On the Quantum Theory of Molecules

Transcription

1 O the Quatum Theory of Molecules M. Bor a, J.R. Oppeheimer b a Istitute of Theoretical Physics, Göttige b Istitute of Theoretical Physics, Göttige Abstract It will be show that the familiar compoets of the terms of a molecule; the eergy of electroic motio, of the uclear vibratio ad of the rotatio, correspod systematically to the terms of a power series i the fourth root of the ratio of electro mass to average uclear mass. The treatmet yields amog other thigs a equatio for the rotatio, which represets a geeralizatio of the treatmet of Kramers ad Pauli top with built-i fly-wheel. Furthermore, there appears a justificatio of the cosideratios of Frack ad Codo o the itesity of bad lies. The relatioships are illustrated for the diatomic molecule. Itroductio The terms of molecular spectra are usually made up of parts of various orders of magitude; the largest cotributio comes from the electroic motio about the uclei, the follows the cotributio of the uclear vibratio, ad fially that from the uclear rotatio. The basis for the possibility of such a classificatio obviously rests i the comparative magitudes of uclear ad electroic masses. From the stadpoit of the old quatum theory, which computed statioary states with the aid of classical mechaics, this is the cocept applied by Bor ad eiseberg [1]; it was show that the eergy terms appear as the terms of icreasig order with respect to the ratio m/m, m is the electroic mass ad M a average uclear mass. Thereby, however, uclear rotatio ad vibratio both appear i the secod order, which cotradicts empirical fidigs for small rotatioal quatum umbers. ere the problem will be approached aew from the stadpoit of quatum mechaics. 1 It the becomes ecessary to make our developmet with respect to m/m 1/4 rather tha with respect to m/m, so as to obtai the atural order of eergy terms. The cosideratios also become much simpler ad more trasparet tha i the old theory. The uclear vibratios correspod to terms of secod order ad the rotatios to fourth order i the eergy, while the first ad third order terms vaish. The absece of the first order terms is related to the existece of a equilibrium positio of the uclei, i which the electroic eergy for statioary uclei is at a miimum. The fourth order terms for the rotatioal motio illustrate the geeralizatio of the treatmet of Kramers ad Pauli [] i which the the behaviour of a molecule is compared to that of a top with a built-i fly-wheel. I order to determie the eigefuctios ad thereby the trasitio probabilities oly to the zeroth approximatio, the eergy calculatio must be carried out to terms of fourth order rotatios. Oe obtais expressios for the probabilities of simultaeous jumps of electroic, vibratioal ad rotatioal quatum umber through which the represetatios developed by Frack [3] ad elaborated by Codo [4] may be give precise iterpretatio. Published origially as Aale der Physik, 84, Traslated by S M Blider with emedatios by Bria Sutcliffe ad Wolf Geppert 1 Through the discussio of the basis of this work with us, Dr. P. Jorda has helped us with valuable commets, for which we express our thaks.

2 The approximatios to higher tha fourth order will ot be treated i this work; they correspod to couplig amog the three basic types of motio. A calculatio of this effect is oly meaigful for simultaeous cosideratio of all degeeracies of electroic motio for statioary uclei, especially the eiseberg resoace degeeracy which arises from the equivalece of electros also possibly of some uclei ad i diatomic molecules, from the degeeracy of the eigerotatio about the iteruclear axis; these complicated cosideratios will be forgoe here. As a example we will cosider diatomic molecules i detail, usig ot oly the geeral method but also aother utilisig the separatio of variables i which the rotatio becomes sigificat eve i the zeroth order approximatio, as Bor ad ückel [5] have doe it i the older quatum theory. Part I. Notatio ad Defiitios We deote the mass ad rectagular coordiates of the electros by m, x k, y k, z k ad of the uclei by M l, X l, Y l, Z l. Lettig M be ay average value of the M l, we set ad m 1/4 κ = 1 M M l = M µ l = m κ 4 µ l ; the µ l beig dimesioless umbers of order of magitude 1. Let the potetial eergy of the system be Ux 1, y 1, z 1, x, y, z,... ; X 1, Y 1, Z 1, X, Y, Z,... = Ux, X 3 we deote by x the totality of electroic coordiates ad by X, that of the uclear coordiates. The fuctio U depeds oly o the relative positios of the particles; however, we make o use of its particular form Coulomb s law. The kietic eergy of the electros is represeted by the operator T E = h 8π m the symbol deotes the sum which arises from the above expresio by cyclic permutatio of x, y ad z. x The kietic eergy of the uclei is The total eergy is represeted by the operator x X l k x k T K = κ 4 h µ 8π l. 5 m Xl = + κ T E + U = x, x ; X T K = κ 4 1 X. 7 We itroduce ow, i place of the rectagular coordiates of the uclei, 3N 6 fuctios ξ i = ξ i X 8

3 which deote the relative positios of the uclei with respect to oe aother, ad 6 fuctios θ i = θ i X 9 which determie the positio ad orietatio of the uclear cofiguratio i space. Oe ca i a symmetrical fashio itroduce the rectagular coordiates X l, Ȳ l, Z l of the uclei relative to the istataeous pricipal axes of iertia; betwee these there are 6 relatios: M l X l = M l Ȳ l Z l =... Oe ca thus express the X l by the 3N 6 idepedet parameters ξ 1, ξ,... : l X l = X l ξ,... There the exist trasformatios betwee the origial ad the ew coordiates, of the form X l = X + α xy θ, φ, ψȳ l ξ; 1 y X, Y, Z are the coordiates of the ceter of mass ad the α xy are the coefficiets of the orthogoal rotatio matrix, ad are thus kow fuctios of the Euleria agles θ, φ, ψ. The quatities X, Y, Z, θ, φ, ψ are the fuctios deoted by θ i i 9. By 1, the X l are determied as fuctios of the θ i ad ξ i ; by solvig, oe obtais the expressios 8 ad 9. This trasformatio does ot, of course, separate the eergy ito parts correspodig to traslatio, rotatio ad relative motio of the uclei. owever oe ca separate 1 ito three parts: l 1 = ξξ + ξθ + ; 11 ξξ is liear homogeeous i the ξ i ξ j ; ξθ cotais the ξ i ; is idepedet of all derivatives with respect to the ξ i. Oe ca make further geeralizatios about these operators. If we apply the etire operator 1 to a arbitrary fuctio f ξ of the relative uclear coordiates ξ i, the resultig quatity 1 f ξ must be idepedet of the positio i space, hece of the θ i. I particular, i ξξ the coefficiets of the ξ i ξ j caot deped o the θ i. I cotrast, these do appear i ξθ, associated with the ξ i, the ξ i, θ i ad We will cosider these operators explicitly for diatomic molecules. The mechaical problem we must solve is θ i ; i associated with θ i θ j the θ i, ξ i ad θ i. + κ 4 1 Wψ =. 1 We will show that ay arbitrary solutio which correspods to a combiatio of uclei ad electros formig a stable molecule ca be foud by a developmet i a power series i κ. Part II. Electroic Motio for Statioary Nuclei If oe sets κ = i 1 oe obtais a differetial equatio i the x k aloe, the X l appearig as parameters: } { x, x ; X W ψ =. 13 It is of physical sigificace that this solutio is i geeral made usig ambiguous fuctios; compare [6]. 3

4 This represets the electroic motio for statioary uclei. We assume this eigevalue problem is solved. The eigevalues deped oly o the fuctios ξ i of the X i ; the oe ca use the coordiate system defied by the pricipal axes of iertia, i.e. let X l = X l ξ. I this system of axes, the eigefuctios deped, besides o x k, oly o the ξ i ; however, if oe trasforms back to the arbitrary space-fixed axes, the θ i agai become ivolved. We desigate the th eigevalue ad the correspodig ormalized eigefuctio as W = V ξ ψ = φ x; ξ, θ 14 so that the idetity } { x, x ; ξ, θ V ξ φ x; ξ, θ = 15 is valid. ere we assume that V is a odegeerate eigevalue. As a matter of fact, this is ever the case sice the idistiguishibility of the electros itroduces the resoace degeeracy, discovered by eiseberg ad Dirac; for diatomic molecules there is a additioal degeeracy of the agular mometum about the axis. But sice we are cocered here oly with the systematics of the approximatio procedure, we will ot cosider these degeeracies. Their cosideratio would result i secular equatios i the higher approximatio. The most importat goal of our ivestigatio is the proof that the fuctio V ξ plays the role of a potetial for the uclear motio. For this we must have several auxilliary formulas which will be derived ow. It is ecessary to show that the matrix correspodig to the derivative of the operator x, x ; ξ, θ with respect to ξ i, for costat x, x ca be related to the derivative of the fuctio V ξ. Istead of takig the derivative with respect to the ξ i directly, we replace the ξ i by ξ i + κζ i ad differetiate with respect to κ; the coefficiet of a power of κ is the a homogeeous polyomial i ζ i, these coefficiets beig derivatives with respect to ξ i. Thus we write V ξ + κζ = V + κv 1 + κ V +..., 16 ad correspodigly a V b V 1 = c V = 1 = V ξ V ζ i ξ i i i j ζ i ζ j V ξ i ξ j, = + κ 1 + κ φ = φ + κφ 1 + κ φ Oe ca ow develop the quatities φ 1, φ i the eigefuctios φ x; ξ, θ, settig a φ 1 = b φ = u 1 φ, u φ. 19 4

5 Thus u r is a homogeeous polyomial of the rth order i ζ i, for istace u 1 = u = ζ i i ζ i ζ j i j φ φ φ ξ i dx φ dx. ξ i ξ j These itegrals, i which dx deotes the volume elemet i cofiguratio space, are idepedet of the orietatio of the uclear system i space, hece idepedet of the θ i ; oe ca thus evaluate them i the pricipal axis system. If ow, F deotes ay operator o the x i, we defie the rth order matrix elemet of F φ Fφ r dx = F r. 1 For r = this becomes the usual matrix elemet F = F = φ Fφ dx. I geeral, by 19, Usig 15 for κ = F r = V Furthermore, we obtai by substitutig 16 ad 18 i 15, the followig idetities: r u r F. 3 = u r V V. 4 a V φ V 1 φ = b V φ + 1 V 1 φ 1 + V φ = Multiplyig these by φ ad itegratig over the x i, by virtue of 4 we fid: a u 1 V V + 1 V1 δ = b u V V + 1 V 1 + V δ = From these oe ca compute the 1,,..., ie the matrix elemets apply these formulas. 3 ξ i, 6 ξ i ξ j,.... We will later Part III. Settig-up the Approximate Equatios A arbitrary cofiguratio of electros ad uclei caot always be treated by a geeral approximatio procedure. We will here cosider oly states which correspod to a stable molecule. We will begi with the followig questio: 3 The classical aalogue to the simplest deductio from these formulae, amely the idetity 1 = V 1 which follows from 6a for =, is foud i [7]; compare especially with 4, formula 11. 5

6 Is there a system of values of the relative uclear coordiates ξ i such that the eigefuctios ψ of the eergy operator 6, i so far as they deped o the ξ i, have values sigificatly differet from zero oly i a small eighbourhood of this set? This wave-mechaical requiremet correspods obviously to the classical coditio that the uclei udergo oly small oscillatios about the equilibrium cofiguratio; the ψ is the probability of fidig a certai cofiguratio of give eergy. We cosider, as the uperturbed system, the electroic motio for a arbitrary but heceforth fixed uclear cofiguratio, ξ i. We the develop all quatities with respect to small chages of the ξ i, which we desigate by κζ i ; we presume the that the domai of oscillatio is such that κ is close to zero, a assumptio which is oly justified by its success. We have the as i 18, part II, the developmet x, ; ξ + κζ, θ = x + κ 1 + κ +..., 7 a = x, x ; ξ, b 1 = i ζ i ξ i, 8 c = 1 i j ζ i ζ j ξ i ξ j, ζ ad from 11 sice ξ = 1 κ κ X, X = κ4 κ ζζ + 1 κ ζθ + = κ ζζ + κ3 ζθ + 1 ζζ + κ ζθ + ζζ a ζζ = ζζ ξ, ζ i ζ j b 1 ζζ = ζζ i ζ i ξ i a ζθ = ζθ ξ, θ, ζ, θ b 1 ζθ = ζθ i ζ i ξ i a = ξ, θ, θ i θ j b 1 = i ζ i ξ i

7 The argumets ξ i are hereafter to be cosidered costats. The total eergy operator is the = + κ 1 + κ + ζζ + κ ζθ + 1 ζζ + κ ζθ + ζζ The succeedig terms all have the same form ad ca be formed from the term i κ 4 by icreasig the superscript by 1. We also develop the desired eigefuctio ad eergy parameter with respect to κ : We the obtai the followig approximatio equatios: ψ = ψ + κψ 1 + κ ψ +... W = W + κw 1 + κ W a W ψ = b W ψ 1 = W 1 1 ψ c W ψ = W ζζ ψ + W 1 1 ψ1 d W ψ 3 = W 3 3 ζθ 1 ζζ ψ +W ζζ ψ1 + W 1 1 ψ e W ψ 4 = W ζθ ζζ ψ +W 3 3 ζθ 1 ζζ ψ1 +W ζζ ψ + W 1 1 ψ Part IV. Solutio of the Approximate Equatios of zeroth ad first Order: Equilibrium of the Nuclei The zeroth order equatio 35a describes the electroic motio for statioary uclei as discussed i Part II. From the ormalized eigesolutio φ x; ξ, θ belogig to the eigevalue V = V ξ, we fid the geeral solutio i the form: ψ = χ ζ, θφ x; ξ, θ 36 χ is a, as yet, arbitrary fuctio of the argumets ζ i, θ j ; this must be icluded i order to eable solutios of the followig approximatio equatios. The followig approximatio equatio 35b W ψ 1 = W 1 1 ψ 37 is soluble oly whe the right-had side is orthogoal to ψ relative to the electroic coordiates x i. 4 This gives the coditio { 1 W1} χ ζ, θ = 38 4 We defie the orthogoality of two fuctios f x ad gx by f xgxdx =. 7

8 1 is the diagoal matrix elemet of the operator 1 relative to the x i, thus by 8b a homogeeous liear fuctio of ζ i. This must however, by 38, be costat, sice χ same beig true for ψ. Thus it follows that From 6a ad 17 we have however Thus: W 1 =, 1 = V1 = ζ, θ caot vaish idetically without the 1 =. 39 i ζ i V ξ i. V ξ i =. 4 The validity of cotiuig our approximatio procedure requires that the relative uclear coordiates ξ i must ot be arbitrarily chose, but must correspod to a extremum of the electroic eergy V ξ. The existece of this is therefore the coditio for the existece of the molecule, a law which is usually assumed to be self-evidet. We will show later that it must ecessarily be a miimum as well. The fuctio χ ζ, θ remais, as yet, udetermied. Settig i 37 W = V ξ, W 1 = ad ψ = χ φ we fid the equatio which determies φ 1 V ψ 1 = 1 φ χ. 41 A solutio of this by 5a is ψ 1 = χ φ 1 φ 1 is the fuctio 19a defied by 18. The geeral solutio is obtaied by addig a solutio φ of the homogeeous equatio with the yet udetermied factor χ 1 ξ, θ: ψ 1 = χ φ 1 + χ 1 φ. 4 Part V. Solutio of the Approximate Equatios of secod ad third Order:Nuclear Vibratio We ow reach the approximatio equatio 35c, which after substitutio of the solutios for the lower order approximatios is V ψ = W ζζ χ φ 1 χ φ 1 + χ 1 φ. 43 I order for this to be solvable, the right-had side must agai be orthogoal to φ ; usig the otatio of part II this yields, because of 39: { + ζζ + 1 } W χ =. It follows from 6b with V 1 Sice ζζ = that = V. 44 by 3a is see to be idepedet of the x k we fid: { ξξ + } V W χ =. 45 Notig the meaigs of ζζ uclear vibratio: ad V give by 17c ad 3a we see that 45 represets the equatio for harmoic ζζ ξ, ζ i ζ j + 1 V ζ i ζ j W ξ i ξ j i j 8 χ =. 46

9 This equatio shows that the fuctio V ξ plays the role of a potetial eergy for the uclei, up to terms of d order. For the existece of a stable molecule there is a further coditio that the extremum of V ξ determied by 4 must be a miimum; the the quadratic form V must be positive defiite, thereby all degrees of freedom ζ i stable ad oscillatig about the equilibrium cofiguratio are possible. It is kow that the equatio for the vibratio 46 is separable through a liear trasformatio of the ζ i to ormal coordiates η i. If σ s ζ be the ormalized eigesolutio of 46 belogig to the eigevalue W s, the geeral solutio is a W = W s, χ = χ s, b χ s = ρ s θσ s ζ. 47 The idex s thus represets the set of vibratioal quatum umbers. ρ s is a, as yet, udetermied fuctio of the θ i, the itroductio of which is ecessary for the cotiuatio of the procedure. It is kow that σ s ζ is a liear combiatio of products of orthogoal ermite fuctios for the idividual ormal coordiates η i ; these fuctios have the property that they approach zero very rapidly expoetially outside the limit of classical vibratio. So our substitutio of ξ +κζ is justified sice it ideed leads to a solutio, with regard to the ξ-oscillatio withi the limit, which vaish with κ. We apply the further property of the orthogoal ermite fuctios that they are either eve or odd fuctios of their argumet. Let Φ be ay operator o the ζ i. We ca the costruct the correspodig matrix Φ ss = dζ is the volume elemet i the space of the ζ i. I order to solve equatio 43 we substitute o the right-had side, usig 45, 43 the becomes: The geeral solutio is σ s Φσ s dζ 48 W s ζζ χ s = V χ s ; V ψ = V χ s φ 1 χ s φ 1 + χ 1 s φ. 49 ψ = χ s φ + χ 1 s φ 1 + χ s φ, 5 χ s deotes a ew, udetermied fuctio of the ζ i, θ j ; this is easily verified usig the idetities 5. We ow cosider the approximatio equatio of 3rd order 35d; after substitutio of the already determied quatities, this becomes: V ψ 3 = W 3 3 ζθ 1 ζζ χ s φ + W s ζζ χ s φ 1 + χ 1 s φ 1 χ s φ + χ 1 s φ 1 + χ s φ. 51 We may cosider the right-had side as a developmet i the φ ; we write V ψ 3 = W 3 χ s φ F 3 φ, 5 F 3 = F 3,1 χ s + F 3, χ 1 s + F 3,3 χ s ; 53 9

10 the F are operators o ζad θ, ad a F 3,1 = 1 b F 3, = ζζ W s + 1 1, 54 we ca say about F 3,3 oly that it is a homogeeous fuctio of 3rd degree i the ξ i ad the /ζ i. If 5 is solvable, we must have W 3 χ s F 3 = because of 53 ad 54a, by 54b ad 44 F 3, χ 1 s = W 3 F 3,3 χ s, 55 F 3, = ζζ V W s. Thus 55 is the ihomogeeous equatio correspodig to the vibratio equatio 45; sice 45 has the ormalized solutio σ s belogig to the eigevalue W s, 55 is solvable oly whe the right-had side multiplied by σ s has a vaishig itegral over ζ-space. This gives, usig 47b, a differetial equatio for ρ s θ: F 3,3 W 3 ρ s =. ss owever, F 3,3 is odd i the ξ i ad /ζ i so the diagoal elemet of the ζ-matrix must vaish. Whe oe trasforms to the ormal coordiates η i, σ s becomes a sum of products of orthogoal ermite fuctios, F 3,3, a polyomial of odd order i the η i ad /η i, so that every term cotais at least oe of η i or /η i i a odd power; therefore every term i the ζ-matrix vaishes. It follows therefore ad ρ s remais, as before, udetermied. Now we may solve: S 1 s is the followig operator with respect to the θ i : W 3 = 56 χ 1 s = S 1 s ρ s 57 Fially the solutio of 5: ad by 53, this has the form: S s 1 = s ψ 3 = F 3,3 σ ss s W s W s F φ 59 V V ψ 3 = G 3,1 χ s φ + G 3, χ 1 s φ + G 3,3 χ s φ, 6 G 3, F 3, =. 61 V V 1

11 Notig 54 we see that G 3,1 is a umber, G 3, a differetial operator with respect to the ζ i ad G 3,3 a operator with respect to the ζ i ad θ i. By 6a, Part II G 3,1 χ s φ = 1 φ χ s V V = u 1 φ χ s = φ 1 χ s thus ψ 3 = φ 1 χ s + G 3, χ 1 s φ + G 3,3 χ s φ. 6 Part VI. Solutio of the Approximate Equatios of fourth ad higher Order: Rotatio ad Couplig Effects After substitutio of the quatities already determied, the 4th order approximatio equatio 35e becomes: V ψ 4 = W ζθ ζζ χ s φ ζθ + 1 ζζ χ 1 s φ + χ s φ 1 + W s ζζ χ s φ + χ 1 s φ 1 + χ s φ 1 φ1 χ s + G 3, χ 1 s φ + G 3,3 χ s φ. We develop agai the right-had side i the φ : here we have V ψ 4 = W 4 χ s φ F 4 φ, 64 F 4 = F 4, χ s + F 4,3 χ 1 s + F 4,4 χ s ; 65 F 4, = ζζ W s ad is idetical with F 3, 54b. While F 4,3 is of odd order i the ζ i, /ζ i, F 4,3 is of eve order. The itegrability of 64 requires: W 4 χ s F 4 = ; this meas that by 65 F 4, χ s = W 4 F 4,4 χ s F 4,3 χ 1 s. 67 The left side agrees agai with the vibratio equatio 45 because of 66. The right-had side must also be orthogoal to σ s. Substitutig the expressios for χ s ad χ 1 s from 47b ad 57, ad usig the symbol [ ΦF 3,3 Φ 1 ss = σ s ΦS s 1 ss ]ss dζ =, 68 s W s W s 11

12 we fid { F 4,4 + F 4,3 1 } ss ss W4 ρ s =. 69 This equatio determies fially the fuctio ρ s θ, hece the motio of the pricipal axes of iertia: the traslatios ad rotatios. The pricipal term of the operator i 69 is the oe which cotais the secod derivative with respect to the θ i ; a glace at 63 shows that it arises from s φ, the term correspods i F 4,4 to = χ φ φ... dx, 7 i the place of the dots we have to put i the fuctio which is operated upo. Sice the operator 7 is idepedet of the ζ i, the diagoal elemets of the correspodig s-matrix are idetical with it. Physically the fact that the complicated operators appear istead of the simple operators idicates a couplig betwee the top motio of the uclei ad the electroic motio. These are, as we will later see for the case of the diatomic molecule, the same effects that Kramers ad Pauli [] have tried to demostrate usig the assumptio of a fly-wheel built i to the top. Thus there are terms i 69 that cotribute to the operator ζθ ; these correspod to a couplig of the top motio with agular mometa which are a cosequece of uclear vibratio. Fially, there are terms which do ot cocer the θ I ; these are the additios to the vibratioal eergy of order κ 4. Sice the traslatios ca always be separated i a trivial fashio, we cosider oly the rotatios. If r be the rotatioal quatum umber, we have for the solutio of 7 W 4 = W 4 sr; ρ s = ρ srθ. 71 The oe ca solve 67 ad fially also 64. It is of o use to write out the formulae explicitly. Clearly, the procedure may be cotiued; however othig ew of sigificace will appear. The higher approximatios describe coupligs amog rotatios, vibratios ad electroic motios. Quatum umbers other tha the oes already itroduced do ot eter. We summarize ow the cosequeces of our solutios. The most obvious result is that i order to determie completely the eigefuctios to th order it is ecessary to solve the approximatio differetial equatios to 4th order; we have ψ sr x, ζ, θ = φ x, ξ, θσ s ζρ srθ φ is the eigefuctio for electroic motio for statioary uclei, σ s that for uclear vibratio, ad ρ sr that for rotatio. Thus are defied the vibratioal coordiates ζ i from a equilibrium cofiguratio ξ i which is defied by the requiremet that i this cofiguratio the electroic eergy V ξ is a miimum. The determiatio of the three fuctios φ, σ s ad ρ sr yield the eergy to 4th order: W sr = V + κ W s + κ 4 W 4 sr +... ; 73 V is the miimum value of the electroic eergy which characterizes the molecule at rest, W s is the eergy of uclear vibratio, ad W sr 4 cotais alog with additioal terms for the vibratioal eergy the rotatioal eergy. I this approximatio to κ 4 the three basic types of motio are separated ; the couplig amog them ivolves terms of higher powers of κ. Give 7 we ca ow calculate trasitio probabilities itesities of bads. The electrical momet of a molecule M cosists of a uclear part P ad a electroic part p; the x-compoet is: P x = l e l X l M x = P x + p x, p x = e. 74 k x k 1

13 ece from the set of matrix elemets with respect to x k, ζ i ad θ j ; p x = p x φ φ dx 75 is a fuctio of the ζ i ad θ j, the the are fuctios of the θ j, fially p x s = p x s σ s σ s dζ P x s s = p x sr s r = P x sr s r = P x σ s σ s dζ 76 p x s s ρ srρ s r dθ P x s s ρ srρ s r dθ 77 are umerical costats which determie the radiatio ad the trasitio probability for sr s r. We ca iterpret this step by step procedure as follows: for every electroic trasitio, there correspods a virtual oscillator with momet p x ; from this oe obtais the matrix p x s which correspods to a system of vibratioal bads s trasitios from s s, by a rule somewhat differet from the ordiary oe i which oe uses oe eigefuctio of the lower ad oe of the upper electroic level equatio 76. We repeat the procedure for the lie of the bad correspodig to the trasitio r r. The method of evaluatio of the itesity of vibratioal bads cotaied here is first give by Frack [3] ad further developed by Codo [4]. These are determied by variatio of the fuctios V ξ ad V ξ; oly i the eighbourhood of their miima are the correspodig eigefuctios σ s ad σ s sigificatly differet from zero; their product is so oly whe these regios overlap. Whe the fuctio V ξ chages oly slightly i a electroic trasitio, the bads correspodig to a small chage of s will be itese; however if V ξ chages greatly i the trasitio, a overlap of the itervals i which σ s ad σ s do ot vaish becomes possible oly whe the differece s s is large. These relatios are quatitatively discussed by Codo. Similar cosideratios apply for the rotatios mutatis mutadis. Part VII. Special Case of the Diatomic Molecule As a example we will briefly treat the diatomic molecule. Besides the resoace degeeracy, which is a cosequece of the idistiguishibility of the electros, there is a additioal degeeracy sice correspodig to every eergy value there are two possible modes of motio i which the agular mometum about the iteruclear axis is oppositely directed. Sice we are ot cocered here with the fie structure of bads, we will ot cosider this degeeracy; we limit our cosideratio to cases i which the agular mometum about the axis vaishes or whe the electroic eergy is idepedet or oly slightly depedet o the agular mometum compoet. For two uclei we have oly oe ξ coordiate, the uclear separatio, ad five θ coordiates: the coordiates of the ceter of mass X, Y, Z, ad the polar coordiates of the iteruclear axis θ, ω. The kietic eergy of the uclei becomes T K = κ 4 h 8π m { + µ ξ + µ } ξ ξ ξ ξ θ 78 1/4 m κ = ad µ = M 1 + M 79 M 1 + M M 1 M 13

14 ad Thus: = θ = X + 1 si θ Y + Z, ω + 1 si θ θ si θ θ. 8 ξξ = h 8π m µ ξ, ξθ = h 8π m = h 8π m Substitutig ξ + κζ for ξ ad developig i κ, we fid: ζζ = h 8π m µ ζ, µ ξ ξ, 81 + µ ξ θ. p ζζ =, p = 1,,... 8 ζθ = h µ 8π m ξ ζ, 1 h µ ζθ = 8π m ξ ζ ζ, = h 8π + µ m ξ θ, 1 = h 8π m The uclear separatio is determied by the equatio µ ξ 3 ζ θ. 84 V = V ξ =. 85 The equatio for uclear vibratio is { h 8π m µ ζ + 1 } ζ V ξw χ =. 86 If we set a = 8π m h µ W b = 8π m h µ V η = ζb 1/4 87 we have [8] { } a η + b η χ =. 14

15 The eigevalues are with eigefuctios s is the sth ermite polyomial. The eergy of the vibratios is thus: a/b 1/ = s + 1 s =, 1,,..., σ s = exp η / s η, or κ W s = a h κ µ 8π m = s + 1 h 4π h κ µ = s + 1b1/ 8π m κ 4 µ m V κ W s = s + 1 hν 88 with 1 κ 4π 4 µ m V = V 4π M 1 M = ν 89 the frequecy of the oscillator. We set up ow the equatio 69 for the rotatio, eglectig ay detailed estimatio of the correctio to the vibratioal eergy. Sice ζθ by 81 does ot cotai derivatives with respect to the θ j, we eed cosider oly the term i 69; all remaiig terms we iclude i the costat C s. The rotatioal equatio 69 is the: { } + C s W 4 ρ s =. 9 Sice we have dropped the traslatioal part from, we have by 7 ad 84 for a arbitrary fuctio f θ: f θ = h µ φ 8π mξ θ φ f dx ad by 8 Thus θ φ f = φ θ f + f θ φ 1 + si θ { f = h µ 8π mξ θ f + f +. si θ f ω φ φ ω φ θ φ dx φ f dx + ω θ If we write θ i the form: θ = θ + ctg θ θ + 1 si θ ω we see that it is coveiet to itroduce the followig otatio: Θ = Θ = φ φ θ dx, Ω = φ dx, Ω θ = φ 15 φ φ f ω + φ f. θ θ φ φ ω dx, φ ω dx. φ } θ dx. 91

16 These quatities are the diagoal matrix elemets of p θ, p ω, p θ ad p h ω aside from a factor, h respectively; π 4π the first two deote the average value of the electroic agular mometum about their correspodig Euleria agle; the secod two, the average of the square of the agular mometum of electroic motio. We write the for 9 explicitly: { θ + Θ θ + Θ + ctg θ θ + Θ 1 si θ ω + Ω ω + Ω + 8π mξ h µ W 4 C s } ρ s = 9 This is very similar to the equatio of Kramers ad Pauli for a rotor with a built-i fly-wheel; the differece is essetially that they use the squares of the average values Θ ad Ω, istead of the average of the squares Θ ad Ω. The depedece of the quatities i 91 o the agles θ ad ω may be established by elemetary cosideratios if it is assumed that for this purpose the diagoal elemets of the quatum mechaical matrix may be replaced by the correspodig classical averages. Oe may decompose the motio of the electroic agular mometum vector ito a irregular variatio without average rotatios ad a superimposed uiform rotatio about the molecular axis. We represet the variatio i the average by a costat vector; this rotates uiformly about the axis. This exhibits the same behaviour as a symmetric top with agular mometum compoets with respect to the top-fixed coordiate system havig values L, M ad N. From this we may express the compoets of the agular mometum i the θ, ω directio as follows: Θ = L cos γ M si γ Ω = L si θ si γ + M si θ cos γ + N cos θ, γ is the agle of the eigerotatio about the axis. Averagig over γ, we fid: 5 Θ = Ω = N cos θ Θ = 1 L + M Ω = 1 L + M si θ + N cos θ. We idetify N with the quatum umber ρ which gives the agular mometum about the axis, ad 1 L + M ad 1 N with the averages p ad p of the total electroic agular mometum perpedicular ad parallel to the axis; sice N is costat, p = p. We have fially: Θ = Ω µ = p cos θ Θ = p Ω = p si θ + p cos θ. 93 This result requires aturally a rigorous quatum mechaical verificatio; presumably p is replaced by pp + 1. I the eigevalue problem 9, the quatity 8π mξ h µ W4 is equal to a umerical fuctio of the rotatioal quatum umber r, say g s r; the rotatioal eergy is thus: κ 4 W sr 4 = h µκ 4 8π mξ g sr = h 8π J g sr, 94 J = m µκ 4 ξ = M 1M M 1 + M ξ, 95 5 Compare, for istace [9] 16

17 the momet of iertia of the uclei at equilibrium. A discussio of the higher approximatio is meaigless uless we cosider the degeeracies; we will ot attempt this here. We will ow show briefly that oe ca treat the diatomic by a completely differet perturbatio procedure; the classical aalogue of this treatmet was carried out by Bor ad ückel [5]. The motio of the electroic system is cosidered to be uperturbed ot for statioary uclei but rather for uiform rotatio of the uclei. Part VIII. Idepedet Treatmet of the Diatomic Molecule. We go back to equatio 1, ad rewrite, substitutig 11: { + κ 4 ξξ + ξθ + W } ψ =. Diatomic molecules have the peculiarity that ξθ is geerally idepedet of the θ. I this case, the method eables separatio from the traslatios ad rotatios. From 81, droppig the traslatioal terms: h µ 8π m κ4 ξ + ξ ξ + 1 ξ θ W ψ =. 96 We set Y r is a spherical fuctio of rth order which satisfies the equatio: thus we fid for Ψ r the coditio h µ 8π m κ4 ψ = Y r θ, ωψ r x; ξ, 97 θ Y r + rr + 1Y r = ; ξ + ξ ξ rr + 1 ξ W Ψ r =. 98 We agai substitute ξ + κζ for ξ; thus cosiderig vibratios about the state of uiform rotatio. Deotig the eergy of this state as: R = h µκ rr + 1 = h rr π m ξ 8π J ad settig W = E + R, 1 we fid for 98 + κ 1 + κ + E Ψ r = 11 = 1 = 1 + ζr = + 1 ζ R h µ 1 8π m ζ 3 = ζ3 R h µ 8π m ζ ζ... ; 17

18 ,... are the operators give earlier. All the formulas of Part II are valid without modificatio. The approximatio equatios are: a E Ψ r = b E Ψ 1 r = E 1 1 Ψ r c E Ψ r = E Ψ r + E 1 1 Ψ 1 13 r, 1 The first has the solutio:... E = V ξ, Ψ r = Ψ r = σ r ζφ x; ξ, 14 V ξ ad φ x; ξ are the previously itroduced fuctios ad σ r ζ is, to begi with, arbitrary. The coditio for itegrability of 13b is E 1 1 σ r ζ =. Now, by 6a Part II: ece, as before Part IV, 1 = 1 + ζr = V 1 + ζr = ζ ξ V + R. E 1 =, ξ V + R =. 15 This coditio obviously states that for the uperturbed rotatio, equilibrium must prevail betwee the cetrifugal force ad the quasi-electric force, which, as a cosequece of the electroic motio, resists a displacemet of the uclei. The cetrifugal force is: p r 1 M 1 M ξ = + 1 h rr + 1, 3 M 1 M 4π ξ 3 the quatum mechaical value h π rr + 1 for the agular mometum is substituted for pr ; by 99 ad 95, this agrees with R. From relatio 15, how to calculate the equilibrium separatio ξ r ; depeds o the rotatioal quatum umber r. For small values of the rotatioal eergy R, oe ca develop ξ r i powers of β, : we fid: 6 ξ r = ξ + 1 β = κ 4 µ h 1 rr + 1 = + 1 h m 4π M 1 M β ξ 3 V 3 ξ 7 V 1 + ξ 6 rr + 1; 16 4π V V β Sice β is of order κ 4, we will use by systematic procedure oly as may terms of this set as correspod to the order of the approximatio i the perturbatio method. Sice we cosider this agai, we will shortly see that this is the same method as before, oly simplified by the previous cosideratio of the rotatio. The solutio of 13b is: this correspods to 4; ad the coditio for itegrability of 1c: { } E σ r =. Ψ 1 r = σ r φ 1 + σ 1 r φ 18 6 Oe ca easily deduce this formula from the cited work of Bor ad ückel. 18

19 This is, however, the vibratio equatio { h µ 8π m ζ + 1 } ζ V + R E σ r =. 19 Thus, as i Part VII: κ E rs = s + 1 hν r, 11 the frequecy, ν r = V 4π M 1 M + R 111 still depeds o the rotatioal quatum umber r, from the R. Further, as i Part VII, σ rs = exp η / s η 11 with η = ζb 1/4, b = 8π h m µ V + R. The procedure may be cotiued i the usual fashio. We fid E 3 =, while E 4, besides the deviatio from the harmoic vibratio law, cotais a couplig with the electroic motio. A thorough cosideratio of the formulae would, however, be beyod the scope of this work, which demostrates oly the priciple of the developmet; also the calculatio of the higher approximatios is meaigful oly whe the degeeracies are take ito accout. [1] M. Bor ad W. eiseberg A. d. Phys []. A. Kramers Zeitschr. f. Phys ;. A. Kramers ad W. Pauli jr. ibid [3] J. Frack Tras. Faraday. Soc. 195 [4] E. Codo, Phys. Rev ; Proc. Nat. Acad [5] M. Bor ad E. ückel Phys. Ztschr [6] F. ud Ztschr. f. Phys [7] W. Pauli A. d. Phys [8] S. E. Schrödiger A. d. Phys , [9] F. Klei ad A. Sommerfeld Theorie des Kreisels 1 p 18 19