A NEW APPROACH IN THE RAYLEIGH - SCHRÖDINGER PERTURBATION THEORY FOR THE ROVIBRATIONAL PROBLEM

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1 Lebanee Scence Journal, Vol., No., A NEW APPROACH IN THE RAYLEIGH - SCHRÖDINGER PERTURBATION THEORY FOR THE ROVIBRATIONAL PROBLEM M. Korek Faculty of Scence, Berut Arab Unerty, P.O.Box - Rad El Solh, Berut 7 89, Lebanon fkorek@cybera.net.lb (Receed February Accepted May ) ABSTRACT For a tranton J 'J' (J'=J±) the egenalue and the egenfuncton of the two condered tate can be expreed repectely n term of one arable m (tranton number), relatng thee two tate, a ( ) m, E = e m = ( ) Ψm = ه φ m = and E 'm = e = ' m, Ψ 'm = φ = ' m, where m=[j'(j'+)-j(j+)]/, and the coeffcent e, φ, e ' and φ ' are gen by analytcal expreon. Th new expanon n the perturbaton theory permt a drect calculaton of many factor n pectrocopy that are expanded n term of m a the lne ntente, the wae number of a tranton, the Herman-Wall coeffcent,... etc. The numercal applcaton how that the preent unconentonal approach n the perturbaton theory prode a mple and accurate method for the calculaton of the egenalue and the egenfuncton for the two condered tate of the molecule CO and for the calculaton of the Herman-Wall coeffcent for the ground tate of the molecule HCl. Keyword: quantum mechanc, perturbaton theory, nfrared tranton, Herman - Wall coeffcent

2 INTRODUCTION In the conentonal Raylegh-Schrödnger perturbaton theory (RSPT), the egenalue E and the egenfuncton Ψ of a robratonal tate (J) are gen repectely by J J E = E + B λ D λ + H λ + L λ +... J () () () () () Ψ = Ψ + Ψ λ + Ψ λ + Ψ λ + Ψ λ +... J where and J are repectely the bratonal and rotatonal quantum number, λ=j(j+), B the rotatonal contant, D, H,L,... are the centrfugal dtorton contant (CDC), ( ) Ψ the pure bratonal waefuncton and Ψ are the ( ) Ψ -rotatonal correcton. In the tranton J 'J' the egenalue and egenfuncton of the tate ('J') are gen repectely by () () E ' J' Ψ ' J' = E + B λ' D λ' + H λ' + L λ' +... ' ' ' ' ' () () () () () = Ψ + Ψ λ' +Ψ λ' + Ψ λ' + Ψ λ' +... ' ' ' ' ' ( ) ( ) Where λ'=j'(j'+). For the condered tranton, the wae number (Herzberg, 9) and (Bally, et al., 997) and many factor n pectrocopy are expreed n term of the tranton number m=(λ'-λ)/, uch a the Herman-Wall rotatonal factor (Herman and Wall, 9) F ' (m) = + C ' m + D ' m + E ' m + H ' m () the lne ntente (Bally et al., 997) hc Ln(Ime ) = Ln(N,J ) Bm(m + ) + con tan t (6) KT the Hönl-London factor, the Enten coeffcent,.... The ue of the λ and λ'-expanon of EJ, Ψ J and E'J', Ψ 'J' (), (), (), () for the calculaton of thee factor, that are expanded n term of m, lead to a mathematcal complexty f hgh accuracy requred. Th tudy preent a new expanon of the egenfuncton and the egenalue n the perturbaton theory n term of the tranton number m a

3 Lebanee Scence Journal, Vol., No., ( ) Ψ m = φ m (7) = where the coeffcent ( ) φ and E m ( ) = e m (8) = ( ) e are well expreed by analytcal expreon. Ung the preent unconentonal approach n the perturbaton theory the calculaton of the factor n pectrocopy, that are expanded n term of m, can be calculated drectly by ung Ψ m and Ψ ' m (wthout pang by the ntermedate calculaton of Ψ J and Ψ ' J ' ) where the complexty n the mathematcal calculaton of thee factor can be aoded and hgher degree of precon can be reached. The numercal applcaton to the ground tate of the molecule CO and HCl howed the aldty and the hgh accuracy of the preent formulaton. THE THEORY In the Born-Oppenhemer approxmaton, a robratonal tate (,J) of a datomc molecule characterzed by the egenalue and the egenfuncton of the radal E Ψ J J Schrödnger equaton d Ψ (r ) / dr + {K [E U (r )] λ / r } Ψ = (9) J J J where U(r) the rotatonle potental, r the nternuclear dtance, λ=j(j+) and K= µ / h a known contant for the condered molecule. For the nfrared tranton J 'J', the ntegral number λ and λ' are related to the tranton number m a m=(λ'-λ)/, thu λ and λ' are expreed n term of m n the form (Korek and Kobe, 99) and (Korek, 997) Λ = a m = () where J'=J±, a =, a = + and a = for > for the two condered tate, a = for the lower tate and a = + for the upper tate. By ung the boundary condton of the wae functon Ψ m (r) (7)(for a gen robratonal tate J), one can fnd (Kobe and Korek, 98)

4 φ (r) r r for () For the unnormalzed wae functon Ψ m and at any arbtrary orgn r o we take wthout any lo of the generalty of the problem Ψ (r ) = φ (r ) = and φ (r ) = for > () m Analytc expreon of the coeffcent e In order to pa from the conentonal λ-expanon of ΨJ and E J (() and ()), n the perturbaton theory, to the m-expanon of thee functon we replace (7), (8) and () n (9), we obtan φ" m + {K[ e m u(r) ] a m /r } φ m = = = = = () nce th equaton atfed for any alue of m, one can wrte where d φ dr ε = + K ε φ = () e U n = γ n φ ( n) () ( n ) γ n = Ke + a n / r for n () wth γ = For = the conentonal pure bratonal Schrödnger equaton gen by () d φ () + K ε φ = (6) dr

5 Lebanee Scence Journal, Vol., No., and for > we obtan from () a et of dfferental equaton where the oluton ge the coeffcent e, φ, ( ) e ' and φ ((7) and (8)) for the two condered tate accordng to ' the alue of a () ( n a = - for the lower tate and a =+ for the upper tate ()). If () multpled by ( ) φ and we ntegrated between an arbtrary orgn r o and nfnty, one obtan r o d ( φ () φ dr () d φ dr φ )dr = ( n) () γ n φ φ dr r o n = (7) by ung (6) n (7), we can wrte dφ dr = ( n) () γ φ φ dr n n = r = f the ntegraton of (7) repeated between r o and zero and by ung the contnuty of ( n ) φ, one obtan n = γ n r o φ ( n ) φ ( ) dr = n = r o γ n φ ( n ) φ ( ) dr ( n) () () ( n) γ n φ φ dr = < φ γ φ > = (8) n = n = By replacng () n (8), one can fnd for a gen alue of that n ( ) ( n ) ( ) a ( n ) e < φ φ > = < φ n φ n = n = Kr > (9)

6 6 Thu, for a gen robratonal tate J, analytc expreon are obtaned for the new contant ( ) ( ) ( n) e n term of mple defnte ntegral of the form < φ φ > and ( ) a n ( n ) < φ φ >. Kr Analytc expreon for the functon ( ) φ For the oluton of the pure bratonal Schrödnger equaton (6) the pure bratonal wae functon can be expanded a () () () φ = b α (x) () n = where x=r-r e, and () α n (n=, ) are the pure bratonal canoncal functon (Kobe and Korek, 98). Ung the boundary condton () one may wrte n n ( ) b = lm x x ( ) α ( x ) ( ) α ( x ) () ( ) b = To calculate the functon (Pkono, 969) ( ) φ () replaced by the Volterra ntegral equaton φ ( r r ) ( ) ' ( ) ( ) (r) = φ (r o ) + (r r o ) φ (r o ) + (r t) ε (t) φ (t)dt + (r t) γ n φ (t) dt r o n= r o () th equaton equalent to () n the ene that any oluton of () oluton of () and ( ) ce era. Ung the canoncal functon approach, a rotaton harmonc can be expanded (Courant and Hlbert, 966) a φ ( ) φ = b n α n n = ()

7 Lebanee Scence Journal, Vol., No., 7 where the canoncal functon where γ n gen n term of ( ) e and a n α n, n th work, are gen by α () = α (a) ( ) α (x) = R (x) (b) p p= x ( n) () R (x) = (x t)( γ n φ ) φ (t) dt (c) n= x ( n) R (x) = (x t)( γ n φ )R (t) dt (d) p p n= and the coeffcent bn are gen by b = lm x x α (x) α (x) (a) b ( ) = (b) b ( ) = φ (ro ) =. (c) Snce b ( ) =, () can be wrtten a Thu, the functon expreon. ( ) φ = b n αn (6) n= ( ) φ n the new m-expanon are well expreed by analytcal

8 8 APPLICATION TO THE CALCULATION OF MATRIX ELEMENTS The robratonal matrx element for a tranton J 'J' are gen by ' m M m =< Ψ m f(r) Ψ ' m > (7) By replacng (7) n (7) one obtan < Ψ >=< m f(r) Ψ 'm ( φ m ) f(r) ( φ m ) > = = ' (8) wth where = H β m (9) = β o = β = H p, /H () H jk p= =< φ (j) p f(r) φ (k) ' > By ung () we obtan the expreon of n n functon α and α' a H jk n term of the canoncal H jk = l= h= (j) (j) (k) h (k) h < b l α l f(r) b' α' > () NUMERICAL APPLICATION To tet the aldty and the accuracy of the coeffcent ( ) e for the calculaton of the egenalue E m of a gen tate J, () replaced ( λ = am + m ) n () and by dentfcaton wth (8), expreon of ( ) e are obtaned n term of B, D, H, L,... (),dered n the conentonal perturbaton theory, a

9 Lebanee Scence Journal, Vol., No., 9 () E () ab () B D () a( D + H ) () D + H L () a(h + L + (6) H + 6L + M e e e = (a) e = (b) = (c) = (d) e = + (e) e = M) (f) e = + N (g) The preent formulaton appled to the ground tate of the molecule CO by ung a Dunham potental (Farrenq et al., 99). We preent n Table the alue of the pure bratonal energy, the conentonal rotatonal contant B and the CDC (Korek and Kobe, 99) for. In Table we ge the coeffcent e ( 6) (the coeffcent e () = E are gen n column of Table ) calculated from (9) (frt entry), compared to thoe calculated from () by takng a = + and by ung the alue of Table (econd entry). An excellent agreement notced between thee alue of e for for all the condered bratonal leel. Th agreement become moderate for () (6) e ande ; thee coeffcent are gen n () n term of the CDC H,L,M and N where M and N are relately mall 6 ( M = x ; N = x ). Snce we ued n th work a double precon PCcomputer, the alue of e and e gen n the frt entry (calculated from (9)) are () (6) more accurate than thoe calculated from () becaue they are wthn the computer precon. To llutrate the preent formulaton for the calculaton of the functon φ ( > ) (6) by a numercal example, the ame Dunham potental ued (Farrenq et al., 99) for the

10 ground tate of the molecule CO. For a gen bratonal leel (gen E ), the computaton of φ can be done by the followng tep: ) For a gen alue of, the oluton of the pure bratonal Schrödnger () () equaton (6) (Kobe and Korek, 98; 98) ge e and φ () () ) by ung thee alue of e and φ n (9) for = and a =± (accordng to the tate needed wth n ) one obtan ) the ue of () () e n (6) ge φ. Th procedure wll be repeated for calculatng the other functon φ by ung alternately (9) and (6),.e., the wae functon Ψm for a gen robratonal tate wll be determned. We preent n Table the wae functon Ψ m (x) by uccee approxmaton a () e S = φ () (a) S = S + φ () m (b) S S = S = S + φ + φ () () m m (c) (d) S = S + φ () m (e) S = S + φ () m (f) for m=-, x=. A ;.6 A ;.9 A (x=r- r e ) for the dfferent bratonal leel =,, the wae functon Ψ calculated from m m φ compared to thoe calculated by = a drect method (Kobe et al., 98). We notce the excellent agreement between thee alue up to een gnfcant fgure, or more, for all the condered bratonal leel and for dfferent alue of x. In Table the calculaton of the ame functon repeated a n Table for x=.6 A and m=-, -, -6. We notce alo the good agreement between

11 Lebanee Scence Journal, Vol., No., Ψ = m = φ m and Ψ calculated by a drect method (Kobe et al., 988), but m th agreement decreae lghtly wth the ncreang alue of m ; th may be explaned by the perturbate apect of the preent formulaton. Ung the laer nduced fluorecence combned wth Fourer tranform pectrocopy, Fellow et al. (99) calculated the (Fellow) (Fellow) egenalue E and the rotatonal contant B for the ground tate X Σ + of the (Fellow) (Korek) molecule RbC up to =. From the preent formulaton E and B hae been calculated for the condered leel of the ame tate of RbC (not howed). The comparon between thee alue how an aerage alue of the relate error ( E - (Korek) (Fellow) (Fellow) (Korek) (Fellow) (Fellow) E )/ E equal.76 % and ( B - B )/ B equal.8 %. Th ery good agreement between the expermental reult and thoe calculated by the preent formulaton how the hgh accuracy of the later. Korek (997) calculated the Herman-Wall coeffcent for the molecule CO, up to the fourth order, by ung the m-repreentaton of the wae functon (7) where the functon φ are expreed n term of the conentonal coeffcent Ψ (). By repeatng the calculaton of thee Herman-Wall coeffcent ung (8) for the ame molecule CO, an excellent agreement between thee coeffcent obtaned up to eght gnfcant fgure or more for the condered tranton (not howed). A another tet, and by ung a Dunham potental (Kobayah and Suzuk, 986) of the molecule HCl, Table preent the alue of the Herman-Wall coeffcent calculated from () (Korek, 997) (8) n th work. The comparon of thee alue to thoe calculated by (Kobayah and Suzuk, 986) how a ery good agreement for all the condered tranton. No comparon for the hgher order coeffcent E and H () wth the other reult, becaue they are gen here for the frt tme. Thu by ung the preent new expanon n term of m for the wae functon and the egenfuncton the calculaton of the Herman-Wall coeffcent can be done eaer and wth hgh precon. CONCLUSION Wthn the frame of the perturbaton theory we expand n th work the egenalue and the egenfuncton of a robratonal tate J (or J ) n term of the tranton number m a E m = Ψ = m = = e m φ m

12 where the coeffcent e and φ are gen by analytcal expreon (9) and (6). The numercal applcaton to the ground tate of the molecule CO howed the excellent accuracy of the preent m-expanon for the egenalue and the egenfuncton. Thu, for a gen potental functon U(r) (ether emprcal or of the RKR-type), th work preent a new alternate method for the calculaton of the egenalue and the egenfuncton for a gen tate (J) of a datomc molecule whch greatly helpful n the calculaton of the dpole moment functon by ung the Herman-Wall approach, the matrx element and the other factor n pectrocopy that are expanded n term of the tranton number m. ACKNOWLEDGEMENT Th tudy wa fnancally upported by the France-Lebanon blateral cooperaton n centfc reearch and technology (CEDRE) programme.

13 Lebanee Scence Journal, Vol., No., Value of the pure bratonal energy TABLE E, the rotatonal contant B and the centrfugal dtorton contant for eeral bratonal leel for a Dunham potental of the ground tate of the molecule CO (Farrenq et al., 99) (all alue are n unt of cn - ) () e = E B D (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6)

14 TABLE : contnued H L M N (-) (-7) (-) (-8) (-) (-7) (-) (-8) (-) (-7) (-) (-9) (-) (-7) (-) (-7) (-) (-7) (-) (-7). 8 9 (-) (-8) (-) (-6) (-) (-7) (-) (-) (-) (-6) (-) (-6) (-) (-6) (-) (-6) (-) (-6).76 7 (-) (-6) (-) (-6) (-) (-) (-) (-6) (-) (-) (-) (-7) (-) (-6) (-) (-6) (-) (-) (-) (-6) -. 9 (-) (-) (-) (-6) (-) (-6) (*) Number between parenthee tand for a multplcaton power of

15 Lebanee Scence Journal, Vol., No., TABLE Value of the Coeffcent e calculated from () (frt entry) compared to thoe calculated from (9) by takng a = + (econd entry) for the Dunham potental of the ground tate of CO (Farrenq, 99) (all alue are n unt of cn - ) V () e () e () e ** (-)* (-) (-) (-) -. 8 (-) (-) (-) (-) (-) (-) (-) (-6) (-6) (-6) (-6) (-6) 6

16 6 TABLE : contnued () e () e (6) e (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-6) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (*) Number between parenthee tand for a multplcaton power of (**) Omtted fgure are dentcal to thoe of the frt entry

17 Lebanee Scence Journal, Vol., No., 7 TABLE Value of the uccee approxmaton of the wae functon Ψ m = S p () compared to Ψ calculated from a drect method (Kobe et al., 98) for m= - and =,, m x ( A ). S S S S S S Ψ at x=. A,.6 m= - A,.9 A. S (x) = = = m m = = o φ m * Ψ.6 S S S S S S = Ψ m φ = o m Ψ m 8.9 S S S S S S Ψ m o = φ = m Ψ m 8 (*) Omtted fgure are dentcal to thoe of the frt entry

18 8 TABLE Value of the uccee approxmaton of the wae functon Ψ m = S p () compared to Ψ m calculated from a drect method (Kobe et al., 98) for m= -, -, -6 and =,, at x=.6 A. x=.6 A m S (x) = = = Ψ m = φ m = o Ψ m Ψ m = φ m = o Ψ m Ψ m = φ m = o * Ψ m (*) Omtted fgure are dentcal to thoe of the frt entry

19 Lebanee Scence Journal, Vol., No., 9 TABLE Value of the Herman-Wall coeffcent () for the tranton = and ' 7 for the ground tate of the molecule HCl Kobayah and Preent Work Suzuk 986 C -.6(-)* -.7 (-)* D.(-). 7(-) E (-6) H (-8) C -.() -.96 (-) D.(-).7 (-) E (-6) H (-8) C.(-).8 (-) D.(-).8 7(-) E (-6) H (-7) C.7(-).7 (-) D.(-). (-) E (-) H (-7) C.9(-).9 (-) D.78(-).77 8(-) E (-) H (-7) 6 C.(-).8 8(-) D 7.88(-) 7.87 (-) E (-7) H (-8) 7 C.(-).9 (-) D.8(-). (-) E (-) H (-7) (*) Number between parenthee tand for a multplcaton power of

20 REFERENCES Bally, D., Camy - Peyret, C. and Lanquten, R Temperature meaurement n flame through CO and C emon: New hghly excted leel of CO. J. Mol.Spectroc.8,. Courant, R. and Hlbert, D. 966 Method of Mathematcal Phyc wley, New York. Farrenq, R., Guelachl, G., Saual, A. J., Greee, N., and Farmer, C. B. 99. Improed Dunham coeffcent for CO from nfrared olar lne of hgh rotatonal exctaton. J Mol. Spectroc., 9: 7. Fellow, C., E., Gutterre, R., F., Campo, A., P., C., Vergè, J. and Amot, C. 99. The RbC X+ ground electronc atae: New pectrocopc tudy. J. Mol Spectroc., 97: 9. Herman, R., and Wall, R., F. 9. Influence of braton-rotaton nteracton on lne ntente n braton-rotaton band of datomc molecule. J. Chem. Phy., : 67. Herzberg, G. 9. Spectra of Datomc Molecule. Van Notrand, Toronto. Kobayah, M. and Suzuk, I Dpole moment functon of Hydrogen Chlorde. J. Mol. Spectroc., 6:. Kobe, H. and Korek, M. 98. Analytc expreon of the rotaton harmonc n the braton-rotaton wae functon of a datomc molecule. Int. J. Quant. Chem., :. Kobe, H., Dagher, M., Korek, M., and Chaalan, A. 98. A new treatment of the braton-rotaton egenalue problem for a datomc molecule. J. Comp. Chem., : 8. Kobe, H. and Korek, M. 98. Egenalue functon aocated wth datomc rotaton and dtorton contant. J. Phy. B: At. Mol. & Opt. Phy., 8:. Kobe, H., Kobe, M. and El-Hajj, A On tetng dfference equaton for datomc egenalue problem. J. Comput. Chem., 8: 8. Korek, M. and Kobe, H. 99. Datomc centrfugal dtorton contant for large order at any leel: Applcaton to the XO + -I tate. Can. J. Chem., 7:. Korek, M., and Kobe, H. 99. New analytcal expreon for the Herman-Wall coeffcent of nfrared tranton up to the thrd order. J. Quant. Spectroc.Rad. Tran., : 6. Korek, M Analytcal expreon for the hgh-order Herman-Wall coeffcent of a datomc molecule. Can. J. Phy., 7: 79. Pkono, N Calcul dfferentel et ntegral. Mr. Mocow.