in question. Hence, (D (17) EF' (ÇfJ)O -. EE The components of the poiarizabflity are - (-kF (p3) EF'I° that for Suppose a given vibration of a molecule all of the partial derivations of the components of the polarizability with respect to the norail coordinates of this particular vibration vanish, i. e., the line is forbidden by polarizability theory ( for a free molecule or ion the line would also be forbidden for the hyper-Rainan effect because of the dependence tipon concentration ). C intensity See appendix 7 ). In this case the intensity of the emitted light will be dependent the partial derivatives of the conents upon of the hyperpolariz- ability with respect to the normal coordinates of this particular vibration provided they do not vanish. The components of the hyperpolarizability are given by (19) /8Fr'f" = ______ L In the hyper-.Rann effect the induced dipole mon*nt of a molecule iy be represented as pIJ (20) EF') It" f' In turn can be expanded in terms of the normal co The fi, ,, 9 ordinates of the molecule, = /,i#i (21) TheFfcan +(:'1) be shown to be all zero for molecules with a center of symmetry. r be non-vanishing for The molecules with no center of syn*netry which will give rise to a pure rotational spectrum. If one separates EF into the field, due to the light source, EF? and the field due to the neighboring ions and molecules,ZEF , EF (22) Ei becomes: ¿ -1- y be represented by EF f23) where o5 Th is thè frequency of the incident light. ¿t.E, may be represented as cosc') (24) where , is a normal frequency of a neighbor and time independent part of the surrounding field. EFDis the The normal Fil co. diute, y siso b. written as) sor1 ii ths re C.ining () Q°fr = PF fr.queney of the Z j:" I (") *E° COS C OS Qk° i.c.i. i. sti.n. i(t Cus E( t r7,t)(E°'COS2117- Cosirt)J jp (27) Cos(1T7) these eqstiosst , E*nding u this beccíss =II2 F (è o t' ) Qk " CÜ5 2Cos,?i1U' E(cos)(Cos71Zz') -i-E; ¿1&, E;iì(cos)(ccs)(co5't) *-dEi0 £i, (c.5 g l:ff'oilØ CF» -E;E1 ir t)(Ccs 1p-ir ') Cos1TZ1 (cos'rrvt)(cos-ir i,t) (Cos 2ir7t)(cosii1t)(co iri-t)(cos 'rrVt)(cos f7/) ir) ar4t)] 12 Thus, the oscillating dipole due to the hyper-Rarn effect will have frequencies of the following types: A Rann apparatus is usually only designed to observe slight shifts from the exciting frequency. involve and Hence, only the terms which are of importance. ?- Terms of the type which involve frequencies 1tZ.1 are dependent upon Terris of frequencies ,r seems logical to consider the than (28) terms. LJC, terms as being much smaller This being the case, 2: f: are dependent upon E,L1E,', It O f" r' + )7 for observable lines in the hyperRaman effect. Depolarization Ratios: The two ( quantities 12, p. 43-47 ) most conmionly measured are the relative intensities of the Raman shifts and their depolarization ratios. The depolarization ratios will now be discussed. For the laboratory fixed axes, the total radiation emitted per unit solid angle in the ¡ direction is given by 77,ì (29) c - t see appendix Consider 6 ) that the direction of propagation of the incident 13 incident light coincides with the Incident T Polarized Light 4I Incident 2 Polarized Light -+f7 y -_-//7 (cBS ' 1 axis. ) ' A 1(OBS.1.) (CBS. J-) g::: ; lIght is plane polarized in the 1f the incident and the total scattered intensity is axis, this total scattered I1OBS. li). j direction observed along the T intensity is If the incident light is represented by Zpolarized and the total scattered intensity is observed along the T axis, this total scattered latensity is represented by IT(OBS. in this second case the scattered light J. ). If is polarized parallel to the Z axis, the scattered intensty is represented by If we now consider only lines which are Lcr- 111 bidden in the f irst-order-Ran effect, but allowed in tk byper-Rann effect, the intensities according to equation (17) and (29) and the above (30) 'T ii) -1- = definitions are YFF'(C'XF)( L, PZFFfx&Fx +&F»'X&pK F)FttF] -t- 14 2 TT (oBs.i) = r'(E2z (F (os,L c3 EF)(S4'Z [_III FP' where yFF'EE' FF' r-F' is the electric vector of the external light (E, ¿ ) source and (EEy) is the electric vector acting the molecule in question due to the molecules These equations apply only which are on surround which to a single radiator; for it. N free to assume al]. orientations with respect to the observer's axes with equal probability, the equations are to be multiplied by found as the It N and the average of each expression radiator is allowed to can be shown C see appendix i ) of transformation of Pp i' = Nnv consider, j be all orientations. that the general f orrt1as assume in laboratory fixed axes to in molecular f 1d axes are given by (31) is to -íJ 15 (32) _-4 = = E,1 z (E 4Ef F F" If the incident light from the source is such that then, pp'" , If we now consider 'X&'"x A p''X + On1y(EF' type terms to be those cb1ch produce the hyper-Raman effect ( see equation 28 ), be reduced for our purposes to (p'f"f'X (33) Upon expansion this gives, = 13pkX (s- i- ) P- * /'FXZK4&z + 13FyXeXe,Y However, (L\ ElàEf/) (34) ¡3FF'F" ... - = Then (35) -, p, Hence, (36) (see next page) -I- i,y may -4 - [2 ( ' /3 ' E -i- z it is equation (36) From FXF A'c' FLX' GL] apparent that the intensity of the exciting light is proportional to can be The termL . computed by introducing sorne assumptions regarding the intermolecular forces F F is needed. ( =¿ see appendix 7 "L" Since . FG But ) Also only the F'F' "FLLL'GL" p< O, A4 - 2. FX' (37) )('j "JLL'L" cI and average value of reduces to +2 I The depolarization ratio is 'J "defined as the ratio of the scattered intensity which is polarized perpendicular toi, that is, in the direction of propagation of the incident light, to the intensity paralled tol." incident light is natural ( ratio ny ( 12, p. 47 ) " If the unpolarized ) , the dep.iarization be computed by considering the scattersd light to represent the sum of the intensities of the observations oflde parallel and perpendicular polarized bern. Th* t part to the incident electric vector of a of the light from the parallel observation, being unpolaxi*i, outributes one-half Its 17 intensity to the scattered light polarized, respectively, parallel and perpendicular to I(o&_I,,(oBS.J) (38) i11(os.i)-- ±SIT0U) IT(oES.)/) t 12, p. 4? ) Nowlet Then (39)p 21 f3y -1-- I -f-Z & I.Mf3 Tf3'" 2 A°r "1 This may be written more concisely as (40) ¡on= 2 F 1 ¡XF f± zF F Y2'yi í' 18 zh S1MIflY TIE POINT GROUP First it will be necessary to determine which of the ten distinct s exist as linear couthinations which transform as ce*tain of the synmetry species of the group. One ny cbsørve the following transformations under the operations of the group: Xty'Ztl: (41) ZU; VY'Z' iii7)c*)..rm: X 'Y 'Z' (iz) (y) (-*)= XU X'Y'Z' 2ì(z)(.y)(_s)m; ( See appendix 2 fnce, this shows /)Q(2 4YZ lW (.y) (s) -ZU Z'Y'Z'°4s)(y)(Z)'ZYZ Z 'VZ' for the character table of that In a like manner 360). XY'Z'(X)(7)cÍI)-= it transforms as ( 12, easily that may be and I!Yyz transform as 3 , y, shown p. 359- and izzY transform as The E character per unshifted atom ( see appendix 3) for equals the nuther of coordinates. This is also the total number of linear conJ.inations of the coordinates transform as the various species ( 12, shows will p. 106). Hence, in the infrared there are three cothinations; in six combinations. This which the Ranun there are that the ten byperpolarizability components listed are all of the ones for this point group. Zn o*der to illustrate that there are molecules which allow vibrations which are inactive in the ordinary Raman effect to becoue active in the hyper-Raman effect, consider the ethylene molecule. By (12, standard methods p. 106-113 ), it is found that the vibrational normal coordinates are of species: The f ellowing table summarizes the nuthar of frequencies respectively in infrared absorption, the ordinary Raman, and the hyper-Raman effects. n(() 1! ï n(O iL n(() byoer-R 3 Rl o 02 1 g B z A i u 2 2 a2 2 2 B I i B 'u n (Y) IR n (Y) n) R is the number of is the nuther of infrared modes. Raman modes. hyper-R is the nunther of hyper-Raman modes. active Hence, by observing the frequencies and are k character table the infrared in the hyper-Raman effect alled an additional fundanental frequency is is hyperRain effect which kainan effect. or allowed seen to be Next the of the B1 lt ( p. observable group h point Let species. in the not al1ovd In either the infrared 106409 ) quantities for the various species f will be tabulated. First consider the Q represent a normal coordinate for this u species. 2, Then from the character table for ah in appendix it follows that: (42) -o but all other For simplicity of notation let (43) - 5 nc., since the order of the 4= cartesian subscripts is irrelevant, non-vanishing terms for these species are 1 /ß33) 3/3') A3/') 4W , f3i13) From equation ( 23) and F /32) /32} pzF evaluated. (44) (45) (3 f'ii3 .: /XF + 3333 need to be y')3 (45) continued -+f33I :iff#3 * = The 3/lY3X + + and other averages /k similarly needed averages are calculated in appendix 4. (46) z, /3 t lcdi /3'"X! f333 w f /3333 ¿3223 /3(3 +1313,/33,1 = Tb, - freni : (47) V!X3Y3X/ + (//3 p333 esth*s _ Z3 p333 + 13 th $Et, the value Y! XJ-Yxa f2fl3ZZ)<3 ìì * 3 * e:z (44) , (45) , and (46), . , i/ /dZ 3 -1- °tfFi * ?i /3A:23 7' 3 f3333 * //3 f223) needed which is given by -t (46):; The first term on the right-4iand side of equation (48) be expanded. Thus, may ) 22 (49) + + fS233 f3ß zì3) /3))3 I%I! ±?.Z3) +Az3(1z From the average values of the direction cosines, ($0) i: %i f3/3I 3ff3 -+ 33 + The second term on the right-hand side of equation (48) ny be expanded to gIve 7!.Ç3il3 f3333 Il) + * 7j31)3Z?-3 ¡73Z:5(3333 -J- f313 +j32Z3 Thus, troni equations (48), (50), and (51), (52) (3)3 + +f9113333 -- : - F 7 (3.Êg -1 7 33 : + ;:(R3/:3333 /3ìì3 (23 Us ing equation +4 1.4/3)3 (53) & - Ci1f33 The values of f3)3= X + íì3 33 33 7 values ofÇ3. L and I3333 . For siUcity, øence, zC4x?+ 4Y-,.Zz (54) C il X Il yz 4Zz + XY+YZ ± Xz] will experience a relative extreimim value when p (55) -e-- 3J3333 iit. restricted as shown by determIning misim fS22.3: ,c333 4f!33+ are the nuxinim and /3 -=- W - - let Lflx2 A (56) Hyz + + indicated Now, performing the 4z differentiation partial L22X-y+ZJ 2_1e.X-Y-ZJ (57) xy+xz-i-yzJ -F AZ -zCy-,>zJ 2C2Z-yXJ 32 A Three sets of solutions (56) ,4z. - obi ;ained from these equations: A{x1v, Z3yJ, B{x=--4y1z31] Cy-4x23XJ '4/l3. Solution A gives R, are o = 6/7. 4/13 ,s Solutions B and C both gii found to be the absolute tninir*nn and 6/7 was found to be the absoluta nxiitm of Similar solutions for are obtained for the B maining species yz species the ten ,g ' non-vanishing to the one for the 81u and B species ifl2h for which' O; Of&2h. species The re 's appear is &. For this while the derivatives of the other nine of are zero. 's f. is before, become (ß1123, i' let IZ3 32, *2/,3, In this case the required terms for the determination of (59) LI f3 ççi, f3' / - The 3 i°k are = 4 _3_ ¡3I/, Efl 2 Iz3 'iL' -j ß/t3 -I( AppI7tnØ e*ation (40) to the eqisation given in (59) (60) A sury gronp and the appendix 5. of the , Ti vedous species et this andO&Qb point gxonps will be found in iii, As In the case of the gr°u, LIii let us deterTdfle which cf the ten distinct linear cothinations of the atad with the various species of the group. similar to thea transforn as , as F'm -'xyy ( ,zx' + ( ¶' *7* J3 ' are associ» In a process of equations (41) it can be shown that g,75 and 's transforms as 1:;'rn m- transforms as E 3 transforms f-' P') ( transform as ) P'zyy i 8, - lu J' and ' (see appendix 3). Bausana is a representativa molecule of this group and a procedure alMiar to that which of £2h ts ehylene carried out will be carried out for this molecule. ,' - The z.'.axis is a six.. fold symmetric axis. 7- I, The normal vibrations of C lg *2g+ Elg± 4Ef Ij,, + are: Ø+ 2u *1U Then, since the infrared activa species are ordinary the AZu, Ra* *CtIVe species are hyper-*as aøtive species B, Ein, sad E, the and and 1%. 2u Eit the while are, as indicated above, nueers of activa frequencies are as given in the following tablee n (Y) (Y) nCi) hyjer-R IR 2 B E 1g A lu '2u Blu B 2u E lu SI E Ja Hence, the infrared frequncies are allowed in the hyper-Raman effect and six add1t11 fuds*ental frequencies are allowed also which are not in either the infrared or Raman spectrum. xt the observable quantities, species of the consider the B1 6h point group will be calculated. species. for this species. i2, for the various Let Q represent a normai coordinate Hence: 0 (61) and ali other of the lisiar group are zero. First coinat1esi of the From this it is found that (62) see next page for this (62) q- = and ali othez --7';1 's vanish, O Lat =- Hence the non-vanlshiig te for this species y be 112 represented Ç (63) 1i2t/1zi1 (222' 'I! thiS case, /21. /5 (Z2 O JL"" ,L''&!:/f L 0r o L ipp1ying equation RO), to the equations of (63), (C)//7 = The species B2 to that of B. has linear combination of the This yields that for B / 's similar is given by (65) 1lso ¡Ç the species of the degenerate pair E acts identically as the ) ¡ of the species of the point group = (66) for the E2 species. Next, consider the 's of the A2 of species. U Q represents a nornd coordinate for this species, then Thus, it is found that IL O) (68) and o - (67) all of the other (69) o are zero ter this 's - species. [at t3 Hence, the non-vanisbing terms for this species may be re- presented P131' I'311' I223' ß232' ll3' P"s are exactly those obtained in the Bi These the 2bPOiflt /,Y/33 _/8 -A ,L. /3 A f'223 7?1l3. ,c?' to ,G' &3 _ 1'1i3 and ,c:333 PN As before - ,G'J , J;'37 7(- the maxinuzm and mininnim values of restriction on the range of //333 But, for this species 46h This reduces E z: YO) sain species of of This leads directly to group. Ei/- 4ß3 for the species 322h p put a useful For simplicity, let ,,9. Hence, y::&Y '4 731Z the relative extreniiru values are found by setting 7 __ (71) = This leads to the following set of 2S.- (7e) x z(7+y2_ay)(4+2y) 4y Z3x4y+Z (Z3x2+4yZy)2 Two sets of solutions are (Z3x24yz+Zxy) obtained Af3x/] Solution A gives 4/13 /2x) Y 23xa#4yZ+Zxy (73) equations: from these equations: 5f2v->] _; Solution B 4/13. 6/7. gives is found to be the absolute mininnim and 6/7 is found for species A to be the absolute maxiurnm of of the group 196h' The remaining species ofL96h which of (s's associated with it is E1. has llaerar If Q be a combinations norn1 coordinate for this species, then o 0 (74) while (75) (yy(xxy) o=13zz __Q and so forth. Thus, it is (76) see next page found that (76) è' /9X and al]. 's are zero for this species et the other particular normal coordinate. Hence, te non'-vanishing and Let = /&/J3 ) (77) C),' LÌLY K j6? = Pi terms for 18X/,V::: this species ny be re bJ111. P122 2zl' I-2l2' ì33' 1'3l3' ß$$ set is equivalent to the set of non-vanishing 1$ ef presented This the iu species of the A? cg: ['ßiz-ii ?3 for the E' (78) point group. This leads dt*ectly to 2//-3J/_fi//Gz7 But, for this species 6h This reduces fL = I / p / 2 7 Li to /'/i Again the relative extreimam values of by setting - /& Piz j 11/ species of [) lll3'?l22* 2h where / f x9111 P/il i3J will be determined and Hence (79) (80) f17 Z , bXz -'úox81 ?L. O%299y2/2x/ _______________ . 31 (80) continued Two sets of solutions are obtaIned from these equations: SolutIon A 0.363 gins f = 0.363. Solution B gins )r 0.732. s found to b. the absolute ainiz*a and 0.732 was found to be the absolute waxioum of A suwnery of the for the species E. for the various species of this group will be found in appendix 5. Lc4 ¿y 14'Ò y /4-O i rr LP1ER VII TIE 0,h SYMETRY in the case of the 2h POINT GROUP group, let us determine which of the ten distinct linear coInations of the are associated with the various species of the group. In a process similar to that shown by equations (14), it can be shown that ( , /2? 1' and -t and ( f transform as the species 3 3 f33,77 transform as the species 7I fByyz transforms as the species L.u, and - xxx ,c,cy' ( transforms as the specIes xyy (3xyz (,(3,- 3(3,yy(3yyy (see appendix 2). fl rbon dioxide is a representative molecule of this group and a procedure similar ta that for ethylene of 2h will be carried out for this molecule. - y., The normal vibrations of 2 -z the z axis is the molecular syietric axis are of the Thus, since dipole monnt is of species polarizabllity is of species are as given in the table. *7Tg*Ag (see next page) and ordinary the active modes n«) nI) hpru4 I - i z_ I i -* For a linear triatomic sjitric molecule, such as carbon dioxide, there are four fundamental molecular with vibration. The frequencies of totally synetric vibration associated is active only in the Raman effect, comparing only the Raman and infrared spectra, while on the same basis the anti symmetric stretch of the molecule associated with the species is active only In the infrared spectrum, and the de- generate bending motion, associated with the also active only in the infrared spectrum. 1TU species, is Thus, it is readily seen that the existence of linear coninatIons of the associated with both active and 1's insure that the Infrared fundamentals will also be active in the hyper-Raman effect. Next, consider the species flu. consideration of either Bi or B2 of the point group 6b will show that 6/7 for this species also. II If the the fl's of the species of the species A Qn -- (70) z f for the ( 01 S6h + 7 23f3//3 hare compared with of it is apparent that , 33 -fri -t- //3j43 species and that the range of is given by ± L (81) 13 The last species of the fs Ehin which linear cothinations of are associated is j7. A comparison of the this species with those of the species Ei 0e 's of of the point group ShOWS that (78) fn - for the species '7T' 4(s1ßì -1- 5O/3, 1 /333 -3ßìiììj -1-- The range of n again being given by o.3 (82) A sury of the : f O's 4 o73 for the various species of this group will be found in appendix 5. HAP1ER VIII .ji, ('JI retical expectation for the hyper-Raman to be d. The theo colsis utratlon dep**. of the this subject is y$ work on Much lines should be determined for cases of ion-ion, ion-dipole, dipole-dipole, etc., interactions in detail much more than in appendix 7. The quantum mechanical aspect of this work should be worked out to show the approach. flowever, the experinental side of the theory can not be overlooked. Although the values ratios for the Z been nasured and 7T species fall within and is offered funda*etal for carbon disuluide have A mode of vibration is allod effect. This predicted range, very interesting prediction by the hyperpolarizability theory. effect which is not Raman for the depolarization the theoretically cases should be studied. ny more usual validity of the classical For ethylene a alld in the byper-Raman in either the infrared or the mode corresponds to the species. Theoretically this line should have a depolarization ratio of 6/7. In the case of benzene even more apparent should be the confirmation of the theory. Allowed fundamental modes of vibra tion in the hyper-Rainan effect correspond to the species B1«, B2«, and E2. All six of these allowed lines should have depolarization ratlos of 6/7. This theory should not be considered as a contradiction or a rival theory of those previously presented in papers by Evans and Bernstein, and Welsh, However, it should be considered as a complementary theory describing the same phenomena only considered from a slightly different point of visw. 37 BiBLIOGRAPHY 1. Buckingham, A. D. and J. A. Popie. Theotetical studies of the Keri effect. I. Deviations from a linear polarization law. The Proceedings Section A 68: 9O69O9. 1955. of the Physical Society 2. Goulson, C. A., Allan Maccoil and L. E. Sutton. The polarizability of the molecules in strong electric fields. Transactions of the Faraday Society 48: 106-113. 1952. 3. Evans, J. C. and effect. the H. J. Bernstein. intensity in the Rain interaction on V. The effect of intermolecular Raman spectrum of carbon disulfide. of chemistry 34: 11274133. Canadian Journal 1956. 4. Glasstone, Samuel. Textbook of physical chemistry. Second ed. Princeton, New Jersey, D. Van Nostrand, 1958. 1320 p. Irbert. Classical machanics. Reading, Massachusetts, Addison-Wesely, 1969. 399 p. 5. Goldstein, 6. Gray, Peter and T. C. Waddington. Fundamental vibration frequencies and force constants in the aside ion, Transactions of the Faraday Society 53: 901-Q08. 1967. 7. Herzberg, Gerhard. Infrared and polyatomic molecules. New York, 632 p. Raman spectra of D. Van Nostrand, 1945. 8. Lindsay, Robert Bruce. Concepts and mathods of theoretical physics. New York, D. Van Nostrand, 1951. 515 p. 9. Sewell, Geoffrey L. Stark effect for a hydrogen atom in its ground state. Proceedings of the cambridge Philosophical Society 45: 678-679. 1949. 10. Sherbini, M. A. El. 11. Third-order terms in the theory of the Stark effect. The London, Edinburgh, and Dublin Philosophical MagazIne and Journal of Science 13: 24-23. 1932. il. L, P. E. Pashler and B. P, Stoicheff. Density Raman spectrum of carbon dioxide. Canadian Journal of Physics 30: 99.110. 1952. Weisch, H. effects in the 12. Wilson, E. Bright Jr., John Courtney Decius and Paul C. Cross. Molecular vibrations. New York, MGraw-Hul1, 1955. 388 p. s, I GEFERAL FOR11IAS OF TRANSFORMItTION FOR TiE 1* the hyper-.Ratwi effect (i) is given ,,U PF F'F'1 E1 ¡3 .. by EF» also (2) E) Is exe lE1, E2, light along the tbe electric vector of the Incident tbe asscules ss and (E,, E, IQ is the electric vector et the Incident light along the 1abor are the direction cosines between atory fixed sass. the two fIeld syst, 1* is guss (3 Now, L product of two th «, Y cononents of the electric Z system In terne of the (1, 2, 3) system b7 F E' - ¿T conthining equations (1) and (2) (4) ,.4/p F'F" 1ut,,pi* the U.: (S) ,X1F Y. Z system - in the (1, 2, 3) system. 1A( Let is gi'n by (6) ,«f -Z (,flFF'F» F F''F' Then, solving for -:2X ,41i since ;y Ay (7) 3X are the elennts of an orthogonal matrix. thßgF ExpS*sion of the determinant gives a) _/u - [ ( (A)(FF1 E + [2: (. E'J(z3y 3ZZ) F1'](-3 :zxy) FF' 'a' -- /SZFF E 'F'P) E Hence, (9) ß/oaod 3z -:L3YZZ) 4 S2x) t;:l fPF'1F = F" /FF ' ''(zx3y F Ill/F P X>i) ' Thus, in generai, (10) ,s F3"1 1FF'F" -= 2: F '1ci1 zi& E Ag I i I i 81g 6(zx) 6(yz) i 6(117) i I i 1 1 "i I i 4 4 B29 t 4 i -1 1 4 14 B3g I 4 4 1 1 1 .i1 Au I i I I iii 4 I I i I .1 1 4 1 4 1 1 B« Ay ig 4 1 1 4 1 4 1 .1 * c(xx,efy,c1zz Rz OIxy R7 IZX 'z Au Blu BZu 3u 441 .1 Ç 4 6b *19 E 2C 2C3 C2 34 3G2" i 2S ZGb* -*1r.J--JuF 1 1 Aagli Big i 4 1 1 1 1 11 1 1.14 .1 1 4 1 4 i 1*1 1 4-1 1 0 0 2 *1 4 0 0 0 0 24 4 2 0 0 1 ¡ 1 1 1i-14 1 4 1 i14 020 14 EIg 2 1% 2isl *i i i I 1 1 1 1 1 444 4 2 i j Ji -4 4 4 1 1 1 1 1 444 144 4 4 4 4 1 1 14 141.1 114 14 14 1.14 414 114 21.1.2 00 «241 20 0 1 B Elu E 1 2442 f) 0 .21 140 0 oh ;± 1L4 g $... J. I ... I i I ... i i I.. 1 I i ... D&rj 2, i ç O O 2 2COS?4S .. .. .,. e. Se e. j ... iø'I 4 ... 4 tel ,, i 2eos4. O 2 i.. i ... «4.4 j.( 2 2ccs. 04 2 2Cû2. O 2 4cos24. o 2Cob. O 2 2. O e e. e. S. ecc e. - ce. (T.T) e. T nrt& 1:IT:ur ruM M _ L 1LId1 n t- j- i 1 t 1J . . r o T - , 1±i , - I ;1Y7* z; -:TA (J3yzz, 1JL_,_L J r: u ßrn) - ----- ( f Lt yJ3 *72 fyyy3ßxxy) ( yz, 0ZX) (xiry'xy) i. I ilr. (Rx,Ry) O ::L: t: oXtc%Y)cZ I O 2 2 - 1 ¿__1( coCz i 1, 1W ølmuutr Tc átaia tsaasf*t1e.X*, of a nøods only to nata hat dsftnition, i, a., (12, p.92) (I) In ii1ch R 1$ d1 s a NsIt*$*Iat 1*1W $t$iat = I ** 1. 1k then add up jmønl 1« torna of the tnansfcat1on. for u t1*s 4* a.s lbs Z *ds. fosnatlan Is &Ç; 1 (3) siw y' N& X - I X + cas Y y 2'Z mats, x , k -= i+ec ton a general sotary reflection, S 2# X - X' - cas y' - s1N4x 2'-Z XS -H + -- HIM., (5) ,; - ±1-t- zco' S 'N 4 y Thun. the txans». positive sign is used for proper rotations, III$S where the rotations, for itçroper and the negative sIgn Is used rot*tInes, rotary reflections. This concept can be extended for the transformatIon of tk kind which would be Involve the product of two coordinates This the character of a transformation for polarizability components. Thus, fe general for a rotation the trans formation is (6) - (X ( yi)2 ( (z')2 csd x - 51A/d v) SIW& =- X- X/)(y))(C0 (x')(') (c (y(Z') ¿N 1y)(5wdX cos\/) MY)(Z) X + COSd y)(Z) where hence, (7) Nci co ( + i) as consider a general rotary reflection, fies that S. The only changes of the rotation (8) x')(2') - - (coSo =. hence, (3e* *ezt page) - (3/iVO ;x X -1- sìivd co&d y)C7) )(.J 47 (9) ;2 COSO -)) Thus, (10) cOsc - caonents The ±ì) (,2CoS hpnp*krizabi1ity transform as of the the product of three coordinates. llenes, the character of a transformstion for the hyperpolarlzabiUty components ny be found in the sams vy in as In the previous two cases. gsza1 for a rotation, 11-» U9-:? ((s) X (smoL) ')3 ((sINC() X !- (CßSO) (z') (X)a < U') ') is y) 3 ) 3 z3 ¿:(so )2= («$ (X')2(z') (X c, ue transformation Thus, (Z ')k:: (T92(z') (SINd) )'(srncx+sc X X (SThc ((SoL) X (SINs) ( (Cß$ X (SI$ r) z X -- «os )2 z X+ coos:'6 = (sx (V) «.)2= ) ((SINcô (X') (Y') (Z')= ((sb x r)((szro x-4cosô 'r) )2 z i) (SmO r) sir,c*. z-t. «xso bnce, (12) xíf))z Y,) 2CO.cd- -I) 2 cosc (4co,s-- r) z Now consider a general rotary reflection, changes from that of the rotation (13) (Z')3- are -z3 iZ,= - (wosd z - sm (V)2 Z9 cx' (j The only 4. 'z' ((sc X z (SINd + scb (SThc r) 2 z r) 2 z r)(siNc x-toeso Hence, (14) Thus, X = co 4CoSo ±2c&4-I) r)z The Eulerian ang'es are fIasd as th $kr.. successive engles of rotation b7 asans of which a gives cartesian coor y be transforn*d to another (S, p. 107.109 dinate system Here the system as shown in Goldstein will be used, 's Ç)sica1 ). chanics Hence, the Eulerian angles are detlned by the following three rotations: C' (1) where , 5 and i , represent column matrices ; sents the coordinates of the repre nttia1 cartesian coordinate system and Z' represents the coordinates of the final system, D, C, and (2) B are the ntrices D = Jcos4 sxr o (-S1N4 cos4 e o i s 0 ko /1 c!o o 0 .sin& fcosq' o The product natrix AB cos& si/' o o o i Bf.4ZNV' t. sme will furnish the direction cosines which connect the initial and final cartesian coordinate systems. a) / sV-' s.ws sx sxr -SlN Y'szNçía.00se S NbSC/)SIN&SZN A $lNS SINeSIN cos& There are only four distinct types of products ci direction cosines which need to be evaluated. y;) fxi These are X/y2) X!YI For cu,terian angles the average of a function, f ( X2YZ '4' ), given by (4) (S)&) 8ffff(e)iìded Hence, (5) '/5 z /I5 - is L1 ') TIE P3U TiE POINT SUMMARY OF OF .,!4 _-:-- 81u iip7+1I;7 GROUPS OF TIE SPECIES 2h'D6h' !__._-__ - f?zzz±xz v- -1*_ -.-- WT -,L-r* Z')YY Bi ç3 rpi-Ii Bi m)f?:?z 6dI 4I1$rt 6/7 . ?u7,yzz 4/13F1, i i: ;Y;m z$ Y11 YYYY - p*vy pu1 !(r . T-37z -- T 11 p yzz 413;771/3?77 7 2L4*41.P f - u?xy* _-__ 4;Yz g44ii B2« B2 ii a f f;x . TIE VJRIOUS RANGE OF A? '4 : 6/7 6/7 +1L n ___zzç zzz+zzzç/pzxxg tìrJr Ta flZ0 ¿/9 ç) XXZ ¿/9 __ (zL/jxxEy*stz) rn ¿19 sa 73 6k ---- _J _uz, íz yz --- -.-- - 23Z49 ¡; 1 ; ) L' 1 7cxy (,Lz Iry*, 1'zy* '- ß ßíz LITI L.M L 6/1 -- ;) P'x2 4(5-i8).. 3/z P*í s- --- 2 J- ZZZ -i) *TEI "Consider a linear current a1on .i.nt the Z axis with ita center at the r- QOL d length L zi1a." (8, p. 416) /1' Let the current flowing through the element be given by (1) I = 10SIHwt The magnetic induction1 and the electric field strength it may be written in terms of a vector potentialTand a scalar potential Thus, -3 B7XA 1E+-= -.3 (2) A solution of and4 may be given as a retarded potential: (3) -; Ä; where - , ç()j C 1 Hence, b neu ny that the etectzic field Intensity cttai*d by evaluating A F() and (i). for cur problem A=A2Z1 (4) Az j1,rN cren and S is the w(t,R/Cl setii Tkapese is z : it sui easdietor. "R Is tt* distance Eroe P to tke (8,p. 416). rn Z axis is directed along the since (5) Is Jint of length da.' túeaastreespac.sothit udijC, Th.$.lstionofAsft.ri.$.nhf* bec A (6) z tas tis $r or w(t.iI.) sisathe øs$ r»/Z. Ii epatica! coordinates sise (7) - ø=VXA Thus, sisee (Q) fl r =0 ; e= Cr srneL oes (uts.kr) SIN (utii4tr)j The scaler potential is given by the Lorsata relation (9) t iba sølution of Ç trom the puavias*$yiMaed*** s OES & cOS(wt.kr) (10) + we LQ k S9 SXN(wt-kr) wer Now the components of the electric field intensity may be solved Thus, for. ______ (11) F SIN(wt-kr) j2GDS(wt'u.kr)] SIN(wt-kr) - L151114k2oe8(wtwkr) wr S(wt-kr) LSJ for the oscillating dipole becomes The Poynting vector (12) The time average o -CF (13) - e) ,,. ,L,l -;:ZJoAIc FThe total radiation may be obtained by integrating the Poynting vector over a surface of a sphere of radius r. rsinødd4. element of area is given by (14) .11 pZ o . rx ;i:7).ra î I1' and e d& d4 J _ The Mter performing integration, the resultis vhere oeS(wtu,kr becs* the Poynting vect*r - ( - J 3C this APPENDIX 7 TIE C*W1DEN OF TIE DTXENSITY OF A UYPER-.RAMAN UPON ION ODNOERTION UT'E In the Debye-Hückel theory the electrostatic potential, cL at a given point in the vicinity of a positive , ) (1) çb:= zet D JC for a ± -for a ri or ion is given by ( 4, p. 956-.958 ) a.ths mere ( is the distance from the ion, constant, is the charge D positive ion negative ion is the dielectric Z on one electron, is the nunther of charges on the ion, and K is given by K2 = (2) YìZf' 4tn constant where k is t and n is the nuner of each kind of ion per cc. The electric field at a distance r finding (3) e E11 ny be obtained by Hence, ____ = - However, the electric field at this point + kì) due to the ion itself is given by (4) cj' - ____ Thus, the electric field on the ion i$*SIf due to the sur-. rounding atmosphere is U J\ these is sn*11 (6) = = 'and -4(J' .4(J\ e y be expanded as + SSD r Using these e*pansions, zCe (7) Thus,! j [-e]= 2eAL way be given by (6) This reduces to (9) Equation tA () of the text gives (10) = snd equation (29) (il) :- d L the text gives 2--- T - = # This waans that the intensity in the hyper-Rawan effect is proportional to ( )2 a constant _______ (12) qDg- +2 nzzJ L The above argunent ignores in contrast to indeterminate asj-+o the con1icìtion that is apparently (ne resolution of this difficulty ney involve the assumption of finite radius for the Ions.