# Photochemistry of hydrogen-bonded aromatic pairs: Quantum dynamical calculations for the pyrrole-pyridine complex.

**ABSTRACT** The photochemical dynamics of the pyrrole-pyridine hydrogen-bonded complex has been investigated with computational methods. In this system, a highly polar charge-transfer state of (1)pipi* character drives the proton transfer from pyrrole to pyridine, leading to a conical intersection of S(1) and S(0) energy surfaces. A two-sheeted potential-energy surface including 39 in-plane nuclear degrees of freedom has been constructed on the basis of ab initio multiconfiguration electronic-structure data. The non-Born-Oppenheimer nuclear dynamics has been treated with time-dependent quantum wave-packet methods, including the two or three most relevant nuclear degrees of freedom. The effect of the numerous weakly coupled vibrational modes has been taken into account with reduced-density-matrix methods (multilevel Redfield theory). The results provide insight into the mechanisms of excited-state deactivation of hydrogen-bonded aromatic systems via the electron-driven proton-transfer process. This process is believed to be of relevance for the ultrafast excited-state deactivation of DNA base pairs and may contribute to the photostability of the molecular encoding of the genetic information.

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**ABSTRACT:**Conical intersections (CIs) play an important role in nonadiabatic molecular processes. Characterizing and localizing them is important for describing and controlling electronic energy flow in molecules. It is known that no CI appears in free diatomic systems. In earlier works (Moiseyev et al 2008 J. Phys. B.: At. Mol. Opt. Phys. 41 221001, Sindelka et al 2011 J. Phys. B.: At. Mol. Opt. Phys. 44 045603) it was pointed out that CIs can be formed both by standing and running laser waves even in diatomics. The energetic and internuclear positions of these CIs depend on the laser frequencies, while the strength of their nonadiabatic couplings can be modified by the field intensities. In this work, we calculate the topological or Berry phase of the light-induced CI in the Na2 molecule. The presence of this phase is a clear fingerprint of the laser-induced CI. In addition, we perform a detailed study of the wavepacket propagation and discuss effects which reflect the significant presence of the laser-induced CI.Journal of Physics B Atomic Molecular and Optical Physics 08/2011; 44(17):175102. · 2.03 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this perspective article, we focus on the photochemistry of five-membered nitrogen containing heterocycles (pyrrole, imidazole and pyrazole) in clusters. These heterocycles represent paradigmatic structures for larger biologically active heterocyclic molecules and complexes. The dimers of the three molecules are also archetypes of different bonding patterns: N-H···π interaction, N-H···N hydrogen bond and double hydrogen bond. We briefly review available data on photochemistry of the title molecules in the gas phase, but primarily we focus on the new reaction channels opened upon the complexation with other heterocycles or solvent molecules. Based on ab initio calculations we discuss various possible reactions in the excited states of the clusters: (1) hydrogen dissociation, (2) hydrogen transfer between the heterocyclic units, (3) molecular ring distortion, and (4) coupled electron-proton transfer. The increasing photostability with complexity of the system can be inferred from experiments with photodissociation in these clusters. A unified view on photoinduced processes in five-membered N-heterocycles is provided. We show that even though different deactivation channels are energetically possible for the complexed heterocycles, in most cases the major result is a fast reconstruction of the ground state. The complexed or solvated heterocycles are thus inherently photostable although the stability can in principle be achieved via different reaction routes.Physical Chemistry Chemical Physics 07/2011; 13(26):12123-37. · 4.20 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we explore the formation and the photophysical properties of the scarcely studied open hydrogen bonded aggregates of 7-Azaindole, 7AI. Thus, we have analyzed the influence that the increase of the 7AI concentration and the decrease of the temperature have on the 7AI photophysics. To help the interpretation of the results, the 7AI-Pyridine system has been used as the model for the analysis of the photophysical properties attributable to the open Npyrrolic - HNpyridinic hydrogen bonded aggregates. Also, the hydrogen bond interactions have been studied by means of the atom in molecule approach from the Bader theory. Experimental and theoretical results support that the formation of open hydrogen bonded aggregates, (-7AI-)n with n ≥ 2 can efficiently compete with that of the profusely studied centro-symmetric cyclic dimer (7AI)2. Moreover, these aggregates suffer a proton-driven electron transfer process that strongly quenches their room temperature fluorescence and, therefore, masks their presence in the 7AI solutions. Therefore, because most of the studies on the 7AI photophysics have been interpreted without considering the existence of such aggregates and, more important, ignoring its quenching process, many conclusions obtained from these studies should be carefully revised.Journal of Fluorescence 07/2013; · 1.79 Impact Factor

Page 1

Photochemistry of hydrogen-bonded aromatic pairs:

Quantum dynamical calculations for the

pyrrole–pyridine complex

Zhenggang Lan†‡, Luis Manuel Frutos†§, Andrzej L. Sobolewski¶, and Wolfgang Domcke†

†Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany; and¶Institute of Physics, Polish Academy of Sciences,

PL-02668 Warsaw, Poland

Edited by F. Fleming Crim, University of Wisconsin, Madison, WI, and approved April 17, 2008 (received for review February 8, 2008)

The photochemical dynamics of the pyrrole–pyridine hydrogen-

bonded complex has been investigated with computational meth-

ods. In this system, a highly polar charge-transfer state of1??*

character drives the proton transfer from pyrrole to pyridine,

leading to a conical intersection of S1 and S0 energy surfaces. A

two-sheeted potential-energy surface including 39 in-plane nu-

clear degrees of freedom has been constructed on the basis of ab

initio multiconfiguration electronic-structure data. The non-Born–

Oppenheimer nuclear dynamics has been treated with time-de-

pendent quantum wave-packet methods, including the two or

three most relevant nuclear degrees of freedom. The effect of the

numerous weakly coupled vibrational modes has been taken into

account with reduced-density-matrix methods (multilevel Redfield

theory).Theresultsprovideinsightintothemechanismsofexcited-

state deactivation of hydrogen-bonded aromatic systems via the

electron-driven proton-transfer process. This process is believed to

be of relevance for the ultrafast excited-state deactivation of DNA

base pairs and may contribute to the photostability of the molec-

ular encoding of the genetic information.

conical intersection ? excited-state hydrogen transfer ?

nonadiabatic transition

A

the functionality of hydrogen bonds in the electronic ground

state have been investigated with powerful experimental and

computational methods for decades and are thus quite well

understood (1), much less is known about the role of hydrogen-

bond dynamics in excited electronic states of chemical or bio-

chemical systems. Fluorescence quenching of aromatic chro-

mophores by protic solvents and fluorescence quenching in

intermolecularly or intramolecularly hydrogen-bonded aromatic

systems are well known phenomena, but are still poorly under-

stood at the atomistic level (2–4). One reason for our limited

knowledge of excited-state hydrogen-bond dynamics is the ex-

tremely short time scale of some of these processes (presumably

of the order of 10 fs or less). Another reason is the difficulty of

performing accurate ab initio electronic-structure calculations

for excited states of complex polyatomic systems.

It has recently been proposed that electron-driven proton-

transfer processes along hydrogen bonds could play a decisive

role for the ultrafast excited-state deactivation of biological

molecules and supermolecular structures, such as DNA base

pairs, peptides, or UV-protecting pigments (5–7). The compu-

tational studies suggest that proton-transfer processes driven by

charge-transfer (CT) states of1??*,1n?*, or1??* character

provide barrierless access to conical intersections (8) of the

excited-state and ground-state potential-energy surfaces, where

ultrafast internal conversions take place. This particularly effi-

cient mechanism of energy dissipation could be essential for

photostability of the molecular encoding of the genetic infor-

mation of life (9). Recent experimental results for DNA base

siswellknown,hydrogenbondsareofuniversalimportance

in chemistry and biochemistry. Although the structure and

pairs or biomimetic models thereof seem to support this

conjecture (10–12).

Although ab initio calculations of electronic excitation ener-

gies, minimum-energy reaction paths, and energy profiles, as

well as minima of conical intersection seams can provide valu-

able insight, a true mechanistic understanding requires the

computational treatment of the nuclear dynamics of the photo-

chemical process. Such calculations are challenging because of

the large number of nuclear degrees of freedom, the large excess

energy provided by UV photons, and extremely strong non-

Born–Oppenheimer effects at conical intersections (8). Very

recently, a few ab initio on-the-fly trajectory simulations have

been performed on hydrogen-detachment and hydrogen-

transfer processes in biomolecular systems (13, 14). Although

such simulations can provide useful mechanistic insight, they

have limitations because of the rather significant de Broglie

wavelength of the proton, the approximate treatment of the

nonadiabatic dynamics at the conical intersections, the inevita-

ble compromises with respect to the accuracy of the ab initio

methods, and the very limited number of trajectories that can be

calculated.

In this work, we describe the first attempt of a fully quantum

mechanical treatment of nonadiabatic photochemical dynamics

of a hydrogen bond in a biomimetic system. We adopt the

hydrogen-bonded pyrrole–pyridine aromatic pair (15) as a

model of the Watson–Crick base pairs in DNA. Multiconfigu-

ration ab initio methods have been used for the characterization

of the potential energy (PE) surfaces of the relevant electronic

states. An approximately 39-dimensional analytic PE surface of

the reaction-path-Hamiltonian type (16) has been constructed.

We report the results of reduced-dimensional time-dependent

quantum wave-packet calculations and calculations in the frame-

work of multilevel Redfield theory (17, 18). In the latter ap-

proach, the few most important nuclear degrees of freedom are

explicitly taken into account, whereas the many weakly coupled

degrees of freedom are treated approximately in perturbation

theory and the Markovian approximation.

Results and Discussions

One-Dimensional Potential-Energy Surfaces. The equilibrium struc-

ture of the hydrogen-bonded model system, planar pyrrole–

Authorcontributions:Z.L.,L.M.F.,A.L.S.,andW.D.designedresearch;Z.L.,L.M.F.,andW.D.

performed research; Z.L., L.M.F., and W.D. contributed new reagents/analytic tools; Z.L.,

L.M.F., and W.D. analyzed data; and Z.L., L.M.F., and W.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

‡To whom correspondence should be addressed. E-mail: lan@ch.tum.de.

§Present address: Departamento de Quı ´mica Fı ´sica, Universidad de Alcala ´, 28871 Alcala ´ de

Henares (Madrid), Spain.

This article contains supporting information online at www.pnas.org/cgi/content/full/

0801062105/DCSupplemental.

© 2008 by The National Academy of Sciences of the USA

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Page 2

pyridine, is displayed in Fig. 1. The NH distance R of pyrrole is

defined as the reaction coordinate for the hydrogen transfer.

Fig. 2 gives an overview of the potential-energy surfaces of the

three lowest electronic states of pyrrole–pyridine as functions of

the hydrogen-transfer coordinate R (15). It can be seen that the

hydrogen atom is bonded to pyrrole in the electronic ground

state. The lowest locally excited (LE) singlet state of the complex

is of1??* character and1B2symmetry. The lowest singlet CT

state is of1B2symmetry and1??* character. It is optically dark,

i.e., it cannot be excited directly from the S0 state by a one-

photontransition.ThepotentialenergyoftheCTstateintersects

the energies of the LE state and the S0 state at 2.4 a.u. and

3.9 a.u., respectively, see Fig. 2. The structures of pyrrole–

pyridine at the ground-state equilibrium geometry and the

conical intersection can be found in supporting information (SI)

Text and Tables S1 and S2. The CT–S0energy crossing visible in

Fig. 2 becomes a conical intersection when the vibrational modes

of B2symmetry, are taken into account.

Two-Dimensional Quantum Wave-Packet Dynamics. As the simplest

nontrivial model, we consider the two-dimensional model in-

cluding the reaction coordinate R and the effective coupling

coordinate Qc

tential-energy surfaces of this two-dimensional model are dis-

played in Fig. 3 a and b, respectively, as functions of R and the

effective coupling coordinate Qc

potential-energy surfaces are smooth functions of nuclear ge-

ometry, see Fig. 3a. The double-cone shape of the adiabatic

surfaces of the CT–S0conical intersection can be clearly seen in

Fig. 3b.

In the present system, the diabatic coupling constants ?iof all

coupling modes are found to be rather small. Therefore, ?c

relatively small. This implies that the probability of electronic

population transfer is low. For clarity, we therefore consider in the

following only P1

ground state. The population probability of the CT state remains

near unity on the time scale of a few picoseconds.

When the lowest vibrational level of the electronic ground

state is vertically placed into the CT state, we observe that

oscillatory diabatic population transfer takes place with a period

eff(see Methods). The diabatic and adiabatic po-

eff. As expected, the diabatic

effis also

d, the population probability of the electronic

of ?500 fs, see Fig. 4a. The lack of irreversible electronic

population transfer is the result of the restriction to just two

nuclear degrees of freedom, the weak coupling at the CT–S0

conical intersection, and the rather small CT–S0energy gap in

R

Fig. 1.

state.

Structure of the pyrrole–pyridine complex in the electronic ground

0

5

10

1 2 3 4

Potential Energy (eV)

CT

LE

So

R (au)

Fig. 2.

lowest electronic states of pyrrole–pyridine: diabatic ground S0 state, CT

(1??*) state and LE (1??*) state.

One-dimensional PE functions for hydrogen transfer of the three

−2024

1

2

3

4

5

0

2

4

6

8

10

Qeff

c

R (au)

Potential Energy (eV)

−2

0

2

4

1

2

3

4

0

2.5

5

7.5

10

Qeff

c

R (au)

Potential Energy (au)

a

b

Fig.3.

asfunctionsofthehydrogen-transfercoordinateRandtheeffectivecoupling

coordinate Qc

20 ?c

Diabatic(a)andadiabatic(b)PEsurfacesoftheS0andCT(1??*)states

eff. For sake of clarity, the diabatic coupling is exaggerated (? ?

eff).

0

0.03

0.06

0.09

0 1 2 3 4

Probability

Time (ps)

0

0.03

0.06

0.09

0 1 2 3 4

Probability

Time (ps)

a

b

Fig.4.

the two-dimensional wave-packet calculations (a) and the three-dimensional

wave-packet calculations (b). The Inset of a is the diabatic population of the

S0state within the first oscillation period.

ElectronicpopulationprobabilityofthediabaticS0state,obtainedby

12708 ?

www.pnas.org?cgi?doi?10.1073?pnas.0801062105Lan et al.

Page 3

the hydrogen-transferred complex, see Fig. 2. The superimposed

weak and rapid oscillations (period of ?19 fs, see Inset in Fig.

4a), reflect the time scale of vibrational motion along the

hydrogen-transfer coordinate. In this two-dimensional model,

the conical intersection clearly represents a major bottleneck for

the radiationless deactivation of the CT state.

Three-Dimensional Quantum Wave-Packet Dynamics. To get an im-

pression of the effect of the so-called tuning modes on the reaction

dynamics, we add the effective tuning mode Qt

three-dimensional conical intersection (see Methods). The addition

of a third mode leads to a qualitative change of the electronic

population dynamics. The population probability of the diabatic S0

state now increases monotonically within the first picosecond. The

regularoscillationsofthetwo-modemodelarereplacedbyirregular

fluctuations of the electronic population probability. The time

average of P1

4a to ?0.06 in Fig. 4b. The weak rapid oscillations arising from the

oscillatoryhydrogen-transferdynamicsarestillvisible,butaremore

irregular than in Fig. 4a.

eff, resulting in a

d(t) over the first 4 ps has increased from ?0.03 in Fig.

Multilevel Redfield Dynamics. The reduced density matrix of the

system Hamiltonian, which includes the reaction coordinate R

and the effective coupling mode Qc

time, employing a Redfield tensor that has been constructed

from the remaining 19 vibrational modes of A1symmetry (see

eff, has been propagated in

Methods). The coupling of the reaction coordinate with the bath

of the remaining A1vibrational modes has a significant impact

on the electronic population dynamics, see Fig. 5a. After an

initial transient behavior, the population probability of the S0

state increases linearly with time, as expected for an incoherent

rate process. Rapid fluctuations are still visible but die out on a

time scale of ?100 ps. The existence of these transient fluctu-

ations reflects the fact that the data in Fig. 5 have been obtained

from a truly microscopic dynamical theory rather than from an

approximate rate expression.

These features can be understood as follows. The system-

bath-coupling slowly drains the vibrational energy out of the

hydrogen-transfer vibrational motion. The large excess energy

of the system is thus transferred to the A1modes on a time scale

of a few hundred picoseconds. The vibrational damping thus

reduces the recurrence of the vibrational wave packet to the

conical intersection, resulting in the lack of transfer of elec-

tronic population to the CT state.

According to these calculations, the internal-conversion dy-

namics from the CT state to the S0state takes place on a rather

longtimescaleofafewhundredpicoseconds.Onereasonforthe

slow radiationless decay of the CT state is the relatively weak

diabatic coupling at the CT–S0 conical intersection. Another

reason is the unfavorable location of the CT–S0conical inter-

section in a shallow secondary well of the S0potential-energy

surface (see Fig. 2), where the density of states of the S0surface

is rather low. As far as we can tell, our computer simulation of

the radiationless decay dynamics should be qualitatively correct,

which implies that the rather slow radiationless decay dynamics

isapropertyofthisparticularpyrrole–pyridinecomplex.Indeed,

existing calculations of the reaction-path potential-energy pro-

files of related singly or doubly hydrogen-bonded aromatic

systems, e.g., indole–pyridine, the 2-aminopyridine dimer (11),

or the guanine–cytosine Watson–Crick base pair (5) indicate a

topography of the CT–S0 conical intersection that is more

favorable for rapid radiationless decay.

To obtain insight into the role of the interstate coupling strength

at the conical intersection and the strength of the system-bath

coupling, we have performed additional reduced-density-matrix

propagations, varying the parameters of the ab initio-based model.

First, we increase the interstate coupling by multiplying ?c

a factor of 2. In this modified model, the rate of internal

conversion increases by approximately a factor of 2, see Fig. 5b.

The interstate coupling at the CT–S0conical intersection is thus

the rate-limiting factor in this model.

In a further calculation, we have increased the system-bath

coupling strength by a factor of 2. Because the Redfield tensor is of

second order in the system-bath coupling strength, it increases by

a factor of 4. As a result, the internal conversion rate increases

by a factor of ?4 (Fig. 5c), compared with the original model

(Fig. 5a). The ground-state population probability begins to satu-

rate at ?70 ps, see Fig. 5c. The damping of the hydrogen-transfer

dynamics is thus another rate-limiting process in this system.

In the original and the modified model, the interstate coupling

strength and the system-bath-coupling strength are rather low. As

aresult,theinternalconversionprocesstakesplaceonaratherlong

time scale (a few hundreds picoseconds). In this limiting case, the

internal-conversion rate is approximately linearly and quadratically

dependent on the corresponding coupling parameters. This simple

relationshipisnotexpectedtoholdformoregeneralsituationswith

strong interstate coupling at conical intersections. However, the

present results are useful for the understanding of the general

mechanism of the internal-conversion dynamics in biomolecular

systems.

effby

Conclusions

We have investigated the nonadiabatic dissipative dynamics of

the pyrrole–pyridine hydrogen-bonded complex, which is trig-

Probability Probability

0

0.2

0.4

0.6

0.8

0 50

Time (ps)

100 150

Probability

a

b

0

0.2

0.4

0.6

0.8

0 50

Time (ps)

100 150

Probability

c

0

0.2

0.4

0.6

0.8

0 50

Time (ps)

100 150

Fig. 5.

from the reduced-density-matrix propagations within the original model (a),

for increased interstate coupling strength (b), and for increased system–bath

coupling strength (c).

Electronic population probability of the diabatic S0state, obtained

Lan et al.

PNAS ?

September 2, 2008 ?

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no. 35 ?

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CHEMISTRY

SPECIAL FEATURE

Page 4

gered by the photoinduced electron-driven proton-transfer pro-

cess. The potential-energy surfaces of the relevant electronic

states have been characterized by ab initio electronic-structure

calculations at the CASSCF level. A 39-dimensional model of

potential-energy surfaces has been constructed, which is based

on ab initio energy gradients along the hydrogen-transfer reac-

tion coordinate and ab initio vibronic-coupling constants at the

conical intersection of the CT state with the S0state.

The nonadiabatic quantum dynamics of this multidimensional

model system has been investigated, employing time-dependent

wave-packet and reduced-density-matrix methods. Assuming

vertical electronic excitation of the CT state, we have explored

the time evolution of the population probabilities of the CT state

and the S0state. To gain insight into the microscopic mecha-

nisms of the radiationless decay dynamics of this system, we have

performed time-dependent quantum wave-packet calculations

involving the two or three most relevant nuclear coordinates.

To reveal the effect of the remaining, more weakly coupled,

vibrational degrees of freedom, we have adopted a system-bath

approach (Redfield theory) with ab initio determined Redfield

tensor elements. The results illustrate the evolution from a

quasiperiodic electronic population dynamics (for the case of

two nuclear degrees of freedom), via a stochastically fluctuating

electronic population dynamics (for the case of three vibrational

degrees of freedom), to a nonradiative rate process, when all 39

degrees of freedom of the model are taken into account.

The pyrrole–pyridine hydrogen-bonded complex has been

chosen as a representative model for the investigation of the

ultrafast hydrogen-bond photochemistry of DNA base pairs (10,

12). Pyrrole–pyridine can also serve as a model for fluorescence

quenching through intermolecular hydrogen bonding between

aromatic chromophores in solution (4). The radiationless decay

dynamics of pyrrole–pyridine has been found to be compara-

tively slow (of the order of a few hundred picoseconds). The

rather weak vibronic coupling at the CT–S0conical intersection

and the relatively slow damping rate of the hydrogen-transfer

mode have been identified as the origin of the relatively slow

internal-conversiondynamicsinpyrrole–pyridine.Theextension

of the present methodology can be used to treat the radiationless

decay dynamics of the guanine-cytosine and adenine-thymine

Watson–Crick base pairs, for which extremely fast internal

conversion rates are expected (5, 10, 12).

Methods

Ab Initio Calculations. The ab initio calculations have been performed at the

complete-active-space self-consistent-field (CASSCF) level with the 6–31(G)d

basisset.Theactivespaceincludesall?and?*orbitals.TheCASSCFgradients,

frequencies, and PE surfaces have been calculated with the Gaussian 03

package (19).

ConstructionofthePotential-EnergySurfaces.Ourgoalistheconstructionofthe

potential-energy surfaces of the nonadiabatically coupled S0, LE, and CT states,

includingallrelevantin-planevibrationalcoordinatesofthesystem.Toavoidthe

singular derivative coupling at the conical intersections (8), we construct qua-

sidiabaticpotential-energysurfaces.WeadopttheNHdistanceRofpyrroleasthe

reactioncoordinateofthesystem.Theremainingin-planevibrationaldegreesof

freedomaretreatedapproximatelyinthespiritofthereaction-path-Hamiltonian

approach (16). We focus on the conical intersection of the CT state with the S0

state,becausethedynamicsatthisintersectionisdecisiveforthetimescaleofthe

internal conversion process. The LE–CT conical intersection remains to be char-

acterized.

The planar pyrrole–pyridine complex has C2v symmetry. The symmetry

species of the normal modes are

? ? 20A1? 7A2? 11B1? 19B2.

[1]

The 20 A1modes consist of the reaction coordinate R and 19 so-called tuning

modesoftheCT–S0conicalintersection,whereasthe19B2modesareso-called

couplingmodes.ThemodesofA2andB1symmetryarenotinvolvedwhenthe

potential-energy surfaces are described in the so-called linear-vibronic cou-

pling model (20) and will not be considered in what follows.

Because the proton-transfer reaction implies large-amplitude motion in

the reaction coordinate R, the linear vibronic-coupling parameters of the

tuning and coupling modes and the vibrational frequencies have to be con-

sideredasfunctionsofthiscoordinate.TotakeintoaccounttheRdependence

of the intrastate linear coupling constants ?i

represent them as fourth-order polynomials of R. For the interstate vibronic-

couplingconstants?i,ontheotherhand,weadopttheirabinitiovaluesatR?

Rc, where Rc? 3.9 a.u. is the location of the CT–S0conical intersection. This

simplification of the model is appropriate, because the non-Born–Oppenhei-

mer dynamics at conical intersections depends essentially on the ?iat this

geometry. The dependence of the vibrational frequencies of all nonreactive

modes on the electronic state and the reaction coordinate is neglected in the

present linear-vibronic coupling model.

The intrastate coupling parameters, ?i

gradients of the adiabatic S0 and CT energies with respect to Cartesian

displacementcoordinates(8).Theyaretransformedtodimensionlessground-

state normal coordinates, by using the L-matrix (21) of the latter. Analytical

functions ?i

fourth-orderpolynomialsofRforeachofthe19tuningmodesofA1symmetry.

Theinterstatecouplingparameters?iareobtainedbytheprojectionofthe

nonadiabatic coupling vector (22) [or h vector (23)] on the ground-state

normal coordinates of B2symmetry, see the discussions in refs. 8 and 24. The

resulting diabatic potential model includes the large-amplitude proton-

transfer coordinate R, 19 linearly coupling tuning modes of A1symmetry, and

19linearcouplingmodesofB2symmetry.Thedependenceofthe?i

reaction coordinate R (see SI Text and Fig. S2 for characteristic examples)

results in a significant coupling of the tuning modes with the reaction coor-

dinate.Thecouplingmodes,ontheotherhand,arestronglycoupledwiththe

tuningmodesandthereactioncoordinateattheCT–S0conicalintersection.As

a result, all 39 in-plane nuclear degrees of freedom in this model are coupled

to each other. We refer to SI Text for a more detailed description of the

39-dimensional potential-energy surface.

(1), ?i

(2)of the tuning modes, we

(k)(k ? 1,2), are obtained as the

(k)(R) are obtained by a least-squares fit of the ab initio data to

(k)(R)onthe

Treatment of the Time-Dependent Nuclear Dynamics. In this work, we did not

attempt to perform time-dependent quantum wave-packet dynamics calcu-

lations with the inclusion of all 39 vibrational modes of the potential-energy

surface.Suchacalculationwouldbeextremelychallenging.Thereexistseveral

concepts that allow the reduction of the computational problem to a smaller

number of effective nuclear coordinates. One concept is the definition of

effective modes or cluster modes (25, 26). For a symmetry-allowed conical

intersection, as is the case here, these effective modes are just the gradient-

difference and nonadiabatic coupling (or g, h) vectors (22, 23). The atomic

displacement vectors associated with the g and h vectors are illustrated in Fig.

S1. Within the linear-vibronic-coupling model, the effective tuning and cou-

pling modes are given by (see refs. 20 and 25)

?eff???

i

eff??

i

Qt

??i

??t

effQt,i,

[2]

Qc

?i

?c

effQc,i,

[3]

with the coupling strengths and frequencies

eff???

??t

i

???i?2, ???i? ?1

2?i

?2?? ?i

?1?,

[4]

?t

eff??

eff???

eff??

i?

??i

??t

eff?

2

?t,i,

[5]

?c

i

?i

2,

[6]

?c

i?

?i

?c

eff?

2

?c,i.

[7]

Within the effective-mode approximation, the 39-dimensional dynamics

problem is reduced to a three-dimensional problem, involving the reaction

12710 ?

www.pnas.org?cgi?doi?10.1073?pnas.0801062105Lan et al.

Page 5

coordinate R, as well as the effective tuning and coupling modes. In this

approximation, the anharmonic couplings arising from the R dependence of

the ?i

any contribution in the branching space of the conical intersection. The

effective coupling coordinate Qc

B2modes with frequencies ?1,720 cm?1.

To account for the nonseparability of the reaction coordinate and the

tuning modes Qt,i, we have used multilevel Redfield theory (17, 18). Adopting

a system-bath model, the nuclear coordinates of the system Hamiltonian are

chosen as the reaction coordinate R and the effective coupling coordinate

Qc

The remaining A1modes are considered as a bath composed of harmonic

oscillatorswithdimensionlesscoordinatesQt,i,momentaPt,i,andfrequencies?t,i:

Hb??

i

(k)are ignored. The B2modes with frequencies ?1,720 cm?1do not give

effthus can be constructed by considering all

eff, resulting in a two-state two-mode model.

?t,i

2

?Pt,i

2? Qt,i

2?.

[8]

The system-bath coupling is given by

Hsb

t? ??1

d???1

d??

i

?i

?1??R?Qt,i? ??2

d???2

d??

i

?i

?2??R?Qt,i.

[9]

The ?i

estingly, the ?i

R dependence, that is

(k)(R) are fourth-order polynomial functions of R (see SI Text). Inter-

(k)(R) for different tuning modes have approximately the same

?i

?k??R? ? gi

?k???k??R?; k ? 1,2,

[10]

with

??k??R? ? a?k?R4? b?k?R3? c?k?R2? d?k?R ? e?k?.

[11]

With this approximation, the system-bath coupling becomes

Hsb

t??

k?k

2

??d??R???k

d? ??1

d??

i

gi

?k?Qt,i.

[12]

Thecouplingbetweenthesystemandthebathiscompletelydeterminedby

the spectral function of the bath. In the above approximation, the spectral

functions read

2?

i

Jk??? ??

?gi

?k??2??? ? ?t,i?; ?k ? 1 or 2?.

[13]

Wereplacethe?functionsbyLorentzfunctionstoobtaincontinuousspectral

functions. The full-width half-maximum of these Lorentz functions is 13 meV

(see Fig. S3 for a characteristic example).

It is straightforward to construct the Redfield tensor for this system-bath

coupling model (27). Note that the elements of the Redfield tensor are not

empirical parameters but are determined from the ab initio calculations.

The preparation of the initial state by the absorption of an UV photon is

approximatelydescribedasverticalexcitationfromthelowestvibrationallevelof

the electronic ground state to the CT state. As mentioned above, the actual

electronicexcitationtakesplacefromtheS0statetotheLEstate,fromwhichthe

wavepacketswitchestotheCTstate(seeFig.2).Wehavesimplifiedtheproblem

by assuming direct electronic excitation to the CT state.

Thewavepacketsarepropagatedonthetwocoupledsurfacesbyusingthe

split-operator method, as discussed (28). We have used 64 grid points from

0.83 a.u. to 4.83 a.u. for R. Ten and 6 harmonic-oscillator basis functions are

usedfortheeffectivetuningandcouplingcoordinates,respectively.Thewave

packets are propagated for 4 ps with a time step of 0.1 fs.

We use the split-operator method for the short-time propagation of the

reduced density operator (29):

?s?t ? dt? ? e?iLsdt?2eDdte?iLsdt/2?s?t?,

[14]

wherethesystemLiouvillesuperoperatorLsandthedissipativesuperoperatorD

describethereversiblesystemdynamicsandtheirreversibledissipativedynamics,

respectively. The short-time propagator e?iLsdt/2is evaluated in the eigenstate

representation.TheoperatorDintheeigenstaterepresentationistime-local(17).

Therefore,theshort-timepropagationgovernedbyDleadstoasystemoflinear

differentialequationsforthematrixelementsofthereduceddensitymatrix(29).

The fourth-order Runge–Kutta method is used to evaluate the short-time prop-

agator eDdtfor every time step. In this way, we can disentangle the propagation

of the fast system dynamics and the slow dissipative dynamics. It permits us to

propagate the reduced-density matrix for a very long time duration with high

numerical stability.

The numbers of the grid points and harmonic basis functions, which are

used to generate the matrix representation of Hamiltonian, lead to a

system Hilbert space of dimension 768. This space is truncated to 500 basis

functions in the propagation of the reduced density matrix. The reduced

density matrix is propagated for 150 ps with a time step of 50 fs.

The observables of primary interest of the present study are the time-

dependentpopulationsprobabilitiesoftheelectronicstates(8).Adiabaticand

diabatic electronic population probabilities are defined as the expectation

values of the corresponding projection operators with the time-dependent

wave packet. Although the adiabatic electronic population probabilities, Pi

are the observables that most directly reflect the electronic decay dynamics,

theircomputationisveryexpensivewhenthesystemHilbertspaceislarge.We

therefore consider the diabatic population probabilities, Pi

tive discussion of the nonadiabatic dynamics of the multimode non-Born–

Oppenheimer system.

a,

d, for the qualita-

ACKNOWLEDGMENTS. We thank Michael Thoss and Dassia Egorova for many

useful discussions. This work was supported by the Deutsche Forschungsge-

meinschaft (DFG) through a research grant and the DFG-Cluster of Excellence

‘‘Munich Centre of Advanced Photonics’’ (www.munich-photonics.de). The

Leibniz Rechenzentrum der Bayerischen Akademie der Wissenschaften is

acknowledged for providing an ample amount of computing time. L.F. ac-

knowledgesapostdoctoralgrantoftheAlexandervonHumboldtFoundation

and support given by the ‘‘Ramo ´n Cajal’’ Program.

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