J. Phys. Chem. 1996, 100, 16451-16456
16451
Effects of Solute Electronic Structure Variation on Photon Echo Spectroscopy Badry D. Bursulaya and Hyung J. Kim* Department of Chemistry, Carnegie Mellon UniVersity, 4400 Fifth AVe, Pittsburgh, PennsylVania 15213-2683 ReceiVed: April 3, 1996; In Final Form: July 8, 1996X
Two-pulse photon echo spectroscopy for dipolar solutes in water is studied via equilibrium and nonequilibrium molecular dynamics (MD) simulations. By extension of the valence-bond description employed in our earlier MD work [Bursulaya, B. D.; Zichi, D. A.; Kim, H. J. J. Phys. Chem. 1995, 99, 10069; 1996, 100, 1392], both the solute ground- and excited-state electronic polarizabilities are incorporated into the simulations. By modulation of the static transition frequency distribution and solvation dynamics, the state-dependent solute electronic structure variation is found to have a significant influence on the echo; its relaxation behavior with pulse delay is slower for a polarizable solute than for a nonpolarizable one. Despite their similar overall effects, however, the ground- and excited-state polarizabilities modulate the echo via rather different mechanisms. Equilibrium and nonequilibrium MD yields similar results for a nonpolarizable solute but discernible differences for a polarizable solute.
Introduction Solvation is critical for theoretical understanding of many condensed phase processes, e.g., transport phenomena and chemical reactions, due to its strong modulation of their free energetics and dynamics.1 While the information on the solutesolvent coupling and its dynamic behavior can in principle be extracted from the solute optical line shape, it is usually obscured by a static broadening, arising from an inhomogeneous distribution of the local solvent environments. As a result, ordinary linear spectroscopy is sometimes not suitable for probing the system dynamics, such as solvent motions responsible for the solute electronic coherence decay. To circumvent this difficulty, various types of nonlinear spectroscopy are often used.2,3 One well-known technique is photon echo (PE); through a dephasing-rephasing process, it can sufficiently reduce the inhomogeneous contribution and thus is widely used to study solvation dynamics and their influence on optical dephasing.3-11 In many theoretical studies of nonlinear spectroscopy, the solute molecule is usually assumed to be nonpolarizable.3-6 As a result, its electronic structure variation (e.g., the induced dipole moment and oscillator strength changes) with the fluctuating solvent environment is not included. However, since many probe molecules are characterized by large transition dipole moments and high polarizabilities,7-11 its proper account is needed for accurate interpretation of experimental findings. Recent MD simulation studies show that the solute electronic polarizability can affect in a nontrivial fashion both the free energetics and dynamics in solution, including the solvation structure and dynamics and the solute-transport properties.12 Thus one would naturally expect a significant polarizability influence also on nonlinear spectroscopy. To characterize and quantify its effects, we have conducted several MD simulations in water. In this letter, we present our initial results for twopulse PE spectroscopy. A detailed analysis including threepulse PE will be published elsewhere.13 Theory As in our previous MD studies of solvation, we use a quantum mechanical description for the solute electronic structure and a classical description for a nonpolarizable but polar solvent.12 X
Abstract published in AdVance ACS Abstracts, October 1, 1996.
S0022-3654(96)01001-5 CCC: $12.00
The Hamiltonian for the combined solute-solvent system using a point dipole approximation for the solute charge distribution14 is given by
H ˆ )H ˆ ° + Hsolv + WLJ - pˆ ‚e b
(1)
where H ˆ ° is the solute electronic Hamiltonian in vacuum, Hsolv the solvent Hamiltonian, WLJ the solute-solvent short-range interactions,15 pˆ the solute electric dipole operator and b e the electric field at the solute arising from the solvent. For convenience, we use the vacuum adiabatic basis for the solute electronic structure, consisting of the eigenstates |i〉 of H ˆ °; i.e., H ˆ °|i〉 ) E°i|i〉 where i ) 0, 1, ..., in the order of increasing energy eigenvalues. We note that the last term in eq 1 corresponding to the solute-solvent electrostatic interactions is, in general, not diagonal in this basis. This term thus introduces solventdependent electronic coupling among |i〉’s, which is absent in vacuum. As a result, the solute energy eigenstates in solution are given by mixtures of the vacuum adiabatic states.16 Since this mixing varies with b e due to the solvent-dependent coupling, the solute charge distribution and dipole moment fluctuate with the surrounding solvent environment; it is thus electronically polarizable in our theoretical description.12 In the two-pulse PE in solution, the optical dephasing of an electronically coherent state, initially prepared by the first pulse, is subsequently compensated by rephasing via the second pulse. In the impulsive limit, the photon echo signal SPE(τ) is given by3
SPE(τ) ∝ ∫0 dt |R(3)(t,0,τ)|2 ∞
(2)
where τ is the time separation between the two pulses and R(3)(t,0,τ) the third-order solvent response function to an external electric field. In the classical framework employed here for the nuclear degrees of freedom, R(3)(t,0,τ) relevant for the PE is given by3-6,17
〈
R(3)(t,0,τ) ) pge(τ + t) peg(τ) peg(τ) pge(0) × τ i τ+t exp - [∫τ dt′ ∆E(t′) - ∫0 dt′ ∆E(t′)] p
(
© 1996 American Chemical Society
)〉
g
(3)
16452 J. Phys. Chem., Vol. 100, No. 41, 1996
Letters
i i peg,ge(t) ) exp Hrt peg,ge exp - Hrt p p i i ∆E(t) ) exp Hrt (Ee - Eg) exp - Hrt p p
( ) ( )
(
TABLE 1: Parameters for EN Solutes
)
(
)
where g and e label the solute ground and excited electronic states in solution, |g〉 and |e〉, involved in the echo,16 Eg,e and peg,ge are their energies and transition dipole moment, < >g denotes an ensemble average with an initial equilibrium distribution on the ground electronic free energy surface, and Hr is a reference Hamiltonian.3,5 In this letter, we will consider two different reference Hamiltonians for PE, viz., the propagation of nuclear dynamics on the ground electronic free energy surface via equilibrium MD and on the arithmetic mean of the |g〉 and |e〉 surfaces via a nonequilibrium method.5b For later use, we also consider a second-order cumulant expansion3,4a R(3) C (t,0,τ) of the solvent response function in eq 3:
RC(3)(t,0,τ) ) exp[-i〈ωeg〉g(t - τ)] exp[-2g(t) - 2g(τ) + g(t + τ)] (4) g(τ) ) 〈δωeg2〉g∫0 dt (τ - t)C(t) τ
C(t) ) 〈δωeg(t) δωeg(0)〉g/〈δω2eg〉g
(5)
where ωeg and 〈ωeg〉g denote, respectively, the electronic transition frequency between |g〉 and |e〉 and its average, δωeg ) ωeg - 〈ωeg〉g, and C(t) is a normalized time correlation function of ωeg.18 With R(3) C (t,0,τ), the echo signal in eq 2 simplifies to C (τ) ∝ ∫0 dt exp[-4g(t) - 4g(τ) + 2g(t + τ)] SPE ∞
(6)
Models and Simulation Methods As in our previous studies,12 the simulation cell is comprised of a single electrically neutral rigid diatomic solute molecule immersed in 256 rigid water molecules. All solvent and solute bonds are constrained with the SHAKE algorithm.19 The SPC/E water model is used to compute solvent interactions.20 The solute atoms are fixed at a separation of 2.5 Å and interact with the solvent through Lennard-Jones (LJ) and Coulomb potentials. The solute LJ parameters, σ ) 4.0 Å and /kB ) 200 K, and mass, m ) 80 amu, are identical for each constituent atom and remain fixed for all solute models considered here. The partial charges on the solute molecule are varied within the models discussed below. We consider two distinct types of solute models, i.e., an electronically nonpolarizable (EN) model characterized by vanishing transition dipole moments and a quantum mechanically polarizable (QP) model with atomic charges that fluctuate in response to the electrostatic interactions with the solvent. By employing a two-state description, two different EN models are explored: EN1 with the excited-state partial charges q1 ) (0.89e and dipole moment µ1 ) 10.7 D and EN2 with q1 ) (0.75e and µ1 ) 9.0 D, where e is the magnitude of the electronic charge. Their ground-state electronic structures, q0 ) (0.45e and µ0 ) 5.4 D, are identical. Thus each state of the EN models is characterized by different but fixed charge distributions.21 As for the QP solutes, we have studied two different models: QP1 and QP2. For the former, the first excited state is more polarizable than the ground state. By
solute
µ0 (D)
µ1 (D)
E°1 - E°0 (cm-1)
EN1 EN2
5.4 5.4
10.7 9.0
25 000 25 000
TABLE 2: Parameters for QP Solutes A. Vacuum Hamiltonian solute
E°1 - E°0 (cm-1)
E°2 - E°0 (cm-1)
E°3 - E°0 (cm-1)
R°0 a
R°1 b
QP1 QP2
23 000 25 000
39 000 35 000
40 000
0.057 8.12
3.21 3.48
(
B. Dipole Operatorc
5.4 0.36 0 pˆ ) 0.36 8.4 2.28 0 2.28 13.2
(
3.6 2.4 0 4.8
2.4 8.4 2.4 0
0 2.4 10.8 0
)
(for QP1),
4.8 0 0 10.8
)
(for QP2)
a Ground-state electronic polarizability in vacuum (Å3). b First excited-state electronic polarizability in vacuum (Å3). c The dipole matrix elements are evaluated in the vacuum adiabatic basis (units: D). All components are along the direction of the molecular axis.
contrast, the ground-state polarizability for the QP2 solute is higher than its first excited state. We use the three-state and four-state descriptions for QP1 and QP2, respectively.22 The dipole moments of the vacuum adiabatic states as well as the transition dipole moments are represented by partial point charge pairs centered on the atomic sites.23 We have adjusted the QP parameters [Table 2] so that their average ground- and excitedstate dipole moments in water are, respectively, 5.4 and 10.7 D for both QP1 and QP2. These average values coincide, respectively, with the fixed dipole moments of the ground and excited states of EN1. The simulations were conducted in the canonical ensemble at 298 K using the extended system method of Nose´.24 Periodic, truncated octahedral boundary conditions25 were employed. The central containing cube of the truncated octahedron is 25 Å in length, yielding a system density of 1.0 g/cm3. The long-range electrostatic interactions were computed with the Ewald method.26 The trajectories were integrated with the Verlet algorithm27 and a time step of 1.0 fs. At each step of the QP model simulations, the proper adiabatic free energy surface (either |g〉 or |e〉)16 was followed by diagonalizing the Hamiltonian matrix in solution to yield the charges on the polarizable solute. Equilibrium simulations were carried out with 20 ps of equilibration, followed by a 200 ps trajectory from which averages were computed. In the nonequilibrium simulations, the ground-state solute initially in equilibrium with the solvent was promoted instantaneously to an electronic surface corresponding to the arithmetic mean5b of the ground and first excited-state potentials and the ensuing solvent relaxation was monitored for 100 fs. To obtain reliable nonequilibrium statistics, we have generated for each solute model two different ground-state equilibrium trajectories, from which initial configurations for nonequilibrium simulations, separated by 250 fs in time, were collected.28 Thus the resulting PE signal is an average over 1600 distinct nonequilibrium trajectories.29
Letters
J. Phys. Chem., Vol. 100, No. 41, 1996 16453
TABLE 3: Simulation Results in Watera solute
〈µg〉gb
Rgc
〈µe〉ed
Ree
〈µe〉gf
〈δωeg2〉g1/2 g
ωsh
EN1 EN2 QP1 QP2
5.4 5.4 5.4 ( 0.0 5.4 ( 0.7
0.0 0.0 0.1 11.9
10.7 9.0 10.7 ( 0.48 10.7 ( 0.24
0.0 0.0 9.6 4.7
10.7 9.0 9.0 ( 0.25 9.3 ( 0.36
207 142 146 225
65.5 65.5 64.4 48.4
a 〈 〉 and 〈 〉 denote, respectively, the equilibrium ensemble average with the ground and first excited solute electronic states. b Equilibrium g e average and fluctuations of the solute ground state dipole moment (units: D). c Equilibrium average polarizability (units: Å3 ) of the solute ground state along its molecular axis. d Equilibrium average and fluctuations of the solute excited state dipole moment (units: D). e Equilibrium average polarizability (units: Å3) of the solute excited state along its molecular axis. f Average and fluctuations of the solute excited-state dipole moment (units: D) calculated with the ground-state equilibrium ensemble distribution. g Units: ps-1. h Solvent frequency (units: ps-1 ) associated with the inertial solvation dynamics on the ground electronic free energy surface.
Figure 1. Photon echo signal SPE(τ) for the QP and EN solutes in water: EN1 (‚ ‚ ‚); EN2 (O); QP1 (s); QP2 (- ‚ -). SPE was determined directly from the simulations by integrating eq 2 up to t ) 250 fs with eq 3. The initial SPE value is normalized to unity for all solute models. In the inset, ln SPE is displayed. The results for the C cumulant expansion SPE [eqs 4-6] are almost indistinguishable from SPE and thus are not shown.
Results and Discussion The simulation results are summarized in Table 3 and displayed in Figures 1-4. We begin with the PE signal in Figure 1. SPE(τ) there was calculated directly from the groundstate equilibrium simulations via eqs 2 and 3. Though not shown there, we mention that the cumulant expansion results, C i.e., SPE calculated with eqs 4-6, are in excellent accord with SPE. For all cases studied, SPE exhibits an initial rise during the first ∼5 fs, followed by a rapid decay with a characteristic time of 5-10 fs. The initial rise is attributed to the finite inhomogeneous broadening,4b,30 while the ensuing relaxation arises from the dynamical contribution. One of the salient features is that even though the respective equilibrium average values of the ground- and excited-state dipole moments in water are exactly the same for the QP1, QP2, and EN1 models [Table 3], their SPE relaxation behaviors differ considerably. While the EN1 and QP2 echoes show a similar initial increase, the subsequent decay for the polarizable QP2 is slower than that for the nonpolarizable EN1. Compared to these models, QP1 is characterized by significantly slower SPE; also its peak height at τ = 5 fs is lower. It should be noted that despite the close similarity in their ground-state electronic character (both the average dipole and polarizability) and excited-state equilibrium dipole moment, the QP1 and EN1 solutes show considerably different SPE behavior; in fact, the QP1 echo agrees much better
Figure 2. (a) Absorption spectrum for the QP2 solute as a function of δωeg (units: cm-1). The filled circles (b) denote the line shape determined via eq 7, while the solid line is its static approximation result with I(t) ) 〈exp[i∆E(0)t/p]〉g. Thus the dynamical contribution is completely neglected in the latter. (b) Comparison of the absorption spectra for the EN and QP solutes in the static approximation: EN1 (‚ ‚ ‚); EN2 (O); QP1 (s); QP2 (- ‚ -). Since δωeg denotes the deviation of ωeg from its equilibrium value, all four spectra are centered at δωeg ≈ 0.
with that of EN2, whose excited state dipole moment is ∼1.7 D smaller than the corresponding EN1 and equilibrium QP1 value. This clearly illustrates the importance of the excitedstate electronic polarizability Re in PE. Similar comparison of the QP2 and EN1 echo signals exposes the significance of the ground state polarizability Rg.
16454 J. Phys. Chem., Vol. 100, No. 41, 1996
Letters
Figure 4. Comparison of the equilibrium and nonequilibrium simulation results for SPE(τ): equilibrium EN1 (‚ ‚ ‚); nonequilibrium EN1 (- - -); equilibrium QP1 (s); nonequilibrium QP1 (- ‚ -). The equilibrium echo signals are calculated by averaging over two 200 ps trajectories, while the nonequilibrium average is computed from 1600 trajectories, each with duration of 100 fs after the initial FranckCondon excitation from the ground state.
Figure 3. (a) Time correlation function C(t) of electronic transition frequencies [eq 5]. The inertial dynamics responsible for the rapid initial drop become slower with increasing ground-state polarizability. (b) C Photon echo signal in the cumulant expansion, SPE (τ) [eqs 4-6], with 2 the neglect of the 〈δωeg 〉g variation: EN1 (‚ ‚ ‚); QP1 (s); QP2 (- ‚ -). In the numerical calculations, 〈δωeg2〉g1/2 ) 225 ps-1 is employed for all models.
To gain insight into the inhomogeneous and dynamical broadening, we have studied the absorption spectrum corresponding to the |g〉 f |e〉 electronic transition. In Figure 2a, the QP2 spectrum obtained by the Fourier transform of the time domain lineshape function I(t)31
〈 (pi ∫ dt′ ∆E(t′))〉
I(t) ) exp
t
0
g
(7)
is displayed.32 For comparison, the static approximation result with the neglect of the dynamical broadening is also exhibited. The excellent agreement between the two indicates that the spectral line is broadened mainly by the static ωeg distribution.5 In Figure 2b, the MD results for the static absorption spectra for the QP and EN solutes are compared. We first consider the QP1, QP2, and EN1 solutes. Despite their identical equilibrium dipole character, their spectral line shapes differ significantly. While the absorption spectrum for QP2 is somewhat broader than that for EN1, the corresponding QP1 line width is much narrower than those for the two former solutes; the results for the full width at half-maximum are 1800,
2800, and 2600 cm-1 for the QP1, QP2, and EN1 models, respectively. The origin of this rather striking feature is the polarizability difference ∆R (≡Re - Rg) between |g〉 and |e〉, which varies with the solute models. As explained in ref 12a, the instantaneous solute electronic structure adjustment to the local polarization environment tends to reduce the force constant associated with the solvent fluctuations. As a result, the electronic free energy profile along the solvent polarization coordinate becomes softer with increasing solute polarizability. Thus, in the case of QP2 whose Rg is higher than Re (∆R ) -7.2 Å3 ), the ground-state free energy curve is considerably wider than the excited state. This results in a relative broadening of its absorption spectrum, compared to that of EN1.33 By contrast, the QP1 ground state is less polarizable than its excited state (∆R)9.5 Å3 ), so that the former curve is much tighter than the latter. Hence, the QP1 absorption line width is narrower than that of EN1.34 Furthermore, since the ∆R magnitude difference between EN1 and QP1 is larger than that between EN1 and QP2, the line shape difference between the former two solutes is more pronounced than that between the latter two. Thus ∆Rsboth its magnitude and signsplays a primary role in the differing static broadening, i.e., the 〈δωeg2〉g variation with the solute models. Turning to the EN2 spectrum in Figure 2b, its comparison with QP1 suggests that the Re effects on absorption for a polarizable solute could be mimicked by appropriately changing the excited-state dipole moment of the corresponding nonpolarizable solute; in the present case, the reduction of µ1 from 10.7 D (EN1) to 9.0 D (EN2) reproduces the QP1 spectrum.35 It should be noticed that the latter value is the same as 〈µe〉g for QP1 [Table 3], viz., the mean excited-state dipole calculated with the ground-state equilibrium distribution, corresponding to the average for the Franck-Condon (FC) excited states. Since the solvent configuration relevant for the more dipolar FC excited states is actually in equilibrium with the less dipolar ground electronic state, the solvent-induced polarization via Re tends to reduce the FC state dipole moment, compared to 〈µe〉e associated with the equilibrium solvation excited states [Table 3].36 Since the nonpolarizable EN2 model effectively takes
Letters
J. Phys. Chem., Vol. 100, No. 41, 1996 16455
account of the FC state dipole decrease and thus the lowering of the solute-solvent coupling measured by 〈δωeg2〉g, it better reproduces the QP1 lineshape than EN1. However, we also point out that in the case of QP2, a similar dipole scaling would yield a poor result due to the neglect of the Rg effects on absorption.37 Returning to Figure 1, we notice that the initial slope of SPE is strongly correlated with the absorption line width; i.e., the broader the absorption spectrum, the more abrupt the initial echo rise. From eqs 4-6 above, one can show that the initial slope is determined completely by the absorption frequency fluctuation via13
|
1 dSPE(τ) ) SPE(0) dτ 0
xπ8 〈δω
2 1/2 eg 〉g
(8)
With the 〈δωeg2〉g values in Table 3, eq 8 yields 0.23, 0.36, 0.33, and 0.22 fs-1 for QP1, QP2, EN1, and EN2, respectively. This is in good agreement with the trend found in Figure 1. Another feature we observe is that the τ value at the SPE maximum tends to coincide with the inverse line width.30 (The QP2 solute shows a somewhat large deviation from this trend due to its slow solvation dynamics [see Figure 3a].)38 Thus the echo behavior for small τ, viz., the initial slope and peak position, is closely related to the absorption line width. In view of the discussion above, we therefore conclude that ∆R is mainly responsible for the differing short-time SPE behavior among the solutes with the same dipole character. We also notice that due to the effective inclusion of the reduced solute-solvent coupling, the EN2 model well reproduces the initial QP1 echo dynamics (actually entire relaxation; see below). To further characterize the role of Rg and Re in PE, we analyze our MD results via eqs 4-6. These equations are quite revealing in that both the solvation dynamics gauged by C(t) and the solute-solvent coupling 〈δωeg2〉g can be important in the echo relaxation. We first consider the former contribution.5a In Figure 3a, the MD results for C(t) are displayed. As in our previous studies,12 the solute polarizability has dramatic influence on solvation dynamics. To be specific, comparison among the EN1, QP1, and QP2 solutes shows that the rapid initial relaxation of C(t) via inertial dynamics, which is well described by a Gaussian exp[-ωs2t2/2], becomes slower as Rg increases. This has been attributed to the reduction of the solvent force constant (and also to the enhancement of the solvent effective mass), arising from the instantaneous solute charge adjustment to the fluctuating solvent configuration (see above).12a It should also be mentioned that Re little affects C(t) because solvation dynamics occur on the ground electronic free energy surface. Therefore, the C(t) dynamics for the EN1 and QP1 solutes are very similar due to their nearly identical ground-state electronic character [Table 3], while those for QP2 are much slower than the former. C(t) for EN1 and EN2 are identical since their ground-state charge distributions as well as their LJ parameters are the same. The MD results for the solvent frequency ωs are 65.5, 64.4, and 48.4 ps-1 for the EN1 (EN2), QP1, and QP2 solutes, respectively. With this in mind, we proceed to C (τ) in Figure 3b. To separate the polarizability modulation SPE C via C(t) from that via 〈δωeg2〉g and to explicitly quantify of SPE the former, we have used the same 〈δωeg2〉g1/2 ) 225 ps-1 (i.e., the QP2 value) for all the solutes there, rather than their actual values in Table 3. This way, we can eliminate the contribution arising from the differing 〈δωeg2〉g aspect with the solute models, which is mainly due to ∆R as mentioned above (except for C for the QP1 and EN1 solutes are EN2). The resulting SPE almost indistinguishable due to their similar C(t) behavior. By
contrast, the echo decay for QP2 is considerably slower than those for the former models. Thus if the 〈δωeg2〉g variation with ∆R is completely neglected, we find a general parallel between C and C(t) dynamics; to wit, as Rg increases, C(t) the SPE C . This also becomes progressively slower and so does SPE reveals that insofar as the polarizability influence on SPE via solvation dynamics is concerned, the most dominant factor is Rg since C(t) is little affected by Re. C The polarizability effects on SPE through 〈δωeg2〉g, on the other hand, can be illustrated graphically by comparing Figures 1 and 3b. The most notable feature there is that the relative echo behaviors for QP1 and QP2 in Figure 3b are the opposite of those in Figure 1. As analyzed above, with the neglect of C decay for QP1 would the 〈δωeg2〉g variation, the resulting SPE be very similar to that for EN1 and thus faster than that for QP2 [Figure 3b]. However, with its proper account, the relative time scales of the two polarizable models become reversed compared to those in Figure 3b and the QP1 echo becomes actually slower than both QP2 and EN1 [Figure 1]. We thus conclude that the slow relaxation of the QP1 echo arises mainly from its small absorption frequency fluctuations [Figure 2b] and not from C(t) dynamics. This also explains why EN2 with reduced 〈δωeg2〉g reproduces the QP1 echo extremely well in Figure 1. The importance of 〈δωeg2〉g similar to this has been previously noted and analyzed in a model study of PE.6b The major difference is that in the current work, the state-dependent solute electronic structure variation gauged by ∆R is responsible for the 〈δωeg2〉g modulation. This is to be contrasted with the C relaxation is mainly dynamic in QP2 solute whose slow SPE origin, i.e., its C(t) is much slower than those of the other solutes;39 therefore, Rg plays a primary role as pointed out above. These rather different mechanisms for polarizability influence thus clearly indicate that both Rg and Re are essential to the accurate understanding of the echo dynamics. Finally, to examine the dependence of SPE on the reference Hamiltonian Hr in eq 3, we have performed nonequilibrium simulations by propagating the system dynamics on the surface corresponding to the arithmetic mean of the |g〉 and |e〉 (free) energy levels.5b The MD results for the EN1 and QP1 solutes are displayed in Figure 4. While the equilibrium and nonequilibrium simulation methods yield almost identical SPE for the nonpolarizable EN1 solute, there is a discernible discrepancy between the two predictions for the polarizable QP1 model; the nonequilibrium simulations tend to predict a slower overall echo behavior for QP1 than the equilibrium ones. This is somewhat reminiscent of the breakdown of linear response for inertial solvation dynamics in the presence of a polarizable solute, found in our earlier MD studies.12 However, the degree of discrepancy is much milder here for PE than for solvation dynamics. Though not presented here, we have also investigated the effects of the oscillator strength variation with the fluctuating solvent environment on PE. We have found that it has almost negligible influence on SPE.40 Before we conclude, we briefly pause here for perspectives. The short-time echo dynamics analyzed above, in particular, the initial SPE rise and peak, are too fast to be resolved with current time-domain spectroscopic techniques. However, since the time scale for the initial echo behavior increases with decreasing 〈δωeg2〉g [eq 8 and ref 38], sufficiently weak solutesolvent coupling could allow experimental access to the shorttime SPE dynamics.8 The required reduction of 〈δωeg2〉g can be accomplished by lowering the solvent polarity or using a probe molecule with a small |g〉-|e〉 dipole difference; the solvation stabilization decrease with increasing solute molecular size could also be exploited.41 Also, the existence of multiple vibronic
16456 J. Phys. Chem., Vol. 100, No. 41, 1996 modes that can be excited coherently via ultrafast laser pulses could introduce an oscillatory structure in SPE, arising from quantum beats.7 To summarize, we studied the solute electronic polarizability effects on the two-pulse photon echo spectroscopy in water. We found that the solute electronic structure variations associated with both the ground and excited states have important consequences through modulation of the static absorption frequency distribution and solvation dynamics. The details, including the three-pulse echo, will be published elsewhere.13 Acknowledgment. This work was supported in part by NSF Grant No. CHE-9412035. We would like to thank one of the referees for useful comments. References and Notes (1) See, e.g.: Hynes, J. T. AdV. Chem. Phys. 1985, 36, 573. Berne, B. J.; Borkovec, M.; Straub, J. E. J. Phys. Chem. 1988, 92, 3711. (2) For recent reviews, see, e.g.: Zinth, W.; Kaiser, W. In Topics in Applied Physics; Kaiser, W., Ed.; Springer: Berlin, 1993; Vol. 60. Kobayashi, T. AdV. Chem. Phys. 1994, 85, 55. (3) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University Press: New York, 1995. (4) (a) Yan, Y. J.; Mukamel, S. J. Chem. Phys. 1991, 94, 179. (b) Fried, L. E.; Bernstein, N.; Mukamel, S. Phys. ReV. Lett. 1992, 68, 1842. (5) (a) Walsh, A. M.; Loring, R. F. Chem. Phys. Lett. 1991, 186, 77. (b) Shemetulskis, N. E.; Loring, R. F. J. Chem. Phys. 1992, 97, 1217. (6) (a) Cho, M.; Scherer, N. F.; Fleming, G. R.; Mukamel, S. J. Chem. Phys. 1992, 96, 5618. (b) Cho, M.; Fleming, G. R. J. Chem. Phys. 1993, 98, 2848. (7) (a) Becker, P. C.; Fragnito, H. L.; Bigot, J. Y.; Brito Cruz, C. H.; Fork, R. L.; Shank, C. V. Phys. ReV. Lett. 1989, 63, 505. (b) Bigot, J.-Y.; Portella, M. T.; Schoenlein, R. W.; Bardeen, C. J.; Migus, A.; Shank, C. V. Phys. ReV. Lett. 1991, 66, 1138. (8) Nibbering, E. T. J.; Wiersma, D. A.; Duppen, K. Phys. ReV. Lett. 1991, 66, 2464; Chem. Phys. 1994, 183, 167. (9) (a) Yang, T.-S.; Vo¨hringer, P.; Arnett, D. C.; Scherer, N. F. J. Chem. Phys. 1995, 103, 8346. (b) Vo¨hringer, P.; Arnett, D. C.; Yang, T.-S.; Scherer, N. F. Chem. Phys. Lett. 1995, 237, 387. (10) de Boeij, W. P.; Pshenichnikov, M. S.; Wiersma, D. A. Chem. Phys. Lett. 1995, 238, 1. (11) Tokmakoff, A.; Zimdars, D.; Urdahl, R. S.; Francis, R. S.; Kwok, A. S.; Fayer, M. D. J. Phys. Chem. 1995, 99, 13310. (12) (a) Bursulaya, B. D.; Zichi, D. A.; Kim, H. J. J. Phys. Chem. 1995, 99, 10069; (b) 1996, 100, 1392. (13) Bursulaya, B. D.; Kim, H. J., to be submitted. (14) For simplicity, a point dipolar solute will be assumed throughout, even though the extended charge distributions were used in the actual simulations. (15) We assume that WLJ does not depend on the solute electronic states in the current formulation. (16) To distinguish the adiabatic states in vacuum and in solution, we will denote the former as |i〉 ) |0〉, |1〉, ..., corresponding to the vacuum ground, first-excited,... states, while the latter will be represented as |g〉, |e〉, .... (17) The polarization induced in the solvent is in two different directions, B2 - B k1, where B k1 and B k2 are the wave vectors associated with the B k1 and 2k first and second pulses, respectively. Since the signal in the direction of k1 is usually regarded as a photon echo, we will also adopt this 2k B2 - B convention in this letter. (18) C(t) is a microscopic quantity widely used to characterize equilibrium solvation dynamics and is accessible via time-dependent Stokes shift measurements under the assumption of linear response.42 (19) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J. Comput. Phys. 1977, 23, 327. (20) Berendsen, H. J. C.; Gigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269.
Letters (21) Since there is no solvent-induced electronic coupling for the EN models due to vanishing transition dipole moments, their vacuum and solution-phase adiabatic states are identical in our description, i.e., |g〉 ) |0〉 and |e〉 ) |1〉. (22) A two-state description with a nonvanishing transition dipole moment yields a negatiVe electronic polarizability for the (first) excited state. To avoid this, we need, at least, one more state that is higher in energy and electronically coupled via a nonvanishing transition dipole to the first excited state. (23) Thus, the QP solutes are polarizable along the molecular axis.12 (24) Nose´, S. J. Chem. Phys. 1984, 81, 511. (25) Adams, D. J. Chem. Phys. Lett. 1979, 62, 329. (26) Heyes, D. M. J. Chem. Phys. 1982, 74, 1924. (27) Verlet, L. Phys. ReV. 1967, 159, 98. (28) The statistical error analysis43 of our previous MD results for the radial distribution functions for similar systems indicates that the MD configurations separated by J200 fs are not strongly correlated.12b (29) We have found that the echo calculation essentially converges with ∼400 nonequilibrium trajectories for both polarizable and nonpolarizable solutes. (30) Walmsley, I. A.; Kafka, J. D. In Contemporary Nonlinear Optics; Agrawal, G. P., Boyd, R. W., Eds.; Academic Press: Boston, 1992. (31) Lax, M. J. Chem. Phys. 1952, 20, 1752. Kubo, R. AdV. Chem. Phys. 1969, 15, 101. Heller, E. J. Acc. Chem. Res. 1981, 14, 368. (32) Thus the transition dipole moment fluctuations are not included in eq 7. We nonetheless believe that these will not influence the spectra significantly (see below). (33) The solvent force constants associated with the |g〉 and |e〉 electronic curves for EN1 are about the same because both states are nonpolarizable. (34) With the same reasoning, we would expect that the steady-state emission spectra for the EN1, QP1, and QP2 solutes would show exactly the opposite trend of the absorption; to be specific, the spectral line width for emission will be the largest for QP1 and smallest for QP2. The MD results, though not presented here, confirm this. (35) This is in contrast with the general parallel between the effects of increasing polarizability and of growing dipole moment observed for the equilibrium solvation structure and solute rotational dynamics in ref 12b. (36) The degree of the FC state dipole reduction for QP2 is smaller than that for QP1 because Re for the former solute is weaker than that for the latter. (37) The excited-state dipole decrease to fit the 〈µe〉g value of QP2 would lower 〈δωeg2〉g compared to the EN1 solute, just like in the QP1 case. However, since the actual 〈δωeg2〉g for QP2 is higher than that for EN1, this dipole reduction would further increase the absorption line-shape difference between the QP2 and EN models. From this observation and ref 34, we also conclude that in contrast to the absorption case, the EN2 solute will not reproduce the steady-state emission spectrum of QP1. (38) If the τ value at the echo maximum is comparable to 〈δωeg2〉g-1/2, the echo peak occurs approximately at13
τ ∼ 〈δωeg2〉g-1/2[ln(〈δωeg2〉g/ωs2)]1/2 where ωs is the solvent frequency in Table 3. (39) Strictly speaking, the QP2 echo behavior relative to the EN1 signal is determined by two opposing contributions from 〈δωeg2〉g and C(t). Since the QP2 absorption spectrum is broader than that of EN1 [Figure 2b], the former factor would tend to make the QP2 echo faster than that of EN1. However, this is completely opposed and dominated by the slow C(t) C relaxation actually slower than that dynamics of QP2, which make its SPE of EN1. (40) However, this could have significant consequences for three-pulse PE,3,4,6,7b,9b,10 which actually populates the excited states. (41) The lowering of the solute-solvent coupling 〈δωeg2〉g will be signalled, in general, by a decrease in Stokes shift in fluorescence spectroscopy. (42) For reviews, see, e.g.: Simon, J. D. Acc. Chem. Res. 1988, 21, 128. Barbara, P. F.; Jarzeba, W. Acc. Chem. Res. 1988, 21, 195. Maroncelli, M.; MacInnis, J.; Fleming, G. R. Science 1989, 243, 1674. (43) Fincham, D.; Quirke, N.; Tildesley, D. J. J. Chem. Phys. 1986, 84, 4535.
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