DIATOMIC MOLECULAR SPECTROSCOPY WITH STANDARD AND ANOMALOUS COMMUTATORS

Transcription

1 Intenational Science Pess Diatomic ISS: Molecula Spectoscopy with Standad and Anomalous Commutatos REVIEW ARTICLE DIATOMIC MOLECULAR SPECTROSCOPY WITH STADARD AD AOMALOUS COMMUTATORS CHRISTIA G. PARIGGER AD AMES O. HORKOHL The Univesity of Tennessee Space Institute Tullahoma Tennessee U.S.A. Abstact: In this eview we addess computation of diatomic molecula specta. An oveview of the theoy is discussed based on symmeties of the diatomic molecule. The standad quantum theoy of angula momentum fully accounts fo the otational states of the diatomic molecule. Details ae elaboated in view of standad vesus anomalous commutatos fo geneation of a synthetic spectum. Specific example specta ae pesented fo selected diatomic molecules in view of diagnostic applications in lase-induced optical beakdown spectoscopy. Keywods: Molecula Specta (PACS 33.0) Intensities and Shapes of Molecula Spectal Lines and Bands (PACS 33.70) Plasma Diagnostics (PACS 5.70) Lase-Induced Optical Beakdown.. ITRODUCTIO Although Van Vleck's evesed angula momentum technique [] is little used today the anomalous angula momentum commutatos [] upon which it was based still appea in cuent texts [3 5]. In this eview we show that the standad quantum theoy of angula momentum fully accounts fo the otational states of the diatomic molecule. We find that the commutatos which define angula momentum ae not changed in a tansfomation fom a laboatoy coodinate system to one which otates with the molecule and the seemingly anomalous behavio of the otated angula momentum opeatos ± and ± is simply the esult of thei opeating on the complex conjugate of otation matix elements. A otation is a unitay tansfomation and theefoe peseves the commutation elationships which define angula momentum. Also opeation of the aising and loweing opeatos on standad angula momentum states is the same in all coodinate systems connected by pope otations. Two appoaches ae pesented to show that pope otations peseve the angula momentum commutatos. The fequently applied nonstandad behavio of the aising and loweing opeatos on diatomic states is illustated as a esult of the opeatos acting on elements of the otation opeato matix instead of standad angula momentum eigenfunctions. A specific element of the otation matix cannot epesent a standad angula momentum state because the otation matix element caies two magnetic quantum numbes. A fundamental popety of angula momentum is that the squae of the total angula momentum and only one of its components ae constants of motion. An equation giving opeation of the aising and loweing opeatos on the otation matix is deived. Some matix elements of the otational Hamiltonian ae calculated to demonstate that standad esults can be obtained without esoting to Van Vleck's agument [] that the unexpected behavio of ± and ± is the esult of the anomalous commutatos. Specifically we show below that: ± ( + ) ( ± ) ±... () D M ± αβγ ( + ) ( ) ( αβγ ) D M... () otice the molecule-fixed opeato ± acts on the state whose magnetic quantum numbe is efeenced to the molecule-fixed z axis as expected but that + lowes Intenational Review of Atomic and Molecula Physics () anuay-une 00 5

2 Chistian G. Paigge and ames O. Honkohl when opeating on the complex conjugate of the otation matix element while aises of D αβγ. The fully standad fomalism of angula momentum is applicable without modification to the theoy of diatomic specta as will be demonstated hee: ( i) An exact equation fo the total angula momentum states of the diatomic molecule will be deived. (ii) The Hund's cases (a) and ( b) basis sets will be obtained fom the exact equation. (iii) The effects of + + and on DM αβγ and its complex conjugate will be calculated. (iv) Some case ( a) matix elements of the otational Hamiltonian will be evaluated to show that the standad angula momentum fomalism gives the accepted esults. Also Hönl-London factos fo all n-photon a a and b b tansitions obeying the S 0 selection ule will be calculated. Applications will be illustated in the study of molecula specta ecoded following lase-induced optical beakdown.. Review of Standad and Anomalous Angula Momentum Commutatos The customay stating point fo the quantum theoy of angula momentum is the commutation fomula fo the catesian components of the angula momentum opeato i j j i i ijk k... (3) whee + i j k in cyclic ode ijk i j k not in cyclic ode. 0 if any indices equal The above commutato popety usually defines the angula momentum opeato. Coodinate tansfomations leave the angula momentum opeato definition invaiant. The consevation law fo angula momentum is fundamental. The definition of angula momentum Eq. (3) is of couse invaiant unde specifically spatial tanslations and otations. Futhemoe Eq. (3) is invaiant unde coodinate invesion and time evesal. Van Vleck's evesed angula momentum method stats with Eq. (3) but then utilizes change of sign of i fo angula momentum when a tansfomation of coodinates to a system attached to a otating molecule is made M i j j i i ijk k... (4) Hee the pimed index denotes a otated coodinate. This equation containing the evesed sign of i is known as Klein's [] anomalous commutation fomula. Two appoaches ae debated in this wok namely an opeato and an algebaic appoach without utilizing Klein's anomalous commutation fomula. Each appoach begins with the standad commutato fomula Eq. (3). We point out that in pinciple Eq. (4) can be utilized in building angula momentum theoy. Howeve in analogy to distinction between ight- and left- handed coodinate systems diffeent signs occu. We only use the standad sign as indicated in Eq. (3) in computation of a molecula diatomic spectum [6] i.e. without esoting to use of Klein's anomaly and Van Vleck's evesed angula momentum method. Hee the evesal of the sign in Eq. (3) is biefly investigated fo an unitay and an anti-unitay tansfomation. The Eule otation matix is a eal unitay matix (see Eq. () below). The deteminate of the Eule otation matix is + meaning that the sign of vectos is peseved unde otations. A spatial otation of coodinates is a pope tansfomation. Convesely the invesion o paity opeato constitutes an impope otation this tansfomation cannot be descibed exclusively in tems of the Eule angles. Howeve angula momentum is a pseudo o axial vecto peseving the sign of unde impope otations. The paity opeato is also unitay and Eq. (3) is peseved by the paity opeato. Time evesal (time invesion o evesal of motion) changes the sign of and it complexconjugates the imaginay unit due to time evesal being anti-unitay. Thus Eq. (3) is invaiant unde time evesal. As shown in texts (e.g. Messiah [7]) the time evesal opeato has been designed to be anti-unitay fo the vey pupose of peseving the sign of i in commutation fomulae.. Effect of Unitay Tansfomation on Angula Momentum Commutatos In a bief eview we show that unitay tansfomations peseve commutation fomulae. Conside the opeatos A B and C which satisfy the commutation fomula AB BA ic... (5) and subject these opeatos to the unitay tansfomation U. That is A U AU... (6) 6 Intenational Review of Atomic and Molecula Physics () anuay-une 00

3 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos and A U AU... (7) with simila equations holding fo B and C. The opeato U is unitay i.e. U U so U AUU BU U BUU AU o U ABU U BAU i U CU... (8) AB BA ic... (9) The above textbook discussion (e.g. Davydov [8]) confims that the angula momentum quantum commutatos Eq. (3) ae peseved unde unitay tansfomations. The Eule otation matix Eq. () below is easily demonstated to be unitay. An algebaic appoach is used in the following to show that the commutato Eq. (3) emains invaiant when pope otation of coodinates is applied. The laboatoy efeenced is tansfomed to the otated coodinate system by application of the otation matix D (αβγ) D (αβγ)... (0) whee α β and γ ae the Eule angles and D (αβγ) is an othogonal matix whose deteminant is + Goldstein[9] D (αβγ) cos α cos β cos γ sin α sin γ sin α cos β cos γ + cos α sin γ sin β cos γ cos α cos β sin γ sin α cos γ sin α cos β sin γ + cos α cos γ sin β sin γ cos α sin β sin α sin β cos β... () The Eule angles and the matix D (αβγ) used hee ae those nomally used in quantum mechanics Messiah [7] Davydov [8] Goldstein [9] Rose [0] Bink and Satchle [] Tinkham [] Vashalovich et al. [3] Gottfied [4] Baym [5] and Shoe and Menzel [6]. This same set of Eule angles is also used by some authos of books on the theoy of diatomic specta udd [7] Mizushima [8] Bown and Caington [3] and Lefebve-Bion and Field [4]. Evaluation of the angula momentum commutation fomulae in the otated system of coodinates gives i j j i i ijk k... () This esult is obtained fom Eqs (3) (0) and (). The calculation is simplified somewhat if one notes that fo an othogonal matix the cofactos i.e. signed mino deteminants ae equal to the coesponding matix elements of D (αβγ) labeled m ij i.e. m ij is its own cofacto. Fo example x y y x and since i [(m m 3 m 3 m ) ( y z z y ) + (m 3 m m m 3 ) ( x z z x ) + (m m m m ) ( x y y x )]... (3) m m 3 m 3 m m 3 m 3 m m m 3 m 3 m m m m m (4) the ight side of the Eq. () educes to i z. Compaison with Van Vleck's [] teatment of Klein's anomalous fomula given between his Eqs (4) and (5) shows that we do not equie evesal of sign in ou appoach. The anomalous sign in Eq. (4) does not eveal a novel aspect of the natue of diatomic molecules. otewothy is that the anomalous commutation fomula emains today a time honoed tadition in the theoy of molecula specta e.g. see Refs. [ 5 4 ]. Klein's anomalous commutatos ae means by which matix elements of vaious opeatos in the molecula Hamiltonian in paticula those expessed in tems of angula momentum aising and loweing opeatos ae obtained..3 Effect of Raising and Loweing Opeatos on Standad States M and on Elements of the Rotation Matix ( αβγ ) D M The angula momentum aising and loweing opeatos have the following effects on the standad M states ± M C ± ( M) M ±... (5) whee C ± ( M) ( + ) M ( M ± ) ( M ) ( ± M )... (6) This geneal equation is of couse applicable to the diatomic molecule. Howeve as a esult of appoximation one deals with appoximate diatomic eigenfunctions. Contained in Van Vleck's method is his discovey that the above standad esults ae not diectly Intenational Review of Atomic and Molecula Physics () anuay-une 00 7

4 Chistian G. Paigge and ames O. Honkohl applicable to appoximate diatomic eigenfunctions. Typically two magnetic quantum numbes occu fo appoximate diatomic eigenfunctions M and in Hund's case (a) o M and Λ in case (b). In moden notation appoximate diatomic angula momentum states ae epesented by elements of the otation matix DM ( αβγ ) which cay two magnetic quantum numbes one moe than allowed by the natue of angula momentum. Only and one of its components by usual convention z commute with the Hamiltonian. It will be impotant in ou appoach to find the effects of the aising and loweing opeatos on elements of the otation matix while applying standad theoy. The otated aising opeato + + x + i y... (7) lowes the quantum numbe on Hund's case (a) states. See fo example Van Vleck [] udd [7] Mizushima [8] Feed [9] Kovacs [0] Hougen [] Caington et al. [] Zae et al. [3] Bown and Howad [4] and Lefebve-Bion and Field [4]. Similaly the otated aising opeato + lowes the Λ quantum numbe on case ( b) kets. Ageement between eigenvalues of Hamiltonian matices built using these esults and expeimentally measued tem values has fimly established thei coectness. Klein's anomalous commutatos ae often efeenced in debating the eason why + lowes and why + lowes Λ. Howeve of inteest will be the following equation ( αβγ ) * ± M αβγ D M C D... (8) which will be deived below. ote that + lowes that aises and that an unexpected minus sign occus. The natue of angula momentum does not allow M and both to be igoously good quantum numbes. This is equivalent to stating that z and z do not commute. Accoding to definition of angula momentum Eq. (3) z does not commute with x o y but it is pehaps not obvious that z and z fail to commute. One can easily show Gottfied [4] that: [ z z ] i sin (β) β... (9) i sin (β) [ sin (α) x + cos (α) y ]... (0) whee β / β is the angula momentum opeato fo otation about the fist intemediate y-axis. In geneal z DM αβγ cannot epesent a state of angula momentum of a molecule o any othe system. The otation matix connects two diffeent states of angula momentum. * and z do not commute. Thus. AGULAR MOMETUM OPERATORS Angula momentum opeato epesentations in tems of Eule angles ae elaboated. A otation povides a paticulaly simple way of expessing a component of angula momentum. The thee Eule otations give the following thee components α i α z... () β i... () β γ i γ z.... (3) Each of these opeatos is efeenced to a diffeent coodinate system. That is α z in the laboatoy system β y in the fist intemediate system and γ z in the fully otated system. Applying the fist Eule otation to the vecto opeato esults in x y z cos α sin α 0 x sin α cos α 0 y 0 0 Fom this one finds z... (4) y β sin α x + cos α y... (5) giving β in tems of the laboatoy coodinates of. Similaly using the full otation matix Eq. () one can expess γ in laboatoy coodinate system D (αβγ)... (6) z γ cos α sin β x + sin α sin β y + cos β α... (7) whee the substitution z α has been made. Equations (5) and (7) can be inveted fo x and y : cos x i cos cot sin α α β α + α β sin β γ... (8) 8 Intenational Review of Atomic and Molecula Physics () anuay-une 00

5 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos sin y i sin cot cos α α β + α + α β sin β γ z... (9) i.... (30) γ The method in obtaining these esults included evaluation of β and γ in tems of the laboatoy components of. Similaly x y and z can be obtained by expessing α and β in tems of the components of. The invese of the full otation matix is applied to the otated vecto D (αβγ) D (αβγ)... (3) to find α in tems of the otated coodinates of z α sin β cos γ x + sin β sin γ y + cos β γ.... (3) The Eule β-otation is taken about the fist intemediate y-axis meaning that the fist intemediate and second intemediate y-axes coincide. Thus β can be evaluated in fully otated coodinates by applying the invese of the γ otation matix to x y zz cos γ sin γ 0 x sin γ cos γ 0 y 0 0 y y β theefoe we find z... (33) β sin γ x + cos γ y.... (34) The two equations in two unknowns ae inveted as befoe. We find fo x and y cos x i cos cot sin γ γ β + γ γ β sin β α... (35) sin y i sin cot cos γ γ β + γ + γ β sin β α z... (36) i... (37) γ The aising and loweing opeatos ae then constucted using the esults above: + iα ie cot β + i + α β sin β γ... (38) + iα ie cot β i + α β sin β γ iγ ie cot β + i γ β sin β α iγ ie cot β i. γ β sin β α... (39)... (40)... (4) These geneal esults also apply to systems composed of any numbe of paticles. A modification o bette simplification is equied fo a system consisting of a single paticle (o two paticles since the two-body eduction can always be applied to a system of two paticles). The thid Eule angle γ is supefiuous fo a single paticle i.e. / γ 0. Choosing the fist Eule angle to be azimuthal angle φ and the second Eule angle to be the pola angle θ then Eqs (8 30) educe to the familia textbook equations fo the angula momentum opeatos of a single paticle. Compaison of Eqs (8 30) with Eqs (35 37) shows the components of angula momentum ae changed by a coodinate tansfomation. Howeve Eq. (3) defines the angula momentum in tems of its catesian components. This defining elationship among the components emains invaiant although the individual components diffe. We note that Eqs () (38) and (39) agee with udd's [7] Eq.(.) but (3) (40) (4) diffe in sign fom udd's Eq. (.3) pesumingly due to the use of the anomalous commutato fomula. It appeas that udd obtained his otated opeatos in a manne which guaanteed they would obey Klein's anomalous commutation fomula. 3. AGULAR MOMETUM STATES OF THE DIATOMIC MOLECULE Fundamental symmeties including special geometical symmeties of the diatomic molecule allow us to deive a geneal equation fo the angula momentum states of the molecule. In the laboatoy system of coodinates the eigenfunction fo the diatomic molecule can be witten as Ψ nm (R R... R R a R b ) R R... R R R... (4) a b nm Hee R R... R ae the spatial coodinates of electons and R a and R b ae nuclea coodinates. is the Intenational Review of Atomic and Molecula Physics () anuay-une 00 9

6 Chistian G. Paigge and ames O. Honkohl quantum numbe fo the total angula momentum that includes spin M is the quantum numbe fo the z-component of and n epesents all othe equied quantum numbes. The symbol n eflects the complexity of the system. Moeove the angula momentum quantum numbes and M contain also electonic and nuclea spins theeby adding complexity. In pinciple the total eigenfunction fo the system consisting of the molecule and the adiation fields should be constucted so that enegy linea and angula momentum is fomally conseved. Howeve the pobability of quantum tansitions is contolled by matix elements connecting initial and final states. The inteaction occus at some intemediate instant. One can conside independent states of the molecule and the adiation field fo the initial and final states of the complete system. The system state at these times is the poduct of two independent states. Consevation laws can be met by equiing that conseved quantities lost o gained by the molecule ae coespondingly gained o lost by the adiation field. 3. Consevation of Enegy The fist fundamental symmety consideed is consevation of enegy. Enegy is the quantity conseved unde a tanslation of time. The evolution opeato geneates a tanslation in time of the eigenfunction. Given an eigenfunction at time t 0 one can find the eigenfunction at some late time t by applying the evolution opeato R R... R R a R b U t t nmt 0 0 R R... R Ra R nmt b... (43) The evolution opeato is unitay. Also U (t t). Consevation of enegy and aspects of the evolution opeato ae not futhe discussed hee. They wee mentioned to demonstate the common featues they shae with the following consevation laws and thei associated unitay opeatos. 3. Consevation of Linea Momentum The second fundamental symmety consideed is the homogeneity of space that is consevation of linea momentum. The total linea momentum is conseved unde a tanslation of coodinates. Again an opeato is devised to poduce the spatial tanslation. Fo homogeneous space a tanslation of coodinates to some new oigin R 0 : i R i R 0... (44) leave the state of the molecule unchanged. Tanslation meely takes us to anothe vantage point fom which to view the molecule. As was the case of the time tanslation opeato the spatial tanslation opeato T (R 0 ) is defined by its effect upon the eigenfunction: R R... R Ra Rb T R0 nm... a b nm.... (45) The tanslation opeato is unitay like the unitay evolution opeato. The unitaity of T (R 0 ) can be exploed by expanding into Taylo seies the oiginal eigenfunction about R 0. One finds:... a nm b R R... R R R ir0 P / a b e nm... (46) whee P is the total linea momentum of the molecule. A compaison of Eqs (45) and of (46) yields: T (R 0 ) i R 0 P /... (47) e The Schödinge (o x ) epesentation of the adjoint tanslation opeato (in pactice its complex conjugate) can be intepeted as the total linea momentum eigenfunction. This can be elucidated by ewiting Eq. (45) as: Ψ nm (R R... R R a R b ) ψ (... ) T... (48) R0 nm a b An analogous situation occus with the tempoal evolution of quantum states. One defines the unitay evolution opeato by its action on the eigenfunction. Yet the evolution opeato satisfies the time dependent Schödinge equation again showing the inteplay of eigenfunction and opeato. A simila connection aises a when the otation of coodinates is consideed. A sum ove matix elements of the otation opeato become the angula momentum eigenfunctions of the system. Befoe exploing consevation of angula momentum we fist find the optimal oigin of coodinates about which to make otations. The geometical symmety of the diatomic molecule guides us in finding this optimal oigin that is how to choose R 0. A geometical symmety is implied fo a diatomic molecule. Thee ae pecisely two nuclei and thei motion 30 Intenational Review of Atomic and Molecula Physics () anuay-une 00

7 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos can be educed to that of a single fictitious paticle having educed mass µ µ mamb m + m a b.... (49) when we choose the location of R 0 to be the cente of mass of the nuclei R cmn R 0 R cmn mar a + mbr b.... (50) m + m a This two-body eduction eplaces the six coodinates of R a and R b with the thee coodinates of R cmn and the thee coodinates of b R a R b... (5) the intenuclea vecto. ote that T (R cmn ) epesents the total linea momentum not the linea momentum of only the nuclei. The total eigenfunction can now be witten: Ψ nm (R R... R R a R b ) ψ (... ) T... (5) Rcmn nm in which the intenal eigenfunction (... ) ψ nm is a function of spatial coodinates not the total Because we ae inteested hee in only the intenal states of the molecule the total linea momentum P is set equal to zeo giving T (R cmn ). The cente of mass of the nuclei is the oigin of the intenuclea vecto and this vecto which defines the intenuclea axis is an axis of symmety of the diatomic molecule. Any axis about which a otation is made is an axis of symmety because space is isotopic but this otation does not necessaily induce a dynamical vaiable. While it is tue that the atio of nuclea to electonic masses is lage meaning that R cmn nealy coincides with the cente of total mass and that nuclea motions ae appoximately sepaable fom electonic motions the motivation fo choosing the oigin to lie on the intenuclea axis is that the otation about this axis can be used as a dynamical vaiable of the molecule not the simplifying appoximations which esult fom the lage atio of nuclea to electonic masses. In passing it is noted that one who egads consevation of linea momentum as a fomal nicety which can be skipped in pactice will miss the mass polaization tem which appeas in the Hamiltonian. This tem is compaable in magnitude to the tems dopped in the Bon-Oppenheime appoximation see fo example Bunke [5] o Bansden and oachain [6]. 3.3 Consevation of Angula Momentum The thid fundamental symmety consideed is the isotopy of space that is consevation of angula momentum. As befoe the opeato associated with this symmety the otation opeato (αβγ) is defined by its effect on the eigenfunction:... whee fo example: ( αβγ ) nm... nm... (53) D (αβγ)... (54) with simila equations holding fo.... Below one will see examples showing that the otation opeato poduces tanslations of angula coodinates just like the evolution opeato tanslates time and the tanslation opeato tanslates Catesian coodinates. The otation opeato is unitay like the tanslation opeato but angula momentum states ae discete wheeas linea momentum states ae continuous. The effect of the otation opeato can also be expessed as a unitay tansfomation of the kets:... ( αβγ ) nm D Μ... (55)... n ( αβγ ) in which the matix elements of the otation opeato ae defined by: D Μ ( αβγ ) αβγ M.... (56) The quantum numbe M epesents the z-component of and the quantum numbe epesents the z-component of. Equation (53) is witten in tems of the eigenfunction... nm showing otated coodinates and the quantum numbe M which is efeenced to the laboatoy coodinate system. Similaly Eq. (55) is witten in tems of the eigenfunction... n showing laboatoy coodinates and the quantum numbe which is efeenced to the otated Intenational Review of Atomic and Molecula Physics () anuay-une 00 3

8 Chistian G. Paigge and ames O. Honkohl coodinate system. Equations (53) and (55) seve to define the otation opeato and its matix elements. Howeve we seek an identity with tanspaent intepetation as follows: Opeate on the laboatoy αβγ (i.e. in Eq. (53) eplace eigenfunction with the otation opeato with its adjoint) and follow this with opeation by ( αβγ ) to obtain:... n... (57)... n DΜ ( αβγ ) in which the eigenfunctions ae now eithe efeenced puely to the laboatoy coodinate system o otated coodinate system. This equation eveals the effect on the eigenfunction fom a otation of coodinates. In geneal the otation of coodinates changes the diection of the z-axis and theefoe a new magnetic quantum numbe appeas. A geneal popety of angula momentum is that only one of its components is a constant of the motion. If M is a good quantum numbe cannot be. A vesion of Eq. (57) can be witten in which is a good quantum numbe but then the equation contains a summation ove all M (see Eq. (7) below). 3.4 Inclusion of Special Symmety fo Diatomic Molecules An oigin of coodinates lying on the intenuclea axis was chosen above. This choice allows us to take advantage of the axial symmety of the molecule. Equation (57) is valid fo abitay Eule otations. Let us choose specific Eule angles which simplify the analysis. Fist we wite the intenuclea vecto in tems of its spheical pola coodinates: (θφ).... (58) The Eule angle α and the azimuthal angle φ both epesent counteclockwise otations about the z-axis. If φ is the azimuthal angle of the intenuclea vecto in the laboatoy coodinate system afte the Eule otation α the azimuthal angle has a new value φ φ α. We ae at libety to choose the value of α which makes φ 0 (i.e. α φ). Similaly we ae at libety to choose β θ. These choices foce the z-axis to coincide with the intenuclea axis. The Eule angles α and β now seve dual puposes. They ae paametes of the otation of coodinates and they ae dynamical vaiables of the molecule. Second we model the two nuclei as point masses lying on the intenuclea axis. Hence the thid Eule angle γ cannot epesent motion of the nuclei. If it is to be made a dynamical vaiable γ must be made an electonic coodinate. This is accomplished by letting the thid Eule angle γ descibe otation of one of the electons say the th electon about the intenuclea axis [7]. The molecule-fixed catesian coodinates x y z of the th electon ae eplaced by the cylindical coodinates ρ χ ξ x ρ cos (χ )... (59) y ρ sin (χ )... (60) z ζ... (6) whee because the laboatoy otation χ and γ ae both counteclockwise otations about the same axis χ χ γ... (6) we may choose χ γ that is choose χ 0. Equation (57) can now be ewitten to ead... nm... (63)... ρζ n DM ( αβγ ) whee the scale is the intenuclea distance. Pimes on ρ and ζ have been dopped fo the same eason that the intenuclea distance is not pimed. Like ρ ae ζ ae scala invaiants distances whose value is the same in all coodinate systems. The ket n is not an angula momentum state but it is a function of and. This behavio is analogous to the adial eigenfunction of the hydogen atom which is not an angula momentum eigenfunction but does depend on the obital angula momentum quantum numbe l (see Eq. (64) below). With the chosen Eule angles the otation matix element has eplaced the angula momentum eigenfunction of the diatomic molecule. The eigenfunction... ρζ n is not a function of the Eule angles. If one of the nuclei and all but one of the electons wee emoved fom Eq. (4) then following the above pocedue one would find sepaating off the electon's spin: nlm nl l + 4π D ( αβ ) l* m0 0 ψ ( θ β φ α) nl Y.... (64) lm 3 Intenational Review of Atomic and Molecula Physics () anuay-une 00

9 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos The change of vaiables indicated in Eq. (58) is familia and θ and φ ae dynamical vaiables of the poblem. We demonstated above that γ can be made a dynamical vaiable of the diatomic molecule. The diffeence between Eqs (64) and (63) is that γ is not elevant to a spheically symmetic hydogenic atom but γ is elevant to the axially symmetic diatomic molecule. 3.5 Inclusion of Spin In accod with standad diatomic notation Eq. (63) should be witten in tems of the quantum numbes F M F and F that in the standad notation epesent the total angula momentum F R + L + S + T... (65) whee R is the total obital angula momentum of the nuclei L is the total obital angula momentum of the electon S is the total spin of the electons and T is the total spin of the nuclei. Howeve Eq. (63) follows the usage of in the liteatue of angula momentum whee and M and epesent the total angula momentum. Deivation of Eq. (63) was based on the isotopy of space and theefoe holds only if M and epesent the total angula momentum including all obital momenta and all spins. The electonic and nuclea spin in Eq. (63) can be evealed by application of the invese Clebsch- Godan seies to the otation matix element Eq. (80) below. 3.6 Rotated Diatomic Angula Momentum States Equation (63) shows that the Eule angles ae dynamical vaiables of the diatomic:... nm... ρζαβγ nm... (66) If analysis of the molecule had begun with the intoduction of genealized Lagangian coodinates the Eule angles φ θ and χ would have been intoduced fist as dynamical vaiables and only late as paametes of coodinate otation. This oute takes one to the equation:... ρζ φθ χ nm ρζ φ θ χ n DM ( αβγ ) (67) whee φ φ α... (68) θ θ β... (69) χ χ γ.... (70) Choosing φ α θ β and χ γ makes φ θ χ 0 theeby emoving all angula dependence fom... n ρζ on the ight side of Eq. (63). Multiplication of Eq. (67) by D M ( αβγ) and summation ove M yields:... ρζ φθ χ n M ρζ φ θ χ nm DM ( αβγ )... The appeaance of DM... (7) αβγ on the ight side indicates that it oiginated fom application of the invese of the otation opeato to the left side:... ρζ φθχ n whee... ρζ φθχ αβγ n... (7)... ρζ φθχ γ β α n... (73) φ φ + γ... (74) θ θ + β... (75) γ χ + α.... (76) As befoe we choose α β and γ to emove all angula dependence fom the eigenfunction of the ight side of Eq. (7). Placing the obseve in the molecula coodinate system is a evesal of motion. Because the ode of Eule otations about diffeent axes is significant (e.g. (αβγ) (βαγ)) evesal of motion also equies that the ode in which the otations ae taken be evesed i.e. (αβγ) ( γ β α). Because the signs of the Eule angles ae inveted one might mistakenly believe that the sign in font of i in Eqs ( 3) would also be inveted thus implying that the anomalous commutatos Intenational Review of Atomic and Molecula Physics () anuay-une 00 33

10 Chistian G. Paigge and ames O. Honkohl hold in the molecule-fixed coodinate system but evesal of motion (also called time evesal) is anti-unitay. The time evesal opeato is nomally witten as the poduct of a unitay opeato and the complex conjugation opeato. Thus in evesal of motion changes in the signs of the Eule angles ae canceled by complex conjugation and the sign in font of i in Eqs ( 3) emains unchanged. 3.7 Discussion of Angula Momentum States of the Diatomic Molecule The significance of Eq. (63) is that it shows the total angula momentum is exactly sepaable in the diatomic eigenfunction. The sepaation is independent of the Bon- Oppenheime appoximation and is possible fo vey few simple systems such as the diatomic molecule and hydogen/hydogenic atom (see Eq. (64) above). The summation ove in Eq. (63) insues that only one magnetic quantum numbe M is good. An element of the D-matix is a combination of the states M and. o one tem in the sum on the ight side of Eq. (63) contains a standad angula momentum eigenfunction. The equation ± C± ( ) ± indeed holds but this esult is not diectly applicable to Hund's case (a) o (b) eigenfunctions. Fo the diatomic molecule the angula momentum eigenfunction is eplaced by a sum ove matix elements of the otation opeato quantities which ae obviously defined in tems of angula momentum but which nevetheless ae not angula momentum eigenfunctions because z and z do not commute. Symbols such as ψ M ( αβγ ) and αβγ M (viz. symbols fo Hunds case ( a) basis states) ae best avoided due to difficulties of econciling these symbols with the fomalism and physics of angula momentum. The α- and γ-dependence of the otation matix element is clea D ( αβγ ) M e d β e... (77) im α i γ M possibly tempting one to take the matix element apat and put the γ tem with the eigenfunction... n calling the esult the electonic-vibational eigenfunction. Thee is histoical pecedence fo sepaating the total eigenfunction into electonic vibational and otational pats but this scheme does not fit the pesent thus fa exact fomulation. Fom a pactical standpoint one who takes the otation matix element apat fo example to utilize fomulae of the kind: π π π D j D αβγ αβγ sin β j* mω mω 8π dα dβ dγ δ δ δ + m m j j ωω and the invese Clebsch-Godan seies: M D αβγ j j m j m j ω j ω j... (78) j m j m M j ω j ω j ( αβγ ) ( αβγ ) D D... (79) j mω mω may find that it is indeed best to not sepaate the D-matix. 4. HUD'S CASE (a) AD (b) BASIS FUCTIOS The Hund's case (a) eigenfunction is constucted fom Eq. (63) by omitting the summation ove and enomalizing the esult though use of Eq. (78):... nm SΣ π DM ρζ n SΣ αβγ... (8) It is notewothy that the Hund case (a) basis identifies both M and as so-called good quantum numbes. The value of a basis may be associated with the ease in accuately constucting the eigenfunction. Howeve one must ealize that Hund's case (a) eigenfunction does not contain standad M o angula momentum states. Application of the aising and loweing opeatos to case (a) eigenfunction will eveal nonstandad esults. The Hund case (a) basis cannot igoously epesent a physical state. Howeve let us investigate its appoximate physical significance of a model fo a diatomic molecule whee nuclea spin effects ae not impotant. Fo this model the total angula momentum is the sum of L the electonic obital S the electonic spin and R the nuclea obital angula momenta. o L + R + S... (8) + S... (83) 34 Intenational Review of Atomic and Molecula Physics () anuay-une 00

11 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos whee is the total obital angula momentum: L + R.... (84) This last equation a definition is futhe debated: Accoding to the evesed angula momentum method and R ae examples of momenta which ae expected to obey Klein's anomalous commutation fomulae but L obeys the standad fomulae. The opeato which otates when applied to the ight side of the above equation must teat L and R diffeently. Constuction of such an opeato maybe difficult. In compaison a fomalism in which all angula momentum opeatos obey the same commutato fomulae in all coodinate systems is pefeed and offes simplicity. A Clebsch-Godan expansion of the D-matix of Eq. (8) esults in:... nm SΣ π S M M S S D ρζ n SΣ M SM M ΛSΣ S D * * M Λ M S Σ S αβγ αβγ... (85) Both M SM S M and ΛSΣ ae Clebsch- Godon coefficients. The quantum numbes M and M S ae efeenced to laboatoy coodinates Λ and Σ to otating coodinates. The Clebsch-Godan coefficients vanish unless M M + M S and Λ + Σ. The two D- matices in this equation have the same set of Eule angles meaning that and S ae pefectly coupled in accod with the definition of Hund's case (a). The Hund case ( a) basis is a useful physical appoximation only when the coupling and S is stong. Convesely fo weak (o negligibly small) coupling between and S constuction of anothe basis set is indicated. In the absence of electonic and nuclea spin Eq. (63) eads:... nm Constuction of the Hund case (b) basis begins by omitting the summation ove Λ and e-nomalizing the esult. Subsequently the spinless equation is multiplied by the electonic spin states SM S and the appopiate Clebsch-Godan coefficient. Finally the esult is summed ove all laboatoy magnetic quantum numbes:... n M Λ S π S M S S M ρζ n SΣ M SM M D Λ * S M ( αβγ )... (87) giving Hund's case (b) eigenfunction. States constucted fom Clebsch-Godan coefficients give a diagonal epesentation of the Hamiltonian only if the two angula momenta that ae combined do not inteact. Such noninteaction would pefectly match Hund's case (b) (viz. weak o negligibly small electonic spin-obit inteaction). As one expesses the spin ket in the otated system: S* SM Σ ( α β γ ) S S S DM S Σ S S S... (88) Σ S whee α S β S and γ S ae the Eule angles fo otation of the electonic spin the following compaison between cases (a) and (b) can be made: Fo Hund's case (a) the spin Eule angles ae equal to the Eule angles descibing the otations of the molecule but fo Hund's case (b) the spin angles ae completely independent of α β and γ. * S* Consequently in case (b) D Λ ( αβγ ) and D Σ ( α β γ ) M M S S S S cannot be combined in a Clebsch-Godan contaction to build the M states as in case (a). Tansfomations between cases ( a) and ( b) ae accomplished using: a b + + ΛSΣ... (89) M Λ... (86) Λ *... ρζ n D ( αβγ ) b a + + ΛSΣ... (90) Intenational Review of Atomic and Molecula Physics () anuay-une 00 35

12 Chistian G. Paigge and ames O. Honkohl These coefficients can be obtained fom Eqs (78) (80) (8) and (87). 5. EFFECTS OF RAISIG AD LOWERIG OPERATORS O D M ( ) Opeation of the + + and on the D-matices will be exploed hee. Substitution of Eq. (63) into Eq. (5) gives: ± M... ρζ n SΣ D αβγ C M... ρζ n ± * S D M ± Σ αβγ... (9) We mentioned above that the eigenfunction... n has no angula dependence. Theefoe ± acts only on the D-matix elements. A tem by tem compaison of the left and ight sides of Eq. (9) gives: ± D M αβγ * ( ) C M D M ± ± αβγ... (9) One ecognizes that opeation by ± gives an expected esult when applied to the complex conjugate of the D-matix. The complex conjugate of the D-matix is given by: M D ( αβγ ) D M fom which one finds: + D M αβγ αβγ... (93) M M ( ) D ( αβγ ) + M M ( ) C ( M ) D ( αβγ ) + M + ( ) C M D M αβγ.... (94) In the last step we used C + ( M) C ( M) as can be infeed fom Eq. (6). This esult illustates that + lowes M-states when applied to the otation matix elements. ote also the minus sign which appeas in font of C ( M). Similaly: D M αβγ ( ) C+ M D M + αβγ... (95) ow we investigate action of + and. Rotation of coodinates is a unitay tansfomation; theefoe if Eq. (5) holds in the laboatoy coodinate system then:... k n ± C±... k n ±... (96) must hold in the otated coodinate system. Substituting Eq. (7) into both sides of Eq. (96) and poceed as befoe allows us to obtain Eq. (8) and D αβγ ± M ( ) C± D M ± αβγ... (97) Equations (8) (94) (95) and (97) ae found by consistently applying standad commutato algeba. In compaison Van Vleck's evesed angula momentum method would utilize diffeent identities that ae difficult to econcile with ou standad fomalism. ote the unexpected signs and appaent anomalies in these equations (e.g. in Eq. (95) one sees that aises M and intoduces a minus sign). These esults lose thei mysteious appeaance as one ecalls that the D-matix elements cannot be teated as standad angula momentum eigen-functions. Standad angula momentum eigenfunctions show only one but not two magnetic quantum numbes. Expessions like D M ± αβγ show seveal levels of complexity: Fist complex conjugation is in effect pompting consideations whethe the oles of the aising and loweing opeato ae evesed. Second and moe impotantly thee ae two magnetic quantum numbes pesent. The Appendix shows that evaluation of commutatos acting on DM αβγ matix elements athe than on physical eigenfunctions geneates elations that one might vey well label anomalous. 6. THE BOR-OPPEHEIMER APPROXIMATIO Electonic... ρζ n and vibational v n basis functions ae defined and Eq. (63) is ewitten to ead: 36 Intenational Review of Atomic and Molecula Physics () anuay-une 00

13 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos... nm... n υn M n v D αβγ... (98) whee a pimed summation indicates summation ove discete states and integation ove continuum states. The fist ode Bon-Oppenheime appoximation (e.g. see Ref. [6]) consists of omitting the summations ove n and v n. A diffeent electonic eigenfunction is obtained fo each value of the intenuclea distance. That is the electonic eigenfunction becomes a paametic function of which is denoted by witing the electonic eigenfunction as... ρζ; n... nm M... ρζ; n v D αβγ... (99) The physical justification fo omitting the summations fom Eq. (99) is of couse the smallness of the electonic mass in compaison to the nuclea mass. In the following the subscipt on v n will be dopped. 7. HUD CASE (a) HAMILTOIA MATRIX ELEMETS Some example matix elements of the Hamiltonian ae calculated hee. Standad esults will be obtained by application of standad methods. One of the many tems in the Hamiltonian is the kinetic enegy of otation of the nuclei called the otational Hamiltonian: H ot 4π cµ B () R R B () ( L S) n n L+ L + L L+ B () + + Lz + S ( L L L + + z z ) ( S S S + z z ) ( L S L S L S + + z z ) (00) The tansfomation of the opeatos fom laboatoy to molecula coodinates is made because ou investigation of the fundamental and geometical symmeties has shown that the eigenfunction is patially sepaable in the otating coodinate system Eq. (63). A pagmatic comment is that calculation of matix elements is easie if the opeatos ae expessed in the otating coodinate system. Revesal of motion is not an issue hee. The steps shown above meely take advantage of invaiance unde pope otations. A familia esult fom Van Vleck's evesed angula momentum method would be that opeation of + S on Hund's case (a) eigenfunction lowes both and Σ. One's intuition might be that + should be a aising opeato o if + indeed lowes why then is S not a aising opeato? The opeato + lowes when applied to the case ( a) eigenfunction Eq. (44) because the opeato acts on the complex conjugate of the D-matix element not on the angula momentum ket. The opeation of S is completely standad it lowes Σ because it acts on the spin ket SΣ. Thus the contibution of + S to the Hamiltonian matix is B v C ( ) C (S Σ) δ ( +). The minus sign caied by + S in Eq. (00) is canceled by the minus sign in Eq. (8). The phase conventions i.e. sign conventions which ae pat of Van Vleck's method poduce the same esults but fo diffeent easons. Equation (8) is usually not used in Van Vleck's method leading to occuence of the opposite sign on the ight. Usually a minus sign is intoduced fo the action of S ± on the standad ket SΣ. The intoduction of this paticula sign leads in tun to the same esult as found with standad methods. In passing we note that Eqs () and (3) of Zae et al. [3] cay opposite signs but these two opposite signs cancel out in the matix elements of the poduct + S. Lefebve-Bion and Field [4] use the standad phase convention fo the spin kets thei Eq. (.4.8). The spinotation inteaction: Intenational Review of Atomic and Molecula Physics () anuay-une 00 37

14 Chistian G. Paigge and ames O. Honkohl H SR γ () S γ () ( S) S γ () ( S S ) γ() + S + S + + z SZ S... (0) contains tems having the same dependence on and S as tems in the otational Hamiltonian above (γ () is the spin-otation paamete not the thid Eule angle). Hee γ vc C S Σ δ +. + S contibutes ( ) ( ) This esult agees with Zae et al. [3] but shows a dispaity in sign with Lefebve-Bion and Field [4]. The latte authos show the opposite sign on the ight side of Eq. (8) see thei Eqs (.3.3) and (.3.5). When combining the opposite sign with sign conventions fo the spin kets it maybe difficult to find coect signs fo the spin-otation matix elements. 8. DIATOMIC LIE STREGTHS The pobability of a quantum tansition is contolled by the absolute squae of the tansition moment of the opeato esponsible fo the tansition. The line stength is defined by: S (n n ) M M n M T ( k)... n M... (0) in which T l (k) is the l-th component of the ieducible tenso opeato of degee k. A single pime denotes the uppe state double pimes the lowe. As Eq. (63) holds fo the diatomic eigenfunction: ( k ) (... ) T l k ( k ) k* Tλ (... ρζ ) Dl λ ( αβγ )... (03) λ k must hold fo any diatomic opeato. Hund's case ( a) tansition moments of the components of the opeato T (k) ae theefoe given by: l ( k ) (... ) n M T n M l ( + ) ( + ) k λ k 8π ( k ) (... ) n v T ρζ λ ( ) n v δ S S δ Σ Σ π π π ( αβγ ) ( αβγ ) D D D k* * M lλ M ( αβγ ) sin β dα dβ dγ... (04) The integal ove the Eule angles is the poduct of two Clebsch-Godan coefficients: ( k ) (... ) n M T n M k + + k λ k ( k ) (... ) n v T ρζ n v M kl M kλ λ.... (05) Foming the absolute squae of the tansition moment followed by summation ove all M and M gives the diatomic line stength in Hund's case (a) basis S (nv n v ) S ev (nv n v ) S ( )... (06) whee the electonic-vibational stength is given by: S ev (nv n v ) v R n n v.... (07) The electonic tansition moment is: R nn () ( k ) ( ) n T... ; n.... (08) λ The emaining facto S ( ) is the Hund's case (a) Hönl-London o otational line stength facto: S ( ) ( + ) k ( S S ) δ δ Σ Σ.... (09) With a Taylo's expansion of the electonic tansition moment 38 Intenational Review of Atomic and Molecula Physics () anuay-une 00

15 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos R nn () a 0 + a + a (0) and the Fanck-Condon facto: q (v v ) and the -centoids: j v v j ( v v ) v v v v... ()... () ae defined the electonic-vibational stength can be witten as a poduct of two factos: S ev (nv n v ) a a ( v v ) a ( v v ) ( v ) q v.... (3) The electonic stength is defined: S e (nv n v ) a a ( v v ) a ( v v ) (4) 0 The oveall diatomic line stength is: S (nv n v ) S e (nv nv) q (v v ) S ( ).... (5) The diatomic line stength is the poduct of the electonic stength S e (nv n v ) and two unitless stength factos the Fanck-Condon facto q (v v ) and the Hönl-London facto S ( ). In molecules with small educed mass µ and theefoe lage B () otational distotion of the vibational states gives the Fanck- Condon factos and -centoids and dependence. Equation (5) still holds in the Hund's case (b) basis but the Hönl-London facto is now given in tems of a Wigne 6j-symbol athe than a Clebsch-Godan coefficient: S ( ) ( + ) ( + ) ( + ) Λ k Λ Λ Λ S δ ( S S ). k... (6) Calculation of the total line stength equies obviously knowledge of the ieducible tenso esponsible fo the tansition. Howeve the Hönl-London facto depends only on the degee k of the tenso. Fo k Eqs (09) and (6) give Hönl-London factos fo electic dipole tansitions. Fo k they give the factos fo two-photon o Raman tansitions and so foth. The Hund's cases ( a) and ( b) eigenfunctions will aely diagonalize the Hamiltonian to a satisfactoy degee of appoximation. Since efficient numeical algoithms exist fo the diagonalization of symmetic matices this is a not a seious impediment. Diagonalization of the Hamiltonian gives the tems values and the othogonal matix U which diagonalized the Hamiltonian. The equation fo the line stength must be modified to include the matices U and U but this is again a mino complication. Of pactical concen is the detemination of tem combinations that will indeed lead to spectal lines fom the uppe tem values fo a fixed and lowe tem values fo a fixed. A well established pocedue fo finding allowed and fobidden lines is based on invoking paity selection ules. An algoithm which coectly gives the paity of states fo many diffeent types of states is difficult to devise. Howeve the paity selection ules can with no loss of igo be eplaced by the angula momentum selection ules. A tansition fo which the Hönl-London facto vanishes is fobidden. One using the paity selection ules must wite a new compute pogam each time he encountes a diffeent type of tansition. One using the angula momentum selection ules can wite a single pogam which handles a wide vaiety of tansitions see Honkohl et al. [8]. 9. APPLICATIOS I STUDY OF MOLECULAR SPECTRA FOLLOWIG LASER-IDUCED OPTICAL BREAKDOW Selected diatomic molecula specta ae computed by use of so-called line stength files. Fo isolated molecula tansitions the Fanck-Condon factos ae found by solving the adial Schödinge equation numeically and the Hönl-London factos ae obtained by numeical diagonalization of the otational and fine-stuctue Hamiltonian Honkohl et al. [8]. Fo the computation of a diatomic spectum typically the tempeatue and spectal esolution is specified and in tun non-linea fitting outines may be used to infe the spectoscopic tempeatue. Fo a ecent summay on lase-induced beakdown spectoscopy see Miziolek et al. [9] Cemes and Radziemski [30] and Singh and Thaku [3]. Of inteest ae diatomic molecules such as AlO C CH C CF + st eg st Pos nd Pos H O OH. Accuate line-stength files ae in use fo Intenational Review of Atomic and Molecula Physics () anuay-une 00 39

16 spectoscopic analysis with the so-called Boltzmann Equilibium Spectum Pogam (BESP) Honkohl and Paigge [3]. The BESP is used fo example in the analysis of lase-induced optical beakdown and laseinduced fluoescence. Figues 7 show selected synthetic specta fo OH AlO C Swan band oveview (po-gession) and selected sequences ν 0 ± of the C Swan Band and C espectively. Chistian G. Paigge and ames O. Honkohl Fig. 3: Computed spectum of C swan d 3 Π g a 3 Π u band pogession T 8000 K fo a spectal esolution of ν ~ cm. Indicated ae the ν 0 ± sequences of the C pogession Fig. : Computed spectum of the A Σ X Π uv band of OH T 4000 K fo spectal esolutions of ν ~ 3 cm (top) and ν ~ 0. cm (bottom) of the ν 0 sequence Fig. 4: C swan d 3 Π g a 3 Π u band ν + sequence T 8000 K ν ~ 6 cm Fig. : Computet spectum of the AlO B Σ + X Σ + band T 4000 K fo a spectal esolution of ν ~ 3 cm. Indicated ae the ν 0 ± Sequences of the AlO pogession Fig. 5: C swan d 3 Π g a 3 Π u band ν 0 sequence T 8000 K ν ~ 6 cm 40 Intenational Review of Atomic and Molecula Physics () anuay-une 00

17 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos that individual matix elements of the otation opeato ae not standad angula momentum eigenfunctions. It is then clea why the standad fomulae fo the aising and loweing opeatos do not apply to Hund's cases a and b diatomic eigenfunctions. The standad fomulae must be eplaced by equations which give the effects of the aising and loweing opeatos on the complex conjugate of D-matix elements. The igoous teatment of diatomic molecula specta allows us to accuately compute molecula specta. Applications include diagnostics of molecula ecombination specta following lase-induced optical beakdown. Fig. 6: C swan d 3 Π g a 3 Π u band ν sequence T 8000 K ν ~ 6 cm Fig. 7: Computed spectum of the C violet B Σ + X Σ + band T 8000 K fo a spectal esolution of ν ~ 6 cm of the ν 0 sequence These synthetic specta have been instumental in the analysis of lase-induced optical beak-down and in geneal analysis of diatomic molecula specta see fo example Refs. [33 40]. 0. COCLUSIOS The fully standad methods allow us to compute diatomic specta. A pope otation of coodinates peseves the commutation fomulae which ae customaily taken to be the definition of angula momentum. We compaed and contasted ou appoach with Klein's anomalous angula momentum commutatos and Van Vleck's method. The fully standad method yields esults consistent with expeimental data. An essential point is APPEDIX A APPLYIG AGULAR MOMETUM COMMUTATORS TO ROTATIO MATRIX ELEMETS Geneal elations fo commutatos ae typically obtained by applying the commutato to an abstact ket descibing a physical state. Altenatively in the Schödinge epesentation the commutatos ae obtained by applying diffeential opeatos to physical eigenfunctions. In this appendix we demonstate how two anomalous esults may occu. We apply [ x y ] and [ y y ] to the otation D αβγ. matix elements M The commutato [ x y ] is evaluated using: [ x y ] + + ( + ) i i +... (A) and Eq. (8) which fo convenience is epeated hee D αβγ ± M * ( ) ( αβγ ) C D... (A) M Successive application of the opeatos in the otated fame of efeence yields the intemediate esult: i D + M ( αβγ ) i ( + C ( ) C ( ) ( ) ( )) C + C+ D M αβγ... (A3) which afte inseting (compae Eq. (6)) Intenational Review of Atomic and Molecula Physics () anuay-une 00 4

18 Chistian G. Paigge and ames O. Honkohl C ± ( ) ( ) ( ± + ) leads to: x y DM αβγ i DM... (A4) αβγ.... (A5) It might be tempting to conclude the anomalous commutato elations Eq. (4) in the otated molecula fame fom this identity (Eq. (A5)). Howeve the otation matix elements contain two quantum numbes M and one too many to epesent a physical eigenfunction. Similaly [ y y ] is evaluated using: [ y y ] (A6) and applying it to the otation matix elements D M ( αβγ ). ote y acts on (see Eq. (8)) while y acts on M (see Eq. (9)). The intemediate step is given hee: y y DM αβγ which educes to + C + D M αβγ + 4 ( ) C M D M 4 ( ) αβγ C+ ( ) D M + ( αβγ ) + 4 C M D M 4 ( ) αβγ C+ ( ) + D M + ( αβγ ) 4 C M D M 4 ( ) αβγ + + C ( ) D M ( αβγ ) 4 C M D M 4 ( ) αβγ +... (A7) y y DM αβγ (A8) Again one might be tempted to falsely conclude that Eq. (A8) is a geneal commutato elation that also applies to angula momentum states. REFERECES [].H. Van Vleck Rev. Mod. Phys. 3 3 (95). [] O. Klein Z. Physik (99). [3]. Bown and A. Caington Rotational Spectoscopy of Diatomic Molecules Cambidge Univ. Pess Cambidge (003). [4] H. Lefebve-Bion and R.W. Field The Specta and Dynamics of Diatomic Molecules Elsevie Amstedam (004). [5] P.R. Bunke and P. ensen Fundamentals of Molecula Spectoscopy IOP Publishing Bistol 005. [6].O. Honkohl and C. Paigge Am.. Phys (996). [7] A. Messiah Quantum Mechanics and oth Holland Amstedam (96). [8] A.S. Davydov Quantum Mechanics Pegamon Pess Oxfod (965). [9] H. Goldstein Classical Mechanics nd ed. Addison- Wesley Reading (98). [0] M.E. Rose Elementay Theoy of Angula Momentum ohn Wiley and Sons ew Yok (957). [] D.M. Bink and G.R. Satchle Angula Momentum Oxfod Univ. Pess Oxfod (968). [] M. Tinkham Goup Theoy and Quantum Mechanics McGaw-Hill ew Yok (964). [3] D.A. Vashalovich A.. Moskalev and V.K. Khesonskii Quantum Theoy of Angula Momentum Wold Scientific Singapoe (988). [4] K. Gottfied Quantum Mechanics Addison-Wesley Reading (989). [5] G. Baym Lectues on Quantum Mechanics Benjamin/ Cummings Reading (969). [6] B.W. Shoe and D.H. Menzel Pinciples of Atomic Specta Addison-Wesley Reading (968). [7] B.R. udd Angula Momentum Theoy fo Diatomic Molecules Academic Pess ew Yok (975). [8] M. Mizushima The Theoy of Rotating Diatomic Molecules ohn Wiley & Sons ew Yok (975). [9] K.F. Feed. Chem. Phys (966). [0] I. Kovacs Rotational Stuctue in the Specta of Diatomic Molecules Ameican Elsevie ew Yok (969). 4 Intenational Review of Atomic and Molecula Physics () anuay-une 00

19 Diatomic Molecula Spectoscopy with Standad and Anomalous Commutatos [].T. Hougen The Calculation of Rotational Enegy Levels and Rotational Line Intensities in Diatomic Molecules.B.S. Monogaph 5 U.S. Govenment Pinting Office Washington D.C. (970). [] A. Caington D.H. Levy and T.A. Mille Adv. Chem. Phys (970). [3] R.. Zae A.L. Schmeltekopf W.. Haop and D.L. Albitton. Mol. Spectosc (973). [4].M. Bown and B.. Howad Mol. Phys (976). [5] P.R. Bunke. Mol. Spectosc. 8 4 (968). [6] B.H. Bansden and C.. oachain Physics of Atoms and Molecules nd edition Peason Education Limited London (003). [7] G. Hezbeg Molecula Specta and Molecula Stuctue I. Specta of Diatomic Molecules Van ostand Pinceton (950) pp. 3. [8].O. Honkohl C.G. Paigge and L. emes Appl. Opt (005). [9] A.W. Miziolek V. Palleschi and I. Schechte eds. Lase- Induced Beakdown Spectoscopy (LIBS) Fundamentals and Applications Cambidge Univesity Pess ew Yok Y (006). [30] D.E. Cemes and L.. Radziemski Handbook of Lase- Induced Beakdown Spectoscopy Cambidge Univesity Pess ew Yok Y (006). [3].P. Singh and S.. Thaku eds. Lase Induced Beakdown Spectoscopy Elsevie Science ew Yok Y (007). [3].O. Honkohl and C.G. Paigge Boltzmann Equilibium Spectum Pogam (BESP) The Univesity of Tennessee Space Institute Tullahoma Tenn USA 00 view.utsi.edu/besp/. [33] C.G. Paigge.O. Honkohl and L. emes Int.. Spectoscopy (00). [34].O. Honkohl L. emes and C.G. Paigge Spectoscopy Dynamics and Molecula Theoy of Cabon Plasmas and Vapos in Advances in the Undestanding of the Most Complex High-Tempeatue Elemental System L. emes and S. IleEds. Wold Scientific 009. [35] C.G. Paigge Lase-induced Beakdown in Gases: Expeiments and Simulation in Lase Induced Beakdown Spectoscopy A.W. Miziolek V. Palleschi and I. Schechte Eds. Chapte 4 Cambidge Univesity Pess ew Yok Y (006). [36] L. emes A.M. Keszle.O. Honkohl and C.G. Paigge Appl. Opt (005). [37] C.G. Paigge G. Guan and.o. Honkohl Appl. Opt (003). [38] I.G. Dos C. Paigge and.w. L. Lewis Opt. Lett (998). [39] C. Paigge D.H. Plemmons.O. Honkohl and.w.l. Lewis. Quant. Spectosc. Radiat. Tansfe (994). [40].O. Honkohl C. Paigge and.w.l. Lewis. Quant. Spectosc. Radiat. Tansfe (99). Intenational Review of Atomic and Molecula Physics () anuay-une 00 43