.. HE JOURNAL OF CHEMICAL PHYSICS
VOLUME 48, NUMBER 2
15 JANUARY 1968
Intramolecular Radiationless Transitions
MORDECHAI BIXON AND JOSHUA JORTNER
Department of Chemistry, Tel-Aviv University, Tel-A ViII, Israel
(Received 7 August 1967)
In this paper we consider a theory for intramolecular radiationless transitions in an isolated molecule. The
Bom-oppenheimer zero-order excited states are not pure in view of configuration interaction between
nearly degenerate zero-order states, leading to the broadening of the excited state, the line shape being
Lorentzian. The optically excited state can be described in terms of a superposition of molecular eigenstates,
and the resulting wavefunction exhibits an exponential nonradiative decay. The linewidth and the radiationless lifetime are expressed in terms of a single molecular parameter, that is the square of the interaction
energy between the zero-order state and the manifold of all vibronic states located within one energy unit
around that state. The validity criteria for the occurrence of an unimolecular radiationless transition and for
exponential decay in an isolated molecule are derived. Provided that the density of vibrational states is large
enough (i.e., exceeds the reciprocal of the interaction matrix element) radiationless transitions are expected
to take place. The gross effects of molecular structure on the relevant molecular parameters are discussed.
I. INTRODUCTION
nels of radiationless transitions and to express the
mean lifetime in the form
Radiationless electronic relaxation processes in polyatomic molecules are of considerable interest in molecular spectroscopy, photochemical reactions, biological
processes, and laser technology. Radiationless transitions involving a change in the electronic state of the
bound molecule are divided into two categories: transitions between states of the same multiplicity are referred
to as internal conversion, while transitions between
states of different multiplicity are called intersystem
crossing. A vast number of experimental studies of
radiative process in polyatomic organic molecules in
solution and in solid matrices lead to the following
conclusions1- a:
l/T= (l/To)
ki •
i
(a) The emitting level of a given multiplicity is the
lowest excited state of that multiplicity. A famous
exception to this rule involves the azulene molecule.4 •6
Another violation of this rule is exhibited by the ferrocene6 sandwich compound, which shows emission
from the second absorption band.
(b) The character of the emission spectrum is independent of the exciting wavelength.
(c) Both intersystem crossing and internal conversion are first-order processes. The decay mode of excited
singlet states is exponential characterized by a mean
lifetime T which is lower than the radiative lifetime TO
calculated from the oscillator strength. The fluorescence
quantum yield cp determined independently is consistent
with the relation CP=T/TO' It is customary to define a
set of unimolecular rate constants k i for various chanP. Seybold and M. Gouterman, Chem. Rev. 65, 413 (1965).
M. Kasha, Discussions Faraday Soc. 9, 14 (1950).
3 M. Kasha, Radiation Res. Suppl. 2, 243 (1960).
4 M. Beer and H. C. Longuet-Higgins, J. Chem. Phys. 23, 1390
(1955).
i G. Binsch, E. Heilbronner, R. Jankow, and D. Schmidt, Chern.
Phys. Letters 1, 135 (1967).
6 D. R. Scott and R. S. Becker, J. Chem. Phys. 35,516 (1961).
1
2
+L
(d) The energy gap law: It is generally established
that the rate constants for radiationless transitions
decrease with increasing the energy difference between
the two states. This phenomenon is general for all molecular relaxation processes (i.e., vibrational relaxation).
(e) The intramolecular isotope effect: Deuteration
greatly decreases the rate of radiationless transitions,
the effect being manifested in the case of a large energy
gap. This effect is well established for intersystem
crossing between triplet states and the ground state,7-10
and was also observed in the case of internal conversion
in azulene,ll where the fluorescence quantum yield from
the second singlet is enhanced by deuteration. As
deuterium substitution affects only the vibrational
frequencies, it was inferred that the change in the
Franck-Condon vibration overlap integrals is a major
factor determining the rate of the process.12
(f) Environmental effects on radiationless transitions are not yet fully understood. For some cases
(i.e., anthracene) the singlet lifetime and the fluorescence quantum efficiency are identical in the gas phase
and in solution. Some other observations are of general
interest. (1) The heavy-atom effect in the enhancement
of intersystem crossing, which is interpreted in terms
7 C. A. Hutchison Jr. and B. W. Mangum, J. Chern. Phys. 32,
1261 (1960).
8 M. R. Wright, R. P. Frosch, and G. W. Robinson, J. Chem.
Phys. 33, 934 (1960).
g R. E. Kellogg and N. C. Wyeth, J. Chem. Phys. 45, 3156
(1966).
10 W. Siebrand and D. F. Williams, J. Chem. Phys. 46, 403
(1967).
11 G. D. Johnson, L. M. Logan, and I. G. Ross, J. Mol. Spectry.
14, 198 (1964).
12 G. W. Robinson and R. P. Frosch, J. Chem. Phys. 37, 1962
(1962).
715
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716
M.
BIXON AND J.
JORTNER
of the increase of the spin-orbit coupling. 8 (2) Tem- must be responsible, at least in part, to the phenomena
perature effects on radiationless transitions in solid in dense media.
matrices '3 provide evidence for coupling between the
Current theories of radiationless transitions are
molecular electronic states and the lattice phonons. based on the implicit assumption that the coupling
(3) Level inversion effect: Crystal-field effects in pure between the molecule and the medium is essential for
molecular crystals may lead to the inversion of the the occurrence of the radiationless process. This coupling
order of exciton levels relative to the molecular states. A is required to provide a sink for the dissipation of
dramatic effect is observed in crystalline anthracene molecular excitation energy, and to secure energy
where the fluorescence quantum yield of the order of conservation restrictions by demanding that the energy
cf>=0.9 compared to the value cf>=O.3 for the isolated spectrum of the coupled system is almost continuous.
molecule."
Robinson and Frosch12 .22 considered zero-order states
of the coupled system to be nonstationary states whose
Although most of the available experimental results time development is determined by intramolecular
concern systems of interacting molecules, sufficient time-independent terms (i.e., vibronic coupling and
experimental data are now available which conclusively spin-orbit coupling). These authors and also Hunt,
demonstrate that radiationless transitions appear in McCoy, and Ross23.24 feel that the rate-determining step
isolated molecules. The relevant evidence is as follows: in radiationless transitions is an intramolecular process.
The medium provides a manifold of states nearly
degenerate with the excited state and a sink for vibra(a) Resonance fluorescence is not observed when
tional relaxation processes. An extreme attitude conanthracene,'5 naphthalene,'6 and tetracenel7 are excited
cerning environmental effects was set forward by
to the second singlet in a rarified gas. These experiments
Gouterman25 who claimed that in the absence of enwere carried out at such gas densities where the calcuvironmental perturbations, radiationless transitions
lated duration between successive collisions is much
will not take place in an isolated molecule.
longer than the calculated radiative lifetimes of the
In view of the rather extensive experimental data
molecule. The observed fluorescence resembles the
concerning radiationless transitions in rarified gases,
emission resulting from excitation to the first excited
one should inquire whether radiationless transitions
singlet but it is slightly shifted to the red and is diffuse.
should occur in an isolated molecule. An atom or a
(b) The mean lifetimes of the first excited singlets,
molecule in a stationary state cannot make transitions
after extrapolation to low density, are about 25%
to other states which are induced by "small terms in
l
of the calculated radiative lifetime in anthracene 8,l9
the
molecular Hamiltonian."12 A molecular system can
and less than 50% in the case of perylene. 20
make transitions only by coupling with the radiation
( c) The fluorescence yield of benzene in the gas
field, so that all time-dependent transitions between
phase21 is cf>=0.2, being independent of pressure at
stationary states are radiative in nature. However, this
sufficiently low pressures. After excitation to the first
conclusion does not apply if the molecular system is
excited singlet, at such low pressures, part of the mole"prepared" by some experiment in a nonsteady state
cules appear in a triplet state as is evident from their
of the system's Hamiltonian.
ability to induce cis-trans isomerization of 2-butene.
In previous work on this subject, transition rates
between Born-Oppenheimer states were estimated. The
The rates of radiationless transitions in dense media Hamiltonian H of the system was divided into two parts
are of the same order of magnitude as the rates observed H = Ho+ V, the initial excited state was assumed to be
in rarified systems. Therefore, it may be concluded an eigenstate of Ho, and the transitions to other eigenthat the mechanism which operates in isolated molecules states of Ho are induced by the perturbation term V.
There is no a priori justification for locating the excited
molecule in a zero-order state. This approach is similar
13 R. E. Kellogg and R. P. Schwenker, J. Chern. Phys. 41, 2860
to the Coulson-Zalewsky theory of predissociation,26
(1964).
14 R. E. Kellogg, J. Chern. Phys. 44, 411 (1966).
where it was assumed that the system is "prepared"
16 P. Pringsheirn, Fluorescence and Phosphorescence (Interscience
in a Born-Oppenheimer state, and transitions are then
Publishers Inc., New York, 1949), p. 271.
16 R. J. Watts and S. J. Strickler, J. Chern. Phys. 44, 2423
forced to the manifold of continuum dissociative states.
(1966).
17 R. Williams and G. J. Goldsmith, J. Chern. Phys. 39, 2008
(1963) .
18 K. H. Hardtl and A. Scharrnann, Z. Naturforsch. 12a, 715
(1957).
19 W. R. Ware and P. T. Cunningham, J. Chem. Phys. 43, 3826
(1965).
20 W. R. Ware and P. T. Cunningham, J. Chern. Phys.44, 4364
(1966) .
21 G. B. Kistiakowsky and C. S. Pararnenter, J. Chern. Phys. 42,
2942 (1965).
22 G. W. Robinson and R. P. Frosch, J. Chem. Phys. 38, 1187
(1963) .
23 G. R. Hunt, E. F. McCoy, and I. G. Ross, Australian J. Chem.
15, 591 (1962).
24 J. P. Byrne, E. F. McCoy, and I. G. Ross, Australian J. Chem.
18,1589 (1965).
26 M .Gouterman, J. Chem. Phys. 36,2846 (1962).
26 C. Coulson and K. Zalewski, Proc. Roy. Soc. (London) A268,
437 (1962).
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717
INTRAMOLECULAR RADIATIONLESS TRANSITIONS
The questions we attempt to answer in the present
work are as follows:
briefly restate the general formulation of the problem
using the Born expansion which avoids the tedious
power series expansion originally applied by Born
and Oppenheimer. The total Hamiltonian for the
molecule is displayed in the conventional form
(a) What information on the electronically excited
vibronic states of an isolated polyatomic molecule can
be obtained?
(b) What is the effect of the dense distribution of
the Born-Oppenheimer vibronic states on the stationary states of the polyatomic molecule?
(c) What is the line shape for optical excitation of
the polyatomic molecule?
(d) How is the molecular excited state "prepared"
by optical excitation from the ground state, and is it
meaningful to consider time development of such
excited states?
(e) What are the molecular parameters determining
the experimental behavior of an isolated molecule?
H=T(q)+T(Q)+U(q, Q)+ V(Q),
(1)
where q= {qi/ represents the set of electronic coordinates and Q= {Qk\ corresponds to the set of nuclear
coordinates with masses {Mk }. The electronic kinetic
energy operator T(q) and the nuclear kinetic operator
T(Q) are represented in the usual form
T(q)
=-
L: (FN2m) (iJ2jiJqJ
2) ,
(2)
j
and
T(Q) = -
In the present paper, the problem of radiationless
transitions in isola ted molecules is considered. We
adopt the method of configuration interaction, similar
to that used in the treatments of predissociation27 and
autoionization. 28 In the first stage of the calculation,
general expressions for the eigenstates of the molecules
are derived, considering the breakdown of the BornOppenheimer separability conditions. The time evolution of an optically excited state is then considered.
Finally, we derive the necessary restricting conditions
for the occurrence of a radiationless transition in an
isolated molecule in the gas phase.
L: Wj2M
k)
(iJ 2jiJQk2 ).
(3)
k
Finally, U(q, Q) is the total electronic potential energy
and V(Q) is the potential energy of the nuclei.
The electronic wavefunctions at a fixed nuclear configuration are chosen to satisfy the partial Schrodinger
equation
(4)
[T(q)+U(q, Q)]'P,,(q, Q) =En(Q)'Pn(q, Q),
where En (Q) corresponds to the electronic energy at
this nuclear configuration. The exact molecular wavefunction 1/;(q, Q) is now expanded in the form
1/;(q, Q) = L: 'Pn(q, Q)xn(Q).
II. MOLECULAR EIGENSTATES
(5)
n
The conventional method of calculating molecular
structure and molecular dynamics involves the BornOppenheimer adiabatic approximation. 29 We shall
The vibrational wavefunctions are then displayed by
a series of coupled differential equations
[T(Q)+Er(Q)+V(Q)+('Prl T(Q) I'Pr)-Wr]Xr(Q)+L: [('Pr IT(Q)
.;<'r
1'P.)-2L: (1i2j2Mk )('Prl
(iJjiJQk)
l'Ps)
k
The adiabatic approximation is now introduced by neglecting the coupling terms in Eq. (6). The molecular
wavefunctions now reduce to the simple product terms
(7)
The approximate wavefunctions of the adiabatic approximation are characterized by the following off-diagonal
matrix elements between different electronic states:
(1/;nr I H I1/;n.) = Wn,.or.,
(if;mr I HI 1/;".)=
J
Xm.*(Q) ('Pm I T(Q) I 'Pn}Xn.(Q)dQ-2
(8a)
~ 2:k JXm.*(Q) <'Pm Ia~k I'Pn> iJ~k Xn.(Q)dQ.
(8b)
The adiabatic approximation is applicable only provided that the energy difference between the vibronic states is
large relative to the nuclear matrix elements connecting these states. From the electronic Schrodinger Equation (4)
(a) O. K. Rice, Phys. Rev. 33, 748 (1929); (b) R. A. Harris, J. Chern. Phys. 39, 978 (1963).
U. Fano, Phys. Rev. 124, 1866 (1961).
29 M. Bom and K. Huang, Dynamical Theory of Crystal Lattice (Oxford University Press, New York, 1954).
27
28
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718
M. BIXON AND J. JORTNER
it follows that
(9a)
(9b)
for all Qk. In the case of close lying vibronic states
belonging to different electronic configurations, it is
expected that the adiabatic approximation will completely fail. This breakdown of the Born-Oppenheimer
approximation is well known in the case of a degenerate
electronic state (the Jahn-Teller effect),30--32 in the
case of nearly degenerate states (the pseudo-JahnTeller effect), and in the case of widely separated
electronic states giving rise to vibrationally induced
electronic transitions ( the Herzberg-Teller effect) .33,34
We shall now turn our attention to the Born--0ppenheimer levels constructed from two electronic states
A and B. Let 1/1. represent a vibronic state corresponding
to the electronic state A to which optical excitation
from the ground state is allowed. The energy of this
vibronic state will be denoted by E •. Let us now denote
by 11/l;} the set of vibrationally excited vibronic states
corresponding to the electronic state B which are
characterized by almost the same energy as E •. In the
case of quasiresonance between the state 1/1, and the
set of states {1/Ii) the adiabatic approximation breaks
down. The manifold of states {1/1;} becomes more
densely spaced with increasing the energy gap between
the two electronic states A and B in view of the large
number of the possible combinations of the fundamental
vibrational frequencies. The "background" of states
l~;) consists of a dense distribution of vibronic states.
In view of the small energy denominators which determine the vibronic coupling matrix elements (8b) we
thus expect an appreciable mixing of the state 1/1. with
the background manifold of vibronic states. This
situation of near degeneracy resembles the situation
encountered in the problem of the pseudo-Jahn-Teller
effect; however, in the present case, a large number of
nearly degenerate states has to be coupled to a single
state. It is, of course, out of the question to attempt to
solve the problem exactly, and a simple mode calculation will be first applied, based on the following
assumptions:
(a) The states {1/1.) are uniformly spaced. The manifold of vibronic states is approximated by equidistant
IOH. A. Jahn and E. Teller, Proc. Roy. Soc. (London) A161,
220 (1937).
31 W. Moffit and W. Thorson, Phys. Rev. 108, 1251 (1957).
32 H. C. Longuett-Hi~gins, U. Opick, M. H. L. Pryce, and R. A.
Sack, Proc. Roy. Soc. (London) A244,1 (1958).
33 G. Herzberg and E. Teller, Z. Physik Chern. (Leipzig) B21,
410 (1933).
"A. C. Albrecht, J. Chern. Phys. 33,156,169 (1960).
series of states, with an energy difference E between
consecutive states. For the sake of bookkeeping, the
value i=O is given for the nearest state from below to
1/1,. We set i<O for states lower than 1/1, and i>O for
states higher than 1/1,. Setting a=E,- Eo, the energy
of the state 1/1; is
i=O, ±1, ±2,···.
E.=E.-a+ie,
( 10)
(b) Constant coupling matrix elements. The matrix
elements of the complete Hamiltonian between the
states ~, and 1/1;, V;= (1/1; 1H 11/1,), are assumed to be
constant, independent of the index i. Actually, Vi is a
function of the energy and will strongly decrease with
increasing 1E.-Ei I. For computational reasons Vi
can be assumed to be constant near E. and it can be
taken as constant for all values of i because of mutual
conciliation of the terms involving positive and negative
values of the index i for high values of the index i.
The new wavefunctions of the molecule can be expressed as linear combinations of the adiabatic functions in the form
\(r,,=an~.+
L: binh
(11)
i
The values of the expansion coefficients an, bin and the
new energy levels En are given by the solution of the
eigenvalue problem
E.
an
an
V
V
0
V
=E"
Ei
V
l.
bi"
(12)
bin
0
which is equivalent to the following set of linear
equations:
(E.-En)an+v
L: b.n=O,
(13a)
i
i=O, =1=1, ±2,···.
(13b)
From Eq. (13b) by using Eq. (10) we obtain
bin = - [va,,/ (E,- a+iE- E,,) ].
(14)
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719
INTRAMOLECULAR RADIATIONLESS TRANSITIONS
By substituting into Eq. (13a) it follows that
[E.-En+V2
:t
(En-E.+a-i€)-l]a,.=O,
the definition of "In, one gets
an2=(1+~:t
2
(15)
E 1-00
i=-co
which is an equation for En.
The infinite sum in Eq. (15) diverges if the summation is carried out either on only positive or only on
negative values of i. An important part of the sum
involves the contribution of a small range near E.
because the value of Vi decreases for high energies. By
including all the elements in the sum down to minus
infinity and up to infinity, the mathematical treatment
is greatly simplified.
Combining the elements of the sum in Eq. (15)
corresponding to - i and +i, and using the abbreviation "In = (En-E.+a)/E we obtain
)-1.
(22)
7r2 csc( 7r"fn)
(23)
1
('Yn-i)2
The value of the infinite sum is35
:t ("In - i)
-2 =
i=-oo
so that
a,,2=[1+(rvN) csc2 (7r"fn)]-1.
One can now rewrite Eq. (24) in the alternative form
a n2=[1+ (rv 2j€2) (7r2v2j€2) cot2(7r'Yn)]-I. (25)
+
Rearrangement of Eq. (17) yields
cot(7r"f,,)
= (E/V) [(E.-E,,)/7r].
L: ('Yn2-
i2 )-1=-(2'Yn2)-L(7r/2'Yn) cot(7r'Yn).
(17)
i
Substituting this result in Eq. (16) gives
Substituting the last result into Eq. (15) leads to the
following equation for the eigenvalue En:
(19)
or, using the definition of "In we get
(26)
Squaring the last expression and substituting into Eq.
(25) gives
a,.2=v2[(&- E.)2+ V2+ (W/E) 2]-1.
The summation of the series yields35
(24)
(27)
This equation exhibits the dependence of an2 , the
probability of finding the vibronic state 1/1. in the new
states, as a function of energy. This equation describes
a Lorentzian envelope for the values of an 2 as function
of En. The probability of light absorption into the new
states is proportional to a,,2, therefore this Lorentzian
envelopes a series of absorption lines centered around
the energy E •. The experimental implication is the
broadening of absorption lines yielding a half-linewidth
which is given by
LlE= [v+ (7rV2/E) 2]1/2.
As we shall later demonstrate v€-l»l, so that
LlE = 7rV2/ E.
(28)
(29)
III. TIME DEVELOPMENT OF EXCITED STATES
The new eigenvalues En can now be obtained by a
numerical or graphical solution of this equation. A
particularly interesting feature of these solutions is
that each new eigenvalue En falls between a pair of the
zero-order Born-Oppenheimer levels. This is .a general
feature of generalized perturbation expansions and is
encountered in studies of impurity effects on elementary
excitations (i.e., exciton or phonon states) in molecular
crystals. The oldest example for this rule is exhibited
by Rayleigh theorems on the effect of defects on lattice
vibrations.
The value of the coefficient an is determined from
the normalization condition
a,,2+
L
(bin)2= 1.
(21)
i
We now proceed to consider the state of the system
resulting from light absorption. We consider that the
molecule in the ground state I i'o(O» described by a
time-independent Hamiltonian Ho be subjected to a
radiative perturbation HR(t) for the time interval tl , so
that
HR(t) =0;
t<O; t>tl'
(30)
where E(t) is the electric field intensity at the location
of the molecule at time t and P=e·" where e is the
field polarization vector and" is the molecular dipole
operator. The state vector I i'(t» is obtained from the
molecular ground-state wavefunction I i'o(O» by the
evolution operator U (t, 0) 36 so that
I i'(t»=U(t, 0)
li'o(O».
(31)
Substituting the value of bi" from Eq. (12) and using
As we are interested only in first-order linear radiative processes, the evolution operator can be displayed
86 E. C. Titchmarsh, The Theory of Functions (Oxford University
Press, London, 1958). p. 114.
B6 A. M. Bonch-Bruevich and V. A. Khodovoi, Soviet Phys.Usp. 8,1 (1965).
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720
J. {o R T N E R
M. B I X 0 NAN D
within the framework of first-order time-dependent theory:
U(t, 0) =exp
(-~ Hot) [1-~fl exp
a
Hot') PEU') exp ( -~ Hot') dt'].
(32)
The state vector 1 '1'(t» can now be described as a superposition of the manifold of the states 1 '1'n),
1
'1'(t»= ~ 1 '1'n) exp
En t ltl exp (ih Ent' )E(t') ('1'n
(-hi)
0
1
P 1 '1'o)dt'.
(33)
We have assumed that one-photon absorption is allowed only to the zero-order state 11/18), being forbidden to the
manifold of zero-order states {11/1d. Hence, the dipole matrix element ('1'" 1 P 1 '1'0) for the optical excitation I '1'0)-t
1'1'n) is proportional to the coefficient an, so that
(34)
We are interested in the time development of the amplitude of the state
written in the form
ltl
(1/181 '1'(t) )=--i p.o
11/1.) in the excited state, which can be
s(t, t')E(t')dt',
(35)
Set, t') =:E a,.2 exp[ - (ilft)En(t-t')].
(36)
ft
0
where the kernel Set, t') =s(t-t') is given by
n
From Eq. (27) it immediately follows that
(E - E)2]-1 exp [i
-h En(t-t') ].
s(t, t') = ~ [ 1+ ( 7TV)2
-; + ~
(37)
To provide an evaluation of this sum, it has to be approximated by an integral. The following conditions have to
be fulfilled to make this approximation valid:
(a) The energy levels En are equidistant and can be expressed as
En-E.=ne;
n=O, ±1, ±2···.
This assumption seems to be a good one, because the energy levels Ei were equidistant with a separation e, and
between every two consecutive states of the first approximation there exists a new level En.
(b) The interaction energy has to be much larger than the energy difference between consecutive states:
v»e.
(c) The time up to which our result will be correct is limited by the inequality
1-t'<<li/E.
Under these assumptions, we may write the sum SCt, t') in the form
S(/, I') =exp
(-~ E.(t-t'») ~ [(:V)\(;y n2r1 exp (-~ en(t-t »)
l
(38)
and approximate it by the integral
Set, t') =exp (-~ E.(t-t »)
l
L: [(7)\(;Y Z2r1
exp
(-~ ez(t-t' ) )dZ.
(39)
The integral is a Fourier transform of a Lorentzian and the final result is
Set, t') = exp[- (il/i) E8(t- t')] exp[ - (7Tv2/e/i) (t- t')].
(40)
Substitution of this result into Eq. (35) leads to the following expression for the amplitude:
(1/181 '1'(t) )=-r;i P 80
1 (i E8(t-t') )
0
t1
exp -",
exp (7TV2
- Eft (t-t') ) E(t')dt'.
(41)
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INTRAMOLECULAR RADIATIONLESS TRANSITIONS
We now define a relaxation time
T
in the form
T=Ji/27rv2
(42)
and rewrite the expression for the amplitude of the state 1 if;.)
(if;. 1 'l1(t»
721
= -~ P.o exp ( -
t~:1) exp ( -~ Est) ~tl exp C' ~t1) exp
a
Est') E(t')dt'.
(43)
The probability W8 for locating the molecule at the state if;.) is the square of the amplitude:
1
w.(t, E) =
1(if;sl 'l1(t) 12= :~2 exp ( -
l~t1)
fl fl
expt' +t;T- 2it exp [
-i
Es(t" -I') ] E(t')E(t")dt'dt".
(44)
The probability w.(t, E) is obtained for a definite field E(t).
The radiation field of a conventional light source consists of a superposition of a large number of individual
harmonic waves characterized by random phases.37 The amplitudes and the phases of the field are subjected to
random fluctuations, which are determined by the width Av of the spectral band. The average probability is obtained by averaging w.(t, E) over the distribution of the field. The averaging procedure reduces to the average of
the product E(t') E(t"). This average
cI>(t', t") = (E(t')E(t") )
(45)
corresponds to the second-order correlation function of the field. In the case of usual stationary radiation sources
of constant intensity, the correlation function depends only on the time difference t'-t"; that is, cI>(t't") =
'P( I' - t"). The average probability is given by
1)
Ps02
/ - / 1) tl tl
exp (--TW=h2
10 10 exp (/'+t"-2/
2T
exp
8
(ir"E.(t,,,) cI>(t,-t")dldl,,,.
-I)
( 46)
This equation exhibits an exponential decay of the average probability characterized by a lifetime T. In view of the
restrictive condition (c) on the evaluation of the sum S(/, t')[/-I'«Ui/e) , where 0::;t'::;/1J, the duration of illumination that will give rise to the nonradiative decay should not exceed hie, so that 11<<li/e.
The correlation function can now be expressed in terms of the spectral density I(v):
cI>(t', I") =
8: L:
I(v) exp[iv(t"-I')Jdv.
(47)
The correlation function differs from zero only for time intervals smaller than the width Av of the exciting source
1I'-t" I::; (Av)-1. A stationary interaction mode is established between the radiation field and the molecule after
a time interval of the order of (Av)-1. The radiation from a thermal source is believed to be characterized by a
Gaussian distribution
I(v) =10 exp{ - [(v-v.) / AvJ2},
( 48)
where v. = E /h. This distribution is chosen to be centered around v,. The correlation function is
8
0
cI>(/', t") = 8:1
fLO exp[iv(/"-I')J exp [-(vz'YJ dv,
(49)
which can be approximated by the Fournier transform of a Gaussian
cI>( I', I") = (87rlo/ c) exp[iv.( t" - I') J exp[ - (Av) 2( t" - I') 2].
(SO)
Now imposing the condition that Av»1 v-v. I so that the spectral width of the exciting optical source is appreciably
wider than the linewidth AE [Eq. (29)J, the spectral distribution is constant and the correlation function can be
approximated in terms of a delta function
cI>(t', t")---7(87rlo/c)o(t"-t').
The average probability in this limiting case is given by
87rlo P 802
(I-it) tl tl
(t'+I"-2t1)
wS=-c- fi exp --T- 10 10 exp
2T
exp
37
(i
-Ii E.(t"-t') )o(t"-t')dt'dt"
(51)
(52)
M. Born and E. Wolf, Principles oj Optics (Pergamon Press, Inc., New York, 1959), p. 500.
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722
M. BIXON AND J. JORTNER
and we get
l
(t-It) 1 exp (tf---tl) dt,f
- 81T oP.o2 exp - - w.=--2
ch
leading to the equation
w.=A exp[ -(t-tl )IT]T[l-exp(-tl/T)],
(54)
where
A =8-ilP.No/hh.
This is an ordinary kinetic equation describing the
exponential decay of the state produced by irradiations
for the period t1' The decay is characterized by a nonradiative lifetime T. For short irradiation time w.=
Ah exp[ - (t-t1) IT]. On the other hand, for a stationary state tl»T (but also t1<<ft/E) and we obtain
W.=AT exp[ - (I-h) IT].
(55)
In the present treatment the radiative decay of the
excited state was not taken into account. The radiative
decay can be introduced for irradiation times short
compared to the radiative lifetime TO, that is h«TO. In
this case, we can account for the radiative process by
the Weisskopf-Wigner ansatz38 assuming an exponential
decay of the phase (1/1.1 '1'(t) ) characterized by a mean
lifetime TO' This phenomenological treatment leads to
the result
w.=A exp{- (t-t1) [(liT) + (liTo) ]JT
X{l-exp[ -(t1/T)]J.
(53)
We shall now provide an estimate of the effect of
this mixing and the off-diagonal matrix element which
is responsible for the radiationless process. Let 1/1. be
the excited singlet Born-oppenheimer wavefunction
characterized by the energy E., 1/;T is the second triplet
vibronic state with energy ET and {1/;.} represent the
manifold of vibronic states corresponding to the first
triplet. The part of the Hamiltonian which is not
diagonal in the Born-oppenheimer approximation is
given by
(57)
where H.o is the spin-orbit coupling and HVib is the
vibronic term
Hvib=H- L: I I/Inr)Wnr (1/;nr I,
(58)
where H is the molecular Hamiltonian (1), 1/;nr the
adiabatic wavefunctions, and Wnr the total energy in
the Born-oppenheimer approximation [Eq. (8a)].
The matrix elements of the Hamiltonian between the
zero-order states are
(1/;.1 Hoo I1/;T)=Vso ;
(1/;.1 HVib I 1/;7,)=0,
(1/;.1 Hso 11/;;)= Vsoi;
(1/1.1 HVib l1/;i)=O,
(1/;T I Hso 11/;;)=0;
(1/;T I Hvib Il/li)=Vvib.
(59)
(56)
This equation exhibits the pertinent features of the
problem. The probability decays because of two independent processes: a radiative decay with a lifetime
TO and the radiationless decay characterized by the
lifetime T.
IV. RADIATIONLESS TRANSITIONS IN A
THREE-LEVEL SYSTEM
In some cases, it happens that more than two levels
are involved in the process of a radiationless transition.
This effect is manifested in the enhancement of intersystem crossing when the second excited triplet state
is located just below the excited singlet state. This is
the case for benzene39 and anthracene. 14 In the case
where the second triplet is close to the singlet, the
interaction energy between these states is appreciably
larger than the interaction between the singlet and the
lower triplet state, so that the mixing between the two
former mentioned states is extremely important.
W. Weisskopf and E. Wigner, Z, Physik 63, 54 (1930).
39 S. D. Colson and E. R. Bomstein, J. Chern. Phys. 43, 2661
(1965) .
38
tl
TOT
The spin-orbit coupling matrix element V 80 is much
bigger than the matrix element V oo' in view of the
appearance of a small Franck-Condon vibrational
overlap factor SFC in the latter term. For the sake of
qualitative discussion we set V 00'= V.oSFC •
We shall also set Vvib= VvSFC, assuming that the
Franck-Condon factors are identical for the close lying
states 1/;. and 1/;T.
We can now use first-order perturbation theory to
estimate the mixing between these two states, seeking
a representation which is diagonal in the spin-orbit
interaction. The new wavefunctions are
1/11 = cI/I.+ dl/l T,
(60a)
1/12 = dI/I. - cI/IT.
(60b)
The expansion coefficients are determined by the energy
gap A=E.-ET and by Vso:
C=[(y+A) 12yJl/2,
(61a)
d= [(y- A) 12yJ1I2,
(61b)
where
( 62)
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I N T RAM 0 L E C U LA R R A D I A T ION L E SST RAN SIT ION S
is the energy gap between the states 1/11 and 1/12. The
resulting states now interact with the vibronic states
in the background; the relevant matrix element is
V=(1/Ill H' 11/1.>
transition rate due to the presence of the second triplet
state:
(V /Vo)2=c2+2cd( V,,/V so) +d2(V./Voo )2.
(66)
If ~> V so",l cm-l 40 we can set d= Voo/ ~ so that
= (Cl/t.+d1/lT I Hoo+Hvib 11/1.>
(67)
(63)
so that the square of the coupling matrix element which
determines the time evolution of the radiationless
transition is
(64)
The value of this parameter in the absence of the second
triplet state is
(65)
The ratio between the matrix elements in Eqs. (64)
and (65) provides the enhancement of the radiationless
Vmr,n.= (1/Imr I HI 1/In.) = -!h2
723
As V v"'l()3 cm-t,41 then if ~<1()3 cm-1 which is of the
order of one vibrational quantum, the rate is appreciably enhanced. In the case of degeneracy ~=O we get
(VjVO)2=!+C V v /V. o )+HV,,/V.o )2.
In this case, one obtains (V/Vo)2=lOS.
V. INTERACTION MATRIX ELEMENTS
In what follows, we shall consider the coupling matrix
elements between vibronic states. We can now display
the coupling matrix elements in terms of intramolecular
normal coordinates {Qk} and get
~ f Xmr*(Q) <C{)m(qQ) IiI~k21 C{)n(qQ)
2
-h
U(q, Q) =U(q, O)+U'(Q),
>
xn.(Q)dQ
>
~ f Xm.*(Q) <C{)m(q, Q) IiI~k IC{),,(q, Q) iI~k Xn.(Q)dQ.
The adiabatic electronic functions C{),,(q, Q) correspond
to the eigenfunctions of the electronic Hamiltonian (4).
In the treatment of vibronic coupling problems, it is
customary to expand the electronic potential energy in
the normal coordinates:
(70)
(68)
(69)
ing to draw attention to a classical calculation of the
force constants of excited electronic states of aromatic
molecules,42 using linear and square terms which lead
to wrong results unless cubic terms are included. In
spite of these reservations, we shall adopt the conventional procedure.
The linear approximation leads to the result
where
(73)
where the pure electronic term is given by
]
k= (C{)m(q, 0)
mn
For small enough nuclear displacements, the electronic
wavefunctions can be expanded in terms of harmonic
set at a fixed nuclear configuration Qo=O:
C{),,(q, Q) =C{)n(q, 0)
" (C{),,(q, 0) I U'(Q) I C{)a(q, O)} ( 0)
+~
&(0) - Ea(O)
C{)a q, •
(72)
The conventional procedure is to retain only the linear
term in the expansion (71). It should be noted that
while such a treatment is justified in the case of the
Jahn-Teller problem and the Herzberg-Teller vibronic
coupling problem, it may not be applicable to our case,
where molecular wavefunctions at large nuclear displacements are concerned. In this context, it is interest-
I (ilUjilQk)O I C{),,(q, 0»
&(0) -&(0)
,
(74)
being independent of nuclear coordinates. The second
electronic matrix element in Eq. (69) vanishes in the
linear approximation. In the second-order approximation retaining terms in Q2 these matrix elements can be
displayed in the form
(C{)m(q, Q) I il 2 jilQ2k I C{)n(q, Q) >
(C{)n(q,O)
I (il 2UjilQk2 ) I C{)m(q, O)}
En(O) -&(0)
(75)
40 J. W. Sidman, J. Chern. Phys. 29, 644 (1958).
41 J. A. Pople and J. W. Sidman, J. Chern. Phys. 27, 1270
(1957) .
41 (a) V. Griffing, J. Chern. Phys. 15,421 (1947); (b) S. Ehrenson and M. Wolfsberg, ibid. 45, 3879 ~1966).
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724
M. BIXON AND J. JORTNER
It can be safely assumed that the contribution of the
term involving the nuclear kinetic energy operator is
negligible. The coupling matrix element can be finally
represented in the form
Vmr,n.=h,2
~J...nk
f
Xmr*(Q)
a~k xn.(Q)dQ.
(76)
The molecular vibrational wavefunctions are now approximated in terms of products of harmonic-oscillator
wavefunctions Xm!'( v ,!,) and xn.( Vn,), where Vm!' and
Vn, correspond to the vibrational quantum numbers
as sums of products of electronic terms and FranckCondon overlap integrals. This result is well known
from the pioneering work of Robinson and Frosch and
Ross et at. Recently, Siebrand43 attempted to derive
these relations using similar approximations as employed in the present treatment. However, in Siebrand's
work, only matrix elements of the form
1I
and
Xmr= IT XmJ.l(vmJ.l)
(77)
X,.. = IT X,.,(v n ,).
(78)
Defining now vibrational overlap Franck-Condon
factors:
Fk(mr, ns)
= (Xmk(Vmk) I a/aQk I Xnk(Vnk»
xIT (Xm;(Vmi) I Xni(Vnj).
(79)
iFk
The coupling matrix element can now be written in the
form
V mr,n.=h2 L: Jm,.kFk(mr, ns).
(80)
k
The final stage in the present rough calculation involves the introduction of spin-orbit coupling between
pure spin states. Setting
'Pm(q, 0) ='PmO(q, 0)
+L: Km..;p·l(q, 0),
where 'Pmo corresponds to the pure spin state, and the
spin-orbit coupling matrix element is given by
(82)
where H.o is the spin-orbit coupling Hamiltonian.
Hence we obtain for the vibronic terms
Jmnk=JmnkO+
L:
(Km..,J..,,,kO+Jm..,"OK..,n) ,
(83)
~.,.
where JmnkO corresponds to the matrix element (74)
where the electronic harmonic functions are represented
by pure spin states.
To conclude, we display the general expressions for
the coupling matrix elements. For the case of internal
conversion
Vmr,n.=LJmnkOFk(mr,ns),
(84)
k
while for the case of intersystem crossing, one obtains
Vmr,n.=
L:
VI. DENSITY OF VIBRATIONAL STATES
(81)
~m
Km..,= ('PmO I H.o I 'P..,O)/[E.n(O) - E..,(O)],
were considered, so that the dominant contribution
to the coupling matrix element was not taken into
account, The resulting matrix elements obtained herein
are consistent with Lin's work.44
(b) The nonvanishing of the coupling matrix elements provides a necessary condition for the occurrence
of an intramolecular radiationless transition. For the
case of internal conversion, the selection rules are the
same as those for vibronic coupling. The matrix elements J m"kO will be nonvanishing only for vibrations
Qk which correspond to the same representation of the
molecular point group as the direct product 'PmX'Pn' For
nondegenerate electronic states, there is only one
molecular vibration which scrambles these states. In
this case, the matrix element reduces to a product of a
single electronic term and a Franck-Condon factor. On
the other hand, for the case of intersystem crossing,
selection rules for both vibronic and spin-orbit coupling
have to be simultaneously considered.
Apart from the interaction matrix element, the second
parameter determining the rate of the radiationless
transition involves the spacing of vibronic states E, or
rather p=c1, which corresponds to the density of
vibrational energy levels. The well-known classical
and semiclassical approximations for the density of
vibrational states are not sufficiently accurate.
We adopted here the approximation due to Haarhoff,45 where the density is given by the equation
p= ( -
1
X [ 1- (1+11)2
]130
'
(86)
where n is the number of vibrational degrees of freedom,
(V)Av is the average frequency, Vi are the n frequencies
A, and Po and 11 are defined by the equations
A-1=
L L [Km..,J..,nkOFk('Yt, ns)
~m,n'
2)'12]"
2 )1/2 (1-1/12n)A [( 1+11) ( 1+h(v )Av(1 +11)
2
11
7rn
k
IT (Vi/(V)Av,
i
+Jm..,kOK..,nFk(mr, 'Yt)].
(85)
From these results, we conclude that:
(a) The coupling matrix elements which determine
the rate of the radiationless transition can be displayed
W. Siebrand, J. Chern. Phys. 46, 440 (1967).
S. H. Lin, J. Chern. Phys. 44,3759 (1966).
4i P. C. Haarhoff, Mol. Phys. 7, 101 (1963).
43
44
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INTRAMOLECULAR RADIATIONLESS TRANSITIONS
where
0:2= (,,2)Av/ (II )Av2,
'l/=E/EO;
E is the energy interval and EO is the zero-point energy.
These equations enable us to compute the density of
states and to check the consistency of the conditions
V»E and t<<Ii/ E. We shall consider now some specific
examples.
In the case of intersystem crossing in anthracene, the
energy interval between the lowest excited singlet
IB2u and the lowest triplet 3B2 ,. is about 12 ()()() cm-I •
By using the vibrational frequencies of the molecule,
one can calculate the density of states p=5.10Io cm
(i.e., states per energy interval of 1 cm-I ) for this
energy gap. The radiationless transition lifetime is
r=5.10-9 sec so that r<<Ii/E=0.25 sec. From these data
using Eq. (42) we get v=6.10-7 cm-I which is much
larger than E (2.10-11 cm- I ). Similar calculations on
naphthalene yield for the internal conversion between
the second IB 2u and the first IBau excited singlets which
are separated by 3400 cm-t, the value p = 2.103 cm. The
transition lifetime is lower than 10-12 sec and therefore
the condition r«h/E = lQ-8 sec is fulfilled. The intersystem crossing to the ground state has a lifetime
of ",,2 sec and energy interval of 20 ()()() cm-I • The
density of states in this case is p=8.1O+1• cm and again
the condition r<<Ii/E=4XIQ4 sec is maintained.
We shall now consider intersystem crossing between
the lowest excited singlet IB2u and the lowest triplet
aBI ,. in benzene. In this case, the energy interval is
8400 cm-t, the calculated density of states is 7.8XIQ4
cm and h/E=6XI0-7 sec which is comparable with the
lifetime r~10-6 sec, so that deviations from exponential
decay may be expected in this case. It is also interesting
to compare the density of states for the IB2u_3Blu
intersystem crossing in C6H6 (7.8XIQ4 cm) and in
C6D6 (1.25XlO· cm). It is of course expected that in
any polyatomic molecule, the density of states is increased by deuteration, however, the effect is rather
small and the Franck-Condon factors which increase
eAponentionally with the energy gap lead to an appreciable decrease in v, so that the product pv2 is expected to decrease on deuteration.
VII. DISCUSSION
In the present paper we have been concerned with
the problem of whether radiationless transitions occur
in an isolated molecule. Our calculations are based on
the simple model of a two-electronic-level system,
considering configuration interaction between BornOppenheimer states. The following explicit assumptions
are introduced:
(a) The manifold of vibronic levels f,pd is ordered
in an equidistant series.
(b) The coupling matrix elements between ,p. and
{,pi} are constant.
725
(c) One-photon absorption from the ground state
is allowed only to the zero-order state ,p., being forbidden to the states {,pd. This assumption is justified
in view of the extremely small Franck-Condon vibrational overlap integrals between the molecular ground
state and the highly vibrationally excited vibronic
states f,p;}.
From the results of our treatment we conclude that:
(a) Configuration interaction between close lying
Born-Oppenheimer states must be taken into account
in the treatment of radiationless transitions. The idea
that the excited states are not pure was suggested by
Klemperer.21
(b) Configuration interaction leads to the broadening of the zero-order state 1/1., the line center being
unshifted from E •.
(c) The line shape is Lorentzian, the linewidth
being given by 7rV2E- I •
(d) The excited state prepared by optical excitation
by a broad optical source shows an exponential nonradiative decay, characterized by a lifetime r=Eh/27rv2•
The validity criteria are E«V and t<<Ii/E. Although
this result is formally identical with that obtained
from Fermi's golden rule, the derivation is of interest
in view of the restricting conditions obtained herein.
These conclusions are based on a rather oversimplified physical model. An exact solution can only be
obtained provided that the exact Born-oppenheimer
states are known, which is of course an impractical
proposition. Nevertheless, it seems that the exact
distribution of levels is not crucial for the analysis of
radiationless transitions. From other model calculations
for a system of equidistant states with alternating
values of the coupling matrix element, one can show
that the relevant parameter is again v2E-I averaged over
the excited zero-order states. In other words, the
lifetime for the radiationless transition is inversely
proportional to the sum of Vi 2 contributed by all states
located within one energy unit around E •.
(e) An intramolecular radiationless transition can
take place in an isolated molecule provided that the
density of the intramolecular vibrational states around
E. is large enough. The energetic criterion is given by
V»E.
(f) A critical number of intramolecular vibrations
is required to make the intramolecular radiationless
process possible. Some very rough estimates are now
relevant: For an energy gap of 1 eV Robinson and
Frosh' 2 estimate F""10- 4 • For an intersystem crossing
we take v= 103 XF=0.1 cm-l for this gap. Therefore,
E«O.l cm-I so that ",,100 states per cm-l are required
to make the radiationless processes feasible. An estimate
of the vibrational densities of states in polyatomic
molecules characterized by equal vibrational frequencies
of 1000 cm-l yields the following results for this energy
gap: E-1 =0.06 cm for a triatomic molecule, E-1 =4 cm
for a four;-atom molecule, E-I =50 cm for a five-atom
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726
M. BIXON AND
molecule, e-1 =400 cm for a six-atom molecule and
e-1 =4XIQ5 cm for a lO-atom molecule. Thus, a fiveatom molecule is just on the limit of showing internal
conversion. An appreciably higher level density is
required for an internal conversion process. For the
same energy gap, we now take for the spin-forbidden
radiationless process v;:::::;F= 10--4 cm-1 so that E-1»
lO" cm. Therefore while for an energy gap of lO" cm-1
a medium size molecule (five atoms) will show internal
conversion, intersystem crossing will be observed only
in a 10-atom molecule.
(g) Provided that the energetic criterion is fulfilled,
the nonradiative decay will be unimolecular on a time
scale t<<1I,/E.
The experimental implications of the present theory
can be summarized as follows:
(a) Provided that the density of vibrational states
is sufficiently large to obey the energetic criterion
v»e, intramolecular radiationless transitions are expected to occur in an isolated polyatomic molecule. The
experimental observation of radiationless transitions
in benzene, naphthalene, and anthracene in the gas
phase is consistent with the theory, and does not
"contradict the laws of quantum mechanics."21
(b) Radiationless transitions in an isolated molecule
lead to the broadening of absorption bands. An expression for the bandwidth is given by Eq. (29). In view
of rotational broadening, one can look for the effects of
configuration interaction only if the linewidth exceeds
0.5 cm-1,46 which corresponds to radiationless transition
lifetimes shorter than 10-11 sec. Therefore, only internal
conversion processes between excited states are expected to show line broadening due to this effect. It is
well known that the higher excited singlet states of
aromatic molecules except the second singlet state of
azulene are diffuse.
(c) Exponential nonradiative decay is expected on
a time scale t<<1I,/ E. Internal conversion processes in
aromatic molecules satisfy this criterion. On the other
hand, the IB2•.-3B1u intersystem crossing process in the
benzene molecule violates this rule. It will be extremely
interesting to study the radiative decay curves for
this system and to look for deviations from exponential
decay at long times.
(d) The presence of a second triplet state in the
vicinity of the excited singlet greatly enhances the rate
of the intersystem crossing, leading to an appreciable
increase in the apparent coupling term between the
first excited singlet and the first excited triplet.
Some speculations are in order concerning the fate
of an electronically-vibrationally excited state resulting
from a radiationless transition in an isolated molecule.
In the case of internal conversion, fluorescence can
48
G. R. Hunt and 1. G. Ross,
J.
Mol. Spectry. 9, 50 (1962).
J. JORTNER
appear from the lowest electronic state. This fluores·
cence is more diffuse than the ordinary fluorescence
as it originates in transitions between highly vibrationallyexcited states. After the radiative transition, the
molecule remains in the vibrationally excited ground
state. If the molecule stays free, it will emit its excess
energy as infrared radiation. In view of the long radiative lifetime for vibrational transitions, vibrational
relaxation by collisions may take place. A similar
situation also prevails if, as a result of intersystem
crossing, the molecule reaches a vibrationally excited
triplet state. A search for infrared radiation from
rarified gases of organic molecules following uv excitation will be of considerable interest.
The present theory was developed for the case of an
isolated molecule which does not interact with the
solvent. We shall conclude our discussion with some
comments on radiationless transitions of a molecule
imbedded in a solvent. Considering a supermolecule
consisting of the molecule of interest and the close
solvent molecules, we have to discuss now the coupling
matrix elements arising from both intramolecular and
intermolecular vibrations. The vibronic matrix elements
J mn iO will now include also coupling of electronic states
by intermolecular vibrations. It is well known from
previous work on vibronic coupling in aromatic hydrocarbons that intramolecular skeleton vibrations rather
than intramolecular C-H vibration lead to an appreciable coupling between 7r electron states. It is thus
expected that the intermolecular vibration will have a
negligible effect on the coupling of the electronic states
of interest. Therefore, for internal conversion processes,
which are determined by vibronic coupling matrix
elements, the interaction terms between two electronic
states per unit energy interval will remain approximately the same as in the isolated molecule, so that no
appreciable change in nonradiative lifetimes is expected
in this case. Heavy solvent atoms will enhance spinorbit coupling in the supermolecule and will therefore
enhance the intersystem crossing process.
An interesting prediction obtained from the present
theory involves the effect of level shifts in the crystalline state. The first excited singlet 1B 2u state in the
anthracene molecule is located 600 cm-1 above the
second triplet state. In crystalline anthracene the
singlet state is red shifted by about 1800 cm-1 while the
triplet state is hardly affected so that the energy gap
between these close-lying states increases in the crystal.
Therefore the value of '11 is smaller in the crystalline
state and the fluorescence yield in this case is close to
unity.14 By increasing the temperature one gets population of excited vibrational states of the excited singlet
close to the triplet and intersystem crossing is enhanced.
This system exhibits the simplest example of a thermally
activated, radiationless process characterized by an
activation energy which is about equal to the energy
difference between the two electronic states.
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