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On the relation between radiative and nonradiative transitions in molecules
Volume9, rider
CHEMICAL PHYSICSLE3TERS
2
ON THE RELATION
BETWEEN
RADfATfVE
TRANSITIONS
of Ckemistry,
AND
IN MOLECULES
W.SIEBRAND** I)ivision
NaticmuE Research
Council
15 April 1971
NONRADEATIVE
*
of ~&Ez.&z, Otiau~,Canada
Received3 March 1971 The first-order pfztutiation description appropriateEoa giveu radialionless tiaasitiaa is related to
the structure and decay rate of a spectroscopic
sipnat.
Radiationless transitions are observed in&rectly via optical signals. In the special case where they are associated with optical signals decaying exponentially in time, their rate constant can be expressed by the Golden Rule
wherep,(~>)(lllsrl~lm,l-l) states in In} and%;
is the density of
is the per~~~on
connec-
ting the zero’&-order states iti} and i?%). This raises the question of how to choose the appropriate zeroth-order states and thereby the perturbation. In most of the earlier treatments [l3] they were taken to be adiabatic Born-Oppenheimer (ABO) states. We have recently argued [4j that for the observed intersystem crossings in aromatic hydrocarbons pure-spin ABO states may be the appropriate choice. Burland and Robinson [5] have advocated the use of special “crude” Born-Oppenheimer (CBO) states, namely states in which the electronic wavefunction is assumed (almost) independent of nuclear displacem&s associated with the mode(s) inducing the transition, They as well as Sharf and Silbey [6] have argued that the standard ABO matrix elements are too small to account for the observed rate constan& and linewidths and that CBU matrix elements give satisfactory resuits. However, the standard derivation of ABO matrix elements is of questionable validity [73; as a result a reliable quantita$ivecomparison between.ABO and CBO matrix eiement is not available. Recently Sfiarf [8] has attempted to describe radiationless + lseuedaa NRCCNo. 11897. ** Addressafter 1 July 1971:Departmentof Chemistry, Universftyof Watedoo, Waterloo, Ontarfo,Canada. .
.
transitions as due to electron repulsion acting on a zeroth-order state “h which the electrons are partly or wholly uncorreh&ed. In Iow orders of perturbation theory these different representations yield different Fate constants.. Of course they must ultimately converge to the same result when the complete hamiltonian is used and the perturbation expansions are carried through to infinite order [9]. However, one usually has to settle for less, so that the proper choice of representation is of considerable practica.I importance, as shown by the many recent papers on this subject [4-S]. in this Note we propose to base the perkurbation description of radiationless transitions on the nature of the signal that is actually observed. As a xule &e observation involves measuring the time dependence of an electionic emission spectrum, If and OI& if this sigwl as a idole decays
exponentially, the corres~~~g can be written as
rate constant
(2) k e z= k(n,m) ?Z
= 2nfi -1~~~~{~~~~~P~~}~2}, 78
where %?&I@ = E,j&
a=m,n.
(31
Here %f&f = %& i@$~ is the ~mq&?&? moh?a&kr
hamiltonian, P is the dipole operator, and aIL states and operators am? time-independent. Eq(2) is based on the notion that the total hamiltonian of the systeqx contains not only Sr, but a&o the electromagnetic fieldand its interaction with the molecule, here represented by P. The crux of eq. (2) is tilt one and on& one zerothorder state is used as the kiti& state for both the radiative and the nomxdia~ve fmnsition. One em always’find a reprqentation in which \nr)in I_’
_.
’
..
.
-
157
Volume 9, number 2
CHEMICAL PHYSICSLETTERS
&- @)-is a pure state if the observed signal decays +Ponenti~y, since exponential decay implies the validity of a Golden Rule, which in turn implies Im> to be time-independent in the sense that its character does not change while it is being depopulated via the two open decay channels. The ultimate aim of this paper is to find the representation in which I??F)is pure. Clearly this representation would allow a convenient description of radiationless trasitions and also a simple physical interpretation. Since we know that the radiative and nonradiative decay processes described by eq. (2) involve the same initial state Inz}, which is an eigenstate of the a priori unknown zeroth-order hamiltonian 91’ , we can obtain information about 1772;by % . g the optical spectrum w_hich is governed gmatrix elements {nIP]m), where In) represents a set of different eigenfunctions of the same zero&order hamiltonian. In favorab\e cases this information may suffice to identify WM. This would allow us to calculate the nonradiative rate constant (1) unequivocally, notwithstanding the fact that the final states In) are usually not the same for (n p& 1m) and (n IP Im), the important fact being that them are all eigenkmctions of the same hamiltonian, which is now known. Before applying these ideas to observed transitions, two comments are in order. (i) Although most emission processes appear to follow an exponential decay law, this is in part an experimental artifact. If it were possible to observe the decay immediately after excitation, the effect of fast processes such as vibrational relaxation and (high-temperature) spin-lattice relaxation would become apparent in the spectrum as well as in the decay rate. However, under typical experimenfal conditions, these processes have run their course when the actual observation begins. Similarly, if the equipment is sensitive enough to register a signal after a time very much longer than k-1, the effect of very slow processes. such as spin-lattice relaxation near Ool( would become apparent. Of course, eq. (2) only refers to the exponential part of the decay curve which is the only part observed in st+dard measurements. (ii) The restriction of the time-dependent perturbation treatment in eq. (2) to first-order matrix elements is due to the fact that the emission processes under consideration are first order in P.’ This -does not rule out second-order matrix .. elements ,of .the form (?zIPI i)(i lh~ Ins), where h~.is a te.rm in 91,. Two cases are to be dis_:tinguished: If hM belongs to %&, the second:order thaw elements can be removed by a -158 1 ._.
15 April 1971
simple transformation to a new basis set. If hi belongs to ‘Z&, however, the correspondent matrix elements yield an independent contribution. It appears that such contributions are unimpor-
tant in the examples to be discussed presently. As a first application we consider triplet-togrouod state transitions in aromatic hydrocarbons or comparable mtilecules [lo]. The nonradiative rate constant is derived by monitoring the decay of the phosphorescence, the spectrum of which is governed by(nlPlm). of In> and Im> were pure-spin functions, this matrix element would vanish; hence these states must contain components of different multiplicity, so that spin-orbit coupling should be included in %k rather than in Srh. In other words, the intersystern crossing is caused by vibronic coupling rather tha? spin-orbit coupling. This mechanism is readily shown to be identical with the one labeled;KfA in previous papers [4]. The predominance of this mechanism for triplet-to-ground state transitions in naphtbalene and anthracene has been demonstrated on the basis of isotope effect% and spin alignment measurements [4,11]. The same conclusion applies when triplet decay is measured by monitoring the absorption T - T1 instead of the emission T1 - SO. This fo%ows from the fact that the absorption will be followed by rapid internal conversion T,-Tl. Neither the absorption nor the internal conversion will be measurably affected by the degree of spin conlaminztion of T, and Tl. Hence no new decay channels are opened and the old ones remain unaffected. Consider now intersystem crossing from the lowest excited singlet state to the triplet manifold. The corresponding rate constant is usually obtained by monitoring fluorescence decay. Since fluorescence is multiplicity allowed, there is no reason to assume the states IiZ), jm} to involve triplet components. Hence, singlet-to-triplet intersystem crossing can be caused by spin-orbit coupling as well as vibronic coupling, so that all three mechanisms, previously [4] labeled%&, 9fi2& and 9!$$, should be considered. similar considerations apply to internal conversion. Let us expand inz) and In> in adiabatic Born-Oppenheimer states [12] :
14
= c $k, i
Q) A;(Q) ,
(4)
where q and Q represent the sets of electronic’ and vibrational coordinates, respectively. The resulting expression for the transition dipole momait
Volume 9. nllmhr
2
25 A.priL1971
CHEMICALPHYSICSLETTERS
is usually approximated by a single term which in turn is expanded about the nuclear equilibrium configuration Q = Qo (Herzber&-Teller expansion):
(~~(q,Q)i\n(Q)lPf~~(q,Q)n,fQ))
us to distinguish between two types of vibrationdl structure, namely that governed by &IA,> for transitions of purely electronic origin, and that governed by (AnI Q] AIn) for vibrationally induced electronic transitions. if the spectrum shows only structure of the type (&IA,), then @,(q, Qo) may well be the appropriate zero&order state, so that the radiationless transition may be due to the potential energy term U(Q) - U(Qo) in the molecular h~ilto~~. However, if the spectrum shows struchcre of the type (h,l Q]&J, then the zeroth-order state must be vibronicafly contaminated, so that U(Q) U(Qo) will not be the appropriate coupling term. In spectroscopy one usually assumes the vibronic mixing to be due to the linear term in the expansion (6). This would amount to taking adiabatic Born-Oppenheimer states for 1jr> and 1tn), so that the radiationless transition would be due to the nuclear kinetic-energy operator. However, there is little evidence to justify this aFgroach in a quantitative manner. Unfortunately, the real question here as well as in the case of intersystem crossing is a quantitative one. The question is not ifa given state is spin contaminated or vibronicaily contaminated, but to what ext.& it is contaminated. This requirt?s a quantitative interpretation of spectral features which at present is @ill out of reach. Thus we have to settle for a more qualitative answer, e.g. the one implied in the assumption of either zero or complete spin contamination for intersystem crossings. However, such an unambiguous assumption is not available for internal conversion, since complete uibronic mixing in the zeroth;order states rules out internal conversion as a decay channel. Hence, if there is evidence for internal conversion, e.g. from quar.tum yield measurements, and if at tbe same time the spectrum monitored shows vibronic contamination, e.g., structure of the type
then the vibronic mixing in ffie zeroth-order states must be nonzero but ineompbte . One approach exhibiting this property is the conve&ional assumption that the zeroth-order states are adiabatic ~rn-Opp@~~rn~ states. Buriand and Robinson [5] favor the alternative assumption of zero vibrotic ~on~rn~~~on through the inducing mode(s) and essentially complete vibronic contamination through the accepting modes which in general are more sensitive to the e’iectroniz structure. It is not easy to test these assumptions. In general radiative and nonradiative transitions are induced by tierent normal modes since ST& and P have different symmetry properties. In addition it is very difficult tc; distinguish vibronic coupling caused by deviations from the nuclear equiIibrium coafiguration from that caused by non-zero nuclear momenta [7,13]. It appears therefore that the vibronic mechanism of internal conversion remains a subject of conjecture. The same 2ppIies to the pureIy electronic mechanism suggested by Sharf [8], for which no comparison with spectroscopic observations is available to date. I am grateful Henry,
G. Orlandi
for helpful comments by B. R. and G. W. Robin~txt.
REFERENCES 111B. R. Henry and &f. Kasha, Ann. Rev. P&s. 19 f 1968) 161. and references tberein.
[2] 26Bixon and J. Jortner, J. Chem.Pbys.
Chem.
46 (1368)
[31 W. l&odes, J. Chem. Phys. 50 (I.969)288% [4] W. Siebrand, Chem. Phys. Letters 6 0970) 192; B.R.Henry and W.Siebrand. J.Chem. Phys., to be
published. [S] I). M. Burland and G. W..Robfnson, Proc.Katl. Acad. Sci. 66 (1970) 257. [S) B.Sbarf and R.SiIbey, Chem.Phys. Letters 4 (1969) 423: 5 (1970) 314. [7j G. Orlandi and W. Siebrand, Chem. Phye. Letters 8 (1971) 473. [PI B.Sbarf. Chem. Phys. Letters 6 (1970) 364. 191 K. F. Freed and J. Jortner. 3.Chem. Phys. 50 (1969) 2916. [lo] W.&brand, J. Chem.Phys. 47 (X967) 2411. [1X] B. R. Henry aMf W.Siebrand. Chem.Phys. Letters 7 (1970) 533. [lit] M. Born, Nachr. Akad. Wiss. Gattingen. Math.PhysiJc. Ki. {1961) nr. 6; Tizc Loogust-Higgins. Advan. Spectry. 2 (1961) and C. A. Manzaeco. Molectir Iuminescence, ed, E.C. Lim (Benjamin, New York, 1969) p. 631.
[131 R. k Hocbsttasser
159 -
CHEMICAL PHYSICSLE3TERS
2
ON THE RELATION
BETWEEN
RADfATfVE
TRANSITIONS
of Ckemistry,
AND
IN MOLECULES
W.SIEBRAND** I)ivision
NaticmuE Research
Council
15 April 1971
NONRADEATIVE
*
of ~&Ez.&z, Otiau~,Canada
Received3 March 1971 The first-order pfztutiation description appropriateEoa giveu radialionless tiaasitiaa is related to
the structure and decay rate of a spectroscopic
sipnat.
Radiationless transitions are observed in&rectly via optical signals. In the special case where they are associated with optical signals decaying exponentially in time, their rate constant can be expressed by the Golden Rule
wherep,(~>)(lllsrl~lm,l-l) states in In} and%;
is the density of
is the per~~~on
connec-
ting the zero’&-order states iti} and i?%). This raises the question of how to choose the appropriate zeroth-order states and thereby the perturbation. In most of the earlier treatments [l3] they were taken to be adiabatic Born-Oppenheimer (ABO) states. We have recently argued [4j that for the observed intersystem crossings in aromatic hydrocarbons pure-spin ABO states may be the appropriate choice. Burland and Robinson [5] have advocated the use of special “crude” Born-Oppenheimer (CBO) states, namely states in which the electronic wavefunction is assumed (almost) independent of nuclear displacem&s associated with the mode(s) inducing the transition, They as well as Sharf and Silbey [6] have argued that the standard ABO matrix elements are too small to account for the observed rate constan& and linewidths and that CBU matrix elements give satisfactory resuits. However, the standard derivation of ABO matrix elements is of questionable validity [73; as a result a reliable quantita$ivecomparison between.ABO and CBO matrix eiement is not available. Recently Sfiarf [8] has attempted to describe radiationless + lseuedaa NRCCNo. 11897. ** Addressafter 1 July 1971:Departmentof Chemistry, Universftyof Watedoo, Waterloo, Ontarfo,Canada. .
.
transitions as due to electron repulsion acting on a zeroth-order state “h which the electrons are partly or wholly uncorreh&ed. In Iow orders of perturbation theory these different representations yield different Fate constants.. Of course they must ultimately converge to the same result when the complete hamiltonian is used and the perturbation expansions are carried through to infinite order [9]. However, one usually has to settle for less, so that the proper choice of representation is of considerable practica.I importance, as shown by the many recent papers on this subject [4-S]. in this Note we propose to base the perkurbation description of radiationless transitions on the nature of the signal that is actually observed. As a xule &e observation involves measuring the time dependence of an electionic emission spectrum, If and OI& if this sigwl as a idole decays
exponentially, the corres~~~g can be written as
rate constant
(2) k e z= k(n,m) ?Z
= 2nfi -1~~~~{~~~~~P~~}~2}, 78
where %?&I@ = E,j&
a=m,n.
(31
Here %f&f = %& i@$~ is the ~mq&?&? moh?a&kr
hamiltonian, P is the dipole operator, and aIL states and operators am? time-independent. Eq(2) is based on the notion that the total hamiltonian of the systeqx contains not only Sr, but a&o the electromagnetic fieldand its interaction with the molecule, here represented by P. The crux of eq. (2) is tilt one and on& one zerothorder state is used as the kiti& state for both the radiative and the nomxdia~ve fmnsition. One em always’find a reprqentation in which \nr)in I_’
_.
’
..
.
-
157
Volume 9, number 2
CHEMICAL PHYSICSLETTERS
&- @)-is a pure state if the observed signal decays +Ponenti~y, since exponential decay implies the validity of a Golden Rule, which in turn implies Im> to be time-independent in the sense that its character does not change while it is being depopulated via the two open decay channels. The ultimate aim of this paper is to find the representation in which I??F)is pure. Clearly this representation would allow a convenient description of radiationless trasitions and also a simple physical interpretation. Since we know that the radiative and nonradiative decay processes described by eq. (2) involve the same initial state Inz}, which is an eigenstate of the a priori unknown zeroth-order hamiltonian 91’ , we can obtain information about 1772;by % . g the optical spectrum w_hich is governed gmatrix elements {nIP]m), where In) represents a set of different eigenfunctions of the same zero&order hamiltonian. In favorab\e cases this information may suffice to identify WM. This would allow us to calculate the nonradiative rate constant (1) unequivocally, notwithstanding the fact that the final states In) are usually not the same for (n p& 1m) and (n IP Im), the important fact being that them are all eigenkmctions of the same hamiltonian, which is now known. Before applying these ideas to observed transitions, two comments are in order. (i) Although most emission processes appear to follow an exponential decay law, this is in part an experimental artifact. If it were possible to observe the decay immediately after excitation, the effect of fast processes such as vibrational relaxation and (high-temperature) spin-lattice relaxation would become apparent in the spectrum as well as in the decay rate. However, under typical experimenfal conditions, these processes have run their course when the actual observation begins. Similarly, if the equipment is sensitive enough to register a signal after a time very much longer than k-1, the effect of very slow processes. such as spin-lattice relaxation near Ool( would become apparent. Of course, eq. (2) only refers to the exponential part of the decay curve which is the only part observed in st+dard measurements. (ii) The restriction of the time-dependent perturbation treatment in eq. (2) to first-order matrix elements is due to the fact that the emission processes under consideration are first order in P.’ This -does not rule out second-order matrix .. elements ,of .the form (?zIPI i)(i lh~ Ins), where h~.is a te.rm in 91,. Two cases are to be dis_:tinguished: If hM belongs to %&, the second:order thaw elements can be removed by a -158 1 ._.
15 April 1971
simple transformation to a new basis set. If hi belongs to ‘Z&, however, the correspondent matrix elements yield an independent contribution. It appears that such contributions are unimpor-
tant in the examples to be discussed presently. As a first application we consider triplet-togrouod state transitions in aromatic hydrocarbons or comparable mtilecules [lo]. The nonradiative rate constant is derived by monitoring the decay of the phosphorescence, the spectrum of which is governed by(nlPlm). of In> and Im> were pure-spin functions, this matrix element would vanish; hence these states must contain components of different multiplicity, so that spin-orbit coupling should be included in %k rather than in Srh. In other words, the intersystern crossing is caused by vibronic coupling rather tha? spin-orbit coupling. This mechanism is readily shown to be identical with the one labeled;KfA in previous papers [4]. The predominance of this mechanism for triplet-to-ground state transitions in naphtbalene and anthracene has been demonstrated on the basis of isotope effect% and spin alignment measurements [4,11]. The same conclusion applies when triplet decay is measured by monitoring the absorption T - T1 instead of the emission T1 - SO. This fo%ows from the fact that the absorption will be followed by rapid internal conversion T,-Tl. Neither the absorption nor the internal conversion will be measurably affected by the degree of spin conlaminztion of T, and Tl. Hence no new decay channels are opened and the old ones remain unaffected. Consider now intersystem crossing from the lowest excited singlet state to the triplet manifold. The corresponding rate constant is usually obtained by monitoring fluorescence decay. Since fluorescence is multiplicity allowed, there is no reason to assume the states IiZ), jm} to involve triplet components. Hence, singlet-to-triplet intersystem crossing can be caused by spin-orbit coupling as well as vibronic coupling, so that all three mechanisms, previously [4] labeled%&, 9fi2& and 9!$$, should be considered. similar considerations apply to internal conversion. Let us expand inz) and In> in adiabatic Born-Oppenheimer states [12] :
14
= c $k, i
Q) A;(Q) ,
(4)
where q and Q represent the sets of electronic’ and vibrational coordinates, respectively. The resulting expression for the transition dipole momait
Volume 9. nllmhr
2
25 A.priL1971
CHEMICALPHYSICSLETTERS
is usually approximated by a single term which in turn is expanded about the nuclear equilibrium configuration Q = Qo (Herzber&-Teller expansion):
(~~(q,Q)i\n(Q)lPf~~(q,Q)n,fQ))
us to distinguish between two types of vibrationdl structure, namely that governed by &IA,> for transitions of purely electronic origin, and that governed by (AnI Q] AIn) for vibrationally induced electronic transitions. if the spectrum shows only structure of the type (&IA,), then @,(q, Qo) may well be the appropriate zero&order state, so that the radiationless transition may be due to the potential energy term U(Q) - U(Qo) in the molecular h~ilto~~. However, if the spectrum shows struchcre of the type (h,l Q]&J, then the zeroth-order state must be vibronicafly contaminated, so that U(Q) U(Qo) will not be the appropriate coupling term. In spectroscopy one usually assumes the vibronic mixing to be due to the linear term in the expansion (6). This would amount to taking adiabatic Born-Oppenheimer states for 1jr> and 1tn), so that the radiationless transition would be due to the nuclear kinetic-energy operator. However, there is little evidence to justify this aFgroach in a quantitative manner. Unfortunately, the real question here as well as in the case of intersystem crossing is a quantitative one. The question is not ifa given state is spin contaminated or vibronicaily contaminated, but to what ext.& it is contaminated. This requirt?s a quantitative interpretation of spectral features which at present is @ill out of reach. Thus we have to settle for a more qualitative answer, e.g. the one implied in the assumption of either zero or complete spin contamination for intersystem crossings. However, such an unambiguous assumption is not available for internal conversion, since complete uibronic mixing in the zeroth;order states rules out internal conversion as a decay channel. Hence, if there is evidence for internal conversion, e.g. from quar.tum yield measurements, and if at tbe same time the spectrum monitored shows vibronic contamination, e.g., structure of the type
then the vibronic mixing in ffie zeroth-order states must be nonzero but ineompbte . One approach exhibiting this property is the conve&ional assumption that the zeroth-order states are adiabatic ~rn-Opp@~~rn~ states. Buriand and Robinson [5] favor the alternative assumption of zero vibrotic ~on~rn~~~on through the inducing mode(s) and essentially complete vibronic contamination through the accepting modes which in general are more sensitive to the e’iectroniz structure. It is not easy to test these assumptions. In general radiative and nonradiative transitions are induced by tierent normal modes since ST& and P have different symmetry properties. In addition it is very difficult tc; distinguish vibronic coupling caused by deviations from the nuclear equiIibrium coafiguration from that caused by non-zero nuclear momenta [7,13]. It appears therefore that the vibronic mechanism of internal conversion remains a subject of conjecture. The same 2ppIies to the pureIy electronic mechanism suggested by Sharf [8], for which no comparison with spectroscopic observations is available to date. I am grateful Henry,
G. Orlandi
for helpful comments by B. R. and G. W. Robin~txt.
REFERENCES 111B. R. Henry and &f. Kasha, Ann. Rev. P&s. 19 f 1968) 161. and references tberein.
[2] 26Bixon and J. Jortner, J. Chem.Pbys.
Chem.
46 (1368)
[31 W. l&odes, J. Chem. Phys. 50 (I.969)288% [4] W. Siebrand, Chem. Phys. Letters 6 0970) 192; B.R.Henry and W.Siebrand. J.Chem. Phys., to be
published. [S] I). M. Burland and G. W..Robfnson, Proc.Katl. Acad. Sci. 66 (1970) 257. [S) B.Sbarf and R.SiIbey, Chem.Phys. Letters 4 (1969) 423: 5 (1970) 314. [7j G. Orlandi and W. Siebrand, Chem. Phye. Letters 8 (1971) 473. [PI B.Sbarf. Chem. Phys. Letters 6 (1970) 364. 191 K. F. Freed and J. Jortner. 3.Chem. Phys. 50 (1969) 2916. [lo] W.&brand, J. Chem.Phys. 47 (X967) 2411. [1X] B. R. Henry aMf W.Siebrand. Chem.Phys. Letters 7 (1970) 533. [lit] M. Born, Nachr. Akad. Wiss. Gattingen. Math.PhysiJc. Ki. {1961) nr. 6; Tizc Loogust-Higgins. Advan. Spectry. 2 (1961) and C. A. Manzaeco. Molectir Iuminescence, ed, E.C. Lim (Benjamin, New York, 1969) p. 631.
[131 R. k Hocbsttasser
159 -