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Nonadiabatic Dynamics of A˜-State Photodissociation of Ammonia: A Four-Dimensional Wave Packet Study† Wenzhen Lai,‡,§,| Shi Ying Lin,‡ Daiqian Xie,§ and Hua Guo*,‡ Department of Chemistry and Chemical Biology, UniVersity of New Mexico, Albuquerque, New Mexico 87131, and Institute of Theoretical and Computational Chemistry, Laboratory of Mesoscopic Chemistry, School of Chemistry and Chemical Engineering, Nanjing UniVersity, Nanjing 210093, China ReceiVed: September 8, 2009; ReVised Manuscript ReceiVed: October 15, 2009
The nonadiabatic photodissociation dynamics of the A˜-state ammonia (NH3) was investigated using a fourdimensional wave packet model. The branching ratio between the excited NH2(A˜2A1) and ground NH2(X˜2B1) products was obtained as a function of energy for photodissociation mediated by several low-lying vibrational states in the ground electronic state of NH3. The calculated results could not fully account for the experimental observations of strong mode specificity in nonadiabatic dynamics but agree qualitatively with a recent trajectorybased coupled-surface study using the same potential energy surfaces. Several possible sources of inaccuracy are discussed. I. Introduction The photodissociation of ammonia via its first excited electronic state has served as a prototype for nonadiabatic dynamics in polyatomic systems. The A˜rX˜ excitation places the system on a dissociative excited state potential, which leads adiabatically to H + NH2(A˜2A1). In addition, the dissociation of the excited state NH3 may also proceed via a conical intersection between the two lowest electronic states, leading to the lower H + NH2(X˜2B1) asymptote. This nonadiabatic channel is the only pathway available below the H + NH2(A˜2A1) dissociation limit. The A˜rX˜ absorption spectrum of ammonia is known to be dominated by a long progression of diffuse lines corresponding to the excitation in the umbrella mode. This feature stems from a transition from the pyramidal ground X˜ state to the first excited A˜ electronic state which is planar.1 The diffuse spectral lines signify predissociation on the A˜-state potential energy surface (PES), which is quasibound in the Franck-Condon region as shown in Figure 1.2-5 Lifetimes of these spectral features range from hundreds of femtoseconds to a few picoseconds, depending onthero-vibrationalquantumnumbersandisotopicsubstitutions.6-11 Beyond the small barrier in the dissociation coordinate lays the conical intersection between the X˜ and A˜ states,2-5,12 which facilitates nonadiabatic transitions. Products in both the adiabatic and nonadiabatic channels have been detected with various experimental techniques.13-24 The A˜-state photodissociation of ammonia provides clues for understanding many important dynamical issues such as tunneling, isotope effects, mode-specificity, and nonadiabatic transitions. Several recent studies by Crim and co-workers25-28 and by others,29,30 for example, indicated that photodissociation promoted from different vibrational levels in the ground electronic state leads to significantly different absorption spectra †
Part of the “Benoît Soep Festschrift”. * Corresponding author. E-mail:
[email protected]. University of New Mexico. § Nanjing University. | Current address: Institute of Chemistry and The Lise Meitner-Minerva Center for Computational Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. ‡
Figure 1. Potential energy curves for the X˜-A˜ states of ammonia and the (2 + 1) Radau-Jacobi coordinate system used for the study of NH3 photodissociation.
and fragment branching ratios. In particular, Hause et al. have recently measured using a velocity mapping ion imaging technique the translational energy distributions of the H fragment from photodissociation of NH3 mediated by the ground vibrational state (00), the first symmetric stretch overtone (11), and the first antisymmetric stretch overtone (31).28 The H-atom was found to be hot in the first two cases but cold in the third case, which led these authors to conclude that the excitation of the antisymmetric stretch of NH3 results in a significant enhancement of the electronically adiabatic dissociation leading to NH2(A˜). These observations presented an interesting paradigm for controlling chemical reactions via mode selection.31,32 However, the origin of such a mode-specific control of chemical reaction pathways has not been elucidated. Our understanding of the A˜-state ammonia photodissociation has benefitted greatly from ab initio calculations of the A˜-state PES,2,4,5,33,34 which allowed the dynamical studies with various techniques.3,35-40 We note in particular that full-dimensional PESs for both the X˜1A′1 and A˜1A′′2 states of NH3 have recently been reported by Truhlar and co-workers.41,42 These PESs have enabled full-dimensional dynamical studies using both quantum
10.1021/jp908688a 2010 American Chemical Society Published on Web 11/11/2009
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mechanical43 and coupled-state quasiclassical trajectory (CSQCT) methods, in which nonadiabatic transitions are treated semiclassically.44,45 The latest theoretical studies have shed much light on the dynamics of the multidimensional dissociation dynamics but also raised some questions. For example, it was reported that the A˜/X˜ branching ratio obtained from several trajectory-based nonadiabatic calculations did not fully reproduce the experimentally observed mode-specificity.45 While a careful analysis was given by Bonhommeau et al.,44,45 it remains unclear whether the theory-experiment discrepancy is due to classical mechanics used to describe the dissociation dynamics, or the semiclassical treatment of nonadiabatic transitions, or something else. To resolve this issue, we report here a fourdimensional quantum wave packet study of the nonadiabatic photodissociation dynamics of ammonia on the same PESs as in the recent CS-QCT work.45 This publication is organized as follows. The next section (section II) outlines the wave packet theory for describing the nonadiabatic photodissociation process of NH3. The results are presented and discussed in section III. A short summary is given in section IV.
allow nonadiabatic transitions between the two electronic states near the conical intersection. Each element of the potential matrix is fit with an analytic form. The PESs correctly reproduced experimental dissociation energies in both channels, the A˜/X˜ excitation energy for NH3 and the A˜/X˜ excitation energy of the NH2 fragment. The Hamiltonian was discretized using a mixed finite basis representation/discrete variable representation (FBR/DVR)47,48
II. Theory
|j1j2m, p〉 ) (2 + 2δm,0)-1/2(|j1m〉|j2 - m〉 + p|j1 - m〉 |j2m〉),p ) (1 (3)
As in our recent full-dimensional work,43 the nuclear Hamiltonian (J ) 0) is given in the (2 + 1) Radau-Jacobi coordinates.46,47 (In reality, nonzero J values might be accessed by photoexcitation, but small nonzero values of J are not expected to alter the nonadiabatic dynamics discussed here.) As shown in Figure 1, the NH2 fragment is described with Radau coordinates (r1, r2), while the Jacobi coordinate (r3) is defined between H and the center of mass of the NH2 fragment. To facilitate the expensive calculations, we focus on one of the three equivalent dissociation channels by fixing the two radial Radau coordinates (r1 ) r2 ) re), which are essentially the N-H bond lengths in the NH2 fragment. The resulting fourdimensional (4D) Hamiltonian takes the following form (p ) 1)47 2
(
) (
∑ ψip j j m|i3〉|j1j2m, p〉
(2)
312
For the radial coordinate r3, a DVR basis indexed by i3 was employed. The action of the lone radial kinetic energy operator (KEO) onto the wave packet was evaluated by a sine-DVR method49 with a uniform grid with 158 points in the region of [1.2, 17.1] a0. The angular FBR was defined as parity-adapted products of the spherical harmonics
The first three angular KEOs are diagonal with the following matrix elements47,48
〈j1′j2′m', p|jˆi2 |j1j2m, p〉 ) ji(ji + 1)δj1j1δj2j2δm'm, ′ ′
i ) 1, 2 (4)
〈j1′j2′m', p|jˆ1zˆj2z |j1j2m, p〉 ) -m2δj1j1δj2j2δm'm ′ ′
(5)
The matrix of the fourth angular KEO is tridiagonal, and the elements are given as follows
)
1 ˆ2 1 ˆ2 1 ∂ 1 1 j1 + j2 + + + ˆ )H 2µ3 ∂r2 2µ1re2 2µ3r23 2µ2re2 2µ3r23 3 ˆj1zˆj2z ˆj1+ˆj2- + ˆj1-ˆj2+ + + + V(re, re, r3, θ1, θ2, φ) 2 µ3 r3 2µ3r23
|Ψ〉p )
(1)
where µ1, µ2, and µ3 are mass factors for b r 1, b r2, and b r 3, respectively. ˆj1 and ˆj2 denote the angular momentum operators r2, respectively, in the body-fixed (BF) associated with b r1 and b coordinate system defined by b r3. θ1 (θ2) is the polar angle between b r1 (r b2) and b r3, and φ is the relative azimuthal angle r2 in the BF system. The latter (φ) can be used between b r1 and b to describe the umbrella motion in an approximate fashion, with φ ) 180° being the planar configuration. In our calculations, re was fixed at the ground state equilibrium geometry. In this reduced-dimensional model, not all vibrational modes of ammonia can be modeled. For example, the symmetric stretch mode of NH3 is not permitted because of the two fixed N-H bonds. In addition, the degenerate antisymmetric stretch of NH3 is now approximated by the stretch between the dissociating H and the NH2 fragment. These approximations are by no means minor, but we believe that they are sufficiently reasonable to provide valuable insights into the nonadiabatic dynamics. The A˜/X˜ PESs of the ammonia system developed recently by Truhlar and co-workers were used in our calculations.41,42 The diabatic PESs are given in a 2 × 2 matrix constructed using multiconfiguration quasidegenerate perturbation theory, and they
〈j1′j2′m', p|jˆ1+ˆj2- + ˆj2+ˆj1- |j1j2m, p〉 ) δj1j1δj2j2[(1 + ′ ′ δ0,m)(1 + δ0,m')]-1/2 × [λj+1,mλj+2,m(δm',m+1 + pδ-m',m+1) + λj-1,mλj-2,m(δm',m-1 + pδ-m',m-1)] (6) ( ) [j(j + 1) - m(m ( 1)]1/2. In our calculations, 21 where λj,m spherical harmonics (j1, j2 ) 0-20) were used for each rotational angle. The action of the potential energy operator was evaluated using a pseudospectral method between the FBR and DVR.50,51 This method exploits the factorizability of the three-dimensional transformation between the nondirect product angular FBR and a direct product angular grid by replacing it with three sequential one-dimensional transformations
L(θ1:m)
L(θ2:-m)
L(φ)
ψji13j2mp 798 ψRi31j2mp 798 ψRi31R2mp 798 ψRi31R2βp
(7) where R1, R2, and β, respectively, denote the indices of the angular grids along the θ1, θ2, and φ coordinates. The transformation matrices are defined as
Dynamics of A˜-State Photodissociation of Ammonia
LR,j(θ:m) ) √wRΘj,m(θR),
J. Phys. Chem. A, Vol. 114, No. 9, 2010 3123
j ) |m|, ..., jmax ; R ) 1, ..., nθ (8)
Lβ,m(φ) ) √(2 - δm,0)/nφ ×
{
p ) +1 ,β ) p ) -1 1, ..., nφ (9)
cos(mφβ) sin(mφβ)
where Θj,m is the associate Legendre function; nθ and nφ denote the number of grid points in the polar angle θ and azimuthal angle φ, respectively. φβ is equal to (β - 1/2)π/nφ, and θR and wR are, respectively, the abscissas and weights of the corresponding Gauss-Legendre quadrature. We first calculated the low-lying vibrational levels of NH3 on the ground electronic state. To this end, the recursive Lanczos algorithm52 was used. Essentially, a random initial wave packet was updated with a three-term recursion relationship, which reduced the full Hamiltonian to a smaller tridiagonal matrix. The diagonalization of the latter yielded eigenvalues. Since the Lanczos algorithm is well documented (see, for example, ref 53), no detail is given here. Typically, 2000 steps of recursion were needed to converge the results. For the absorption spectra, we employed a Condon model, in which a vibrational eigenfunction of NH3(X˜), |χi〉, was placed on the A˜-state PES via a vertical transition and allowed to evolve. The A˜rX˜ absorption spectrum was then obtained from Fourier transform of the Chebyshev autocorrelation function54
Σ(E) )
∑
1 (2 - δk0)cos(kθ)Ck π sin θ k)0
(10)
Here, the Chebyshev angle is related to energy as E ) cos θ, and the autocorrelation function (Ck ) 〈ψ0|ψk〉) was obtained along the Chebyshev propagation with the A˜-state Hamiltonian55
ˆ scaled |ψk〉 - D|ψk-1〉) |ψk+1〉 ) D(2H
(11)
ˆ scaled|ψ0〉 and |ψ0〉 ) |χi〉. In eq 11, the with |ψ1〉 ) DH Hamiltonian was scaled so that its spectrum lies within [-1,1]. The damping function D applied at the grid edges imposes outgoing wave boundary conditions in all three equivalent dissociation channels. In this work, the damping starts at r ) 3.1a0 for the radial coordinate r3. To converge the absorption spectra, 33 000 steps of Chebyshev propagation were used. For narrow resonances, we have also determined their complex energies (En - iΓn/2) from the Chebyshev correlation functions using our version of low-storage filter diagonalization (LSFD).56 Essentially, a small Hamiltonian near the energy of interest was constructed form the correlation functions and diagonalizedtoobtainpositionsandwidthsofdesiredeigenvalues.57,58 Since we focus here on the nonadiabatic dynamics, we will only calculate the branching ratio between the adiabatic and nonadiabatic product channels, without resolving the numerous NH2 internal states. The A˜/X˜ branching ratio, P(A˜;E)/P(X˜;E), was obtained from the total probabilities in the two channels using a flux-based method from the Chebyshev wavepacket59
〈∑
P(E) ) C(E)Im
k
(2 - δk0)e-ikθψk
|∑
(2 - δk'0)e-ik'θ
[
k'
δ(r3 - rf)
∂ ψ ∂r3 k'
Figure 2. Calculated absorption spectra from the 00, 31, and 32 states of NH3(X˜). The parities (g and u) are indicated in blue and red color, respectively, and they are assumed to have equal weights.
]〉
(12)
where C(E) is an energy-dependent factor that cancels in the branching ratio. In our flux calculations, the dividing surface is placed in the dissociation asymptote at rf ) 15a0. The differentiation in eq 12 can be carried out analytically on the sine basis. The convergence of the probabilities can be achieved with 33 000 Chebyshev steps. III. Results and Discussion We first examine the impact of the reduced-dimensional model on the absorption spectra. To this end, the A˜rX˜ absorption spectra from the ground vibrational state (00) of NH3 are displayed in Figure 2. Like in the 6D case,43 the spectra are dominated by the umbrella progression, due apparently to the pyramidal-to-plane transition. The spectra from the NH3 tunneling pair, labeled by parities g and u, are assigned the same weight and plotted in the same figure. In reality, however, they may have different weights due to different statistics of the para and ortho NH3 species.3 The positions and widths of these predissociative resonances have been determined using LSFD and are listed in Table 1, along with the available experimental data and the earlier full dimensional results obtained on the same PES. Although the overall features of the 4D spectrum are similar to their 6D counterpart, there are some significant differences. For example, the widths are generally narrower in the 4D spectra, particularly for higher states. In addition, the width of the second overtone is narrower than the first, which is at odds with the experimental observation and the earlier fulldimensional quantum calculation. These differences indicate that the efficiency of predissociation and presumably that of energy flow from the umbrella mode to the dissociation coordinate are somewhat hindered in the 4D model, underscoring the multidimensional nature of the dissociation dynamics. The absorption spectra for two antisymmetric stretch overtones (31 and 32) of NH3 are also displayed in Figure 2. These
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TABLE 1: Calculated A˜rX˜ Adsorption Band Positions (ν) and Widths (Γ) for NH3 and Comparison with Available Experimental Data 6D Theo.43
4D Theo. (this work) band 0
0 21 22 23 24 25 26 27 28 29 210
Expt.8
ν (cm-1)
Γ (cm-1)
∆ (cm-1)
ν (cm-1)
Γ (cm-1)
∆ (cm-1)
ν (cm-1)
Γ (cm-1)
∆ (cm-1)
46986 47846 48672 49538 50443 51383 52346 53322 54305 55292 56279
94.3 56.9 40.4 39.0 50.4 58.1 56.2 47.6 37.9 32.2 32.7
0 860 826 866 905 940 963 976 983 987 987
45708 46614 47452 48278 49122 49904 50691 51588 52334 53191 54030
103.9 58.8 67.4 121.2 235.1 394.4 369.7 367.1 280.8 213.4 146.1
0 906 838 826 844 782 787 897 746 857 839
46222 47057 47964 48869 49783 50730 51656 52543 53496 54454 55380
109 62.1 83.6 95.5 150.4 186.2 240.0 275.1 263.9 276.3 273.2
0 838 907 905 914 948 926 887 953 958 926
spectra are significantly different from those from the ground vibrational state (00). First, the envelope of the spectrum shifts to higher energies with the vibrational excitation, due apparently to the internal energy of the overtone states. Second, in addition to the sharp umbrella progression, there is a broad background and some additional lines. The broad background, which is particularly conspicuous for the 32 state, is apparently due to the inner repulsive wall of the A˜-state PES, accessed by the extended wave function at small N-H distances. It should be noted that the 4D spectra are generally more structured than the 6D spectra,43 indicating the smaller phase space in the reduced-dimensional model. On the other hand, it is clear from both the full-dimensional and reduced-dimensional models that the vibrational excitation of NH3(X˜) allows the access of different vibrational phase space regions due to differences in the Franck-Condon factors, in support of the conclusions drawn from vibrational-mediated photodissociation experiments.31,32 To shed light on the nonadiabatic dynamics, we have calculated the A˜/X˜ branching ratio for the direct photodissociation of ammonia (00) and photodissociation mediated by two antisymmetric stretch overtones (31 and 32) of NH3(X˜). The branching ratios for the two different parities are roughly the same, and we will thus focus on the average of the two in the following discussions. The branching ratios are plotted in Figure 3 as a function of the total energy and of the photon energy of the photolysis laser. In the latter case, the energy of the overtone curves is shifted downstream by the vibrational energy (3312 and 6470 cm-1, respectively) relative to the 00 state. From the upper panel of Figure 3, it is clear that the calculated branching ratios show very weak dependence on the vibrational state from which the A˜ state NH3 was promoted, at least for the PESs used in our calculations. In other words, the branch ratio is largely a function of the total energy. There are some structures in the branching ratios, presumably attributable to the predissociative vibrational states in the A˜ manifold. However, the mode-specificity is quite weak, except perhaps at very high energies. These results are in sharp contrast to the experimental observations that the photodissociation of the 31 state produces predominately the NH2(A˜) fragment, while that from the 00 state yields largely the NH2(X˜) fragment.28 However, they are consistent with the recent CS-QCT studies, which found very weak mode-specificity between the symmetric and antisymmetric stretching modes.45 It should be pointed out that the CS-QCT model prepared the trajectories from either quasiclassical conditions or Wigner distribution functions corresponding to certain A˜-state quasibound quantum levels.45 This choice was based on the premise that these quantum levels are disproportionally favored in
vibrationally mediated photodissociation processes. In our quantum calculations, however, we have seen little evidence for the existence of such quasibound states, presumably due to the fact that their lifetimes are too short. In addition, the absorption spectra from vibrational overtones on the X˜ state did not show any sign of favorable excitation and neither did the branching ratios. As a result, we did not attempt calculations mimicking the CS-QCT calculations. In addition, our calculations indicate the adiabatic channel leading to the NH2(A˜) fragment is always the minor channel throughout the energy range. At the experimental energy of ∼50 000 cm-1, our results indicate that there are only about 3% of the product exiting in the NH2(A˜) channel, via the adiabatic dissociation. Thus, the nonadiabatic dissociation, which dissociates to the NH2(X˜) channel, dominates. Our quantum result is consistent with the earlier CS-QCT study,45 which predicted values between 0.3 and 2.7% for the adiabatic channel for photodissociation mediated by the 31 state. The CS-QCT study did not consider photodissociation mediated by the 32 state but did consider that by the 36 state. The latter has approximately
Figure 3. A˜/X˜ branching ratio for photodissociation mediated by the 00, 31, and 32 states of NH3(X˜) as a function of the total energy (upper panel) and as a function of the photon energy of the photolysis laser (lower panel).
Dynamics of A˜-State Photodissociation of Ammonia an energy of 65 000 cm-1, and the CS-QCT values range from 12.7 to 17.0%.45 In our quantum study, the A˜ state population increases and reaches about 30% at 60 000 cm-1. At higher energies, the antisymmetric stretch excitation leads to higher excited state populations. At 65 000 cm-1, for example, the NH2(A˜) population is approximately 55%, 58%, and 59% for the 00, 31, and 32 states, which are somewhat larger than the CS-QCT values. The increasing A˜/X˜ branching ratio with increasing energy is perhaps due on one hand to the availability of more open NH2(A˜) internal states and on the other hand to the increased speed in the dissociation coordinate when passing the cross seam. So overall, the agreement between our quantum model and the earlier CS-QCT model is reasonable. We want to emphasize that our reduced-dimensional model is incapable of simulating the symmetric stretch overtone excitations in NH3. As a result, we can only compare photodissociation mediated by the antisymmetric stretch overtones with direct photodissociation from the ground 00 state. Since roughly the same photolysis laser frequency was used for both the 00 and 31 states in the experiment,28 the total energy of the system is different in the two cases. As a result, it might be more sensible to compare the branching ratios at the same photon energy, rather than the total energy. In the lower panel of Figure 3, the branching ratios are plotted as a function of the photon energy. It can be readily seen that in this case the photodissociation mediated by the antisymmetric stretch overtones produces much more NH2(A˜) at the same photon energy, mainly because the system possesses more energy than direct photodissociation. Nevertheless, it still cannot account for the near exclusive A˜-state distribution observed experimentally for photodissociation mediated by the first antisymmetric stretch overtone.28 It is apparent that the experimentally observed modespecificity in the nonadiabatic dynamics is yet to be explained by theory, quantum mechanical or classical. There are many possible reasons for this failure. For instance, the reduced dimensionality of our 4D model significantly reduced the density of states. In addition, the comparison with the full-dimensional results clearly indicated some important differences in the absorption spectra. The impact of these differences on the nonadiabatic dynamics is not clear because such dynamics has not been investigated with a full-dimensional model. On the other hand, the reasonably good agreement between the quantum and CS-QCT results suggests that the various semiclassical treatments of nonadiabatic transitions in these studies are qualitatively accurate. This left us with the possibility of inaccuracies in the PESs. Indeed, there is some tantalizing evidence indicating that the A˜-state PES requires significant improvement at high energies because the calculated lifetimes for high umbrella overtones do not agree with the experimental trend.43 In addition, the vibrational energies in both the X˜ and A˜ state PESs have shown considerable deviations from known experimental band origins. It should be noted that Bonhommeau et al.45 have performed a sensitivity analysis of the nonadiabatic dynamics with respect to the nonadiabatic coupling potential and concluded that the effect is not significant. Thus, if there were deficiencies in the PESs, they are most likely to be related to intermodal couplings, which in turn control the energy flow among the vibrational modes. IV. Summary In this work, we have examined the nonadiabatic dynamics in the photodissociation of ammonia using a reduced-dimensional quantum model on recent potential energy surfaces and
J. Phys. Chem. A, Vol. 114, No. 9, 2010 3125 their coupling. Our results indicate that the nonadiabatic dissociation pathway leading to the NH2(X˜) fragment is dominant in all the energies, but the adiabatic dissociation channel leading to the NH2(A˜) fragment gains importance with the increasing photon energy. Our quantum mechanical results are consistent with the earlier CS-QCT data, validating their treatments of nonadiabatic transitions. However, the failure to reproduce the experimentally observed mode-specificity by both quantum and CS-QCT models suggests the need to improve the relevant potential energy surfaces to rule out an important uncertainty. Acknowledgment. We thank Don Truhlar and Fleming Crim for useful discussions. This work was supported by the US Department of Energy (DE-FG02-05ER15694) and by Chinese National Natural Science Foundation (20533060 and 20725312) and Chinese Ministry of Science and Technology (2007CB815201). References and Notes (1) Douglas, A. E. Discuss. Faraday Soc. 1963, 35, 158. (2) Runau, R.; Peyerimhoff, S. D.; Buenker, R. J. J. Mol. Spectrosc. 1977, 68, 253. (3) Rosmus, P.; Botschwina, P.; Werner, H.-J.; Vaida, V.; Engelking, P. C.; McCarthy, M. I. J. Chem. Phys. 1987, 86, 6677. (4) McCarthy, M. I.; Rosmus, P.; Werner, H.-J.; Botschwina, P.; Vaida, V. J. Chem. Phys. 1987, 86, 6693. (5) Manz, U.; Reinsch, E.-A.; Rosmus, P.; Werner, H.-J.; O’Neil, S. V. J. Chem. Soc., Faraday Trans. 1991, 87, 1809. (6) Ziegler, L. D. J. Chem. Phys. 1985, 82, 664. (7) Ashfold, M. N. R.; Bennett, C. L.; Dixon, R. N. Faraday Discuss. Chem. Soc. 1986, 82, 163. (8) Vaida, V.; McCarthy, M. I.; Engelking, P. C.; Rosmus, P.; Werner, H.-J.; Botschwina, P. J. Chem. Phys. 1987, 86, 6669. (9) Baronavski, A. P.; Owrutsky, J. C. J. Phys. Chem. 1995, 99, 10077. (10) Henck, S. A.; Mason, M. A.; Yan, W.-B.; Lehmann, K. K.; Coy, S. L. J. Chem. Phys. 1995, 102, 4772. (11) Henck, S. A.; Mason, M. A.; Yan, W.-B.; Lehmann, K. K.; Coy, S. L. J. Chem. Phys. 1995, 102, 4783. (12) Yarkony, D. R. J. Chem. Phys. 2004, 121, 628. (13) Fuke, K.; Yamada, H.; Yoshida, Y.; Kaya, K. J. Chem. Phys. 1988, 88, 5238. (14) Biesner, J.; Schnieder, L.; Schmeer, J.; Ahlers, G.; Xie, X.; Welge, K. H.; Ashfold, M. N. R.; Dixon, R. N. J. Chem. Phys. 1988, 88, 3607. (15) Biesner, J.; Schnieder, L.; Ahlers, G.; Xie, X.; Welge, K. H.; Ashfold, M. N. R.; Dixon, R. N. J. Chem. Phys. 1989, 91, 2901. (16) Woodbridge, E. L.; Ashfold, M. N. R.; Leone, S. R. J. Chem. Phys. 1991, 94, 4195. (17) Mordaunt, D. H.; Ashfold, M. N. R.; Dixon, R. N. J. Chem. Phys. 1996, 104, 6460. (18) Mordaunt, D. H.; Dixon, R. N.; Ashfold, M. N. R. J. Chem. Phys. 1996, 104, 6472. (19) Mordaunt, D. H.; Ashfold, M. N. R.; Dixon, R. N. J. Chem. Phys. 1998, 109, 7659. (20) Loomis, R. A.; Reid, J. P.; Leone, S. R. J. Chem. Phys. 2000, 112, 658. (21) Reid, J. P.; Loomis, R. A.; Leone, S. R. J. Chem. Phys. 2000, 112, 3181. (22) Reid, J. P.; Loomis, R. A.; Leone, S. R. J. Phys. Chem. A 2000, 104, 10139. (23) Reid, J. P.; Loomis, R. A.; Leone, S. R. Chem. Phys. Lett. 2000, 324, 240. (24) Wells, K. L.; Perriam, G.; Stavros, V. G. J. Chem. Phys. 2009, 130, 074308. (25) Bach, A.; Hutchison, J. M.; Holiday, R. J.; Crim, F. F. J. Chem. Phys. 2002, 116, 9315. (26) Bach, A.; Hutchison, J. M.; Holiday, R. J.; Crim, F. F. J. Chem. Phys. 2002, 116, 4955. (27) Bach, A.; Hutchison, J. M.; Holiday, R. J.; Crim, F. F. J. Phys. Chem. A 2003, 107, 10490. (28) Hause, M. L.; Yoon, Y. H.; Crim, F. F. J. Chem. Phys. 2006, 125, 174309. (29) Akagi, H.; Yokoyama, K.; Yokoyama, A. J. Chem. Phys. 2003, 118, 3600. (30) Akagi, H.; Yokoyama, K.; Yokoyama, A. J. Chem. Phys. 2004, 120, 4696. (31) Crim, F. F. Annu. ReV. Phys. Chem. 1993, 44, 397. (32) Crim, F. F. J. Phys. Chem. 1996, 100, 12725–12734.
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