- Email: [email protected]
Breakdown of the born—oppenheimer approximation in radiative transitions
Received 3 November 1972
Evidence demojhstsating an explicit nuclear coordinate dependence bf efectr&c matrix e&&s in the tran&ion moment for phosphorescence to the ground state’vibration~ maRifofd has been found for substituted benzenes by an opticaliy detected microwave double resonance esperimen:. Theoretical expressions for .the experimentai obseW* tions developed within the framework of the Coridon qnd He.rzber,O-Teller approsimations f&l to cxpla?n thk results, implying that these approximations are not general enough to give a good description of radiative transitions in aromatic molecules. Additional tem’is in the Herzberg-Teller expansion and the breakdown of the Born-Opperr heimer approximation are briefly discussed as nltemativcs which can be used to explain the ~x~e~ent~ obsqa!ions
_
1. Introduction The detailed nature of rad~a~ve and radiat~onI~ss transitions from excited‘states in aromatic molecules has received a ~ons~d~rabIe amount of attention in the literature. From a theoretical standpoint: knowiedge bf the CXplicitnuclear dependence of electronic matrix elements is essential. to the’utidetstanding of these phenomena. In treating &is problem two limiting cases arise. The first, Herzberg-Teller coupling [If, uses crude Born-Oppenheimer states 121 as a basis set and expands rJre eIectronic’transition moment in a Taj/lor’s series about an equilibrium nuclear ~o~~guration. Providing ,$I-&the ekktronic transition moment is a siow$y varying function ofnuclear coordinates, only linear terms in normd coordinates are retained. If, on the other hand, the energy separatick between ekctronic states is smail, neither crude, nor.adiabatic Bbrn-Oppe,r&eimer states constitut_= a good description of the “real*’ states and mixing of adiabatic states via the nuclear kinetic energy operator becomes signifkant [3,4]. Both the r&ear coordinate dependence of ftre gradient of the elect&tic potential and nuclear kinetic &ergycan;‘to a great& or Iess,er extent, provide intensity into radiationless and radiative decay channels, Unfortunately, empiricai inform&ion about excited states of aromatic m6lecules is not yet complete enough to provide a dear understanding of the relative i~~#~portance of each case. Afthough the Henberg-Tefler treatment has lorig been considered successful in treating vibronic intensity in aIecTronic&y fo;bidden,transitions, questions have recently .been raised concerning its general appli~ab~ty [S, 61. Freed and Geibart [6] have pointed cut that periurbation theory should be avoided if possible since in many caseS hider ,order terms are not ne~~gib~e. They alsO note that the gross features of electronic relaxation processes arh due. to Fran&-Condon type integrals which fact‘or but of electronic coupling schemes. %he quaIitative features of experimenta! observations are therefore relatively ~nd~~~*dent of the coupling “mechanism” ,.and, as a result, experkmentaf verification of these schemes becomes niuch more d~f~cu~t. Orkkdi and Siebrand f7], and others [8j have proposed that the two cases may be d~stin~~~ed by observing isotopic effects’~n.~b~n~~ ~~en~~t~es~ffowqer; the frequency change . . F Alfred ti. Sloan Fellow. I. ‘. ‘. (, . ._ ,:. ‘.
‘_
‘. .. _. :
~. ., .: __ .’ :;. ;.
I _. \
” ,.
: ‘,‘.,,,
‘..:, ‘. : .. . _, .,. ‘. : . . ., “_
,,..
(’ . .. ‘. .’ 309 ‘. _:
‘.
Vofu’me 18, number 3
CHEMXAL;
PHYSICS
iE?TERS
1 Febwary
1973
‘,
I:
Fig. 1. ~hos~h~rescc~ce from the 3@lu state or‘TCi3 to totally sym. me?rjcvibrations in the ground state manifold. The energy spacing is purely schrmatic.
assbciated with isotopic substitution in large molecules is often too small to cause a substantial change in radiative decay chtinnels, and it therefore becomes difficulr to verify which case is most important. Moreover, when radiative transitions acquire intensit)l from both mechanisms, the interpretation of the isotope effect becomes ambiguous. Recent advances in opticoliy detczted magnetic resonance inay help to alleviate these experimental problems since it is now possible to determine the radiative character of-each zerofield triplet spin subIeve1 in phosphorescence to the ground state. We wish to report an experiment in which the troublesome Franck-Condon factors are effectively “cancelled”, aflowing any nuclear dependence of the electronic matrix elements to manifest itself. In this comnnmidation we wish to show, first, that the ratio I.{/$ illustrated in fig. 1 is expected to be the same for phosphorescence to different normal me&s of the totally symmetric vibrations in the ground singlet manifold when the eiectronic matrix elements are slowly varying functions of nuclear coordjnates. This is the case for Condon [9] and/or Herzberg-Teller [If ap?rox~mations. Secondly, 3:‘ewill demonstrate experimentally that ‘_the above ratio. as measurer! in substituted benzena triplet states is not constant for emission to various totally symmetric modes and that the Iarge variatton in this ratio cannot be accounted for by the above approximations. This observation implies that the electronic matrix elements in radiative transitions depend strongly upon specific nuclear coordinates.
Ii2.’Discu&ion ; ..-Aparticular ti,bronic levei may be expressed as a product of electronic and vibrational.w~vefunctions .. @:is’tf;e A&I excited efcctronic state in the Wth vibrational mode. CJzrd Q represent the set of.electronic coordi.’ ,‘nates, and n&ear normal coordinates, resf”ctiveIy. ct labels either 3 singlet or a particular zero-field,tripIet s&n :subl~~e~;:PhospHoi~scence fr0m.d triplet state to the ground singict state is weakly allowed if the product funct tions_a?e.+ed & a ~~i~to~~~~.~~~ch includes spin-orbit coupling fI,OJ. The resulting pert&bed triplet.state is ,: .$mby, :~ .,. ..;-. : . ..,‘. ..,_‘, .._ ‘,.. .‘.‘,‘,‘“,. (’ ‘., : ._,. : : , _‘. ., j i_: .,; :3$,): ,y.,:::‘-..: ,:.f’::; :[I(.::]: ,.:‘,‘, ,‘:. ,,J .‘-: ‘,, :’ ‘,’ . .;: .. ,.. :.: ‘_L. : ,: _ : _:_,..-:. .:. ‘-’ ,, ..,~, _‘,,:.’ ‘:.:. _i. ;.-.: : ,;.:‘ ((‘._...’ j
‘,;
;; .
“:, ..’
: _, .;,: I ‘..
..::;, _,; (‘: :...
\
._ ,,_ .‘ ,’ _’ ,;
..,, , ‘. [ ,‘:,;
Volume 18, number 3
CHEMICAL PHYSICS LETTERS
1 February 1973
We wish to focus our attention on emission from the lowest excited triplet in the zero point vibrational mode to the ground state vibrational manifold. Since it is possible to kxperimentally distinguish the relative intensity of emission from each of the three zero field triplet spin sublevels, we examine ratics of the squares of the transition momentsM~ and MJ for emission from two triplet spin sublevels rx and ~~ to some vibrational state @by (see fig. 1). For reasons given later we restrict our observations to the set of totally symmetric vibrations in the ground singlet manifold. Assuming that spin-orbit coupling in the ground state is small, the transition moment is given by
We proceed by considering the simplest of all approximations, that due to Condon [9), in which the electronic wavefunctions are evaluated at some fLyed equilibrium nucIear configuration, Q-,. For sirnpkity we consider only the first order spin-orbit perturbation term, ‘and restrict the mixing of a triplet spin sublevel with only one excited singlet state. The coefficients are thus
C,;,,,= (G;,XSnr,,, W,l ~~x~oY(E;o-E~,,,)-
(4)
Substituting eq. (4) into eq; (3) aid using the fact that the electronic wavefunctions are independent of the integration over nuclear coordinates, we obtain the ratio of transition moments in the Condon approximation as:
It should be noted that #,, x~,~,~;f #PIt ~m~,V~,since 7-y and 7y will have different spin-orbit symmetries. When the separation between Ex and ES is large reIative to vibrationa quanta, as is tie case fcr benzene and its dzrivatives, we may approximate Ey,o - E:,,,,, x AE and invoke closure.
The ratio may then be reduced to M;fM;
= (‘xo,lx-;,,/(~,lx~~)~X/Y
where X and respectively. functions xx cance1.h the
= XiY,
(7)
Y are the electronic matri.. elements <9SolPI~~~2)(~~lHSOlbi’)l~X and (~~IPIQ~,,}(Q~,I~~I~~‘!ILE’, Since spin interactions are only smalI perturbations on the sublevel energies, the vibrational waveand XY will be spin independent and therefore virtually identical; thus, the Fraack-Cordon factors above ratio. Because X and Y remain constant for different v within the Cc&on approxiination, the squares of the tr&dtion monzznts should yield a constant ratio for emission to diffwent totally symmeh-ic vibra.’ tional levds in the ground state. Next we consider the effect of Herzberg-Teller coupling on t&above ratid: Expanding the electronic transition moment in’normal coordinates about an equilibrium nucIear configuration:QO,
CHEMiCAL PHYSICS LETTERS
:
1 February 1973
~e.tra~si~~o~ moment now consists of 8 term idcitticaf to the Condon approximation plus a ~e~berg-Te~er coupling term. Invokng the same closure argument for the Herzberg-Teiler term, the ratio becomes
using phosphorescerice micrawave double resonance (P~~~~ techniques [ 11, 12f .’ radiative cont~butions from the individual spi.n.subleveIs to various totally symmetric vibrations in the ground sin&t manifold were deterniined for the @west excited trip-let state of 1,2,4, !5 tetra~~orobe~zene (‘KS). Both It,-TCB and +TCB doped 1% M/M in durene were investigated. The orbitd symmetry of the fewest excited triplet state is 3BIU; thus, the spin-orbit symrn~~~,‘~l3, 141 of the individual spin sublevels, I;-, c-Yand rz tr~sfoRn as BXu, Bzu and A, re. spectively in the coordinate system illustrated ti fig. I. In this state bniy the r, Gnd 7u spin sublevels have electric dipole activity to totalfy ~rnrnet~~ ground state vibrations, The j fifetimes of.the three spin sublevels were determined using a combination of adiabatic fast passage [I51 and microwave induced delayed ~hospho~esce~c~ [12JL The s&nples were placed in a slow wave helix and immersed in liquid He which was pumped to a temperature of I .4°K. The sali& data for both h2-TCB and dYTCB are listed in table 1. The zero field splittings of dJ,-TCB are sir&rto. those deported for hyTCB in neat crysia& f13, It;] ,md in a durene host [14]. The lack of an isotope effect on the spin sublevel lifetimes implies that the principal c~n~ribu~jo~ to the fifetimes is radiative decay to So, The two spin sublevels, rX and r,,, used in dete~i~ing ratios corre~ondin~ to, the above theoretical treatment bwtve lifetin& of 790 and 36 msec respe?tiveIy. The ratios can be easily and accurately determined in the foilow‘. ing manner. After exposing the Sarnplc to the 3 100 a region of 6 hig$ pressure Ng exciting source for approximately five $ecbnds, optical pumping was stopped by means ofan $ectiozGc shutter and the decriy was monitored with a Jarre&-Ash 3f4 -meter Czerney-Turner. spectrometer equipped with. a koied (-XI’%) EM1 6256s p~~otomulti$ier @la&d 90” to the exciting source. 550 msec‘after optical pumping was stopped, an accurate zer~i~tens~~ . ’ base line was estab~i~~d by niean: of another shutter placed between the sarriple and spect%meter, This process Was rep&ted and.&e decay ctirves were stored on a PAR waveform ediictor in aider ‘to acquire high signalto ..:-_ :;noi+,Becausk of the large di~feren~e,in the lifetim& of the 7X and,?), spin sublevels, the intensity after 250 r&c &meS ~mos~‘ent~r~~y from &e~for&iv@ level. A~iaba~~~~~ inverting the popu!ation fromr, into the empty rj, ‘spin sublev’e!,‘and.m6aS3ring..~e changes ~,ph,~sph~r~s~~n~~ intensiQ at various times, allods,tfie relative popu;fs ., : ~.,~on’.~~~~n~~~.~~ be me+r&j’~?-a mantier s~~a~‘t~,~~~t d~v~l~p~~,~y~?~~d~r Waals~and coworkers [X2] . The I* ., 1’ ,,: : “ji2: .,:,._.,:~::‘,.,“(‘....,..~,.,1.“‘-I,:. ‘y._’ I’.’ ‘- ” : ..,,,,,.‘.‘:.;: .,._,. : ‘..: .-. .; ‘.-,.’,,?’.,;,,; ‘- : 1.: .‘. ‘.,;, (.” ....,,, ,( ~~.-::-._ .: ;,
...:c,,, _: ) ‘, ., : t ;_- _.,. ‘. ~’
:., .‘l.‘., _,
,l,
__
Volume 18, number 3
CBEMICALPHYSICSLETTERS
1 February 1973
Table 1 hficrowave zero-field transitions (GHz) and triplet spin sublevel lifetimes (msec) for 1% M/M h2-TCBand d2-TCBdoped in durene at 1.4”K __.~~_~___ D+IEi
~-
II,-TCB d2-TCB
D -IEi ---._
5.429 5.463 _______ __.____
3.683 3.641 _.__~_
21EI _______~__
Tr
1.746 1.822
790 f 30 750 f 40
‘i
-- I;
36+ 1 36 c- I
_--. ._ --~.- --..-... ____..______~__
38r 1 382 1
-~
Table 2
Ratios of radiative rates obtained from ratios of intensities illustrated in fig. 1 _________ __ --___-
/I*-TCB (k_;/kf;)x 10-j ?3 X 10-3
Av
-d2-TCB WJkJJ x 10-3 ?3 x 10-3
I
(cm-’ )
---___-._
-I
AV )
km-1
___--_--__._,__----____-
1169
13 40
32 50 13
2203 1538 1162
689
47
22
670
100
18
“S
-
v.I
58
3070 1549
v3
16
V?
29
Vl
24
0
0
15 ‘4 33 45
100
fraction of inversion was determined by the methods described by Harris and Hoover [IS]. Utilising this information and the phosphorescence decay data to the various totally symmetric vibrations, V, in So, the ratio of the radiative rate constants X
oriented
microcrystalline
samples whose phosphorescence
4. Results and conclusions The steady state rate constant ratio, ?$:/!I$, to each of the observed a, fundamentals is Listed in table 2. Notice first that the ratio is far from constant, the largest difference relative to The (O,c3)band occurs for the 1549 and 1538 cm-l vibrations in h?-TCB and cf7-TCB respectively. Secondly, there are no large isotope effects refiected in the ratios. Unfortunately, &e totally symmetric C-H stretch at 30?0 cm-l in Ir2-TCB could not be located. This vibration would have the largest isotope effect when compared to the corresponding vibration at 23-03 cm-l in d2-TCB. In addition, we should like to point out that the observed effects are most probably not a property of the particular TCB-durene system chosen. Both 1, 2,4, 5 tetrabromobenzene [ 131 and 1,4 dichlorobenzene [ i7 I show anomalous PMDR peaks at 1527 and 1579 cm-l respectively. These bands may be explained in terms of the present data if it is assumed that the relative radiative rates from the two contributing subIeveIs are considerably different from their values to other vibronic bands of the same symmetry. Furthermore, when the vibronic transitions in dichlorobenzene and TCB were examined by quadrature PMDR [IS], the data demonstrated that there was enough variation in the radiative rate constant ratios in phosphorescence to different vibronic levels to cause in some cases large deviations from a constant phase angle for any given set of vibrations of the same symmetry. From.the results of this experiment we may conclude the following. The data demonstrate an explicit dependence of the total transition moment on nuclear motion; The observed ratios differ by as much 3s a factor of 2.4 for Iz,TCB and 2.8 for d,-TCB. Since the Condon and Herzberg-Teller approximations can only account for a change of a few percent, they fail to explain the results, and further refinements in the theory of radiative transi‘,
313 ‘.,..
..’
Volumc~l8,
number 3
1 February 1973
CHEMICAL PHYSICS LETTERS
tions must be considered. At least two extensinns of the above development could be made to provide a theoreticd basis for the experimental observations. Firstly, corrections can be made by explicitly considering the quadratic .terms in the Herzberg-Teller expansion ,&ich have been neglected in the previous discussion. Secondly, and less
trivially, an explicit nuclear coordinate dependence of the electronic transition moment can be brought about by a breakdown of the adiabatic Born-Oppenheimer states via the nuclear kinetic energy operator. This is done by evaluating the coefficients in eq. (3) in second order so that contributions from the nuclear kinetic energy can be made. Further insight into the specific dependence of the electronic wavefunction on nuclear motion may be gained by analyzing the normal coordinate of the vibration in relationship to the positions of the nodal planes in the excited electronic state. The raiio might be relatively unaffected, for example, when the nuclear motion is associated with a direction that keeps the nuclei in a nodal plane of the electronic wavefunction. Examination of these relationships is presentIy,under investigation.
Acknowledgement The authors woulh like to express their appreciation to Professor W.IM.Gelbart for many interesting, helpful, and fruitful discussions. This work was supported in part by a grant from the National Science Foundation and in part by the Inorganic Materials Research Division of the Lawrence Berkeley Laboratory under the auspices of the U.S. Atomic Energy Commission.
References [I] G. Herzberg and E. Teller, Z. Physik. Chcm. B21 (1933) 410; H. Sponer and E. Teller, Rev. ?&d. Phys. 13 (1941) 75. [Zj D.M. Burland and G.W. Robinson, Proc. Nat]. Acad. Sci. 66 (19701257; J. Chem. Phys. 51 (1969) 4548. [3] S.H. Lin, J. Chem, Phys. 44 (1966) 3759;48 (1968) 2732. [S j $1.Bison and J. Jortner, J. Chen?. Fhys 48 (1968) 715; 50 (1969) 4061; 4. Nitzan and J. Jortner. J. Chsm. Phys. 55 (1971 j 1355; 56 (1972) 2079,520O. [SJ B. Sharf, 1. Chcm. Phys. 55 (1971) 1379. 16 J K.F. Freed and W.M. Gelbart, Chem. Phys. Letters 10 (1971) i87; D.F. Heller, K.F. Freed and W.M. Gelbxt, 3. Chem. Phys 56 (1972) 2309. [ 71 G. Orkmdi and W. Siebrand, Cbem. Phys. Letters 15 (1972) 465. {8] S.F. Fischer and E.C. Lim, Chem. Pbys. Letters 14 (1972) 40. 191 E.U. Condon, Phys Rev. 32 (1928) 858. i 10) D.S. McClure, J. Chem. Phys 20 (I 952) 682; A.L. Albrecht, J. Chem. Phys. 38 (1963) 354; W. Siebrand, Chem. Phys. Letters 6 (1970) 192. 11 l] D.S. Tinti, M.A. El-Sayed, A.H. Xl&i and C.B. Harris, Chem. Phps. Letters 3 (1969) 343. [12] J. Schmidt, W.S. Veeman and H.J. van der Wards, Chem. Fhys. Letters4 (1969) 341. [13] A.H.FrancisnndCB.Harris,J.Chcm.Phys. 57(1972) 1050. [ 141 CR. Chen ai@ MA. El-Sayed, Chem. Phys Letters 10 (1971) 307. [IS) C.B. Harris, J. Chem. Yhys. 54 !I9711 972; CR. Harris and R.J. Hoover, J. Chem. Phys. 56 (1972) 2199. [16] A.H. Francis and C.B..Harris, Chem. Phys. Letters 9 (1971) 188. [17) M.J. Buckley, C.B. Harris +nd ,R.hi. Panos, J. i\m. Chem. Sot. 94 (1972) 3692. 1181 C.B. Harrfs,and R.J. Hoover, G-km. Phys. Letters 12 (1971) 75. ‘.
..,“.,,’ . ., . . ‘. “,-, ., ;_.,. ..,. ‘.’ . . . ..I. :.:;.:,. ‘. “~3g,~;‘.--_ y:-‘:: ,;: I.,::.._ ,,._’ .: .. ,’ .. : rT.‘::;.~.: \ “..1 __ ,, ,,._-. : ;.. ,- ;‘,.: .,-__.:_ ;,: ,.., ‘.’,.., ._ ,:,, ‘. ‘,.., ,_ _;: :.::. ,. . ‘.’ _y:.,’‘_:,,._\ .c: .’ :
:
‘, .” -,,
c
,,, .: ‘.
‘. ‘. ,.. ,..,_ _: ” ‘, ;,,,: .,- __, ,;. ,I I...’ _.,‘. .‘. ‘, ‘,’ / ,,.: .. .,: ._,,..,::.
.,,
.:_::,:
_-,,-_ :
.,: .-,.
I,. ,.-
Evidence demojhstsating an explicit nuclear coordinate dependence bf efectr&c matrix e&&s in the tran&ion moment for phosphorescence to the ground state’vibration~ maRifofd has been found for substituted benzenes by an opticaliy detected microwave double resonance esperimen:. Theoretical expressions for .the experimentai obseW* tions developed within the framework of the Coridon qnd He.rzber,O-Teller approsimations f&l to cxpla?n thk results, implying that these approximations are not general enough to give a good description of radiative transitions in aromatic molecules. Additional tem’is in the Herzberg-Teller expansion and the breakdown of the Born-Opperr heimer approximation are briefly discussed as nltemativcs which can be used to explain the ~x~e~ent~ obsqa!ions
_
1. Introduction The detailed nature of rad~a~ve and radiat~onI~ss transitions from excited‘states in aromatic molecules has received a ~ons~d~rabIe amount of attention in the literature. From a theoretical standpoint: knowiedge bf the CXplicitnuclear dependence of electronic matrix elements is essential. to the’utidetstanding of these phenomena. In treating &is problem two limiting cases arise. The first, Herzberg-Teller coupling [If, uses crude Born-Oppenheimer states 121 as a basis set and expands rJre eIectronic’transition moment in a Taj/lor’s series about an equilibrium nuclear ~o~~guration. Providing ,$I-&the ekktronic transition moment is a siow$y varying function ofnuclear coordinates, only linear terms in normd coordinates are retained. If, on the other hand, the energy separatick between ekctronic states is smail, neither crude, nor.adiabatic Bbrn-Oppe,r&eimer states constitut_= a good description of the “real*’ states and mixing of adiabatic states via the nuclear kinetic energy operator becomes signifkant [3,4]. Both the r&ear coordinate dependence of ftre gradient of the elect&tic potential and nuclear kinetic &ergycan;‘to a great& or Iess,er extent, provide intensity into radiationless and radiative decay channels, Unfortunately, empiricai inform&ion about excited states of aromatic m6lecules is not yet complete enough to provide a dear understanding of the relative i~~#~portance of each case. Afthough the Henberg-Tefler treatment has lorig been considered successful in treating vibronic intensity in aIecTronic&y fo;bidden,transitions, questions have recently .been raised concerning its general appli~ab~ty [S, 61. Freed and Geibart [6] have pointed cut that periurbation theory should be avoided if possible since in many caseS hider ,order terms are not ne~~gib~e. They alsO note that the gross features of electronic relaxation processes arh due. to Fran&-Condon type integrals which fact‘or but of electronic coupling schemes. %he quaIitative features of experimenta! observations are therefore relatively ~nd~~~*dent of the coupling “mechanism” ,.and, as a result, experkmentaf verification of these schemes becomes niuch more d~f~cu~t. Orkkdi and Siebrand f7], and others [8j have proposed that the two cases may be d~stin~~~ed by observing isotopic effects’~n.~b~n~~ ~~en~~t~es~ffowqer; the frequency change . . F Alfred ti. Sloan Fellow. I. ‘. ‘. (, . ._ ,:. ‘.
‘_
‘. .. _. :
~. ., .: __ .’ :;. ;.
I _. \
” ,.
: ‘,‘.,,,
‘..:, ‘. : .. . _, .,. ‘. : . . ., “_
,,..
(’ . .. ‘. .’ 309 ‘. _:
‘.
Vofu’me 18, number 3
CHEMXAL;
PHYSICS
iE?TERS
1 Febwary
1973
‘,
I:
Fig. 1. ~hos~h~rescc~ce from the 3@lu state or‘TCi3 to totally sym. me?rjcvibrations in the ground state manifold. The energy spacing is purely schrmatic.
assbciated with isotopic substitution in large molecules is often too small to cause a substantial change in radiative decay chtinnels, and it therefore becomes difficulr to verify which case is most important. Moreover, when radiative transitions acquire intensit)l from both mechanisms, the interpretation of the isotope effect becomes ambiguous. Recent advances in opticoliy detczted magnetic resonance inay help to alleviate these experimental problems since it is now possible to determine the radiative character of-each zerofield triplet spin subIeve1 in phosphorescence to the ground state. We wish to report an experiment in which the troublesome Franck-Condon factors are effectively “cancelled”, aflowing any nuclear dependence of the electronic matrix elements to manifest itself. In this comnnmidation we wish to show, first, that the ratio I.{/$ illustrated in fig. 1 is expected to be the same for phosphorescence to different normal me&s of the totally symmetric vibrations in the ground singlet manifold when the eiectronic matrix elements are slowly varying functions of nuclear coordjnates. This is the case for Condon [9] and/or Herzberg-Teller [If ap?rox~mations. Secondly, 3:‘ewill demonstrate experimentally that ‘_the above ratio. as measurer! in substituted benzena triplet states is not constant for emission to various totally symmetric modes and that the Iarge variatton in this ratio cannot be accounted for by the above approximations. This observation implies that the electronic matrix elements in radiative transitions depend strongly upon specific nuclear coordinates.
Ii2.’Discu&ion ; ..-Aparticular ti,bronic levei may be expressed as a product of electronic and vibrational.w~vefunctions .. @:is’tf;e A&I excited efcctronic state in the Wth vibrational mode. CJzrd Q represent the set of.electronic coordi.’ ,‘nates, and n&ear normal coordinates, resf”ctiveIy. ct labels either 3 singlet or a particular zero-field,tripIet s&n :subl~~e~;:PhospHoi~scence fr0m.d triplet state to the ground singict state is weakly allowed if the product funct tions_a?e.+ed & a ~~i~to~~~~.~~~ch includes spin-orbit coupling fI,OJ. The resulting pert&bed triplet.state is ,: .$mby, :~ .,. ..;-. : . ..,‘. ..,_‘, .._ ‘,.. .‘.‘,‘,‘“,. (’ ‘., : ._,. : : , _‘. ., j i_: .,; :3$,): ,y.,:::‘-..: ,:.f’::; :[I(.::]: ,.:‘,‘, ,‘:. ,,J .‘-: ‘,, :’ ‘,’ . .;: .. ,.. :.: ‘_L. : ,: _ : _:_,..-:. .:. ‘-’ ,, ..,~, _‘,,:.’ ‘:.:. _i. ;.-.: : ,;.:‘ ((‘._...’ j
‘,;
;; .
“:, ..’
: _, .;,: I ‘..
..::;, _,; (‘: :...
\
._ ,,_ .‘ ,’ _’ ,;
..,, , ‘. [ ,‘:,;
Volume 18, number 3
CHEMICAL PHYSICS LETTERS
1 February 1973
We wish to focus our attention on emission from the lowest excited triplet in the zero point vibrational mode to the ground state vibrational manifold. Since it is possible to kxperimentally distinguish the relative intensity of emission from each of the three zero field triplet spin sublevels, we examine ratics of the squares of the transition momentsM~ and MJ for emission from two triplet spin sublevels rx and ~~ to some vibrational state @by (see fig. 1). For reasons given later we restrict our observations to the set of totally symmetric vibrations in the ground singlet manifold. Assuming that spin-orbit coupling in the ground state is small, the transition moment is given by
We proceed by considering the simplest of all approximations, that due to Condon [9), in which the electronic wavefunctions are evaluated at some fLyed equilibrium nucIear configuration, Q-,. For sirnpkity we consider only the first order spin-orbit perturbation term, ‘and restrict the mixing of a triplet spin sublevel with only one excited singlet state. The coefficients are thus
C,;,,,= (G;,XSnr,,, W,l ~~x~oY(E;o-E~,,,)-
(4)
Substituting eq. (4) into eq; (3) aid using the fact that the electronic wavefunctions are independent of the integration over nuclear coordinates, we obtain the ratio of transition moments in the Condon approximation as:
It should be noted that #,, x~,~,~;f #PIt ~m~,V~,since 7-y and 7y will have different spin-orbit symmetries. When the separation between Ex and ES is large reIative to vibrationa quanta, as is tie case fcr benzene and its dzrivatives, we may approximate Ey,o - E:,,,,, x AE and invoke closure.
The ratio may then be reduced to M;fM;
= (‘xo,lx-;,,/(~,lx~~)~X/Y
where X and respectively. functions xx cance1.h the
= XiY,
(7)
Y are the electronic matri.. elements <9SolPI~~~2)(~~lHSOlbi’)l~X and (~~IPIQ~,,}(Q~,I~~I~~‘!ILE’, Since spin interactions are only smalI perturbations on the sublevel energies, the vibrational waveand XY will be spin independent and therefore virtually identical; thus, the Fraack-Cordon factors above ratio. Because X and Y remain constant for different v within the Cc&on approxiination, the squares of the tr&dtion monzznts should yield a constant ratio for emission to diffwent totally symmeh-ic vibra.’ tional levds in the ground state. Next we consider the effect of Herzberg-Teller coupling on t&above ratid: Expanding the electronic transition moment in’normal coordinates about an equilibrium nucIear configuration:QO,
CHEMiCAL PHYSICS LETTERS
:
1 February 1973
~e.tra~si~~o~ moment now consists of 8 term idcitticaf to the Condon approximation plus a ~e~berg-Te~er coupling term. Invokng the same closure argument for the Herzberg-Teiler term, the ratio becomes
using phosphorescerice micrawave double resonance (P~~~~ techniques [ 11, 12f .’ radiative cont~butions from the individual spi.n.subleveIs to various totally symmetric vibrations in the ground sin&t manifold were deterniined for the @west excited trip-let state of 1,2,4, !5 tetra~~orobe~zene (‘KS). Both It,-TCB and +TCB doped 1% M/M in durene were investigated. The orbitd symmetry of the fewest excited triplet state is 3BIU; thus, the spin-orbit symrn~~~,‘~l3, 141 of the individual spin sublevels, I;-, c-Yand rz tr~sfoRn as BXu, Bzu and A, re. spectively in the coordinate system illustrated ti fig. I. In this state bniy the r, Gnd 7u spin sublevels have electric dipole activity to totalfy ~rnrnet~~ ground state vibrations, The j fifetimes of.the three spin sublevels were determined using a combination of adiabatic fast passage [I51 and microwave induced delayed ~hospho~esce~c~ [12JL The s&nples were placed in a slow wave helix and immersed in liquid He which was pumped to a temperature of I .4°K. The sali& data for both h2-TCB and dYTCB are listed in table 1. The zero field splittings of dJ,-TCB are sir&rto. those deported for hyTCB in neat crysia& f13, It;] ,md in a durene host [14]. The lack of an isotope effect on the spin sublevel lifetimes implies that the principal c~n~ribu~jo~ to the fifetimes is radiative decay to So, The two spin sublevels, rX and r,,, used in dete~i~ing ratios corre~ondin~ to, the above theoretical treatment bwtve lifetin& of 790 and 36 msec respe?tiveIy. The ratios can be easily and accurately determined in the foilow‘. ing manner. After exposing the Sarnplc to the 3 100 a region of 6 hig$ pressure Ng exciting source for approximately five $ecbnds, optical pumping was stopped by means ofan $ectiozGc shutter and the decriy was monitored with a Jarre&-Ash 3f4 -meter Czerney-Turner. spectrometer equipped with. a koied (-XI’%) EM1 6256s p~~otomulti$ier @la&d 90” to the exciting source. 550 msec‘after optical pumping was stopped, an accurate zer~i~tens~~ . ’ base line was estab~i~~d by niean: of another shutter placed between the sarriple and spect%meter, This process Was rep&ted and.&e decay ctirves were stored on a PAR waveform ediictor in aider ‘to acquire high signalto ..:-_ :;noi+,Becausk of the large di~feren~e,in the lifetim& of the 7X and,?), spin sublevels, the intensity after 250 r&c &meS ~mos~‘ent~r~~y from &e~for&iv@ level. A~iaba~~~~~ inverting the popu!ation fromr, into the empty rj, ‘spin sublev’e!,‘and.m6aS3ring..~e changes ~,ph,~sph~r~s~~n~~ intensiQ at various times, allods,tfie relative popu;fs ., : ~.,~on’.~~~~n~~~.~~ be me+r&j’~?-a mantier s~~a~‘t~,~~~t d~v~l~p~~,~y~?~~d~r Waals~and coworkers [X2] . The I* ., 1’ ,,: : “ji2: .,:,._.,:~::‘,.,“(‘....,..~,.,1.“‘-I,:. ‘y._’ I’.’ ‘- ” : ..,,,,,.‘.‘:.;: .,._,. : ‘..: .-. .; ‘.-,.’,,?’.,;,,; ‘- : 1.: .‘. ‘.,;, (.” ....,,, ,( ~~.-::-._ .: ;,
...:c,,, _: ) ‘, ., : t ;_- _.,. ‘. ~’
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Volume 18, number 3
CBEMICALPHYSICSLETTERS
1 February 1973
Table 1 hficrowave zero-field transitions (GHz) and triplet spin sublevel lifetimes (msec) for 1% M/M h2-TCBand d2-TCBdoped in durene at 1.4”K __.~~_~___ D+IEi
~-
II,-TCB d2-TCB
D -IEi ---._
5.429 5.463 _______ __.____
3.683 3.641 _.__~_
21EI _______~__
Tr
1.746 1.822
790 f 30 750 f 40
‘i
-- I;
36+ 1 36 c- I
_--. ._ --~.- --..-... ____..______~__
38r 1 382 1
-~
Table 2
Ratios of radiative rates obtained from ratios of intensities illustrated in fig. 1 _________ __ --___-
/I*-TCB (k_;/kf;)x 10-j ?3 X 10-3
Av
-d2-TCB WJkJJ x 10-3 ?3 x 10-3
I
(cm-’ )
---___-._
-I
AV )
km-1
___--_--__._,__----____-
1169
13 40
32 50 13
2203 1538 1162
689
47
22
670
100
18
“S
-
v.I
58
3070 1549
v3
16
V?
29
Vl
24
0
0
15 ‘4 33 45
100
fraction of inversion was determined by the methods described by Harris and Hoover [IS]. Utilising this information and the phosphorescence decay data to the various totally symmetric vibrations, V, in So, the ratio of the radiative rate constants X
oriented
microcrystalline
samples whose phosphorescence
4. Results and conclusions The steady state rate constant ratio, ?$:/!I$, to each of the observed a, fundamentals is Listed in table 2. Notice first that the ratio is far from constant, the largest difference relative to The (O,c3)band occurs for the 1549 and 1538 cm-l vibrations in h?-TCB and cf7-TCB respectively. Secondly, there are no large isotope effects refiected in the ratios. Unfortunately, &e totally symmetric C-H stretch at 30?0 cm-l in Ir2-TCB could not be located. This vibration would have the largest isotope effect when compared to the corresponding vibration at 23-03 cm-l in d2-TCB. In addition, we should like to point out that the observed effects are most probably not a property of the particular TCB-durene system chosen. Both 1, 2,4, 5 tetrabromobenzene [ 131 and 1,4 dichlorobenzene [ i7 I show anomalous PMDR peaks at 1527 and 1579 cm-l respectively. These bands may be explained in terms of the present data if it is assumed that the relative radiative rates from the two contributing subIeveIs are considerably different from their values to other vibronic bands of the same symmetry. Furthermore, when the vibronic transitions in dichlorobenzene and TCB were examined by quadrature PMDR [IS], the data demonstrated that there was enough variation in the radiative rate constant ratios in phosphorescence to different vibronic levels to cause in some cases large deviations from a constant phase angle for any given set of vibrations of the same symmetry. From.the results of this experiment we may conclude the following. The data demonstrate an explicit dependence of the total transition moment on nuclear motion; The observed ratios differ by as much 3s a factor of 2.4 for Iz,TCB and 2.8 for d,-TCB. Since the Condon and Herzberg-Teller approximations can only account for a change of a few percent, they fail to explain the results, and further refinements in the theory of radiative transi‘,
313 ‘.,..
..’
Volumc~l8,
number 3
1 February 1973
CHEMICAL PHYSICS LETTERS
tions must be considered. At least two extensinns of the above development could be made to provide a theoreticd basis for the experimental observations. Firstly, corrections can be made by explicitly considering the quadratic .terms in the Herzberg-Teller expansion ,&ich have been neglected in the previous discussion. Secondly, and less
trivially, an explicit nuclear coordinate dependence of the electronic transition moment can be brought about by a breakdown of the adiabatic Born-Oppenheimer states via the nuclear kinetic energy operator. This is done by evaluating the coefficients in eq. (3) in second order so that contributions from the nuclear kinetic energy can be made. Further insight into the specific dependence of the electronic wavefunction on nuclear motion may be gained by analyzing the normal coordinate of the vibration in relationship to the positions of the nodal planes in the excited electronic state. The raiio might be relatively unaffected, for example, when the nuclear motion is associated with a direction that keeps the nuclei in a nodal plane of the electronic wavefunction. Examination of these relationships is presentIy,under investigation.
Acknowledgement The authors woulh like to express their appreciation to Professor W.IM.Gelbart for many interesting, helpful, and fruitful discussions. This work was supported in part by a grant from the National Science Foundation and in part by the Inorganic Materials Research Division of the Lawrence Berkeley Laboratory under the auspices of the U.S. Atomic Energy Commission.
References [I] G. Herzberg and E. Teller, Z. Physik. Chcm. B21 (1933) 410; H. Sponer and E. Teller, Rev. ?&d. Phys. 13 (1941) 75. [Zj D.M. Burland and G.W. Robinson, Proc. Nat]. Acad. Sci. 66 (19701257; J. Chem. Phys. 51 (1969) 4548. [3] S.H. Lin, J. Chem, Phys. 44 (1966) 3759;48 (1968) 2732. [S j $1.Bison and J. Jortner, J. Chen?. Fhys 48 (1968) 715; 50 (1969) 4061; 4. Nitzan and J. Jortner. J. Chsm. Phys. 55 (1971 j 1355; 56 (1972) 2079,520O. [SJ B. Sharf, 1. Chcm. Phys. 55 (1971) 1379. 16 J K.F. Freed and W.M. Gelbart, Chem. Phys. Letters 10 (1971) i87; D.F. Heller, K.F. Freed and W.M. Gelbxt, 3. Chem. Phys 56 (1972) 2309. [ 71 G. Orkmdi and W. Siebrand, Cbem. Phys. Letters 15 (1972) 465. {8] S.F. Fischer and E.C. Lim, Chem. Pbys. Letters 14 (1972) 40. 191 E.U. Condon, Phys Rev. 32 (1928) 858. i 10) D.S. McClure, J. Chem. Phys 20 (I 952) 682; A.L. Albrecht, J. Chem. Phys. 38 (1963) 354; W. Siebrand, Chem. Phys. Letters 6 (1970) 192. 11 l] D.S. Tinti, M.A. El-Sayed, A.H. Xl&i and C.B. Harris, Chem. Phps. Letters 3 (1969) 343. [12] J. Schmidt, W.S. Veeman and H.J. van der Wards, Chem. Fhys. Letters4 (1969) 341. [13] A.H.FrancisnndCB.Harris,J.Chcm.Phys. 57(1972) 1050. [ 141 CR. Chen ai@ MA. El-Sayed, Chem. Phys Letters 10 (1971) 307. [IS) C.B. Harris, J. Chem. Yhys. 54 !I9711 972; CR. Harris and R.J. Hoover, J. Chem. Phys. 56 (1972) 2199. [16] A.H. Francis and C.B..Harris, Chem. Phys. Letters 9 (1971) 188. [17) M.J. Buckley, C.B. Harris +nd ,R.hi. Panos, J. i\m. Chem. Sot. 94 (1972) 3692. 1181 C.B. Harrfs,and R.J. Hoover, G-km. Phys. Letters 12 (1971) 75. ‘.
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