Ultrafast Nonadiabatic Photodissociation Dynamics of F2 in Solid Ar*, 1

ISSN 1054
660X, Laser Physics, 2009, Vol. 19, No. 8, pp. 1651–1659.

STRONG
FIELD PHENOMENA IN MOLECULES

© Pleiades Publishing, Ltd., 2009. Original Russian Text © Astro, Ltd., 2009.

Ultrafast Nonadiabatic Photodissociation Dynamics of F2 in Solid Ar*, 1 M. Sukhareva, A. Cohenb, Robert Benny Gerberb, and Tamar Seidemana a

Department of Applied Sciences and Mathematics, Arizona State University at the Polytechnic Campus, Mesa, AZ 85212, USA b Department of Physical Chemistry and the Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, 91904, Israel e
mail: t
[email protected] Received March 2, 2009

Abstract—We explore the ultrafast spin
flip dynamics in a diatomic molecule imbedded in a rare gas matrix using the combination of a quantum mechanical and a semiclassical surface hopping method. Specifically, we investigate (1) the extent to which the phenomenon of electronically
localized eigenstates in strongly
cou
pled manifolds survives in the presence of rapid decay and a multitude of electronically coupled states; (2) the ability of the surface hopping method to predict the short time dynamics; and (3) the time range over which frozen lattice models are valid. Our results point to the active role played by a large number of coupled elec
tronic states in the F2/Ar dynamics while substantiating our confidence in the validity of the popular surface hopping approach for the system considered. PACS numbers: 31.50.Gh, 31.15.Xg, 33.80.Gj DOI: 10.1134/S1054660X09150389

*

1. INTRODUCTION Nonradiative transitions in the excited states of polyatomic molecules have been the topic of intensive studies for several decades [1] and continue to gener
ate new and interesting questions for theoretical research [2]. One such question is the concept of elec
tronically localized eigenstates of coupled vibronic Hamiltonians [3] the occurrence of highly vibra
tionally
excited, yet essentially electronically
pure eigenstates, embedded in a sea of strongly electroni
cally
mixed eigenstates. The origin of this phenome
non and its generality are explained in [3, 4] through its equivalence to the problem of scares of unstable periodic orbits. In addition to its fundamental interest, the phenomenon of electronically localized states plays a central role in an optimal control approach to suppression of non
radiative transitions utilizing spe
cifically shaped laser pulses. The studies of [3, 4] used as test cases two
and three
dimensional models of the internal conversions of pyrazine and octatetraene, where the excited state manifold includes two states. An interesting question is thus to what extent this phe
nomenon survives in multidimensional systems and in problems where several excited electronic states play an active role in the dynamics. [5, 6] addressed the dimensionality problem by considering a 24
mode model of the internal conversion of pyrazine. The results of [5, 6] illustrated that the concepts introduced in [3, 4] survive in higher dimensions, at least in as far * In loving memory of Professor Nikolay Delone, a great scientist and gifted teacher, who inspired generations of students to pur
sue research. 1 The article is published in the original.

as one can judge from studies of the internal conver
sion of pyrazine. The effects of a dissipative bath and of a multitude of coupled electronic states, however, remain to be explored. Molecules imbedded in rare gas matrices exhibit rich nonradiative decay dynamics that does not have a counterpart in the limit of isolated molecules. In addi
tion to ultrafast spin
flip transitions that are induced by the host environment, such systems offer also a large manifold of strongly coupled electronic states and multiple modes. As such, they provide an interest
ing ground for exploration of decay dynamics, and one that is accessible to current experiments. One of our goals in the present study is to explore the extent to which electronically localized states survive in a matrix environment, in the presence of an increasing number of coupled electronic states. Photochemical reactions of molecules embedded in rare
gas matrices are a problem of significant inter
est in its own right, whose study has been instructive in providing insights into chemical dynamics in con
densed phases [7–23] (for a review, see [24]). The nature and role of nonradiative transitions in such sys
tems has been addressed in several recent studies [7, 8, 10–15, 19–23]. In condensed media, nonradiative transitions may occur through the interaction of the guest molecule with the host environment, a mecha
nism that does not have an analog in the gas phase. Dynamical simulations that account for nonadiabatic processes in condensed phases, have been carried out mainly using the Surface
Hopping (SH) method of Tully [25]. An obvious advantage of this approach is its ability to handle high
dimensionality and a multitude

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SUKHAREV et al.

of electronic states. Tests of the method by compari
sons with experimental data and quantum models have been published [12, 20]. Nonetheless, the extent to which classical mechanics can address problems where several inherently quantum phenomena such as coherence, dephasing, and 2
level systems play a role, remains to be tested. A second goal of the present work is to compare the results of the SH model with a quantum approach using the photoinduced dynamics of F2 molecules embedded in Ar matrix as an experimentally
relevant model. The questions investigated include: How many quantum states play a role in the dynamics? How does the quantum dynamics evolve in time? What is the role of quantum coherences? To what extent and under what conditions can the SH model mimic the quan
tum motion? Before addressing these questions in Section 3, we introduce our model system (Section 2) and briefly describe our semiclassical (Section 2.1) and quantum mechanical (Section 2.2) theoretical approaches. In Section 4 we conclude this work with an outlook to future research. 2. MODEL AND NUMERICAL IMPLEMENTATION The molecular Hamiltonian is written as a sum of three terms, ˆ = T ˆ (r) + T ˆ (R) + U ˆ ( r, R ), (1) H e

n

ˆ where T e ( n ) is the kinetic energy operator of the elec
tronic (nuclear) subsystem, U is the potential energy, r denotes collectively the coordinates of the electrons and R denotes the nuclear coordinates. Expanding the total wavefunction Ψ(r, R) in an adiabatic basis of electronic wavefunctions Φn(r, R), and inserting the expansion into the Schrödinger equation, we derive a set of coupled differential equations that determine the nuclear wavefunctions χv(R). Explicitly, (m) ⎛ 1 d2 dχ v ( R )⎞ ( n ) 1 C nm ( R )

















⎟ χ v ( R ) ⎜ –











2 + W n ( R ) –

µ dR ⎠ ⎝ 2µ dR m≠n (2)

∑

(n)

(n)

= ε v χ v ( R ),

the system are based on the Diatomics
in
Molecule (DIM) model, [26] which has been used extensively and successfully in numerous studies in the past [8, 10, 27–29]. The method was extended in [30] to include the ionic states in the diatomics
in
ionic
systems (DIIS) model. The latter method was not used in this study since charge transfer between the F and the Ar matrix atoms is not expected to take place. The F2 molecule consists of two p
type electrons, each having an angular momentum of magnitude 1 and a spin projection S along the molecular axis. Thus, the molecule has a total of 36 neutral electronic states (9 singlet states and 27 (3 × 9) triplet states) [10]. Prop
erly symmetrized wave functions are constructed from the explicit two p
type electrons of the F atoms via the Valence Bond (VB) approach. The complete potential is thus ˆ = V ˆ +V ˆ ˆ ˆ U F2 F–Ar + V Ar–Ar + V SO ,

(3)

ˆ represents the F–F interaction, V ˆ where V F2 F–Ar describes the interaction between each of the F–Ar ˆ atom pairs, V Ar–Ar stands for the interaction among the ˆ is the spin
orbit (SO) cou
Ar matrix atoms, and V SO

pling term associated with the two F atoms. This potential is represented in the p
type basis of the two effective electrons using a VB framework. The molec
ular term is taken from an ab initio calculation for the isolated F2 molecule [31] (Fig. 1a). The anisotropic interaction between the p
type F orbital and each of the Ar matrix atoms is taken into account via an inter
action expansion in Legendre polynomials [12, 32]. The SO coupling interaction is approximated by the sum of the SO interactions for a p
electron in the iso
lated F atoms and is thus independent of the F–F dis
tance. After the diagonalization of the Hamiltonian at each nuclear configuration in the course of the dynamics, the isotropic Ar–Ar interaction [34] of all the Ar–Ar pairs is added to each of the 36 eigenvalues, resulting in 36 adiabatic potentials (Fig. 1b). We proceed by outlining our method of computing the kinetic couplings in Eq. (2) and solving for the nuclear dynamics within the semiclassical (Section 2.1) and quantum mechanical (Section 2.2) frameworks.

where Wn(R) is the adiabatic potential energy, µ is the (n)

reduced mass, ε v is the vibrational energy, and the Cnm are kinetic coupling coefficients, Cnm(R) = 〈Φn(r, R)|∇RΦm(r, R)〉. In Eq. (2), we introduced the standard approximation of neglecting a term proportional to the second derivative of electronic wavefunction, Φm(r, R), with respect to internuclear distance, R. Atomic units are used in Eq. (2) and throughout this article. The system potentials used in this paper were pre
sented in detail in [10], therefore only a brief descrip
tion will be given here. The potential energy surfaces of

2.1. Semiclassical Simulations The propagation method used in this part of this study is Tully’s “surface hopping” technique [25]. Within this method, the nuclei initially evolve on one of the adiabatic potential surfaces. This classical prop
agation on the ith potential surface continues as long as nonadiabatic transitions do not take place. At each time step, all 36 adiabatic surfaces Wn(R), n = 1, …, 36 and their corresponding (parametrically R
depen
dent) electronic eigenstates Φn(r1, r2, R) are generated. Since these states are generated “on the fly” along the LASER PHYSICS

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E, eV 4 (a) F2 bare molecule potentials 2

Πu

1

Πu

3

0 X1Σ+g −2

2

4

6

8

10

4 (b) F2@Ar255 potentials along the 〈111〉 axis 2

Πu

1

3Π

0

u

X1Σ+g

−2

2

4

6 R, Bohr

8

10

Fig. 1. (a) Shows the F2 bare molecule potential energy curves excluding the spin orbit coupling interactions. Indicated by arrows 1 +

are the ground state, X Σ g , the attractive 3Πu state, and the initially excited 1Πu state. Among the 36 electronic states of the F2 molecule only two are bound, the ground and the 3Πu states. (b) Shows the potential energy curves of an F2 molecule interacting with a slab of 255 Ar atoms along the 111 direction accounting for spin
orbit interactions. The spin
orbit coupling interaction lifts part of the degeneracy and causes several of the repulsive excited states of the bare molecule to become attractive.

trajectory R(t), they can be written as Φn ≡ Φn(r1, r2, t) for each particular trajectory. The electronic degrees of freedom are described by a wave function of the coor
dinates ri, which evolves in time according to the Schrödinger equation. The expansion of ψ(r1, r2, t) in the adiabatic basis Φn(r1, r2, t) results in a set of cou
pled differential equations for the expansion coeffi
cients An(t) given as [25] iA· n ( t ) = – i

∑A

·

m ( t ) 〈Φ n|Φ m〉

+ A n ( t )W n ( t ).

(4)

m

· 〉 can be Through use of the chain rule, the 〈Φ n|Φ m written as. · 〉 = R· ( t ) 〈Φ ( r, R )|∇ Φ ( r, R )〉 = R· ( t )C , (5) 〈Φ |Φ n

m

n

R

m

nm

where the nonadiabatic coupling terms Cnm (n ≠ m) are given via the Hellman
Feynman theorem as, ˆ ( r , r , R ) |Φ ( r , r , R )〉 〈Φ n ( r 1, r 2, R )|∇ R H 1 2 m 1 2






















































. (6) C nm =




























Wm ( R ) – Wn ( R ) Calculation of the nonadiabatic coupling, like the cal
culation of the adiabatic states, is done “on the fly.” The nuclei are propagated classically on the adiabatic surface Wn using the 5th order Gear algorithm with a time step of 0.01 fs. The use of a very small time step is necessary due to occurrence of the nonadiabatic tran
sitions, which can lead to instabilities for larger steps. LASER PHYSICS

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At each point along the trajectory, all 36 adiabatic potential surfaces are constructed and the nonadia
· 〉 are obtained. Equation (4) batic couplings 〈Φ n|Φ m for the An coefficients n = 1, …, 36 is solved numeri
cally using the Runge−Kutta algorithm along the clas
sical trajectory of the nuclei. A stochastic condition is employed for transition from the current adiabatic potential surface to each of the other surfaces. The condition is based on the changes in the An coefficients and is determined within the “fewest switches” algo
rithm [25]. The initial structure was obtained by replacing the central Ar atom from a slab of 255 Ar atoms in an FCC structure with an F2 molecule located in a monosub
stitutional site. Periodic boundary conditions were employed at the ends of the slab. This structure was next optimized via simulated annealing, followed by an equilibration at 8 K by a classical trajectory for 5 ps. Snapshots of the positions and momenta of the atoms were taken along this trajectory to create the initial ensemble of atomic configurations for the semiclassi
cal trajectories. The photodissociation of the F2 mole
cule in the course of each trajectory of this ensemble was modeled using the classical Franck−Condon ver
tical excitation from the ground into the spectroscop
ically allowed 1Πu state with excitation energy of about 4.6 eV. The atomic configurations for which the verti
cal excitation energy between the ground and the 1Πu state equals the excitation energy were accepted as ini


1654

SUKHAREV et al. |〈Ψv |S0〉|2 1.00 0.75

1.00

(a)

0.75

0.50

0.50

0.25

0.25 0

0 |〈Ψv |S6〉|2 1.00 0.75

1.00

(b)

0.75

0.50

0.50

0.25

0.25

1.00

(c)

0.75

0.50

0.50

0.25

0.25

0
1

(e)

0

0 |〈Ψv |S7〉|2 1.00 0.75

(d)

0

1

2

0 3
1 Ev, eV

(f)

0

1

2

3

2

Fig. 2. (Color online) Projections 〈ψ v|S n〉 of the exact vibrational wavefunction ψv onto the set of vibrational eigenstates of the uncoupled adiabatic electronic state as functions of the vibrational energy, Ev. The left column shows results for the three
state 1 +

model and the right column for the five
state model: (a, a): projections onto the ground state ( X Σ g ), (b, e): projections onto the dark state (3Πu0), (c, f): projections onto the bright state (1Πu).

tial configurations for the photodissociation dynam
ics. A set of 49 trajectories were sampled in this way, each propagated for 1.5 ps. 2.2. Quantum Dynamics Simulations

We note that within DVR one can avoid computing numerical derivatives of the wavefunction in calcula
tion of the matrix elements of the kinetic couplings, (n) (m) 〈χ˜ v | Cnm(R) |∂χ˜ v /∂R〉 , by interpolating the wave
(n) function as, χ˜ v , [35]

(n)

The vibrational eigenfunctions, χ v , and eigenen
(n) εv ,

ergies, of the system represented by the left hand side of Eq. (2) are obtained by diagonalization of the Hamiltonian matrix in the basis of uncoupled eigen
states (corresponding to Cnm(R) = 0). To that end we (n) first calculate the eigenstates χ˜ v for each of the uncoupled adiabatic electronic state, Wn, by diagonal
izing the corresponding Hamiltonian in a discrete variable representation (DVR) [33]. Numerical con
vergence is achieved for a spatial step δR = 8.7 × 10 ⎯3 au on a grid of dimension [R0, Rmax] = [1.059 au– 10 au]. The complete Hamiltonian is next re
diago
(n) nalized in the χ˜ v basis. The vibrational energies, En, in the range –1 to 3 eV, are converged with 320 uncou
pled states for each of the adiabatic electronic states.

(n) χ˜ v ( R ) =

NR

∑ χ˜

(n) ⎛ π v ( R i ) sin c ⎝




( R

i=1

δR

– R i )⎞ . ⎠

(7)

In Eq. (7), sinc(x) = sin(x)/x, Ri is a grid point within the DVR grid, NR is a total number of grid points, and (n) χ˜ v (Ri) are the DVR eigenfunctions. 3. RESULTS AND DISCUSSION It is informative to first explore the electronic nature of vibrational states of the coupled system (2) for different numbers of electronic states. Figure 2 2 shows projections, 〈ψ v|S n〉 , of the complete wave
function, ψv, onto the set of vibrational eigenstates, Sn, of the uncoupled adiabatic electronic state, Wn, as LASER PHYSICS

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ULTRAFAST NONADIABATIC PHOTODISSOCIATION DYNAMICS

functions of the vibrational energy, Ev, for two cases: a three
state model, which consists of the ground 1 + ( X Σ g ), a dark (3Πu0), and a bright (1Πu) state, and a five
state model that includes two additional mani
folds. As is evident in Fig. 2, both the three
and the five
state models exhibit the phenomenon of electron
ically localized states, shown in recent work [3, 4] to be a general feature of molecular systems undergoing nonradiative transitions. It is interesting to observe that the effect survives the introduction of a bath and appears in systems supporting as many as five elec
tronic states. In the three
state model for F2 embed
ded in an Ar matrix the vibrational state |ψv = 192〉 with the energy Ev = 192 = 1.53 eV is strongly localized in the dark state S6 with a projection |〈ψv = 192|S6〉|2 = 0.88, dominated by a single vibrational state of the uncou
pled S6. Similarly, the very next vibrational state |ψv = 193〉 is localized in the bright state with a projec
tion 0.83. Figure 2 clearly illustrates that the inclusion of additional states results in loss of regularity of the vibrational spectra of the excited manifold, leading to less electronic localization character. We note, how
ever, that while the involvement of an increasing num
ber of electronic states enhances the mixing of the bright and dark states in the excited manifold, it pre
serves series of localized states in the ground electronic state (see Figs. 2a and 2d). For example, state |ψv = 258〉 with the energy Ev = 258 = 2.15 eV has a projection |〈ψv = 258|S0〉|2 = 0.88 in the three
state model, and the equivalent state (although with a different vibrational quantum number) in the five
state model, with energy Ev = 424 = 2.15 eV, is somewhat better localized, |〈ψv = 424|S0〉|2 = 0.94. As was illustrated before for the simpler case of a two
electronic state excited mani
fold, the electronically localized states have an inter
esting effect on the wavepacket dynamics in the excited manifold and play a central role in a coherent control approach to suppression of electronic dephasing. We proceed to examine the wavepacket evolution within the three
and five
state models by projecting the ground vibrational state of the S0
state onto the bright state and following the resulting wavepacket in time. As discussed in Section 2.1, our potential energy curves and coupling matrix elements are valid only up to a distance Rmax = 10 au. This sets an upper limit of ca. 50 fs on the duration of the propagation that can be reliably carried out. Figure 3 shows the vibrational populations of the 1 + ground ( X Σ g ), S0, dark (3Πu0), S6, and bright (1Πu), S7, electronic states as functions of time for the three
and five
state models with the initial condition taken to be the ground vibrational state of the ground elec
tronic state, projected onto the bright state. The inset presents the populations of the two additional states included in the five
state model, S1 and S2. It is clear LASER PHYSICS

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0.5

Vibrational populations 1.0

0.4 0.3 0.2

0.8

0.1 0 10

0.6

20

30

40

50

0.4 0.2 0 10

20

30

40

50 t, fs 1 +

Fig. 3. Vibrational populations of the ground ( X Σ g ) (black curve), dark (3Πu0) (red curve), and bright (1Πu) (blue curve) electronic states as functions of time, t, for the case of the three
state model (solid curves) and the five
state model (dashed curves). The inset illustrates the dynamics of populations of the two additional manifolds included in the five
state model, S1 state (black curve) and S2 (red curve).

that the decay dynamics for the case of the F2/Ar sys
tem is strongly modified by the inclusion of a larger excited manifold. The decay of the bright state is more rapid in the five
than in the three
state model, and one observes significant vibrational excitation of the S1 and S2 states after 30 fs. To further analyze the nonadiabatic dynamics in the excited manifold and explore the nature, extent and origin of the differences between the minimal (three
state) and more realistic (five
state) models, we show in Fig. 4 the projections of the wavepacket onto the vibrational eigenstates of the uncoupled electronic states for both models considered. Figures 4a–4c cor
respond to the three
and Figs. 4d–4f to the five
state model. The wavepacket evolution is plotted vs the vibrational energy and time, where t = 0 defines the start of the evolution in the excited manifold with the same initial conditions used in Fig. 3. Certain broad features of the time evolution of the five
state model, Figs. 4d–4f, are correctly repro
duced by the three
state analog, but it is evident that the coupling of five vibrational manifolds gives rise to features that are not exhibited in the minimal model. Most notable is the oscillatory time behavior of the five
state model, which stands in contrast to the smooth time evolution in the three
state analog. Comparison with Fig. 2 traces this effect to the loss of regularity of the three
state model eigenvalue spectra

1656

SUKHAREV et al. ×10−4 5

5

×10−4

(a)

(d) 8

(0) ε˜ v [ eV ]

4

6 3

4 2

2

10 3 5 2

×10−3

(b)

(6) ε˜ v [ eV ]

15

4

×10−3 2.5

(e)

4

6

4

3

4

3

2.0 1.5 1.0

2

2

2

0.5

×10−3 15

(c) 4

×10−3 15

(f) 4

10

(7) ε˜ v [ eV ]

10 3

3 5

5 2

2

0

10

(n) 2

Fig. 4. Projections 〈ψ|χ˜ v 〉

20 30 t[fs]

40

50

0

10

20

30 t[fs]

40

50

of the vibrational wavepacket, ψ onto the vibrational eigenstates of the uncoupled electronic states

1 + (n) as functions of time, t, and eigenenergy, ε˜ v . Panels (a, d) correspond to the ground state ( X Σ g ), panels (b, e) to the dark state

(3Πu0), and panels (c, f) to the bright state (1Πu). The left and right columns provide results for the three
and five
state models, respectively.

(Figs. 2b, 2c) upon enlargement of the electronic manifold by two additional states (Figs. 2e and 2f). Figure 5 presents the results of semiclassical SH simulations for the same initial conditions as used in quantum dynamical calculations shown in Figs. 3 and 4. Depletion of the “bright” 1Πu state to which excita
tion takes place and rise of the “dark” 3Πu0 and of the ground state are noticed within 10 fs of the photoexci
tation in both cases. Furthermore, the evolution of the different state populations is rather similar in the two methods of simulations. This is particularly clear for the 1Πu and 3Πu0 state populations, which become almost parallel near t = 50 fs. In addition, the relative magnitudes of the state
resolved populations are rather similar in the semiclassical and fully quantum simulations. The semi
quantitative agreement between the two methods is exhibited also when com
paring the populations of the three states for the SH simulations with the 5
states quantum model. In par


ticular, the steep depletion of the 1Πu state population, and the shallow minimum observed in the dark state population around t = 30 fs, are in very nice accord with the results of the 5
states quantum mode Significant differences between the SH and the quantum model are observed, however, in the magni
tudes of the population in different states. This dis
crepancy is expected, and results from the different numbers of electronic states included in the two types of simulations, rather than from inadequacy of the SH method to address the present problem. (The SH sim
ulation includes 36 states, whereas the quantum mod
els include 3 or 5 states.) The main conclusions that may be drawn from the combination of Figs. 3 and 5 are summarized as fol
lows: 1. The bright states drop in populations starting already 10 fs after the illumination
long before the F atoms approach and hit the cage boundary. LASER PHYSICS

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ULTRAFAST NONADIABATIC PHOTODISSOCIATION DYNAMICS Population 1.0

Population 0.5 All other singlet All other triplet

X 1Σ+g 3 Πu0 1Π u

0.8

0.4

0.6

0.3

0.4

0.2

0.2

0.1

0

1657

10

20

30

40

50 Time, fs

0

10

20

30

40

50 Time, fs

Fig. 5. Populations of the 1Πu, the 3Πu, and the ground state versus time, calculated semiclassically within the SH method.

Fig. 6. Population of the 31 states excluded from the five
state quantum model but included in the SH simulations. These are grouped into a set of singlet states (blue curve) and a set of triplet curves (red curve).

2. Rapid population buildup in the dark state (3Πu0), corresponding to an ultrafast spin
flip, is evi
dent. This effect was previously observed both theoret
ically and in time resolved pump
probe experiments for several related systems. [13] 3. Significant decay to the ground state is observed at rather early times. The discrepancies between the quantum and the SH model are expected to be mostly due to the migra
tion of population to states that are not included in the

five
state quantum model. We remark that the role of the omitted 31 states becomes more important at times t ≥ 30 fs, when the F atoms hit the cage walls, thereby transferring energy to the Ar atoms and leading to energy redistribution among the electronic states. Fig
ure 6 shows the populations of the triplet and the sin
glet states sets which were not included in our quan
tum models. Evidently, at times longer than the range considered by our quantum model, the populations of the excluded triplet states is important.

0.2

Population 1.00

0.1

0.75 0

0.01

0.02

0.03

0.50

0.25

0

0.01

0.02

0.03 Time, ps

Fig. 7. Comparison of nonadiabatic dynamics of vibratonal populations of electronic states for F2 in static and in dynamical Ar matrix. LASER PHYSICS

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SUKHAREV et al.

Finally, since the quantum mechanical results rely on the validity of the frozen lattice model, it is perti
nent to examine the adequacy of this approximation directly in the context of the semiclassical surface hopping (SSH) calculations. For this purpose, an additional set of SSH calculations were carried out, in which the Ar atoms were frozen in their lattice posi
tions. The full set of the 36 electronic states were employed also in these calculations, which are obvi
ously much less computationally demanding than SSH for the dynamical lattice. Results of the static lat
tice calculations are shown in Fig. 7, where the popu
lations of five of the most populated electronic states are shown vs time, up to t = 30 fs, at which time the F atoms essentially hit the repulsive argon wall. The fig
ure shows also the corresponding populations from the dynamical lattice SSH calculations at selected instances. Not surprisingly, the agreement between the dynamical lattice and the static lattice results is quite good. The main deviation that is seen is for the 3Πu0 state, especially around 20 fs. At that time, the F atoms have not yet struck the argon wall. Inspection of the dynamical lattice SSH calculations shows that the Ar atoms deviate on the average about 0.03 Å from their equilibrium positions within the above timescale. The deviation is small, but it suffices to leave an observable effect on the potential energy surfaces and nonadia
batic couplings. Note that the deviation changes the symmetry around the F–F species, which affects also the interactions. In summary, Fig. 7 shows that the frozen lattice model is not fully quantitative even for t ≤ 30 fs, but is a reasonable approximation which therefore supports the quantum model. 4. CONCLUSIONS Our goal in the work discussed in the previous sec
tions has been two
fold. On the one hand, we explored the extend to which the phenomenon of electroni
cally
localized eigen
states in strongly coupled mani
folds survives in the presence of rapid decay to a mul
titude of electronic states, including the ground state. Our previous study of this phenomenon considered a rather simple model, including only two nonadiabati
cally coupled states in the gas phase. To that end, the case of a diatomic molecule imbedded in a rare gas matrix serves as an interesting model, and one of sig
nificant current experimental interest. In particular, it offers ultrafast spin flip dynamics and a dense mani
fold of electronically coupled states. Our study sug
gests that the phenomenon of electronically
localized states survives in the presence of rapid electronic relaxation and several electronic states, becoming, however, less significant as the number of coupled electronic states grows. Our second goal has been to test the fidelity of the popular surface hopping (SH) model that was often applied in previous studies of similar phenomena to

that discussed here. To that end we made direct com
parisons of fully quantum mechanical, frozen lattice calculations with surface hopping results for the case of F2 in an Ar matrix. Our results suggest the validity of the SH model for the problem considered here. At the same time, they show that the frozen lattice quantum model is useful over a short but experimentally mean
ingful time range. ACKNOWLEDGMENTS We thank Professor J. Manz for helpful discussions. T.S. is grateful to the National Science Foundation (grant number CHE
0616927) for support and to the Weston Foundation for a Visiting Professor Award under the auspices of which this research was carried out. R.B.G. thanks the Deutsche Forschungsgemein
schaft in the framework of the SFB 450 project, “Anal
ysis and control of ultrafast photoinduced reactions.” Our numerical work used in part the resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract no. DE
AC02
06CH11357. REFERENCES 1. For few of the many early reviews see, J. Jortner, S. A. Rice, and R. M. Hochstrasser, Adv. Photochem. 7, 149 (1969); K. F. Freed, in Radiationless Processes in Molecules and Condensed Phases, Ed. by F. K. Fong (Springer, Berlin, 1976), p. 23; S. H. Lin, Radiationless Transitions (Academic, 1980); E. S. Medvedev and V. I. Osherov, Springer Ser. in Chem. Phys., vol. 57 (Springer, Berlin, 1995). 2. T. Suzuki, L. Wang, and H. Kohguchi, J. Chem. Phys. 111, 4859 (1999); V. Blanchet, M. Zgierski, T. Seide
man, and A. Stolow, Nature 401, 52 (1999); D. R. Yar
kony, J. Chem. Phys. 114, 2614 (2001); M. Bargheer, M. Y. Niv, R. B. Gerber, and N. Schwentner, Phys. Rev. Lett. 89, 108301 (2002); H. Koppel, M. Doscher, I. Baldea, H. D. Meyer, and P. G. Szalay, J. Chem. Phys. 117, 2657 (2002);Y. Suzuki, M. Stener, and T. Seideman, Phys. Rev. Lett. 89, 233002 (2002); R. P. Krawczyk, A. Viel, U. Manthe, and W. Domcke, J. Chem. Phys. 119, 1397 (2003); M. Wanko, M. Ga
ravelli, F. Bernardi, T. A. Niehaus, T. Frauenheim, and M. Elstner, J. Chem. Phys. 120, 1674 (2004). 3. M. Sukharev and T. Seideman, Phys. Rev. Lett. 93, 093004 (2004). 4. M. Sukharev and T. Seideman, Phys. Rev. A 71, 012509 (2005). 5. P. S. Christopher, M. Shapiro, and P. Brumer, J. Chem. Phys. 123, 064313 (2005). 6. P. S. Christopher, M. Shapiro, and P. Brumer, J. Chem. Phys. 125, 124310 (2005). 7. A. I. Krylov, R. B. Gerber, and R. D. Coalson, J. Chem. Phys. 105, 4626 (1996). 8. A. I. Krylov and R. B. Gerber, J. Chem. Phys. 106, 6574 (1997). LASER PHYSICS

Vol. 19

No. 8

2009

ULTRAFAST NONADIABATIC PHOTODISSOCIATION DYNAMICS 9. O. Kuhn and N. Makri, J. Phys. Chem. A 103, 9487 (1999). 10. M. Y. Niv, M. Bargheer, and R. B. Gerber, J. Chem. Phys. 113, 6660 (2000). 11. T. Kiljunen, J. Eloranta, J. Ahokas, and H. Kunttu, J. Chem. Phys. 114, 7144 (2001). 12. G. Chaban, R. B. Gerber, M. V. Korolkov, J. Hanz, M. Y. Niv, and B. Schmidt, J. Phys. Chem. A 105, 2770 (2001). 13. M. Bargheer, R. B. Gerber, M. V. Korolkov, et al, Phys. Chem. Chem. Phys. 4, 5554 (2002). 14. M. V. Korolkov and J. Manz, Int. J. Res. Phys. Chem. Chem. Phys. 217, 115 (2003). 15. M. V. Korolkov and J. Manz, J. Chem. Phys. 120, 11522 (2004). 16. M. Gühr, H. Ibrahim, and N. Schwentner, Phys. Chem. Chem. Phys. 6, 5353 (2004). 17. D. Bonhommeau, N. Halberstadt, and A. Viel, J. Chem. Phys. 124, 184314 (2006). 18. M. Fushitani, N. Schwentner, M. Schroder, et al, J. Chem. Phys. 124, 024505 (2006). 19. A. B. Alekseyev, M. V. Korolkov, O. Kuhn, J. Manz, and M. Schroder, J. Photochem. Photobiol. A 180, 262 (2006). 20. M. Bargheer, A. Cohen, R. B. Gerber, M. Guhr, M. V. Korolkov, J. Manz, M. Y. Niv, M. Schroder, and N. Schwentner, J. Phys. Chem. A 111, 9573 (2007).

LASER PHYSICS

Vol. 19

No. 8

2009

1659

21. A. Borowski and O. Kuhn, Theor. Chem. Acc. 117, 521 (2007). 22. A. Borowski and O. Kuhn, J. Photochem. Photobio. A Chem. 190, 169 (2007). 23. A. Borowski and O. Kuhn, Chem. Phys. 347, 523 (2008). 24. V. A. Apkarian and N. Schwentner, Chem. Rev. 99, 1481 (1999). 25. J. C. Tully, J. Chem. Phys. 93, 1061 (1990). 26. F. O. Ellison, J. Am. Chem. Soc. 85, 3540 (1963). 27. I. H. Gersonde and H. Gabriel, J. Chem. Phys. 98, 2094 (1993). 28. P. J. Kuntz, Chem. Phys. 240, 19 (1999). 29. V. S. Batista and D. F. Coker, J. Chem. Phys. 106, 6923 (1997). 30. I. Last and T. F. George, J. Chem. Phys. 87, 1183 (1987). 31. D. C. Cartwright and P. J. Hay, J. Chem. Phys. 70, 3191 (1979). 32. V. Aquilanti and G. Grossi, J. Chem. Phys. 73, 1165 (1980). 33. J. C. Light and T. Carrington Jr., Adv. Chem. Phys. 114, 1958 (2000). 34. R. A. Aziz and M. J. Slaman, Mol. Phys. 58, 679 (1986). 35. R. Kosloff, in Dynamics of Molecules and Chemical Reactions, Ed. by R. E. Wyatt and J. Z. H. Zhang (Mar
cel Dekker, New York, 1996).