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Derivation from coupled channel equations of the resonance raman scattering amplitude for a diatomic molecule
Volume
58. number
CHEMIC.AJ_ PHYSICS
2
DERIVATION
LETTERS
15 September 1978
FROhf COUPLED CHANNEL EQUATIONS
OF THE RESONANCE
RAMAN SCA-fTERfNG
AMPLITUDE
FOR A DIATOM% MOLECULE*
0. AT-ABE& R. LEFEBVRE Lm&xzroirede Photophysique &fok?culaire‘*, Ijnii ersizi PoniSud. 91405 Orsay. France
M_ JACON Latiruroire de Rechercks
Optiques. Un;ienitp de Reimr, 51062 Reizm-Cedex, France Received 18 Agill Revised marmscript retched 12 June 1978
The coupled equatiops of molecular collision theory can be formulated in such a way as to yield the amp!itude for resommce Roman scattering of li&t by ;Ldiatomic molecule. Tao vxknts of tke method can be adopted, characterized by dif-
ferent choices for the potential matrices. _%limolecular potentials. either analytical or numerical, are treated on the same fooziq. Tests of accuracy are presentedfor the case of harmonic potentials intersected by linear potentials, for which an exact anslyticzl solution cm bc given_
I. Introduction The xrttering of a photon by a moIecuIe going from a discrete state n to a discrete state n ’ via a continuum of states E is described by the element Or [Tin’) of the transition operator T=V+VGV_ where G is the total Green function of the system molecule-and field and V :he molecuie-field order in V i: is found [ 1-5j
(1) coupling. To Iowest
(2) Assuming the electronic transition moment to be independent of the nuclear coordinates makes it possibie to separate from (2) a factor depending only on Franck-Condon amplitudes (3)
* Wbrk supported in part by Universiti Pierre et .Mtie Curie, UER 52. ** Ltbomtoire du C.N.R.S. 1%
Volume 58, number 2
CiiRflCAL
PI-IIXCS
LETTERS
15 September 1978
For some models of molecular potentials it is possible to perform this integration analytically [S; _It is also possible to use numerical procedures with the Franck-Condon amplitudes under the integral sign determined from a preliminary calculation of the molecuhu wavefunctions [2J_ We wish to demonstrate in this letter that a different, efficient and accurate procedure consists in the solution of appropriately constructed coupled channel equations-
2_ hfethod \Veconsider for a diatomic molecule of reduced mass p a set of five coupled equations having the form -
+ Vi
1
U,(R) = c
J-‘i
Wii(R>q(R)
(i,j=1,...5),
(4)
with the following specifications for the potentials Q(R): (a) The potentials of channels 1 and 5 can be chosen somewhat arbitrarily, but they must be open at energy ,“. (b) The potentials of channeIs 2 and 4 are those of the initial and final eiectronic states subject to some vertical displacements for reasons to be given below. (c) The potential of charmel 3 is that of the excited electronic state also subject to a vertical displacement. The potentials in charnels 2,3 and 4 include eventually the centrifugal energy. The vertical displacements mentioned under (b) and (cc) are introduced in order to produce the situation depicted in fig. i _ In the example shown in fig. 1, the molecule goes from the zero point level of the ground state to the first vibrationally excited level of this same state via the continuum symbolized by 3 (thus giving the fundamental of the Raman spectrum). The general case with transition from n to n’ is covered by displacing the potentials in Channels 2 and 4 so as to place E, and E,* respectively in the neighbourhood of .!?. The potential of channel 3 is arranged in such a way as to put the continuum state in resonance with the incoming photon at energy En. This lowering of the excited state potential has the effect of producing in general curve crossings with the potentials of channels 2 and 4, as in the case of the description of various other laser induced processes [61. Such curve crossin@ are not shown in fig. 1 which is only of symbolic value. The potentials of the artificial channels 1 and 5 are such as to ensure some overlap between the eigenfunctions of these channels and the initial and fmal vibrational functions respectively. Artificial channels have been previousIy considered in studies of discrete to continuum transition amplitudes [7,8]. Since we are dealing here with a discrete to discrete transition amplitude two artificial channels are introof the S matrix of the coupled duced, the purpose being to deduce the amplitude (3) f rom the element S5l@) channel probiem obtained by a convenient choice of the coupling matrix Wii(R)_ The elements Wq(R) are chosen to be independent of R and to couple only a~&.ceut channels. Two possibilities have been considered-. (a) Symmetric coupling matrix, with Wt2 = W21, FV23= Ws2, etc. (b) Asymmetric coupling matrix [7] with WI2 f 0 but Wzl = 0, etc. In practice all non-zero W’s in both cases, are given a common real value denoted below as $_ * A variant leading also to the determination of the amplitude (1) would consist [9 ] in setting up a four channel problem with cydic couplings-
Fig_ 1. Schematic representation of closed channels and continua involved in-the multichannel problem producing the resonant Raman scattering amplitude. 1 and 5 symbolize the continuous spectra of energies associated with the artificial channeIs, 3 the continuous spectrum of excited molecular levels. En the example &own Raman scattering is from the zero point level of the ground elecuonic state to the first vibrationally excited state via 3.
197
CHEXICAL PHYSICS LEi-l’ERS
Voiume 58, number 2
15 SepteMer 1978
We assume now (as suggested by fe_ 1) that E= En = E,+ and that levels R and n’ are well isolated (in the sense that the widths they may draw from their interactions with the continua az~ much smaller than the level separations)_ This is not a severe restriction siice fl can be chosen at will_ The element Ssl(E) of the S matrix can then be calculated un&r the assumption that in) acts as a doorway state between continuum 1 and 3 and in’) as a doorway state between continua 3 and 5_ This element is to be deduced from S = 1- 2ti T, with f given by
= -2ti
ei~S(~)i5~IVfn)(nfGfn’)
(n’fVf I,!?>ei*l@
,
0
where I I,!?) and IS,!?>represent the distorted waves of channels 1 and 5 wi’J1 asymptotic phases TJ~(,!?)and Y#)_ The phase factors result from the fact that the elements of the S matrix come from au analysis of the asymptotic coupled channel wavefunctions in terms of free waves and not in terms of distorted waves [IO JI The element
J’iGP2 =Pr [(‘z -P2HoP2
(z --PIHoP
-PtRPI)
-P2RP1J
-lPz
,
c6)
where R = VQ(z
- QH,,Q)-1
QV _
(7)
These operator relations are deduced with standard techniques [l 11 from the deftition ators are defmed as Pt = In>?nL
Pz = f?z’>
Q= I --Pi
-Pz
of G. The projection oper-
_
09
The total hamiltonian H has been divided into fig, producing oniy dastic events an-dV responsibfe for inelastic scattering_ Lke has been made of the following relations which result from our particular coupling scheme PiNPI
‘PIHoPI.
P2HP2 =P2HoP2,
P1HP2 = 0,
QHQ=QHoQ-
P)
In the case ofa symmetric coupling matrix, application of (6) gives
4m=yr
f-&!TIVln~l’+
rn(~~=zi(fcI~fvfn>f2
KnfVf3E)i2
E--E
(j&.
9
(10
f fCnlVf3E>f2)
and simik expressions for A&!!?) and widths fl2] only if 8= E,.
fW
and l?,+!!?)_ The quantities (I 1) and (12) are to be interpretedas level shifts
In the case of an asymmetric coupling matrix, there is ‘the much simpler result <“fGfn’)=BzI,~8(EI)j(~-EE,)(~-E,~).
(13)
inspection of expressions (S), (6) and (13) shows that I,,=(&) can be ev&ated after ail the other quantities have been estimated. Ss~(&!?) may be derived directly from the fnre&tarme~ problem_ The quantities A,, X’, , Ati* and fn* can be obtained from two4anneel cakulations which provide also the cou@ngs #tSEiv fn) and can be derived from onechannel cakufations or from analytical formulas valid for simpIe potentials. The case of an qmmetric coupling matrix is even simpler to deal with since the diagonal elements of the S matrix provide the phases of the distorted waves. The structureof formula (13) explains the choice l
198
Vohme
58, number 2
15 September 1978
CHEhliCAL PHYSICS LSFTERS
made for Bin fig. 1. Exploitation of (13) to obtain I,@) requires ,!?to be different from Em and E,, , but these differences must be accurately known. This can be done, as shown by Shapiro 171, by looking first of all for the real poles of S. The complex quantity I,,@) obtained in this way can be decomposed into Inn@‘)
= A,,&)
-
iI’,,n(E) ,
(14)
with the following meaning for the real and imaginary parts
l-+,,*(iT)=+ s
.
(16)
In order to obtain an excitation profde for a given n and n’, we need I,,* as a function of ,!?. This could be done in principle by just varying E while keeping all potentials fured. However the condition 6% E, = E,v is necessary to give a sizeable magnitude to S,1(,#!?). Another procedure is to be preferred: t&is consists in just moving vertically the potential of the repulsive electronic state as a function of photon frequency. This is particularly simple in the case of the asymmetric coupling matrix since auxiliary quantities such as phases and couplings remain the same throughout the calculation. Tests of the accuracy of this method are presented in section 3 by comparison with exact analytical formulas for the case where levels In) and In’) correspond to harmonic potentials and 1E) to a linear potential.
3. Analytically soluble model The mathematical apparatus necessary to treat resonance Raman scattering with harmonic and linear potentials can be derived from the treatment of level shifts and widths for the same potentials by Sink and Bandrauk [13]_ This formulation proceeds via expressions of the Franck-Condon amplitudes iu momentum space. For the purpose of testing the accuracy of the procedure outlined in section 2, we consider series of the type 0 -+ n, the t‘lrst member corresponding to Rayleigh scattering (this case is formally equivalent to the treatment of a level shift and a width), the second (n = 1) being the fundamental of the Raman spectrum; the others the overtones. Apart from some vertical displacements, as explained above, the potentials are V2,4(R)=+J2K
V1,3,5
(17)
3
lu being the reduced mass, w the angular frequency of the harmonic oscillator, F the negative of the slopes of the linear potentials. For simplicity the artificial charmels are given the same potential as the chaiinel associated with the repulsive excited state. The rotational quantum number is ignored, thus n is now the quantum number for an harmonic potential_ The analytical expression for I,,@) is found to be
I,,(E)
=-
c (2mm9L/2(2n,9t/2
x C2a3/2Ai(k)(~o)
[Rio&-&
(8a)‘kCof2~n_k(8n3)&_~(8a3)
k=O Z=O - ~AP(Z~)]
+
ak%@, zryazka/lr=s=o)
,
(18)
with the following meaning for the parameters
199
Vofume 58. number 2
R,
CHEMICAL
is the turning point at energ
LETTERS
15 September
1978
E; @(z, z’) represents the integral
while r (or s) is the au_xilia~r palmeter
H,(X) =
PHYSICS
appearing in the generating form for Hermit: polynomials
an e-2rx e-"/a? I_
wo
.
The last term in (I 2) can be reduced to infinite series of incompiete gamma functions [ 13 3 _Aitk) and Bi(@ represents the kth derivatives of the reguiar and irregular Airy functions [ 143. Tabie 1 collects the results obtained for the real and imaginary parts of ion for n up to 5 for a case with ,u = 75 aston, a slope of 140000 cm-l/& 2GOO cm-l for the quantum of vibration and at energy equal to 999.95
Table 1 RcaI (4) and imxginary (r) parts of tic trzxsition ampIitude ion acmrd~ to three different procedures: (a) and (b) coupled equations with symmetric (a) or asymmetric (b) potentiaI matrices; (c) a.uIyticaI formuhs. Numbers are in cm _---
n
-ion
r0n
(a) ________
0 1 2 3 4 5 ~~__ Tablc 2 The ~uarcs incm-_----
@) _-----
_-
0.8538-5 O-8637-5 0.2068-3 0.2069-3 0.35394 O-3536-4 -0.1403-3 -o-1403-3 O-1068-3 0.1070-3 050284 0.50354 ___________-___
of the moduli of the trsuition .--_ __
__~__
-__ n ______.. 0 I 2 3 4 5
200
Cc)
(a)
UJ)
Cc)
0.8413-S 0.2C68-3 0.3550-4 -0.1402-3 0.1070-3 0.50224
O-2633-3 -0.3758-S -0.1726-3 -0.74264 0.99694 0.1271-3
0.2635-3 -0.2079-5 -0.1728-3 -0.7422-A O-0989-4 O-1272-3
O-2633-3 -0.2040-5 -0.1727-3 -0.74334 0.9971-4 O-1272-3
--
amplitudes IOn according
_.---
to the three procedures
(see legend to table 1). Numbers are
~-
(a) ___-_____O-6940-7 o-4277-7 o-3105-7 02520-7 0.2135-7 0.1868-7
@)
(c)
0.6949-7 0.4281-7 0_3110-7 02520-7 O-2142-7 O-1871-7
O-6942-7 0.4276 -7 0.3107-7 0.2519-7 0.2139-7 O-1869-7
Volume 58, number 2
CHEMICAL PHYSICS LEl-PERS
15 September 1978
cm-l. The two variants of the numerical method have produced columns (a) and (b) while column (c) gives the analytical results. It can be observed that the agreement is generally very good, with errors for the numerical procedures which do not exceed in most cases a few units in the fourth place. Table 2 gives a similar comparison for iiu, I2 which is the anly quantity which can be attained experimentally_ It appears from table 1 that some inaccuracy may result in the smallest of the two quantities A and r when there is large unbalance between the two (forinstance n = 1 where Ael is one hundred times I’uL). However this is of no effect on 11u, I 2 as may be judged from table 2. One may also conclude from these tables that the use of an asymmetric potential matrix is to be preferred since the procedure provides also excellent results while being somewhat simpler.
4. Conclusion The coupled channel equations are treating all potentials in the same way. Thus one may envisage the use of more complicated analytical potentials, or even of numerical potentials derived from spectroscopic data. Work in this direction is being undertaken. Other extensions can be envisaged, such as the introduction of several excited states in order to account for interference effects [ 151 or intensity transfer between coupled states [ 16]_ Another possible use of coupled channel equations is the consideration of more than one nuclear mode since it has been shown that the vibrational states of small polyatomics may be treated via a coupled channel approach [ 171.
References [l] [2] [3] [4]
15)
[6] [7] [S] [9] [lo] [ 111 [12] [13] [ 141 [is] [16] ] 171
M. Jacon. &I. Bejot and L_ Bernard, Compt. Rend. Acad. Sci. (Paris) 273B (1971) 5%. P-F. \Viims and D-L. Rousseau, Phys. Rev. Letters 30 (1973) 961_ D.L. Rousseau and P-F. Wiiiams, J. Chem. Phys. 64 (1976) 3519P-F. WiIliams, A. Fernandez and D-L. Rousseau, Chem. Phys. Letters 47 (1977) 150. hI. Mingardi, IV. Siebrand, D. van Labeke and M. Jacon. Chem_ Phys. Letters 31 (1975) 208; JP. Laplante and A-D. Bandrauk, J. Chem. Phys. 63 (1976) 2592. J.M. Yuan, J-R. Laii and T-F. George, J. Chem. Phys. 66 (1977) 1107, and references therein. M. Shapiro, J. Chem. Phys. 56 (1972) 25820. Atabek and R. Lefebvre, Chem. Phys. 23 (1977) 51. A.D. Bandrauk, private communication. M.S. ChiJd, MoIecuIar collision theo_ry (Academic Press, New York, 1974). L. Mower, Phys. Rev. 142 (1966) 799. U. Fano, Phys. Rev. 124 (1961) 1866. M.L. Sink and A-D. Bar&auk, Chem. Phys. Letters 49 (1977) 508. N. Abramowitz and LA. Stegun, Handbook of mathematical functions (Dover, New York, 1965). P. Baierl, W. Kiefer, P.F. Wiis and D.L. Rousseau, Chem. Phys. Letters 50 (1977) 57. M-2. Zgierski, J. Raman Spectry. 6 (1977) 53. M. Shapiro, Chem. Phys. Letters 46 (1977) 442; 0. Atabek and R. Lefebvre, J. Chem. Phys. 67 (1977) 4983.
201
58. number
CHEMIC.AJ_ PHYSICS
2
DERIVATION
LETTERS
15 September 1978
FROhf COUPLED CHANNEL EQUATIONS
OF THE RESONANCE
RAMAN SCA-fTERfNG
AMPLITUDE
FOR A DIATOM% MOLECULE*
0. AT-ABE& R. LEFEBVRE Lm&xzroirede Photophysique &fok?culaire‘*, Ijnii ersizi PoniSud. 91405 Orsay. France
M_ JACON Latiruroire de Rechercks
Optiques. Un;ienitp de Reimr, 51062 Reizm-Cedex, France Received 18 Agill Revised marmscript retched 12 June 1978
The coupled equatiops of molecular collision theory can be formulated in such a way as to yield the amp!itude for resommce Roman scattering of li&t by ;Ldiatomic molecule. Tao vxknts of tke method can be adopted, characterized by dif-
ferent choices for the potential matrices. _%limolecular potentials. either analytical or numerical, are treated on the same fooziq. Tests of accuracy are presentedfor the case of harmonic potentials intersected by linear potentials, for which an exact anslyticzl solution cm bc given_
I. Introduction The xrttering of a photon by a moIecuIe going from a discrete state n to a discrete state n ’ via a continuum of states E is described by the element Or [Tin’) of the transition operator T=V+VGV_ where G is the total Green function of the system molecule-and field and V :he molecuie-field order in V i: is found [ 1-5j
(1) coupling. To Iowest
(2) Assuming the electronic transition moment to be independent of the nuclear coordinates makes it possibie to separate from (2) a factor depending only on Franck-Condon amplitudes (3)
* Wbrk supported in part by Universiti Pierre et .Mtie Curie, UER 52. ** Ltbomtoire du C.N.R.S. 1%
Volume 58, number 2
CiiRflCAL
PI-IIXCS
LETTERS
15 September 1978
For some models of molecular potentials it is possible to perform this integration analytically [S; _It is also possible to use numerical procedures with the Franck-Condon amplitudes under the integral sign determined from a preliminary calculation of the molecuhu wavefunctions [2J_ We wish to demonstrate in this letter that a different, efficient and accurate procedure consists in the solution of appropriately constructed coupled channel equations-
2_ hfethod \Veconsider for a diatomic molecule of reduced mass p a set of five coupled equations having the form -
+ Vi
1
U,(R) = c
J-‘i
Wii(R>q(R)
(i,j=1,...5),
(4)
with the following specifications for the potentials Q(R): (a) The potentials of channels 1 and 5 can be chosen somewhat arbitrarily, but they must be open at energy ,“. (b) The potentials of channeIs 2 and 4 are those of the initial and final eiectronic states subject to some vertical displacements for reasons to be given below. (c) The potential of charmel 3 is that of the excited electronic state also subject to a vertical displacement. The potentials in charnels 2,3 and 4 include eventually the centrifugal energy. The vertical displacements mentioned under (b) and (cc) are introduced in order to produce the situation depicted in fig. i _ In the example shown in fig. 1, the molecule goes from the zero point level of the ground state to the first vibrationally excited level of this same state via the continuum symbolized by 3 (thus giving the fundamental of the Raman spectrum). The general case with transition from n to n’ is covered by displacing the potentials in Channels 2 and 4 so as to place E, and E,* respectively in the neighbourhood of .!?. The potential of channel 3 is arranged in such a way as to put the continuum state in resonance with the incoming photon at energy En. This lowering of the excited state potential has the effect of producing in general curve crossings with the potentials of channels 2 and 4, as in the case of the description of various other laser induced processes [61. Such curve crossin@ are not shown in fig. 1 which is only of symbolic value. The potentials of the artificial channels 1 and 5 are such as to ensure some overlap between the eigenfunctions of these channels and the initial and fmal vibrational functions respectively. Artificial channels have been previousIy considered in studies of discrete to continuum transition amplitudes [7,8]. Since we are dealing here with a discrete to discrete transition amplitude two artificial channels are introof the S matrix of the coupled duced, the purpose being to deduce the amplitude (3) f rom the element S5l@) channel probiem obtained by a convenient choice of the coupling matrix Wii(R)_ The elements Wq(R) are chosen to be independent of R and to couple only a~&.ceut channels. Two possibilities have been considered-. (a) Symmetric coupling matrix, with Wt2 = W21, FV23= Ws2, etc. (b) Asymmetric coupling matrix [7] with WI2 f 0 but Wzl = 0, etc. In practice all non-zero W’s in both cases, are given a common real value denoted below as $_ * A variant leading also to the determination of the amplitude (1) would consist [9 ] in setting up a four channel problem with cydic couplings-
Fig_ 1. Schematic representation of closed channels and continua involved in-the multichannel problem producing the resonant Raman scattering amplitude. 1 and 5 symbolize the continuous spectra of energies associated with the artificial channeIs, 3 the continuous spectrum of excited molecular levels. En the example &own Raman scattering is from the zero point level of the ground elecuonic state to the first vibrationally excited state via 3.
197
CHEXICAL PHYSICS LEi-l’ERS
Voiume 58, number 2
15 SepteMer 1978
We assume now (as suggested by fe_ 1) that E= En = E,+ and that levels R and n’ are well isolated (in the sense that the widths they may draw from their interactions with the continua az~ much smaller than the level separations)_ This is not a severe restriction siice fl can be chosen at will_ The element Ssl(E) of the S matrix can then be calculated un&r the assumption that in) acts as a doorway state between continuum 1 and 3 and in’) as a doorway state between continua 3 and 5_ This element is to be deduced from S = 1- 2ti T, with f given by
= -2ti
ei~S(~)i5~IVfn)(nfGfn’)
(n’fVf I,!?>ei*l@
,
0
where I I,!?) and IS,!?>represent the distorted waves of channels 1 and 5 wi’J1 asymptotic phases TJ~(,!?)and Y#)_ The phase factors result from the fact that the elements of the S matrix come from au analysis of the asymptotic coupled channel wavefunctions in terms of free waves and not in terms of distorted waves [IO JI The element
J’iGP2 =Pr [(‘z -P2HoP2
(z --PIHoP
-PtRPI)
-P2RP1J
-lPz
,
c6)
where R = VQ(z
- QH,,Q)-1
QV _
(7)
These operator relations are deduced with standard techniques [l 11 from the deftition ators are defmed as Pt = In>?nL
Pz = f?z’>
Q= I --Pi
-Pz
of G. The projection oper-
_
09
The total hamiltonian H has been divided into fig, producing oniy dastic events an-dV responsibfe for inelastic scattering_ Lke has been made of the following relations which result from our particular coupling scheme PiNPI
‘PIHoPI.
P2HP2 =P2HoP2,
P1HP2 = 0,
QHQ=QHoQ-
P)
In the case ofa symmetric coupling matrix, application of (6) gives
4m=yr
f-&!TIVln~l’+
rn(~~=zi(fcI~fvfn>f2
KnfVf3E)i2
E--E
(j&.
9
(10
f fCnlVf3E>f2)
and simik expressions for A&!!?) and widths fl2] only if 8= E,.
fW
and l?,+!!?)_ The quantities (I 1) and (12) are to be interpretedas level shifts
In the case of an asymmetric coupling matrix, there is ‘the much simpler result <“fGfn’)=BzI,~8(EI)j(~-EE,)(~-E,~).
(13)
inspection of expressions (S), (6) and (13) shows that I,,=(&) can be ev&ated after ail the other quantities have been estimated. Ss~(&!?) may be derived directly from the fnre&tarme~ problem_ The quantities A,, X’, , Ati* and fn* can be obtained from two4anneel cakulations which provide also the cou@ngs #tSEiv fn) and
198
Vohme
58, number 2
15 September 1978
CHEhliCAL PHYSICS LSFTERS
made for Bin fig. 1. Exploitation of (13) to obtain I,@) requires ,!?to be different from Em and E,, , but these differences must be accurately known. This can be done, as shown by Shapiro 171, by looking first of all for the real poles of S. The complex quantity I,,@) obtained in this way can be decomposed into Inn@‘)
= A,,&)
-
iI’,,n(E) ,
(14)
with the following meaning for the real and imaginary parts
l-+,,*(iT)=+ s
.
(16)
In order to obtain an excitation profde for a given n and n’, we need I,,* as a function of ,!?. This could be done in principle by just varying E while keeping all potentials fured. However the condition 6% E, = E,v is necessary to give a sizeable magnitude to S,1(,#!?). Another procedure is to be preferred: t&is consists in just moving vertically the potential of the repulsive electronic state as a function of photon frequency. This is particularly simple in the case of the asymmetric coupling matrix since auxiliary quantities such as phases and couplings remain the same throughout the calculation. Tests of the accuracy of this method are presented in section 3 by comparison with exact analytical formulas for the case where levels In) and In’) correspond to harmonic potentials and 1E) to a linear potential.
3. Analytically soluble model The mathematical apparatus necessary to treat resonance Raman scattering with harmonic and linear potentials can be derived from the treatment of level shifts and widths for the same potentials by Sink and Bandrauk [13]_ This formulation proceeds via expressions of the Franck-Condon amplitudes iu momentum space. For the purpose of testing the accuracy of the procedure outlined in section 2, we consider series of the type 0 -+ n, the t‘lrst member corresponding to Rayleigh scattering (this case is formally equivalent to the treatment of a level shift and a width), the second (n = 1) being the fundamental of the Raman spectrum; the others the overtones. Apart from some vertical displacements, as explained above, the potentials are V2,4(R)=+J2K
V1,3,5
(17)
3
lu being the reduced mass, w the angular frequency of the harmonic oscillator, F the negative of the slopes of the linear potentials. For simplicity the artificial charmels are given the same potential as the chaiinel associated with the repulsive excited state. The rotational quantum number is ignored, thus n is now the quantum number for an harmonic potential_ The analytical expression for I,,@) is found to be
I,,(E)
=-
c (2mm9L/2(2n,9t/2
x C2a3/2Ai(k)(~o)
[Rio&-&
(8a)‘kCof2~n_k(8n3)&_~(8a3)
k=O Z=O - ~AP(Z~)]
+
ak%@, zryazka/lr=s=o)
,
(18)
with the following meaning for the parameters
199
Vofume 58. number 2
R,
CHEMICAL
is the turning point at energ
LETTERS
15 September
1978
E; @(z, z’) represents the integral
while r (or s) is the au_xilia~r palmeter
H,(X) =
PHYSICS
appearing in the generating form for Hermit: polynomials
an e-2rx e-"/a? I_
wo
.
The last term in (I 2) can be reduced to infinite series of incompiete gamma functions [ 13 3 _Aitk) and Bi(@ represents the kth derivatives of the reguiar and irregular Airy functions [ 143. Tabie 1 collects the results obtained for the real and imaginary parts of ion for n up to 5 for a case with ,u = 75 aston, a slope of 140000 cm-l/& 2GOO cm-l for the quantum of vibration and at energy equal to 999.95
Table 1 RcaI (4) and imxginary (r) parts of tic trzxsition ampIitude ion acmrd~ to three different procedures: (a) and (b) coupled equations with symmetric (a) or asymmetric (b) potentiaI matrices; (c) a.uIyticaI formuhs. Numbers are in cm _---
n
-ion
r0n
(a) ________
0 1 2 3 4 5 ~~__ Tablc 2 The ~uarcs incm-_----
@) _-----
_-
0.8538-5 O-8637-5 0.2068-3 0.2069-3 0.35394 O-3536-4 -0.1403-3 -o-1403-3 O-1068-3 0.1070-3 050284 0.50354 ___________-___
of the moduli of the trsuition .--_ __
__~__
-__ n ______.. 0 I 2 3 4 5
200
Cc)
(a)
UJ)
Cc)
0.8413-S 0.2C68-3 0.3550-4 -0.1402-3 0.1070-3 0.50224
O-2633-3 -0.3758-S -0.1726-3 -0.74264 0.99694 0.1271-3
0.2635-3 -0.2079-5 -0.1728-3 -0.7422-A O-0989-4 O-1272-3
O-2633-3 -0.2040-5 -0.1727-3 -0.74334 0.9971-4 O-1272-3
--
amplitudes IOn according
_.---
to the three procedures
(see legend to table 1). Numbers are
~-
(a) ___-_____O-6940-7 o-4277-7 o-3105-7 02520-7 0.2135-7 0.1868-7
@)
(c)
0.6949-7 0.4281-7 0_3110-7 02520-7 O-2142-7 O-1871-7
O-6942-7 0.4276 -7 0.3107-7 0.2519-7 0.2139-7 O-1869-7
Volume 58, number 2
CHEMICAL PHYSICS LEl-PERS
15 September 1978
cm-l. The two variants of the numerical method have produced columns (a) and (b) while column (c) gives the analytical results. It can be observed that the agreement is generally very good, with errors for the numerical procedures which do not exceed in most cases a few units in the fourth place. Table 2 gives a similar comparison for iiu, I2 which is the anly quantity which can be attained experimentally_ It appears from table 1 that some inaccuracy may result in the smallest of the two quantities A and r when there is large unbalance between the two (forinstance n = 1 where Ael is one hundred times I’uL). However this is of no effect on 11u, I 2 as may be judged from table 2. One may also conclude from these tables that the use of an asymmetric potential matrix is to be preferred since the procedure provides also excellent results while being somewhat simpler.
4. Conclusion The coupled channel equations are treating all potentials in the same way. Thus one may envisage the use of more complicated analytical potentials, or even of numerical potentials derived from spectroscopic data. Work in this direction is being undertaken. Other extensions can be envisaged, such as the introduction of several excited states in order to account for interference effects [ 151 or intensity transfer between coupled states [ 16]_ Another possible use of coupled channel equations is the consideration of more than one nuclear mode since it has been shown that the vibrational states of small polyatomics may be treated via a coupled channel approach [ 171.
References [l] [2] [3] [4]
15)
[6] [7] [S] [9] [lo] [ 111 [12] [13] [ 141 [is] [16] ] 171
M. Jacon. &I. Bejot and L_ Bernard, Compt. Rend. Acad. Sci. (Paris) 273B (1971) 5%. P-F. \Viims and D-L. Rousseau, Phys. Rev. Letters 30 (1973) 961_ D.L. Rousseau and P-F. Wiiiams, J. Chem. Phys. 64 (1976) 3519P-F. WiIliams, A. Fernandez and D-L. Rousseau, Chem. Phys. Letters 47 (1977) 150. hI. Mingardi, IV. Siebrand, D. van Labeke and M. Jacon. Chem_ Phys. Letters 31 (1975) 208; JP. Laplante and A-D. Bandrauk, J. Chem. Phys. 63 (1976) 2592. J.M. Yuan, J-R. Laii and T-F. George, J. Chem. Phys. 66 (1977) 1107, and references therein. M. Shapiro, J. Chem. Phys. 56 (1972) 25820. Atabek and R. Lefebvre, Chem. Phys. 23 (1977) 51. A.D. Bandrauk, private communication. M.S. ChiJd, MoIecuIar collision theo_ry (Academic Press, New York, 1974). L. Mower, Phys. Rev. 142 (1966) 799. U. Fano, Phys. Rev. 124 (1961) 1866. M.L. Sink and A-D. Bar&auk, Chem. Phys. Letters 49 (1977) 508. N. Abramowitz and LA. Stegun, Handbook of mathematical functions (Dover, New York, 1965). P. Baierl, W. Kiefer, P.F. Wiis and D.L. Rousseau, Chem. Phys. Letters 50 (1977) 57. M-2. Zgierski, J. Raman Spectry. 6 (1977) 53. M. Shapiro, Chem. Phys. Letters 46 (1977) 442; 0. Atabek and R. Lefebvre, J. Chem. Phys. 67 (1977) 4983.
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