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Interference effects in directional photodissociation
Voh:me 29, number 2
1.5 November 1974
CHEMICAL PHYSICS LETTERS
INTERFERENCE
EFFECTS
IN DIRECTIONAL
Shaul )UJkWEL
.-
PMOTODISSOCIATION
and Joshua JORTNER
Departmenr of Chernisrry, Tel-Aviv UnCversiry, Tel-Aviv, I.& and Depart&t of Applied Optic& Soreq Nuclear Research Ceirfer. Yame,
Israel
Rcccived 3 June 1974
We present a sca!tering formnlism for the problem of the angular distribution of the prod-+ in the photofrs& mcntation of dktomics, in s process where both the (zero order) bound excited level and the dissociative continuum cury oscillator strength from the ground state.
Recent ~eoretica1 and experimerrta1 studjes have ehcidated the riature of the angukr distribution of the photodissociation products of diatomics using plane polarized light [l-6]. A semiclassical model for retarded photodissociation was provided by Jonah .[3]; while the present authors have considered the partial erosion of angular anisotropy in a predissociation process [6]. In this note we present a study of the angular distribution in the photofragmentation of diatomics which involve interference between direct photodissociation and predissociation. The level scheme advanced by Fano for the study of optical line shapes in autoionization and pcedissociation [7], was employed. We consider a three electronic level system of a diatomic: a bound ground electronic state Ig,), an excited bound electronic state Is,), and a dissociative state Id,). Interference between potential and resonance scattering originates from radiative coupling of Ig,) both to Is,> and to Id,). The eigenstates of the two bound electronic states are
a-g,s
I~~)=r-‘~~)(q,r)X~(r)[(2Jt1)/47;]”2~~np(~,e,o),
;
-(I)
Aa, u and JM represent the quantum numbers related to the total electronic angular momentum, the vibrational and the rotarional motions of the nuclei, respectively. q are the electronic coordinates while r, 19,g’ correspond to the polar coordinates of the internuclear axis. 9, is the electronic wavefunction, xUJ is the vibratonal wavefunction and l$*,, is the Wigner rotation matrix [S]. The continuous eigenstates of the dissociative state, Id,) are .? y the appropriate.boundary conditions [S] , exhibiting an asymptotic behaviour OF a plane wave in chosen to satis an arbitrary
Kdirection
plus an outgoing
c
XI =(Kr)-i(fp*(q,r)
rnf’
spherical
(2-r + 1) iJ’ exp (-iSJ.)
wave, so that &)iKr)L$.&(p,
19,0)D$.~(@,
0,O)
-
6)
&)(&) is the solution of,the radial equation for the Id,) state, characterized by kinetic energy, E, momentum iiK and normalized_as sin(Kr-4lrJ+6,) at asymptotically large Kr values. 6, is the phase shift and @, 0 are the polar coordinates of K. The states (1) and (2) correspond to the eigenstates of the Born-Oppenheimer hamiltonian Hnb. The free radiation field Hmd is characterized by the zero photon state lvac), and one photon states Ike) where k and e stand for the photon wavevector and polarization, respectively. Since we are inte’rested in photodissociation using plane polarized light we can choose the z direction along e and omit e from the notation for the photon :_ state. .The’total
I
haniiltonian
,.
:.,
for the system is a= ., ;
Ho + H, i,Hht where the zero order hamiltonian ‘. ^
is Ho = HBo f Hd,
169
‘..
.:, .’
:
Yolbme29;numbcr2 .I ‘,‘. ‘.<._.” I : heing’charaiterized
.:
‘:
CH,E,h+2ALPHYSICS L&i-I‘ERS :
...
‘...’i
'-I '... .:,
..-,
15 iidvemtier 1974
by the relevant eigensta’tes Igtlr~;k>,,lsu’~‘M’,vid) and.l&,.viq), k, is-th& intratioleculti coupling and .Hit is the radiation-matter interaction term: The matrix elements of-the electromagnetic interaction between stqtes’are ‘, :- ,. ” .. .., : ,, .,. the bound .. :.
,(sv’J’M’,vaclH,,I~~,k)=R~~)(J:M’n;,~~~~),
“.’
;
..-
,;.
‘;
‘(3).
..,.
_’
represent the,conventional Franck-Condor-t radialintegral, while (3~) gives the angular iniegral which implies the well known’seleetion rules ;i.’ = J, J.-t 1 -andM.’ =M. A,‘= [A,~A,[ is zero for parallel transitions and unity for per: .pendicular transitions. The radiative coupling between the ground state and the dissociative state is
._ ”
..
(4) .,
;
”
.Wheie
.’
:
:
,. .. : 1’ ; .I .,-.
R$fj) =K_‘i’
‘.,and_.
.. .
expf-.i6~~)S~~~?(Kr)~~~(~))~(r)dr
1’
._ ..
.. ,,
&i;tl’A&
Jti/$>
,. ‘. :
1)47+
= r;cJw~~;iwA&,)[(7J’+
. ”
.
.
(4a)
‘,
‘.
_,
:
(4b) ‘_
J.Q&~) is given by &. (3b) by replacing the index s by d. The intramolecular e!oupling H, (r&ponsibIe for th,e preI, -dissociation) Conserves angular moments and the resonance coupling matrix elements are ‘, ,. : c-9: : h hi’ =K’-li/exp(_i6J)SX~~-(iCr:vds(~)x~~l(r)dr, RJ v&
=fIL ;
,..
” .,
“,
‘_ :‘_,
‘.
.I, 1
@j
,:,
,, ‘,
(Sil)
‘,
,,. To e~alu&the~ angular distribution in a “long tinie” experiment rre cbnsider’first the scat&$ cross se&ions :..from &initial ohe photon, contrn~ur$‘st&e Ii? z IgOJM,fi) to the final dissociative continuum channel If> E ._ ]K,vac>which.is given in terms,of thF.reaction rnatrix,qE) [9] ,’ ,.
.‘: ., ” : -1 ‘, (6) ” .:. ., ‘.’ ;. ‘, ‘, -,. . .. ,~.f : .: ,‘, .,. : ’. ., _ ‘. ., ; ,. :where pd is the density of &tes,in.rhe final channel, c isthe
‘:
.; ..
,;+ ”
“. ..‘ :’ , :_ :..,:: ..,; ., ,‘_.:.: : :. :
‘. :
.-,;.
‘, .:.
,.
;.
~
,..
:,_..
,,._ .‘.:I
.’ ,;
“’ ‘, ..::
_:
.: .‘.:” I .‘(7) ..A
. .: ” 1 ,T ;‘-.. .,, 1 ,-‘.,, ,., : 1;: : ::.,_,,. I,,-. .:,‘. .I .:, ,:
.. . .;
CIIEMICAL PHYSICS LETTERS
Voluk~e 29. ilumber 2
L5 Kovcmher 1974
: The reaction
matrix
may. be written
3s:
T(Ej,=~(E).+~(E)~G(E)~~(E)
‘)
where a is the level shift operator takes the form
[ 10, 11 1, G(E) is the Green function
(8) of the system which in t.he~ subspace
(9) The first term in eq. (8) may be regarded as: npn-resonant part whereas the second is the ‘resonant part of T. To evaluate (fl T(E) Ii> one needs the opkratqr QfT& _. On the basis of our previous study of radiative interactions in the Fano problem [ 121, we can safely treat the problem to first order in Hint, whereupon eq. (8) results in-
Let us now ccnsider a single isolated each characterized by an energy E/s operator is diagonal in this basis set Utilizing eqs. (3)-(S) and (8)-( 10)
vibronic level IsLAJ’M’).This levc!k!s a manifold of rotational states IJ’in’, and a width l?Jl -1-2 Im Ls$MlPRplsu”M). We assume that the Ievel shift and that the energies EJp contain the real part of the level shift operator [6]. we get
where ,491 = C(Jj J’lMO) C(JI J’lOh,)
([ia>
..
and ,RJ,, = (4~)“~ pR$+‘R@
.
(1 Lb)
The cross section uJJ,(O, a’, E) for Ii> -+ If) scattering of photons with energy Ei = hkc from a molecule in the Ig0J.M) state.can now be obtained from eqs. (6) and (11). In the common case where &e initial angular distribu-tion of the ground state molecules is isotropic, we have to average the cross section qver the available Ai va!ues,, resulting in the cross section for a single J level (I’)
c~, uAd 0, a, E) .
~~(0, @, E) t (2Jtl)”
This averaging may be performed using the properties After some algebraic manipulations we get? U/(0, E) = (47+)
pd
of the Wigner rotation
R$‘!g)R$
RJ'JRJ,*J +-
T$p
c
J’,J”=J,Jlt
1
matrices
(E-EJ,+firJl)(E-E,,~-:irJl’)
+
c J,;,“=,,J+,
E+l+;irJ~
$j9
and Racah algebra [8].
1, :
where
(13) ..
f Since the angular distrl%ution is independent
on @ wc’cz~n integratevJ(O,
0, E) over 0 resulting in ufi@,E)
=Id4uJ(@,
0, E)
= 2na(~, 4, q.
,.
._
...
171
Vcifutie 29, +-km 2 -,'- ,."
"
,: : .. .CliEnIcAiPiiiYsrCsL~:ERS '." .- " .'.; 1'5N&q&; 1974 .: .' :;:,,.:. ;.::: ~‘:~;':.y,.;;', ., '_ :',., : .._:' ~~~~}~~~~~,~,~); ./_:; -.._:;:.y '-' ,:_, '_ 1 (;, _ .' :,. ,, .,- . '~ ..1, .,
;
; &e;k ~~~‘~r~l~~~,~ is the Clebsch-Gordan ccieffeierrt Ad ~~(~~,~~[~~isthe’f7.nca.h ~~~~~cle~~ [Sj: .(... ‘. : This. result may be furthei ~~~~i~~d by ~~o~~i~~ the-ix&I recoil a~~r~~~rnat~on’[i,, 6f.which rests dr! the Fact
that &hevibratiorial iuavefunctian~deperid’onIy,sfciwIy.dn jr for a rmalf J’ range 3’ “J,J @fjl by RJ;i ‘and &j;“, respectively. Eq.,(13) thus ;tssumes the f&m
* 1. Thus we niay replace,
/ _l&,~nd
,' jRJ 'dg"i2i&gvww2
‘, uJfO;f?)
,,,
~_
,,
" J
I. ,C13-EJ,j~~-EJ~)+f~~‘~~;.,, E-Ej.2*j$:? :‘, ,, tqjysll -IR;iRp y.J _, +1&J12. .‘, 2. I. : 1 f@-&#+@r] ~(&E$~)~+_gr~~J] : : q.Y=J& 1(E-Ej~)‘+@ ,/~,’ .. .’J’,.r’=.r,J*I ‘:t Eip.
..:" ,.'
:
._
-.
.,
'..
,.
.-
&j
(1,3) and (16)’ provide us with tie general result far the directipnal anisatropid ~hot~~ra~~~tati~n under the conditions of m~noc~omat~c excitatipn of a sir&e ground stat.3 IguJ? levef. The angular distribution under .. . . these circumstances is determined by three c~ntr~~tjons~ the firSt term on the r.h.s. of these.equations involves .: the contribution of d&i: Fhotodi~ociation, the second term corresponds to the resonant pkedissociation process wvhiketfle.third tern1 is an interference co~~~,bu~~o~between direct phot~dissoc~ation and resonant predissociat~on. ’ ‘. .To compare these results with ti real life Situation.two further steps should be taken: CA) To account for‘the ener&spread af,the.exciting pulse the cross sections should be averaged over the power spectrum IA(E)i2, where $(E) represents ttie ~rnp~t~d~s of the ‘photon wavepacket [ 133: Thedireclionat, photofragmentatioit ~robabilj~y is W’(O) S+( dEIB(E)j2uJ(0,E)..CB) WJ@) shbuld be thermally averiiged over the ~ound,state.in~tial$ distributicm. .Und& the extceme.conditi~n~ of broad band (or “short time”) exoitatiorr lA(E)12 5 I for all E so that ivy .? J ~EuJ@,,E) and We obtain from eqs., f 13) and (14)
Volume 29. number 2
CHEMICAL PHYSICS LETTERS
LS N&ember
1974
Two concluding remarks should be made. First, our general results eqs. (13) and (16), together with Lhe atisotropic distribution for broad band excitation (eq. (17)] and its special version (201, reduce under proper Limiting conditions to the special results previously derived in this field [3,5,6]. When the resonance intramolecular coupling is switched off I?icS) = 0 [See eq. (5)] only the first term survives in eqs. (I 3) and (16) and our results reduce to the case of direct photodissociation studied by Zare [S]. When the direct radiative coupling to the continuum states vanishes RJrJ (dg) =: 0 [see eq. (4)] only the second term in eqs. ( 13) and (16) is nonzero and the formalism reduces to the case of predissociation of a single resonance originally considered by Jonah within the framework of a semiclassical model [3] and recentIy studied by us [6]. Second, we would Iike to emphasize that the general foml of the angular distribution in the photofra@nentation process, i.e., a(E) f b(E)P2(cosO), for which eq. (19) provides a special example, is independent of the details (i.e., level structure and couplings) of the proHem. TIis general form of the anisotropy is a direct consequence of the dipolar radiative interacticn inherent in our problem. Such general relations, just depending on the order of the multipole of the electromagnetic coupling, are weIl known in nuclear physics [ 141 and in the field of atomic angular correlation spectroscopy [IS].
References [I] R. Bersahn and S.Ii. Lin, Advan. Chem. Phys. 16 (1969) 80. [3_] C. Jon&P. Chandra and R. Bersohn, J.Chem. Phys. 55 (1971) 1903. [3] [4] [S] [6] [7] [8] (91 [IO] [ 111 [12] [ 131
[ 141 [lj]
C. Jonah, J. Chem. Phys. 55 (1971) 1915. G.E. Busch and K.R. Wilson, J. Chem. Phys. 56 (1972) 3638. R.N. Zare, Mol. Phoiochem. 4 (1972) 1. S. Mukamel and J. Jortner, Some Anisotropy Effects in Molecular Photoejection Spectroscopy, J. Chem. Phys., to be published. U. Fano, thys. Rev. 124 (i9St) 1866. M.E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957). B.W. Shore, Rev. hlod. Phys. 39 (1967) 439. L. Mower, Phys. Rev. 142 (1966) 799. C. Cohen Tannoudji, in: Introduction 1 I’Electronique quantique Cours g I’Association Vandoire de Cherchcs en Physique. Sass Fee (1968). S. Mukamel and J. Jorlner, J. Chem. Phys. 61 (1974) 227. J, Jorlner and S. Mukamel, irl: Proceedings of the first International Congess on Quantum Chemistry. eds. R. Daudel and B. Pullman (Reidel, Dordrecht, 1974). K. Siegbahn, ed., Q, p, and 7 ray spectroscopy, Vol. 2 (North-Holland, Amstexdam, 1966) ch’. 19. IJ. Fano, Rev. Mod. Phys. 45 (1974) 553.
173.
1.5 November 1974
CHEMICAL PHYSICS LETTERS
INTERFERENCE
EFFECTS
IN DIRECTIONAL
Shaul )UJkWEL
.-
PMOTODISSOCIATION
and Joshua JORTNER
Departmenr of Chernisrry, Tel-Aviv UnCversiry, Tel-Aviv, I.& and Depart&t of Applied Optic& Soreq Nuclear Research Ceirfer. Yame,
Israel
Rcccived 3 June 1974
We present a sca!tering formnlism for the problem of the angular distribution of the prod-+ in the photofrs& mcntation of dktomics, in s process where both the (zero order) bound excited level and the dissociative continuum cury oscillator strength from the ground state.
Recent ~eoretica1 and experimerrta1 studjes have ehcidated the riature of the angukr distribution of the photodissociation products of diatomics using plane polarized light [l-6]. A semiclassical model for retarded photodissociation was provided by Jonah .[3]; while the present authors have considered the partial erosion of angular anisotropy in a predissociation process [6]. In this note we present a study of the angular distribution in the photofragmentation of diatomics which involve interference between direct photodissociation and predissociation. The level scheme advanced by Fano for the study of optical line shapes in autoionization and pcedissociation [7], was employed. We consider a three electronic level system of a diatomic: a bound ground electronic state Ig,), an excited bound electronic state Is,), and a dissociative state Id,). Interference between potential and resonance scattering originates from radiative coupling of Ig,) both to Is,> and to Id,). The eigenstates of the two bound electronic states are
a-g,s
I~~)=r-‘~~)(q,r)X~(r)[(2Jt1)/47;]”2~~np(~,e,o),
;
-(I)
Aa, u and JM represent the quantum numbers related to the total electronic angular momentum, the vibrational and the rotarional motions of the nuclei, respectively. q are the electronic coordinates while r, 19,g’ correspond to the polar coordinates of the internuclear axis. 9, is the electronic wavefunction, xUJ is the vibratonal wavefunction and l$*,, is the Wigner rotation matrix [S]. The continuous eigenstates of the dissociative state, Id,) are .? y the appropriate.boundary conditions [S] , exhibiting an asymptotic behaviour OF a plane wave in chosen to satis an arbitrary
Kdirection
plus an outgoing
c
XI =(Kr)-i(fp*(q,r)
rnf’
spherical
(2-r + 1) iJ’ exp (-iSJ.)
wave, so that &)iKr)L$.&(p,
19,0)D$.~(@,
0,O)
-
6)
&)(&) is the solution of,the radial equation for the Id,) state, characterized by kinetic energy, E, momentum iiK and normalized_as sin(Kr-4lrJ+6,) at asymptotically large Kr values. 6, is the phase shift and @, 0 are the polar coordinates of K. The states (1) and (2) correspond to the eigenstates of the Born-Oppenheimer hamiltonian Hnb. The free radiation field Hmd is characterized by the zero photon state lvac), and one photon states Ike) where k and e stand for the photon wavevector and polarization, respectively. Since we are inte’rested in photodissociation using plane polarized light we can choose the z direction along e and omit e from the notation for the photon :_ state. .The’total
I
haniiltonian
,.
:.,
for the system is a= ., ;
Ho + H, i,Hht where the zero order hamiltonian ‘. ^
is Ho = HBo f Hd,
169
‘..
.:, .’
:
Yolbme29;numbcr2 .I ‘,‘. ‘.<._.” I : heing’charaiterized
.:
‘:
CH,E,h+2ALPHYSICS L&i-I‘ERS :
...
‘...’i
'-I '... .:,
..-,
15 iidvemtier 1974
by the relevant eigensta’tes Igtlr~;k>,,lsu’~‘M’,vid) and.l&,.viq), k, is-th& intratioleculti coupling and .Hit is the radiation-matter interaction term: The matrix elements of-the electromagnetic interaction between stqtes’are ‘, :- ,. ” .. .., : ,, .,. the bound .. :.
,(sv’J’M’,vaclH,,I~~,k)=R~~)(J:M’n;,~~~~),
“.’
;
..-
,;.
‘;
‘(3).
..,.
_’
represent the,conventional Franck-Condor-t radialintegral, while (3~) gives the angular iniegral which implies the well known’seleetion rules ;i.’ = J, J.-t 1 -andM.’ =M. A,‘= [A,~A,[ is zero for parallel transitions and unity for per: .pendicular transitions. The radiative coupling between the ground state and the dissociative state is
._ ”
..
(4) .,
;
”
.Wheie
.’
:
:
,. .. : 1’ ; .I .,-.
R$fj) =K_‘i’
‘.,and_.
.. .
expf-.i6~~)S~~~?(Kr)~~~(~))~(r)dr
1’
._ ..
.. ,,
&i;tl’A&
Jti/$>
,. ‘. :
1)47+
= r;cJw~~;iwA&,)[(7J’+
. ”
.
.
(4a)
‘,
‘.
_,
:
(4b) ‘_
J.Q&~) is given by &. (3b) by replacing the index s by d. The intramolecular e!oupling H, (r&ponsibIe for th,e preI, -dissociation) Conserves angular moments and the resonance coupling matrix elements are ‘, ,. : c-9: : h hi’ =K’-li/exp(_i6J)SX~~-(iCr:vds(~)x~~l(r)dr, RJ v&
=fIL ;
,..
” .,
“,
‘_ :‘_,
‘.
.I, 1
@j
,:,
,, ‘,
(Sil)
‘,
,,. To e~alu&the~ angular distribution in a “long tinie” experiment rre cbnsider’first the scat&$ cross se&ions :..from &initial ohe photon, contrn~ur$‘st&e Ii? z IgOJM,fi) to the final dissociative continuum channel If> E ._ ]K,vac>which.is given in terms,of thF.reaction rnatrix,qE) [9] ,’ ,.
.‘: ., ” : -1 ‘, (6) ” .:. ., ‘.’ ;. ‘, ‘, -,. . .. ,~.f : .: ,‘, .,. : ’. ., _ ‘. ., ; ,. :where pd is the density of &tes,in.rhe final channel, c isthe
‘:
.; ..
,;+ ”
“. ..‘ :’ , :_ :..,:: ..,; ., ,‘_.:.: : :. :
‘. :
.-,;.
‘, .:.
,.
;.
~
,..
:,_..
,,._ .‘.:I
.’ ,;
“’ ‘, ..::
_:
.: .‘.:” I .‘(7) ..A
. .: ” 1 ,T ;‘-.. .,, 1 ,-‘.,, ,., : 1;: : ::.,_,,. I,,-. .:,‘. .I .:, ,:
.. . .;
CIIEMICAL PHYSICS LETTERS
Voluk~e 29. ilumber 2
L5 Kovcmher 1974
: The reaction
matrix
may. be written
3s:
T(Ej,=~(E).+~(E)~G(E)~~(E)
‘)
where a is the level shift operator takes the form
[ 10, 11 1, G(E) is the Green function
(8) of the system which in t.he~ subspace
(9) The first term in eq. (8) may be regarded as: npn-resonant part whereas the second is the ‘resonant part of T. To evaluate (fl T(E) Ii> one needs the opkratqr QfT& _. On the basis of our previous study of radiative interactions in the Fano problem [ 121, we can safely treat the problem to first order in Hint, whereupon eq. (8) results in-
Let us now ccnsider a single isolated each characterized by an energy E/s operator is diagonal in this basis set Utilizing eqs. (3)-(S) and (8)-( 10)
vibronic level IsLAJ’M’).This levc!k!s a manifold of rotational states IJ’in’, and a width l?Jl -1-2 Im Ls$MlPRplsu”M). We assume that the Ievel shift and that the energies EJp contain the real part of the level shift operator [6]. we get
where ,491 = C(Jj J’lMO) C(JI J’lOh,)
([ia>
..
and ,RJ,, = (4~)“~ pR$+‘R@
.
(1 Lb)
The cross section uJJ,(O, a’, E) for Ii> -+ If) scattering of photons with energy Ei = hkc from a molecule in the Ig0J.M) state.can now be obtained from eqs. (6) and (11). In the common case where &e initial angular distribu-tion of the ground state molecules is isotropic, we have to average the cross section qver the available Ai va!ues,, resulting in the cross section for a single J level (I’)
c~, uAd 0, a, E) .
~~(0, @, E) t (2Jtl)”
This averaging may be performed using the properties After some algebraic manipulations we get? U/(0, E) = (47+)
pd
of the Wigner rotation
R$‘!g)R$
RJ'JRJ,*J +-
T$p
c
J’,J”=J,Jlt
1
matrices
(E-EJ,+firJl)(E-E,,~-:irJl’)
+
c J,;,“=,,J+,
E+l+;irJ~
$j9
and Racah algebra [8].
1, :
where
(13) ..
f Since the angular distrl%ution is independent
on @ wc’cz~n integratevJ(O,
0, E) over 0 resulting in ufi@,E)
=Id4uJ(@,
0, E)
= 2na(~, 4, q.
,.
._
...
171
Vcifutie 29, +-km 2 -,'- ,."
"
,: : .. .CliEnIcAiPiiiYsrCsL~:ERS '." .- " .'.; 1'5N&q&; 1974 .: .' :;:,,.:. ;.::: ~‘:~;':.y,.;;', ., '_ :',., : .._:' ~~~~}~~~~~,~,~); ./_:; -.._:;:.y '-' ,:_, '_ 1 (;, _ .' :,. ,, .,- . '~ ..1, .,
;
; &e;k ~~~‘~r~l~~~,~ is the Clebsch-Gordan ccieffeierrt Ad ~~(~~,~~[~~isthe’f7.nca.h ~~~~~cle~~ [Sj: .(... ‘. : This. result may be furthei ~~~~i~~d by ~~o~~i~~ the-ix&I recoil a~~r~~~rnat~on’[i,, 6f.which rests dr! the Fact
that &hevibratiorial iuavefunctian~deperid’onIy,sfciwIy.dn jr for a rmalf J’ range 3’ “J,J @fjl by RJ;i ‘and &j;“, respectively. Eq.,(13) thus ;tssumes the f&m
* 1. Thus we niay replace,
/ _l&,~nd
,' jRJ 'dg"i2i&gvww2
‘, uJfO;f?)
,,,
~_
,,
" J
I. ,C13-EJ,j~~-EJ~)+f~~‘~~;.,, E-Ej.2*j$:? :‘, ,, tqjysll -IR;iRp y.J _, +1&J12. .‘, 2. I. : 1 f@-&#+@r] ~(&E$~)~+_gr~~J] : : q.Y=J& 1(E-Ej~)‘+@ ,/~,’ .. .’J’,.r’=.r,J*I ‘:t Eip.
..:" ,.'
:
._
-.
.,
'..
,.
.-
&j
(1,3) and (16)’ provide us with tie general result far the directipnal anisatropid ~hot~~ra~~~tati~n under the conditions of m~noc~omat~c excitatipn of a sir&e ground stat.3 IguJ? levef. The angular distribution under .. . . these circumstances is determined by three c~ntr~~tjons~ the firSt term on the r.h.s. of these.equations involves .: the contribution of d&i: Fhotodi~ociation, the second term corresponds to the resonant pkedissociation process wvhiketfle.third tern1 is an interference co~~~,bu~~o~between direct phot~dissoc~ation and resonant predissociat~on. ’ ‘. .To compare these results with ti real life Situation.two further steps should be taken: CA) To account for‘the ener&spread af,the.exciting pulse the cross sections should be averaged over the power spectrum IA(E)i2, where $(E) represents ttie ~rnp~t~d~s of the ‘photon wavepacket [ 133: Thedireclionat, photofragmentatioit ~robabilj~y is W’(O) S+( dEIB(E)j2uJ(0,E)..CB) WJ@) shbuld be thermally averiiged over the ~ound,state.in~tial$ distributicm. .Und& the extceme.conditi~n~ of broad band (or “short time”) exoitatiorr lA(E)12 5 I for all E so that ivy .? J ~EuJ@,,E) and We obtain from eqs., f 13) and (14)
Volume 29. number 2
CHEMICAL PHYSICS LETTERS
LS N&ember
1974
Two concluding remarks should be made. First, our general results eqs. (13) and (16), together with Lhe atisotropic distribution for broad band excitation (eq. (17)] and its special version (201, reduce under proper Limiting conditions to the special results previously derived in this field [3,5,6]. When the resonance intramolecular coupling is switched off I?icS) = 0 [See eq. (5)] only the first term survives in eqs. (I 3) and (16) and our results reduce to the case of direct photodissociation studied by Zare [S]. When the direct radiative coupling to the continuum states vanishes RJrJ (dg) =: 0 [see eq. (4)] only the second term in eqs. ( 13) and (16) is nonzero and the formalism reduces to the case of predissociation of a single resonance originally considered by Jonah within the framework of a semiclassical model [3] and recentIy studied by us [6]. Second, we would Iike to emphasize that the general foml of the angular distribution in the photofra@nentation process, i.e., a(E) f b(E)P2(cosO), for which eq. (19) provides a special example, is independent of the details (i.e., level structure and couplings) of the proHem. TIis general form of the anisotropy is a direct consequence of the dipolar radiative interacticn inherent in our problem. Such general relations, just depending on the order of the multipole of the electromagnetic coupling, are weIl known in nuclear physics [ 141 and in the field of atomic angular correlation spectroscopy [IS].
References [I] R. Bersahn and S.Ii. Lin, Advan. Chem. Phys. 16 (1969) 80. [3_] C. Jon&P. Chandra and R. Bersohn, J.Chem. Phys. 55 (1971) 1903. [3] [4] [S] [6] [7] [8] (91 [IO] [ 111 [12] [ 131
[ 141 [lj]
C. Jonah, J. Chem. Phys. 55 (1971) 1915. G.E. Busch and K.R. Wilson, J. Chem. Phys. 56 (1972) 3638. R.N. Zare, Mol. Phoiochem. 4 (1972) 1. S. Mukamel and J. Jortner, Some Anisotropy Effects in Molecular Photoejection Spectroscopy, J. Chem. Phys., to be published. U. Fano, thys. Rev. 124 (i9St) 1866. M.E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957). B.W. Shore, Rev. hlod. Phys. 39 (1967) 439. L. Mower, Phys. Rev. 142 (1966) 799. C. Cohen Tannoudji, in: Introduction 1 I’Electronique quantique Cours g I’Association Vandoire de Cherchcs en Physique. Sass Fee (1968). S. Mukamel and J. Jorlner, J. Chem. Phys. 61 (1974) 227. J, Jorlner and S. Mukamel, irl: Proceedings of the first International Congess on Quantum Chemistry. eds. R. Daudel and B. Pullman (Reidel, Dordrecht, 1974). K. Siegbahn, ed., Q, p, and 7 ray spectroscopy, Vol. 2 (North-Holland, Amstexdam, 1966) ch’. 19. IJ. Fano, Rev. Mod. Phys. 45 (1974) 553.
173.