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Theory of resonance Raman scattering and fluorescence from strongly
vibronically coupled excited states of polyatomic molecules
G. Stock and W. Domcke
Institut/ur Physikalische und Theoretische Chemie, Technische Universitiit Munchen, D-8046 Garching,
Federal Republic a/Germany
(Received 29 May 1990; accepted 9 July 1990)
A theoretical description of secondary emission from complex absorption bands of isolated
polyatomic molecules is developed. The strong non-Born-Oppenheimer coupling associated
with conical intersections of the multidimensional excited-state potential-energy surfaces is
included in a fully microscopic manner by solving the time-dependent Schrodinger equation
for appropriate model systems incorporating the most relevant electronic states and vibrational
modes. The effect of the large number of remaining vibrational modes and of the weaker
coupling with additional electronic states is modeled by phenomenological relaxation terms
(lifetime broadening and pure dephasing) in the framework of the density-matrix formalism.
Explicit eigenstate-free expressions for absorption, resonance Raman, and fluorescence spectra
are derived via density-matrix perturbation theory. The computational feasibility of the
resulting mixed microscopic/phenomenological theory is demonstrated for a simple threemode model of the vibronic coupling of the SI (mr*) and S2 ( 1T1r*) states of pyrazine. The
effect of excited-state vibronic coupling and ultrafast S2 ..... SI internal conversion on resonance
Raman and fluorescence spectra is analyzed on the basis of these model calculations.
I. INTRODUCTION
As is well known, the secondary emission (SE) of polyatomic molecules following resonant excitation of an electronic transition generally consists of Raman and fluorescence emission, which are of different physical origin. 1-7
Resonance Raman (RR) scattering is a coherent two-photon process, whereas the fluorescence is a two-step process,
where the emission occurs after phase relaxation (loss of
coherence) in the excited state. This phase relaxation (dephasing) may be a consequence of the coupling of the molecule to the environment (pressure broadening in gases, solvation effects in solutions, or phonon coupling in solids) or
may arise from intramolecular relaxation in isolated molecules, either on a single excited electronic potential surface
[intrastate vibrational relaxation (IVR) ] or on vibronically
coupled electronic potential surfaces [internal conversion
(lC) orintersystem crossing (lSC)].
In this paper we shall be concerned with ultrafast and
purely intramolecular electronic excited-state relaxation following resonant excitation of higher lying singlet states
Sn (n)2) of polyatomic molecules. As a general rule, SE
from Sn states of larger polyatomic molecules is strongly
quenched or absent8 as a consequence of rapid IC of the Sn
states into lower singlet states.
In contrast to the SI state, which typically exhibits well
resolved spectra in absorption and emission, at least for excitation near the band origin under supersonic jet conditions,
the higher singlet states typically exhibit largely featureless
absorption spectra and fluorescence is difficult to detect. 8
Therefore very little is known about the dynamical processes
occuring in these short-lived states. Only relatively recently,
with the availability of tunable UV lasers, systematic RR
5496
J. Chern. Phys. 93 (8).15 October 1990
spectroscopy of higher singlet states has become possible. 9 It
has been demonstrated that RR and fluorescence measurements yield valuable information on the vibrational as well
as relaxational dynamics of short-lived excited singlet
states. 10-15
The present work is part of a series of investigations
concerning the spectroscopic and dynamic effects of conical
intersections 16 of adiabatic electronic potential-energy surfaces in polyatomic molecules. It has been shown on the basis of microscopic quantum calculations for a number offewstate few-mode vibronic coupling models that conical
intersections are responsible for ultrafast (femtosecond) IC
processes. 17-23 Spectroscopic effects of conical intersections
have been discussed for photoe~~ctron spectra 17.19.20 and absorption spectra of neutral molecules. 19.21.24 A theoretical
description of femtosecond real-time pump-probe spectroscopy of the dynamics on conically intersecting surfaces has
been worked out recently.25-27 In the present work we address the problem of RR scattering and fluorescence from
conically intersecting excited singlet states in larger polyatomic molecules. In extension of earlier work by Meyer and
Koppel 28 we adopt a Liouville-space density-matrix formulation which allows couplings with additional electronic
states and additional vibrational modes to be included via
phenomenological decay and dephasing constants. This is
essential in order to account properly for the distinction
between RR and redistributed fluorescence emission in complex systems. In contrast to Ref. 28, where the emphasis has
been on the difficult problem of fluorescence emission within
the strongly vibronically coupled manifold in cations, we
shall be concerned with tpe somewhat simpler case of SE
from strongly vibronically coupled excited states (SI ,S2 )
0021-9606/90/205496-14$03.00
© 1990 American Institute of Physics
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5497
G. Stock and W. Domcke: Theory of resonance Raman scattering
into the unperturbed ground state (So),
Vibronic-coupling effects in RR spectra have been extensively discussed within the framework of the classic
Kramers-Heisenberg-Dirac (KHD) formula,29 see Refs.
30-33 and references given therein. More recently, the timedependent reformulation of the KHD formula 34 has been
extended to include vibronic coupling in the excited
states. 35,36
In these works only the pure (coherent) Raman emission has been considered, which arises from SE emission
processes which occur within the total dephasing time T2 of
the system (typically less than 100 fs for larger polyatomic
systems and molecules in solution). On the other hand, the
simultaneous description of both RR and redistributed fluorescence, including the description of line shapes of dispersed SE spectra, has been a subject of considerable interest
for more than two decades. In pioneering works of Huber l
and Hizhnyakov and Tehver2 collisional dephasing and dephasing due to phonon coupling in crystals have been treated, Subsequent investigations of the effect of dephasing on
SE spectra have been based on the model of stochastic modulation of energy levels,1,37-42 on the density-matrix description with phenomenological damping terms,3,4,6,43,44 or on
the so-called transform techniques. 2,45,46
All these formulations which include dephasing effects
are restricted to the case of excited-state dynamics on uncoupled harmonic Born-Oppenheimer (BO) surfaces or to the
case of uncoupled multilevel systems. The main motivation
of the present work is to extend the theory towards a realistic
description of strong non-BO coupling in excited electronic
states. When dealing with femtosecond IC processes, one
cannot a priori invoke perturbation theory for the non-BO
coupling and the Markovian approximation, which would
result in the usual exponential decay law. 47 Our approach is
rather to identify the strongest intramolecular interactions
which are responsible for the fastest decay processes. These
interactions are included in the model Hamiltonian for the
system and the quantum dynamics is treated (numerically)
exactly by solving the time-dependent Schrodinger equation. Additional relaxation effects (arising from the coupling of the electronic states and vibrational modes of the
model system with the remaining states and modes of the
molecule) are included in a phenomenological manner.
As a first application, we present the numerical realization of the theory for a simple three-mode model of the vibronic coupling of the SI and S2 states of pyrazine21,22 [see
Ref. 21 for citations of the extensive literature on
SI (mr* )-S2 ( 1T1T*) vibronic coupling in pyrazine]. While
this simplified vibronic-coupling model is certainly an incomplete model of the photophysics of this moderately large
molecule, its dynamics exhibits several nontrivial features
which are presumably generic for ultrafast IC processes
caused by conical intersections of potential-energy surfaces. 22 Apart from the demonstration of the technical feasibility of the computation ofRR and fluorescence spectra for
strongly coupled model systems with 104 or more energy
levels, we shall be particularly interested in elucidating the
basic effects of conical intersections on RR and fluorescence
spectra.
II. DENSITY-MATRIX DESCRIPTION OF RAMAN AND
FLUORESCENCE SPECTROSCOPY OF VIBRONICALLY
COUPLED SYSTEMS
A. Hamiltonian and relaxation terms
We are concerned with molecular systems where the SE
is affected by a strong vibronic interaction of the excited
electronic states. As a simple but nontrivial model we consider an electronic three-state model system with a few vibrational degrees of freedom. Let IqJo) denote the electronic
ground state and IqJI ), IqJ2) excited singlet electronic states
of a polyatomic molecule. We assume that the excited electronic states IqJI ) and IqJ2) are close in energy and coupled
by the strong intramolecular interaction V12 , whereas the
intramolecular coupling of IqJI ) and IqJ2 ) with the well separated ground state IqJo) may be neglected to a first approximation. In a diabatic electronic representation,48 the model
Hamiltonian H m reads
Hm
L
=
IqJi)Hi(qJil + [lqJI)V12 (Q)(qJ21 +h.c.],
;=0,1,2
(2.1 )
where
H; = TN + V;(Q)
(2.2)
denotes the multidimensional vibrational Hamiltonian in
the diabatic state IqJ;), TN being the nuclear kinetic-energy
operator. The symbol Q represents collectively the vibrational coordinates of the system.
Although being concerned with spontaneous emission,
we may use the semiclassical approach, treating the material
system quantum mechanically and the radiation field classically as an external field. I,3 We assume that either one or
both excited electronic states are coupled to the electronic
ground state via the radiation field, whereas the IqJI )-lqJ2)
transition is dipole forbidden. To simplify the analysis, we
shall often specialize to the case where the lower lying electronic state IqJI ) can be considered as dark in absorption and
emission, i.e., ~I = O. The model then describes the common situation of an optically bright excited state coupled to a
dark background state.
The total Hamiltonian including the interaction of the
model system with the field is thus written as
(2.3a)
L
Hint(t) = -
IqJ;)lL.o°If(t)(qJol+h.c.,
(2.3b)
;= 1,2
where lL.o (i = 1,2) are the nonvanishing transition dipole
moments and If (t) is the electric field. In the steady state the
electric field is given by the superposition of the laser cw
excitation and the spontaneous-emission field, i.e.,
If (t) =
E[e - ;"'/'
+ Ese -
;"',1
+ C.c.,
(2.4)
where WI' Ws are the frequencies and E[, Es are the polarization vectors of the laser field and the spontaneous-emission
field, respectively.
The excited-state dynamics of larger polyatomic molecules can impossibly be treated in a fully microscopic manner. Thus the model Hamiltonian (2.1) is meant to include
only the strongest interactions in the system which deter-
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5498
G. Stock and W. Domcke: Theory of resonance Raman scattering
mine the dynamical evolution on the shortest time scales.
We shall be particularly interested in situations where the
adiabatic potential-energy surfaces of the excited electronic
states exhibit a conical intersection. 16 The additional and
presumably weaker intramolecular couplings due to the remaining vibrational degrees of freedom and additional electronic states will be taken into account here via phenomenological relaxation terms in the density-matrix formalism. 49
In the Schrodinger picture the equation of motion for
the density matrix including relaxation terms is
ifz!....p(t) = [H,p(t)] - ifzr(t).
at
(2.5)
The physical effects to be described by the damping operator
r (t) are best characterized in the basis of eigenstates I"'v) of
the molecular model Hamiltonian H m' First, the vibronic
levels of the exited electronic singlet states IIPI ) and 11P2) are
immersed in the dense manifold of vibrational levels of the
electronic ground state and low-lying triplet states and can
decay by Ie and ISC. In the Markovian approximation,
these effects may be described by a population-decay constant lIT~V) for each vibronic level v of the excited-state
manifold. For the two lowest excited singlet states of pyrazine, which will serve as an example below, it can be concluded from the experimentally observed relatively weak energy
dependence of radiative lifetime and fluorescence quantum
yield within the SI and the S2 bands that T Iv) can be approximately taken as constant over the width of the SI and the S2
absorption band, respectively (see below). The phenomenological population decay can thus approximately be characterized by a single decay constant lIT[ within the energy
range of either electronic state. The value of TI appropriate
for the S2 absorption band is typically much shorter than the
T[ for the S[ band.
A second important relaxation mechanism is the decay
of coherences, that is, off-diagonal elements of the density
matrix. As is well known,49 the interaction of the material
system with a laser pulse leads, in first order in the field
strength, to a coherence of the levels of the electronic ground
state with the levels of dipole-allowed excited electronic
states. The decay of this particular coherence is reflected by
the homogeneous linewidth of the absorption spectrum. 49 It
can again be inferred from experimental data (absorption
profiles of the SI and the S2 states of pyrazine, see below)
that the coherence decay does not vary strongly over the
width of the SI and S2 absorption bands and can thus approximately be characterized by a single electronic dephasing constant lIT2 within the energy range of either state. As
is generally observed in larger polyatomic molecules, 8 the
value of T2 appropriate for the S2 absorption band is much
shorter than the T2 for the S[ band.
As is well known,49 the total dephasing rate lIT2 contains a contribution from the population decay of the excited-state manifold as well as from the so-called pure dephasing rate lIT* according to
lIT2 = 1I(2T[ )
+ lIT*.
(2.6)
A simple physical picture of the origin of the very fast pure
dephasing after S2 excitation is provided by the model of
stochastic modulation of the electronic excitation energy by
fluctuations in the large number of vibrational modes not
included in Hm .37-42 It should be noted that after the initial
ultrafast S2 -SI internal conversion process the microscopic model system possesses a large amount of vibrational energy (= 8000 cm - [ for the example of pyrazine to be discussed below) which will subsequently be distributed over
the large number of remaining vibrational modes. The stochastic-modulation model, which corresponds to the hightemperature limit of the coupling of the system with a heat
bath,37-42 thus appears appropriate.
To take account of these effects, we introduce the following damping operator in the basis of diabatic electronic
states:
nt) = iJ=I
lIT1 1IPi)Pij(t)(lPj l
1.2
+ I
lIT2 [llPo)Poi(t)(lPil+ h .c .],
(2.7)
i= 1,2
with
Pij (t) = (lPi Ip(t) IlPj)'
(2.8)
When Eqs. (2.5) and (2.7) are formally transformed into
the basis of eigenstates of H m' it is seen that this ansatz precisely represents the above-described phenomenological relaxation effects, namely a population decay of all excitedstate vibronic levels with the rate lITI as well as a dephasing
of the coherence between excited-state vibronic levels and
electronic-ground-state vibrational levels with the rate
lIT2 • It should be stressed that we do not assign different
phenomenological relaxation rates to the unperturbed diabatic electronic states. The latter would be inconsistent with
available spectroscopic data. For example, lines in the
S[ (mr*) absorption spectrum which arise by intensity borrowing from the strongly allowed SO-S2 ( 1m*) transition
would thus acquire the large homogeneous linewidth of the
S2 absorption band.
It is clear that there may exist, in general, additional
relaxation effects within the level structure of H m which are
not included in the ansatz (2.7), e.g., pure dephasing of coherences within the manifold of excited-state vibronic levels.
It should be stressed, however, that the model Hamiltonian
H in itself describes relaxational behavior of observables
which pertain to electronic or vibrational subsystems, in particular an ultrafast decay of the population of the higher
excited electronic state and a dephasing of the vibrational
motion (see Refs. 21 and 22 for a detailed discussion). We
believe that the ansatz (2.7) provides a phenomenological
description of the most essential relaxation effects which are
not contained in the microscopic treatment of the problem
and which are of relevance for the description of RR and
fluorescence spectra.
B. Measurable quantities: Rates, cross sections, and
quantum yield
It is well known from classical electrodynamics that the
rate of absorption or emission of photons by a medium, characterized by its polarization P(t), is given by49
W=
-
l/cu( if (t). :t(t) ),
(2.9)
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G. Stock and W. Domcke: Theory of resonance Raman scattering
where If (t) is the electric field and the brackets indicate
time averaging over an optical period. The sign convention is
such that W is negative in the case of absorption (disappearance of photons) and positive for emission (creation of photons). We prefer the concept of the rate of absorption or
emission of photons, because this rate is directly related both
to the .theoreticaltreatment (via the polarization) as well as
to experimentally measurable cross sections.
In the CASe oflinear absorption, the electric cw laser field
If 1 (t) and the linear polarization P( 1) (t) in the steady state
are given by
,z; (
(l.)
I
t) =
P O) (t)
E1e
- iw,l
+ C.C.,
convenient to change to the interaction picture. The operator A '(t) in the interaction representation is related to the
operator A in the SchrOdinger representation by the transformation (henceforth Ii = 1)
A '(t) = Ut(t)AU(t),
where
U(t)=e-
(2.11 )
W(OJ/) = 2 1m l-L*oE/P*(OJ/).
2 a
uA(OJ/) =-1T-OJ/W(OJ/),
2
The Liouville-von-Neumann equation. in the interaction representation reads
i i. P' (t) = [H int (t),p' (t)] - ir' (t),
where
Hint(t)
e
where a is the fine structure constant.
In the case of SE, we have to consider the total electric
field [Eq. (2.4)], consisting of the laser excitation field
(2.10) 'and the spontaneous emission field, yielding the SE
photon rate
+ W(U) (OJs,OJ/),
W(I)(OJ s' OJ I, ) = 2 1m "*OE
p*(I)(OJ s' OJ I,'
)
rs
= 2 1m I-L*OE/P*(U) (OJs,OJ/).
(2.14a)
USE
(OJs;OJ/) =
2
(2.15 )
When considering the total intensity emitted into all vibrational final states, i.e., the SE exCitation profile, we have to
perform the integration over the photon frequencies OJ s'
yielding the total emission cross section
UTE
(OJ/)
= -8 ~-a
9
~c
)2OJ/ 2:(OJ/ -
= +
+
r'(t)
1-L02°E *U22
2:
=
(2.21b)
(t)],
(2.21c)
(t)],
lITII~i)pij(t)<~jl
+ i=2:1,2 lITd I~O)P~i(t)(~il + h.c.].
(2.22)
In the derivation ofEQ.s. (2.21b) and (2.21c) the standard
rotating-wave approxi~ation (RWA)s3' has been employed. Assuming the molecular system to be initially in its
electronic ground state I~o) and the vibrational 81:ound state
10)
= 10,0... ),
the zero-order density matrix is given by
(t) = P' (t = 0)
= 1~0)·10)(01(~01·
(2.23)
The formal integration of Eq. (2.20) yields a system of coupled first-order integral equations
Poo(t) =Poo(O) - i
(2.17)
In order to calculate the pOlarization of the medium, we
have to solve the equation of motion (2.5) for the density
operator P (t). Having in mind to evaluate the time evolution'
of P ( t) using time-dependent perturbation theory, 52 it is
1-Lo2°E *U21
iJ= 1,2
3
C. Calculation of the nonlinear polarization using
density-matrix pertur:batlon·theory
1f*(t)eiwIU~ (t) [I-LOloE*UI1 (t)
and
Ey) 1I(21T) Wy (OJ/),
= UTE (OJ/)/UA (OJ/).
(2,21a)
H ~2 (t) = - 1f*(t)eiwIU~ (t) [1-Lo 1 0E*UI2 (t)
y
where Wy (OJ/) is the total photon rate scattered into a certain final state with the energy Ey •
Finally, we shall consider the excitation-energy dependent fluorescence quantum yield Y F , defined as
YF(OJ/)
H ~I (t)
p'(O)
(2.16)
l~i)H~<~ol +h.c.,
i= 1,2
(2.14c)
!.
~...!!..-)2OJ/OJ: W(OJs,OJ/).
9 U\e c
2:
= -
(2.14b)
P *(I,ll) (OJ s,OJ / ) are the time-independent parts of the nonlinear third-order polarization, depending on the frequency of
the spontaneous emission OJ s as well as on the frequency of
the excitation OJ / [see Eq. (2.28) below] . The corresponding
SE cross section is given by 51
(2.20)
at
(2.13)
3
W(II)(OJ.,OJ/)
(2.19a)
(2.19b)
(2.12)
Including all prefactors, the linear-response absorption
cross section at frequency OJ 1 is given byso
W(I) (OJs,OJ/)
iHml
is the unitary time-evolution operator of the model system
with electronic matrix elements
resulting in the photon ab~orption rate
==
(2.18)
(2.10)
= ..r-p( OJ/ ) e -/ru,t + c.C.,
W(OJs,OJ/)
5499
P ~·(t) =
II
-
il'
o
fdt'(H~IP;o -p~lH;o
(2.24a)
dt' e-1/T,(t-I')(H"/OPOi
-
, H' )
P/O
Oi
(i=1,2),
I
(t)
PDt·
= -
.
l
(2.24b)
,
H'
, -1/T2~I-t')(H' I
Jo d t e
OlPII - Poo 01
t
+ H ~2pil ),
(2.24c)
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5500
, ( )
P02 t
G. Stock and W. Domcke: Theory of resonance Raman scattering
= -
'lldt'
I
o
2
, H'
e -IIT (t-I')(H"02Pn - Poo
02
W(aJ/)
= -
'lldt'
I
o
e -IIT,(t-I')(H"lOPOl
, H' )
- PlO 02'
(2.24e)
(2.240
pij(l) =p;/(I).
Using standard time-dependent density-matrix perturbation
theory, 52 we shall solve the equations of motion successively
up to third order, yielding the populations P;; (I) (i = 1,2)
and the coherence P;2 (I) in even orders and the coherences
Po; (I) (i = 1,2) in odd orders.
D. Linear absorption
In order to evaluate the absorption rate (2.12), we have
to calculate the polarization
P(I)
= Tr[WJ(I)]
= 2 Re{~ol Trv [POI (I)]
+ ~02 Trv [POl (I) p,
(2.25)
where
(2.26a)
POI (I)
=
Uoo (I)POI (I) Uri (I)
+ Uoo (I)P02 (I) UTI (I),
(2.26b)
P02 (I)
=
2:
~.o·E*~ft.E roo dt e -
Jo
/.1=1,2
x e;(w/ Holt (01 (IP; Ie - ;Hml IlPj) 10),
(2.24d)
-
2 Re ..
Uoo (I)POI (I) Ur2 (I)
+ Uoo (I)P02 (I) UT2 (I).
(2.26c)
In Eq. (2.25) the trace refers to a complete set of electronic
and vibrational states (Tr) and to a complete set of vibrational states (Trv), respectively. Note that the vibronic coupling mixes the contributions of the two excited electronic
states to the polarization when we change back to the Schr6dinger representation.
In the case of linear absorption we have to calculate
p(l) (I). Insertion of p(l) (I) into Eq. (2.9) yields the absorption signal
2
W v(I)( aJs,aJ/ )-2R
e I~20·Es 121 ~20·E/ 1 1'1m
1_
00
I
I dt ll3 dt ll2 dt
0
3
0
2
0
IIT2
(2.27)
where €o is the energy of the electronic and vibrational
ground state,
In the derivation of the absorption signal we have employed the Condon approximation, i.e., we have assumed the
transition dipole matrix element ~Oi (i = 1,2) to be independent of the coordinate Q in the adopted diabatic electronic
representation.
E. Secondary emission in the case of a single bright
excited electronic state
In this section we are concerned with the special case
that only the 11P0 )-11P2) transition couples to the radiation
field, the 11P0 )-IIPI) transition being dipole forbidden or
only weakly allowed. This assumption leaves us with the
commonly encountered situation of SE from a bright electronic state coupled to a dark background state, which is a
physically more transparent problem. In this case no coherence PI2 (I) is induced by the laser field and no interference
effects will arise in the emission. Furthermore, the calculation simplifies considerably, because the number of terms
contributing to the SE signal is significantly reduced.
In order to calculate the SE signal, we have to evaluate
Eq. (2.9) using the nonlinear polarization p(3) (I). Restricting ourselves to contributions to p(3) (I) which are resonant
in all optical transitions (RW A), the nonlinear polarization
is obtained in the usual manner by solving successively the
equations of motion (2.24) up to third order and transforming the resulting coherences PoP) (I) back to the Schr6dinger
representation [Eq. (2,26) ]. Due to the special choice of the
damping matrix r(1) in Eq. (2.7), as discussed and motivated above, it is possible to employ the closure relation of the
electronic states (~; lIP; ) (IP; I = 1) and contract the terms
arising from the different electronic potential surfaces, yielding for the SE rates 54
W(aJs,aJ/) =
2: W~I)(aJs,aJ/) + W~II)(aJs,aJ/),
(2.28a)
v
1
l
l)IT2 -(t3l-(t-13+
e2-e
2)IT, -;(W,+Ev )(I-13)
1
e
(2.28b)
(2.28c)
where
ct>yO
(I) = (vi (1P21e -
iHm
l
(2.29)
11P2) 10)
are correlation functions representing the intramolecular excited-state dynamics and €y denotes the energies of the final
vibrational levels Iv) in the electronic ground state. As has been discussed by many authors for three-level systems and
multilevel systems,I-7,41-44 the SE signal consists of two physically distinct components. The first term W (I) (aJ"aJ/) is often
referred to as fluorescence-like emission, as it is directly proportional to the (second-order) excited-state excess population,
while the second term W(I1)(w s 'w/) is referred to as coherent Raman-like emission. It is seen that the proposed model [i.e.,
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5501
G. Stock and W. Domcke: Theory of resonance Raman scattering
the Hamiltonian (2.1) and the damping matrix (2.7) ] yields formally the same SE signal as is obtained for simple uncoupled
multilevel systems, although we are dealing here with a vibronic coupling problem. The whole information on the complex
dynamics of strongly vibronically coupled excited states is incorporated in the intramolecular correlation functions <llvO (t) .
Although our actual computations are based on the "eigenstate-free" correlation functions <llvO (f), it is instructive to
introduce for interpretative purposes the exact eigenstates of H m belonging to the vibronically coupled manifold 1q:>1 ), 1q:>2),
Hm ItPv) = E'vltPv).
(2.30)
It should be stressed that it is not possible to compute these eigenstates even for the simple three-mode model to be discussed
below, as this would require the diagonalization of Hamiltonian matrices of the order of 104 •
Inserting Eq. (2.30) into Eq. (2.28), the time integrations are easily performed, yielding the well-known result for the SE
photon rate scattered into a certain final state l - 7 •38.-43
~ ( 0 I(q:>21 tPa ) ( tPa Iq:>2 ) Iv) (vi (q:>zl tPp ) ( tPp Iq:>2 ) 10)
Wy ( It)s,lt)/ ) -_ 21 1-L20 °Es 121 1-L20 °E/ 12 ~
.
.
a.p (E'O + It)/ - E'a - I/T2 )(E'O + It)/ - E'p + zlTz )
X [11'6(lt)/
+ E'O -It)s -
E'.)
+
(E'a -
lIT*
(
E'p + i/TI ) E'y
+ It)s -
1
E'a - ilT2
+
E'y
+ It)s -
1
E'p
+ ilT2
)] . (2.31)
In this frequency-domain expression the two emission types mentioned above are clearly separated. The first term is recognized as the well-known KHD expression,29 describing the infinitely sharp Raman peaks that occur in the emission spectrum
according to the conservation of energy. The second term accounts for the redistributed fluorescence that disappears in the
absence ofpUfe dephasing (lIT* = 0).
Equations (2.28) and (2.31) completely describe the SE in our theoretical framework for the case of a single bright
excited state. A considerable simplification arises if we are only interested in the total emission intensity scattered into a
certain final state, i.e., the total emission excitation profile, that is obtained by performing the integration over the photon
energies3•7 •55
= 211'1 l-L2o °Es 1211-L~ooEd2 Re
1""
dt21t, dtle -
IIT-(t, -
t,le -
(t,+
tl)/2T,i(""+ €O)(t2 - t\)<llvO (t2 )<ll:«, (tl ).
(2.32)
The corresponding expression in the frequency domain reads
Wy
(TE)(
It)/
)-2 I ° 121. 0'12~ (OI(q:>2ItPa)(tPalq:>z)lv)(vl(q:>zltPp)(tPplq:>2)IO)
11' 1-L20 Es I-Lzo E/ ~
a.p (E'o +It)/ - E'a - i1T2 ) (E'o + It)/ - E'p + i1T2 )
[1+
(E'a -
2i1T*
]
E'p + i/TI )
= W~R)(lt)/) + W~F)(lt)/),
(2.33)
where W~R) (It)/) is the total Raman emission into a certain
final state for an excitation frequency It)/ and W~F) (It)/) is the
corresponding total redistributed fluorescence emission.
Transforming W~R)(lt)/) back into the time-dependent
picture, we finally obtain the time-dependent form of the
(It).-integrated) KHD expression, which has been made popular by Heller et al.: 34
W~R)(lt)/) = 211'11-L2ooEsI21I-LzooE/12
X
11"" dte- tIT2i(OJI+€O,t<llvO(t) IZ.
(2.34)
It is noteworthy that in the derivation ofEq. (2.34) we
did not assume the absence of pure dephasing. Thus the correlation function <llvO (t) is damped with the total dephasing
rate IITz , arising from the population decay as well as from
pure dephasing [Eq. (2.6) ].55 As Eq. (2.34l can also be
derived in the Schrodinger wave-function formalism, the
damping of the correlation function has often been misinterpreted as being due to pure lifetime effects. This explains
inconsistencies in the literature, arising from the interpretation of dephasing times of 10 fs or less as excited-state lifetimes. to
I
F. Secondary emission In the case of two coupled bright
excited electronIC states
So far we have studied in some detail the simplified situation of a single bright excited state coupled to dark background state. Consiciering the more general case where both
the transition Iq:>o )-[q:>1 ) as well as Iq:>o )-1q:>2 ) couples to the
radiation field, we have to solve the equation of motion
(2.20) with the full interaction operator (2.21) (I-LO! #0).
For the sake of brevity, we restrict ourselves here to the calculation of the total (It)s-integrated) emission rates. A
lengthy calculation yields for the total emission excitation
spectrum
W~TE)(lt)/) = 411'Re
L
l-LoioE=-l-LpoErl-LokoEsl-LlOoE/
iJ.k./= 1.2
(2.35)
where
<ll~)(t)
=
(vi (q:>;\e-
iHmt
19'})IO).
(2.36)
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G. Stock and W. Domcke: Theory of resonance Raman scattering
5502
Analogously, the Raman excitation profile [cf. Eq. (2.34)]
is given by
W~R)(ml)
= 21T.I
1=
I foo dte-
tIT2
TABLE I. Vertical electronic excitation energies, vibrational (v, 'V6a ) and
vibronic (v lOa ) coupling constants, and vibrational frequencies of the
S, -S2 vibronic coupling model for pyrazine. All quantities are in electron
volts.
/(OJI+€o)t
1,2 Jo
X [ ....0;·Es .... 20·Ei"<I>~~)(t)
E
+ ....o;.Es .... IO.Ei"<I>~~l)(t)] /2
(2.37)
As it is seen from Eqs. (2.35) and (2.37), we may expect
interference effects due to the interference of the emission of
the two bright electronic states.
III. MODEL SYSTEM AND COMPUTATIONAL METHODS
Recently a conical intersection of the SI and S2 surfaces
of pyrazine has been identified on the basis of semiempirical calculations and the analysis of experimental spectra. 2I
It has been shown that this intersection is responsible for
an ultrafas't (=20 fs) S2 -+SI internal conversion process
and causes a dephasing of the vibrational motion on a time
scale of a few hundred fs.22 We adopt here this model system as a representative of strong vibronic coupling of excited electronic states in polyatomic molecules. For a discussion of the extensive literature on the SI [IB3u (n1T*)]S2 [ IB 2u ( 1T1T*)] vibronic coupling problem we refer to Ref.
21 and the references given therein. The H; and V12 in the
molecular Hamiltonian (2.1) are specified as 21 .22
I(j - J..- mj aQj
a +J..-mjQJ)
2
Ho =
2
2
2
(j= 1,6a,lOa),
(3.1)
H; =Ho +E;
+ I KY)Qj
(i= 1,2,j= 1,6a),
(3.2)
j
VI2 = AQlOa'
(3.3)
where the Q 's are dimensionless normal coordinates and the
m's the associated harmonic vibrational frequencies. QlOa is
the (single) normal coordinate of pyrazine of B Ig symmetry
which can couple SI (IB3u) and S2 (IB2U) in first order, A
being the vibronic coupling constant. The Qj,j = 1, 6a, are
the normal coordinates of the totally symmetric modes VI
and V 6a of pyrazine. EI and E2 are the vertical excitation
energies and KY), i = I, 2,j = I, 6a, represent the gradients of
the excited-state potentials with respect to the A Ig coordinates and account for the shift of the eqUilibrium geometry
of the excited states in these coordinates. The parameters of
the model as determined in Ref. 21 for pyrazine are collected
in Table I.
In pyrazine, the SO-S2 (1T1T*) optical transition is
strongly allowed, the SO-SI transition is weakly allowed,
and the SI -S2 transition is dipole forbidden. The oscillator
strength of the SO-SI transition is about a factor of 10
smaller than the SO-S2 oscillator strength. 56
The computational methods have been described in detail previously,22,26 and are only briefly sketched. It has been
pointed out in Sec. II that the microscopic model Hamiltonian H m enters the calculations only via the correlation
functions <I> ~g) (t), which carry the complete information on
the intramolecular dynamics of the vibronically coupled ex-
v,
V 6a
'B 3U (mr*)
'B2u (1T1r*)
3.94
0.037
- 0.105
4.84
- 0.254
0.149
VIDa
0.262
(J)
0.126
0.074
0.118
cited electronic states. In order to compute these correlation
functions, we represent the model Hamiltonian in a directproduct basis which is built from the electronic states 19'1),
19'2) and harmonic oscillator states Iv) (j= I, 6a, lOa).
Converged results are obtained with a Hamiltonian matrix
of dimension 14 112 (see Ref. 26 for details). The field-free
time-dependent Schrodinger equation with the appropriate
initial condition is solved with a fourth-order predictor-corrector finite-differences method using a time step of 0.04
fs. 22 ,26 The resulting vector of time-dependent coefficients
represents directly the required correlation functions
<I>~g) (t).
The time-dependent propagation of the state vectors of
dimension = 104 and the subsequent multiple time integrations have been performed with a Silicon Graphics Iris
4D/20 workstation. The time-dependent state vectors have
been propagated up to 3 ps (75000 time steps) which is
necessary to obtain converged fluorescence intensities for an
excited-state lifetime of TI = 500 fs. The propagation requires about 5 CPU hours per symmetry (B3u or B 2u ' respectively). The first 40 correlation functions of each symmetry
were stored for subsequent evaluation. The absorption.spectrum [Eq. (2.27)] and the pure Raman signals [Eqs. (2.34)
and (2.37) ] are easily calculated using a standard FFT routine in a few CPU seconds. In this case there is only a single
integral to be performed which converges quickly (the integral is damped with l/T2 ). The double time integrations
required for the evaluation of the total emission intensities
[Eqs. (2.32) and (2.35)] can be performed in a single loop,
using the trapezoidal rule with a time step of 0.32 fs, requiring = 1 CPU minute per final state. The (synthesized) dispersed fluorescence spectra and the fluorescence excitation
spectra (see below) have been computed by wave function
propagation with simultaneous integrations. As we could
not find a trick to eliminate the triple time integration in the
dispersed fluorescence photon rate [Eq. (2.28b)], we solved
the problem numerically by brute force, which required
about 3 CPU hours per final vibrational state.
IV. COMPUTATIONAL RESULTS
A. Absorption spectrum and determination of
phenomenological relaxation rates
To get an impression of how the model is related to reality, we first consider the simplest response of the model
system, that is, the (linear) absorption spectrum. Figure
1 (a) shows the experimental gas-phase absorption spectrum
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5503
G. Stock and W. Domcke: Theory of resonance Raman scattering
of the two lowest singlet states SI (mT*) and S2 (-17"1T*) of
pyrazine adapted from Ref. 57. The SI absorption spectrum
shows well-resolved vibronic bands, and the spectroscopy
and decay dynamics of the SI (m1"*) state have been investigated in considerable detail, see Refs. 56, 58, 59, and references therein. As is typical for higher excited singlet states of
larger polyatomic molecules, 8 the S2 ( 11"11"*) absorption band
of pyrazine is manifestly diffuse, devoid of any resolvable
fine structure. The absorption profile exhibits some structure, though, in the form of three or four irregularly spaced
humps.
In the theoretical framework outlined in Sec. II the homogeneous line broadening of the absorption spectrum is
given by the total dephasing rate lIT2. As is seen directly
from the different linewidths of the SI (m1"*) and S2 ( 11"11"*)
absorption bands, the total dephasing time T2 cannot be regarded as constant within the entire energy range of the SI
and S2 absorption spectra. In the case of linear absorption,
however, it is easy to go beyond the assumption of a constant
dephasing rate. This may be seen as follows. Rewriting Eq.
(2.27) by introducing the exact vibronic eigenstates (2.30),
we obtain the Golden Rule expression convoluted with a
Lorenzian line shape function
W(w/)
= -
2 Re
L
i= 1.2
1~20·ErI2LI (01 (/Pi ItPv) 12
l
>-
I II
~
b
c
340
240
wavelength [nm 1
+r2 (1+e-(E,,-E'h,)IQ)-1
(4.2)
with r l = 0.002 eV, r 2 = 0.021 eV, €thr = 4.35 eV, and
== 0.012 eV. In this way we assign to the eigenstates within
the S2 banda total dephasing time T2 = 30 fs, while
T2 = 300 fs for the eigenstates within the S 1 state (the latter
value is a purely technical parameter which allows a rapid
computation and convenient plotting of the SI absorption
profile; the actual dephasing time for the SI state is considerably larger).
Figure 1(b) shows the absorption spectrum of the twostate three-mode vibronic-coupling model 21 computed with
an energy-dependent dephasing as discussed above. The
spectrum has been obtained by generating a high-resolution
absorption spectrum by evaluation ofEq. (2.27) with a large
value of T2 and subsequent convolution of this spectrum
with a Lorentzian with €v-dependent linewidth as specified
in Eq. (4.2). The ratio of the S2 and S) oscillator strengths
has been taken as 10 to 1.56
Note that this technical trick, which allows us to generate SI and S2 absorption bands with different phenomenological dephasing rates, cannot be transferred to the compu-
a
-
220
We may now generalize Eq. (4.1) by introducing a dephasing rate lIT2 which depends on the energy €v of the vibronic
eigenstates. Motivated by the experimental observation60
that the fluorescence lifetime and the fluorescence quantum
yield of pyrazine show an abrupt decrease near the S2 band
origin, but are approximately constant over the energy
ranges of the SI and S2 absorption bands, we approximate
the energy dependent dephasing rate by a smoothed step
function
=r
S1 Inrr*)
v
(4.1 )
lIT2
a
FIG. 1. Experimental gas phase (Ref. 57) (a) and calculated (b) absorption spectrum of the S, (mr*) and Si ( 1T1r*) states of pyrazine.
tation of SE spectra. If the SI and the S2 absorption bands
are reasonably well separated, however, the primary excitation in a RR or fluorescence experiment will involve vibronic
levels of either the SI or the S2 absorption band with the
corresponding constant phenomenological relaxation rates
(cf. the discussion in Sec. II A). Therefore, consideration of
level-dependent relaxation rates is not required for SE calculations (see below) .
Considering the simplicity of the model, the theoretical
simulation is in reasonable agreement with the experiment
(no attempt has been made to obtain a quantitative fit to the
experiment). The calculation underestimates the density of
levels in the high-energy region of the SI band which arises
from the weak excitation of vibrational modes not included
in the three-mode model. The strongly homogeneously
broadened S2 band exhibits a series of weak and broad
humps. These structures are a consequence of the S) -S2
conical intersection and cannot be assigned in terms of unperturbed vibrational modes on the S2 surface. 21
From the analysis of the absorption spectrum we have
thus obtained an estimate for the total phenomenological
dephasing rate in the energy range of the S2 state. In order to
obtain information on the relative contribution of populationdecay (liT) andpuredephasing (lIT*) to the ultrafast dephasing, we have to consider fluorescence properties
of the system. As will be discussed in Sec. III C below, we
estimate an electronic population decay time of TI = 500 fs
in the energy range of the S2 state from the analysis of the
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G. Stock and W. Domcke: Theory of resonance Raman scattering
5504
fluorescence quantum yield. The result that TI is considerably larger than T2 (500 vs 30 fs) implies that the homogeneous broadening of the S2 absorption spectrum arises almost completely from pure dephasing, i.e., T2 = T *= 30 fs.
(/)
RR spectra of pyrazine in near resonance with the SI or
S2 states have been reported by Hong and Jacobsen 16 and by
Suzuka et al. 57 The former authors have performed a theoretical analysis of the RR spectra in terms of a single-mode
vibronic-coupling calculation. Here we shall present results
obtained with the nonseparable three-mode model of Ref.
2!. Having determined the phenomenological relaxation
rates lITI and lIT2 for the S2 state from absorption and
quantum yield data as discussed above, the model contains
no adjustable parameters.
Figure 2 shows experimental RR spectra of pyrazine
adapted from Ref. 57. The spectrum of Fig. 2(a) has been
obtained with an excitation wavelength AI = 337 nm in solid
pyrazine and is preresonant to the SI state. 57 The spectrum
of Fig. 2(b) has been obtained with AI = 266 nm, approximately resonant with the S2 band origin, in pyrazine vapor.57 It is seen that the preresonant SI Raman spectrum
[Fig. 2(a)] is dominated by the fundamentals of the SI-S2
coupling mode VlOa and the three totally symmetric modes
VI' V 6a , V9a' The excitation ofv6a is surprisingly weak in view
of the fact that V 6a is the dominant progression-forming
mode in the S I absorption spectrum. It is conspicuous that in
the RR spectrum of the S2 state [Fig. 2(b)] the coupling
mode V lOa is completely absent. All peaks can be assigned as
the fundamentals, overtones, and combination levels of the
two totally symmetric modes VI and V 6a •57
The RR spectrum for AI = 337 nm has been computed
from the expression (2.37) for the pure Raman signal (the
detuning of the excitation wavelength from the SI absorption band eliminates any fluorescence emission). As a conse-
a
'At =337 nm
9a
>.(/)
C
C
b
'At = 266 nm
100
....>-
B. Raman spectra
<II
.-
Af=337 nm
1.60
6a
2·1
Stokes shift [cm- 1)
FIG. 2. Experimental Raman spectra ofpyrazine (Ref. 57), obtained (a)
with an excitation wavelength AI = 337 nm in solid pyrazine (preresonant
totheS, state) and (b) with an excitation wavelength AI = 266 nmin pyrazine vapor (resonant with the S2 state).
c(II
.-
2.100
C
1
3.100
A
3000
100+1
\
100.60
2.:0)
1 +60 A
2000
60
A
1000
Stokes shift [cm- 1j
FIG. 3. Calculated pure Raman spectrum, obtained with an excitation
wavelength AI = 337 nm, preresonant to the S, state of pyrazine. To simulate a typical experimental resolution, the Raman lines have been convoluted with a Lorentzian of 5.2 cm ~, FWHM.
quence of the comparatively large oscillator strength of the
S2 state, it is essential to include the contribution of both the
and S2 state to the Raman amplitude. The calculated
Raman spectrum for AI = 337 nm, convoluted with a Lorentzian of 5.2 cm - I FWHM in order to simulate a typical
experimental resolution, is shown in Fig. 3. The calculation
reproduces the expected excitation ofvlOa and 2Xv lOa , but
the intensity of these lines relative to the intensity of the
totally symmetric fundamentals is too strong. This indicates
that the SI-S2 vibronic coupling strength is somewhat overestimated in the model of Ref. 21. The calculation also reproduces the unexpectedly weak excitation of the V 6a fundamental. A closer analysis shows that this arises from a
destructive interference of the SI and S2 Raman amplitudes
at AI = 337 nm.
.
In the calculation of the Raman spectrum at 266 nm we
may neglect the Raman amplitude of the SI state, as this
transition is fairly off-resonant and moreover only weakly
allowed. Because the 266 nm excitation is resonant with the
S2 state, we have to include both Raman and fluorescence
emission according to Eq. (2.28). The resulting theoretical
SE spectrum is shown in Fig. 4 (a). Here and in the following
the Raman lines are represented as Lorentzians with 10.6
cm - I FWHM. In agreement with experiment [Fig. 2(b)]
the coupling mode V lOa is (almost) absent in the 266 nm
spectrum. The calculated excitation of the fundamental and
the overtones of VI is somewhat too strong, indicating that
the coupling constant Kj2) of VI in the S2 state is somewhat
overestimated by the model of Ref. 21.
The presence of incoherent fluorescence emission at
AI = 266 nm is reflected by the smooth underlying background in Fig. 4(a). The pure Raman spectrum computed
from Eq. (2.34) is shown in Fig. 4(b) for comparison. It is
seen that in this case the much simpler and cheaper (by a
factor of = 103 ) computation of the pure Raman intensities
is sufficient for the interpretation of the experimental SE
spectrum. The effect of the SI -S2 vibronic coupling on the
relative intensities of coherent Raman emission and incoherent fluorescence emission will be discussed in more detail
below.
SI
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5505
G. Stock and W. Domcke: Theory of resonance Raman scattering
a
a
d
b
e
Xt = 266 nm
If)
C
.....Q.o
.!:
b
60
2.1
2.1+ 60
2.60
3000
2000
1000
Stokes shift [cm-')
FIG. 4. Calculated SE spectrum (a) and pure Raman spectrum (b) for
approximately resonant excitation of the S, state ofpyrazine. The Raman
lines are represented as Lorentzians with 10.6 cm - I FWHM.
In summary, the two-state three-mode vibronic coupling model of Ref. 21, augmented by phenomenological relaxation terms, yields an acceptable qualitative description
of the Raman spectra for resonant or near-resonant excitation of the SI and S2 states of pyrazine, considering that no
parameters have been adjusted. Improvements of the model
towards a more quantitative description of the RR spectra,
preferably in conjunction with accurate ab initio calculations
ofthe SI and S2 potential-energy surfaces, will be the subject
of future work.
c. Analysis of the effect of vlbronic coupling on Raman
and fluorescence spectra
Having seen that the simple vibronic-coupling model of
Eqs. (2.1) and (3.1)-(3.3) augmentedbythephenomenological nonradiative damping terms (2.7) accounts qualitatively for the main features of the available experimental
absorption and Raman spectra of pyrazine, we wish to analyze now in more detail to what extent the SE of the model
system is affected by the strong vibronic coupling in the excited-state manifold. The most interesting feature of the
model is the conical intersection of the S2 and SI potentialenergy surfaces and the resulting femtosecond S2-81 interval conversion process after S2 excitation. We shall therefore
restrict the discussion in the following to excitation wavelengths which are resonant or nearly resonant with the S2
absorption band.
Let us first consider the excitation profiles of the fundamentals and the combination level of the two totally sym-
35
40
45
35
40
45
excitation energy [1000 cm-')
FIG. 5. Excitation profiles ofthe fundamentals and the combination level of
the two totally symmetric modes VI and V 6a of pyrazine. The left panels
show the excitation profilesofv6a (a), VI (b), and VI + V 6a (c) ofthe vibronically coupled system, the right panels display the corresponding profiles of the uncoupled system (d-f). The full curves represent the total-emission profiles, the dotted lines the pure Raman excitation profiles.
metric modes VI and V6a' In Fig. 5 the excitation profiles of
V 6a (a),v I (b),andv i +v6a (c)ofthevibronicallycoupled
system are contrasted with the corresponding profiles of the
uncoupled (A = 0) system (d-f). The full curves represent
the total-emission profiles, obtained from Eq. (2.32), the
dotted lines the pure Raman excitation profiles, obtained
from Eq. (2.34). In the uncoupled case (right-hand side of
Fig. 5) the total-emission signal has been scaled down by a
factor of 1/4 relative to the pure Raman signal for clarity.
The emission excitation profiles of the coupled system
exhibit irregularly spaced broad structures which are similar
to but more pronounced than the corresponding structures
in the absorption profile [Fig. I (b) ]. As already emphasized above for the absorption profile, these structures cannot be assigned in terms of unperturbed vibrational modes of
the S2 surface. The enhancement of the structure in the Raman profiles as compared to the absorption profile appears
to be a typical effect for strongly homogeneously broadened
excited states, see for example, Refs. 42 and 62. The global
shapes of the total-emission and the pure Raman profiles are
seen to be roughly the same in the coupled system.
The excitation profiles of the uncoupled (A = 0) system, on the other hand; exhibit a more densely spaced and
more regular series of homogeneously broadened peaks
which result from progressions in V 6a and VI' In the uncoupled system the intensity of fluorescence relative to the Ra-
J. Chem.
Phys.,to
Vol.
93,license
No.8, 15
1990
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5506
G. Stock and W. Domcke: Theory of resonance Raman scattering
man intensity is larger by about a factor of 4 than in the
coupled system. It is also seen that in the uncoupled system
the profiles of total emission and pure Raman emission are
notably different.
We now proceed to a comparative discussion of emission spectra for a fixed excitation wavelength, choosing
Al = 256 nm, which corresponds to excitation at the center
of the S2 absorption band of pyrazine. Figure 6 shows the SE
spectrum (a) and the pure Raman spectrum (b) of the coupled system. When comparing these spectra with the corresponding spectra for Al = 266 nm (excitation near the S2
band origin) in Fig. 4, we observe enhanced excitation of
combination levels and overtones of the totally symmetric
modes in Fig. 6. The relative contribution of redistributed
fluorescence to the SE has increased, rendering it more difficult to extract Raman intensities from the SE spectrum.
Figure 7 shows the SE (a) and pure Raman (b) spectra
at Al = 256 nm for the uncoupled system. As has already
been pointed out above, the redistributed fluorescence
strongly dominates in the uncoupled system, leading to a
pronounced and broadly structured background. It is seen
that the intensities of the pure Raman lines, shown separately in Fig. 7 (b), cannot easily be extracted from the actual SE
spectrum in this case. The result of Fig. 7 (a) has immediate
implications for the interpretation of the absorption [Fig.
l(a)] and RR [Fig. 2(b)] spectra of pyrazine: it follows
that the observed large homogeneous broadening of the S2
absorption band cannot be reconciled with the observed low
and flat background of the RR spectrum S7 in a theory which
a
a
At =256 nm
b
2.1
60
....>.
If)
c
....cCII
2.1.60
1.60
c
3000
At =256 nm
2.60
1+2.60
2000
1000
Stokes shift [cm-')
FIG. 7. Calculated SE spectrum (a) and pure Raman spectrum (b) of the
uncoupled system for excitation at the center of the S2 absorption band of
pyrazine. The lowest panel (c) shows a synthesized SE spectrum obtained
by a simplified numerical procedure (see the text). It is seen to be in good
agreement with the full computation in (a).
....'iii>.
c
CII
~
b
2.1
1.60
2.1+60
60
1.2.60
3000
2000
1000
Stokes shift [cm-1)
FIG. 6. Calculated SE spectrum (a) and pure Raman spectrum (b) of the
coupled system for excitation at the center of the S2 absorption band of
pyrazine (A.I = 256 nm).
neglects S) -S2 vibronic coupling. The strong vibronic coupling of the S2 state with the nearly dark S) state leads,
interestingly, to a significant simplication of the SE spectrum in the range of small Stokes shifts (fundamentals and
low overtones and combination levels).
As noted above, the computation of the dispersed SE via
numerical evaluation of the triple time integral in Eq.
(2.28b) is quite expensive, in particular if the complete set
(= 104 ) of final vibrational levels is to be considered. We
therefore propose an approximate, but much less expensive,
computation of the SE spectrum as follows. The total (integrated with respect to (i)s) Raman and fluorescence intensity
for each final vibrational level may be obtained from Eqs.
(2.34) and (2.32), respectively. The fluorescence emission
is then represented by Lorentzians of width lIT2 , while the
Raman emission is represented by narrow Lorentzians determined by some experimental resolution (10.6 cm -) in
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G. Stock and W. Domcke: Theory of resonance Raman scattering
a
Af =256 nm
)( 1/2
20
25
30
35
40
energy [1000 em- 1)
FIG. 8. Dispersed SE spectrum of the coupled (a) and uncoupled (b) model systems of pyrazine, obtained with an excitation wavelength .1/ = 256
nm. The dashed line in (a) displaysthe fluorescence emission of the S2 state
only, whereas the full line represents the SE of both excited states.
the present case). A "synthesized" SE spectrum is then obtained by summing over all final vibrational levels. This procedure reduces the computing time required for the evaluation of the dispersed SE spectrum by a factor of about 103 .
Such a synthesized SE spectrum for the uncoupled system in the range of small Stokes shifts is shown in Fig. 7 ( c ) .
It is seen that the result of the exact numerical computation
[Fig. 7 (a) ] is satisfactorily reproduced. This result justifies
the use of the above-described simplified procedure for the
computation of the complete dispersed SE spectrum, which
otherwise would be prohibitively time consuming (at least
on a small work station as used in the present work).
The complete dispersed SE spectra (including the Rayleigh line) are shown in Figs. 8(a) (full line) and 8(b) for
the coupled and the uncoupled system, respectively. The
pronounced effect of the SI-S2 vibronic coupling on the
overall shape of the SE spectrum is obvious. The emission
spectrum of the coupled system exhibits a strongly red shifted ("'" 11000 cm- I ), broad (FWHM "",7500 cm -1) and
nearly structureless redistributed fluorescence which is
clearly separated from the Rayleigh and Raman-like lines in
the blue part of the spectrum. The pronounced red shift reflects the fact that the redistributed fluorescence emerges
from the relaxed system, that is, from highly excited vibrationallevels of the SI surface. According to the Condon
principle, these fluorescence transitions terminate perferentially in high vibrational levels of the So surface, resulting in
the observed red shift of the spectrum.
5507
In, the computation of the dispersed ft.uorescence spectrum of the coupled system [full line in Fig. 8(a)] the ft.uorescence emission from both excited states has been included [see Eqs. (2.35) and (2.37)]. In the uncoupled system
[Fig. 8(b)] all the SE after S2 excitation arises necessarily
from the S2 state. The spectrum obtained with inclusion of
fluorescence from the (diabatic) S2 component only in the
coupled system is indicated by the broken line in Fig. 8 (a) .
As remarked above, the contribution of the SI oscillator
strength is negligible for the RR lines at excitation wavelengths which are resonant or near resonant with the S2
state. This does not hold, however, for the redistributed fluorescence. As is seen from Fig. 8(a), fluorescence originating from the diabatic SI transition moment contributes significantly to the total SE after S2 excitation, despite the small
transition moment of the S) state. This reflects the fact that
after the ultrafast S2 .... S) IC process most (about 70%) of
the electronic population is found in the diabatic SI
state. 2l ,22
The integrated area of the fluorescence spectrum of the
coupled system is reduced by a factor of about 3 in comparison with the uncoupled system. The SI -S2 vibronic coupling
thus causes not only a redistribution, but also a quenching of
the fluorescence. The mixing of the S2 state with the less
bright SI state diminishes the radiative rate of the coupled
system, and thus impedes the competition of the fluorescence with the nonradiative electronic population decay represented by T 1 • The coherent RR emission, on the other
hand, occuring on a time scale T2 4: TI , is much less affected
bytheSI -S2 vibroniccoupling. TheeffectoftheSI -S2 conical intersection and the resulting ultrafast IC process is thus
particularly apparent in the complete dispersed SE spectrum. The pure RR spectrum, in contrast, is not particularly
sensitive to the strong SI -S2 vibronic coupling, as is seen
from the comparison of Figs. 6 (b) and 7 (b). It appears that
the intensity of the Raman lines is essentially determined by
the gradients (or, equivalently in harmonic models, geometry shifts) of the excited-state potentials along the totally
symmetric modes. This quantitative result for the present
model system is in agreement with earlier qUalitative reasoning based on the time-dependent picture of RR scattering. IO,15,34,35,63 The practical implication is that, independent of complications due to strong vibronic coupling and
fast IC processes in excited states, the gradients ofthe excited-state potentials with respect to the totally symmetric
modes may be directly extracted from experimental RR intensities.
We finally turn to a discussion of the effect of S 1-S2
vibronic coupling on the fluorescence quantum yield. In the
absence of vibronic coupling with nearby electronic states,
the quantum yield for. resonant excitation of the S2 state can
be estimated as8 ,I4,6O
YF
= TIIT}fl,
(4.3)
where Tk ) is the radiative lifetime of the S2 state. From the
oscillator strength of the S2 -So transition of pyrazine 56 we
determine T}fl = 16.3 ns. In the presence of strong S)-S2
vibronic coupling the simple expression (4.3) is not valid
owing to the vibronically induced fluorescence quenching
2
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5508
G. Stock and W. Domcke: Theory of resonance Raman scattering
mentioned above. We have therefore computed the fluorescence quantum yield of the coupled system from the definition (2.17), that is, the ratio of the total SE cross section to
the absorption cross section. Comparison with the experimental result YF ::::: 10 - 5 at the center of the S2 absorption
band of pyrazine 60 allows us to determine the phenomenological model parameter TI for the S2 absorption band of
pyrazine ( YF would be strictly unity for TI = T2 = 00 ). We
thus estimate, as anticipated in Sec. IV A, TI "'" 500 fs. Application of Eq. (4.3), that is, neglecting the strong SI -S2
vibronic mixing, would have resulted in a much shorter population-decay lifetime, namely TI "'" 160 fs. This result illustrates that it is essential to take account of excited-state vibronic mixing in the estimation of phenomenological
non-radiative decay rates. This point has also been emphasized recently by Nickel and Hertzberg for a different molecular system. 14
v. DISCUSSION AND CONCLUSIONS
A theoretical framework for the analysis of Raman and
fluorescence emission from strongly vibronically coupled
excited electronic states of isolated polyatomic molecules
has been outlined. The main emphasis of our work is on a
truly microscopic description of the ultrafast and strongly
non-Markovian IC dynamics of conically intersecting higher excited singlet states of polyatomic systems, based on the
(numerically) exact solution of the time-dependent Schr6dinger equation for suitable multimode vibronic-coupling
models. 18-23 Since any such microscopic model must necessarily be incomplete for larger polyatomic molecules, in particular with respect to the dynamical evolution at longer
time scales, it is necessary to include additional relaxation
effects in a phenomenological manner. Adopting the density-matrix formulation, which allows pure dephasing effects to be incorporated in addition to lifetime broadening
effects,49,52 and assuming phenomenological relaxation
rates which are approximately constant (that is, energy independent) within a given electronic absorption band, we
have formulated a computationally manageable mixed microscopic/phenomenological theory, which should provide
a powerful tool for the description of complex relaxation
phenomena in excited electronic states. The present formulation extends existing eigenstate-free density-matrix formulations of RR scattering and fluorescence 42 .44 by including
explicitly and nonperturbatively the strong non-BO phenomena associated with conical intersections in excited electronic states.
In order to illustrate the general ideas, we have performed numerical calculations of absorption, RR, and dispersed fluorescence spectra for a simple three-mode model
of the vibronic coupling of the SI (mr*) and S2 (1T1r*) states
ofpyrazine which has been proposed earlier. 21 As has been
shown in detail elsewhere,22 this simple model describes an
ultrafast (femtosecond) S2 --+SI IC process. The additional
phenomenological total dephasing (lIT2 ) and lifetime
broadening (lIT I ) rates for the S2 state have been determined by comparison with the experimental S2 absorption
profile and fluorescence quantum yield, respectively. It has
been shown that the resulting model accounts on a qualita-
tive level for the main features of the available absorption
and RR spectra of pyrazine.
The essential effects of excited-state vibronic coupling
on the RR and fluorescence spectra found for the present
model system can be summarized as follows: (i) The coupling mode v lOa' which is the only vibrational mode of pyrazine which can couple SI (mr*) and S2 emr*) in first order,
is strongly active in the RR spectrum oflower (SI ) state, but
is virtually absent in the RR spectrum of the higher (S2)
state. (ii) The shape of the diffuse absorption profile and RR
excitation profiles of the S2 state is strongly affected by the
vibronic coupling: the SI -S2 conical intersection results in
the appearance of widely and irregularly spaced broad structures which cannot be assigned in terms of unperturbed vibrational modes on the S2 surface. (iii) The vibronic coupling causes a suppression of the broad redistributed
fluorescence background in the Raman region of the SE
spectrum observed after S2 excitation. The main part of the
fluorescence from S2 emerges strongly red shifted ( "'" 11 000
cm - 1 ). (iv) The mixing of S2 with the less efficiently fluorescing SI state is reflected by a significant decrease of the
fluorescence quantum yield.
Most of these effects are presumably rather general features of SE from optically allowed excited singlet states
which are coupled via a conical intersection to a lower lying
dark (or only weakly allowed) singlet state in larger polyatomic molecules. The relative intensitivity of the pure RR
signal to the ultrafast IC process of the higher excited state
reflects the fact that the coherent Raman emission is restricted to ultrashort times by the very fast total dephasing of the
S2 -So coherence. The Raman signal carries, therefore, relatively little information on the complex excited-state dynamics. It provides a simple means, however, to determine the
gradients (or shifts of the equilibrium geometry) of the potential energy surface of short-lived excited states with respect to totally symmetric normal coordinates. 10,15,34,35,63
The redistributed fluorescence emission, on the other hand,
arises from the dephased and electronically relaxed system
and is therefore significantly affected by the non-BO dynamics in the excited states. The dramatic red shift of the center
of the fluorescence band predicted by the present quantum
dynamical calculations, for example, reflects most directly
the existence ofa S2 --+SI IC process. We have also seen that
consideration of the excited-state vibronic coupling is essential when estimating electronic population decay times from
experimentally observed quantum yields.
RR scattering and fluorescence emission thus provide
largely complementary information on the excited-state dynamics. The consistent prediction of both RR scattering and
fluorescence emission from short-lived excited states of polyatomic molecules on an ab initio basis is still a major challenge for the theory. The aim of such a theory should be to
include as much of the dynamics as possible in the microscopic model Hamiltonian H m' with electronic energies and
coupling constants derived from ab initio computations. We
believe that a rigorous treatment of the ultrafast non-BO
dynamics associated with conical intersections of excited
state surfaces, though computationally laborious, is essentially for the understanding of the spectroscopy of short-
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G. Stock and W. Domcke: Theory of resonance Raman scattering
lived excited states in polyatomic molecules. Present-day
computer technology allows the treatment of the time-dependent quantum dynamics of vibronic-coupling models
with up to about ten nonseparable vibrational degrees of
freedom (see Ref. 20 for an example). Once the correlation
functions ofEq. (2.29) have been obtained for the appropriate final vibrational levels, the expressions given in Sec. II
provide a comparatively simple means to evaluate the RR
and fluorescence signals, involving only two phenomenological parameters (T\ and T2 ). While the ansatz (2.7) for the
relaxation terms, in particular the assumption of energy-independent T\ and T2 within a given absorption band, is certainly an oversimplification, it may serve as a useful reference for future more ambitious treatments of dephasing and
lifetime broadening effects in photon scattering from polyatomic molecules.
ACKNOWLEDGMENTS
The authors would like to thank Rudolf Schneider for
stimulating discussions and help with the numerical calculations. This work has been supported by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie.
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