F + H, Collisions in the Presence of Intense Laser Radiation: Reactive and Nonreactive Processes BY PAUL L. DEVRIES AND THOMAS F. GEORGE* Department of Chemistry, University of Rochester, Rochester, New York 14627, U.S.A. AND JIAN-MIN YUAN Department of Physics and Atmospheric Science, Drexel University, Philadelphia, Pennsylvania 19 104, U.S.A. Received 5th December, 1978 Two sets of calculations are discussed for F + H2 collisions occurring in the presence of intense laser radiation. The first set, based on a semiclassical theory whereby nuclear degrees of freedom are treated classically, considers collinear reactive collisions in the presence of the 1.06 pm line of a Nd- glass laser. For a high enough intensity the laser alters the vibrational distribution of the HF product.The second set, where all degrees of freedom are treated quantum mechanically, considers the quench- ing of fluorine in its excited spin-orbit state by three-dimensional (nonreactive) collisions with H2 in the presence of a laser whose photon frequency is ~4408 cm-'. For high enough intensity the laser substantially enhances the quenching cross-section. 1. INTRODUCTION There has been considerable interest in the theory of the interaction of intense laser radiation with molecular collision systems [see review of work done at Rochester in ref. (l)]. With high enough intensity, the laser radiation can actually interact with the collision dynamics, even if the frequency of the laser photon is not in resonance with energy levels of the asymptotic collision species. Recently we have obtained some results for both reactive2 and nonreactive3 processes in the F + Hz collision system, which we summarize in this paper.For the reactive processes, discussed in section 2, we restrict ourselves to collinear collisions and consider the case where the fluorine atom is initially in its ground spin- orbit state. We further restrict ourselves to two (semiempirical) potential energy surfaces, where one correlates to fluorine in its ground spin-orbit state and the other to fluorine in its excited spin-orbit state. By integrating classical trajectories for the nuclear degrees of freedom, we investigate how the reaction dynamics is affected by the 1.06 pm line of a Nd-glass laser. For the nonreactive processes, discussed in section 3, we represent the electronic degrees of freedom by a 6 x 6 diatomics-in-molecules Hamiltonian matrix.This problem is simplified by ignoring reactive channels, in which case we are able to con- sider three-dimensional collisions, where all degrees of freedom are treated quantum mechanically. We consider how a laser with photon frequency of ~ 4 0 8 cm-' affects the quenching of fluorine in its excited spin-orbit state to its ground spin-orbit * Camille and Henry Dreyfus Teacher-Scholar ; Alfred P. Sloan Research Fellow.P . L . DEVRIES, T . F . GEORGE AND J . - M . YUAN 91 state. This frequency is sufficiently different from the spin-orbit splitting in fluorine (404 cm-') that radiative transitions cannot occur without the aid of the H2 collision partner. 2.REACTIVE PROCESSES In this section we shall study how the reaction dynamics of the F + H2 collision system can be affected by shining an intense laser field on the collision region. The F + H2 reactive system is especially interesting for this study for several reasons. First, this is a reaction which produces laser emission, namely, the HF-laser. If a second laser shining through the laser cavity could change the branching ratios of the HF vibration states, one would then be able to change the characteristics of the H F laser. Secondly, this is one of the rare systems for which we have ab initio information for ground- and excited-state potential energy surfaces and transition dipole moments as functions of internuclear distances. All our calculations were performed with an initial collision energy of 0.049 eV, relative to the F + H2 (u = 0) state, which is insufficient to access the excited spin- orbit state (which lies at ~ 0 .0 5 eV). Thus, in terms of the electronic-field surfaces employed in this work, reaction can only occur through nonadiabatic transitions between the electronic-field surfaces. Semiclassical trajectory methods developed for electronic transitions in field-free cases can be applied here. We have used the decoupling approximation developed for the Miller-George theory ' in our calcula- tiorx2 The field-free ground electronically diabatic surface is the semiempirical Muckerman V surface.6 The field-free excited electronic surface is obtained by fitting parameters to data based on GRHF-CI calculation^.^ Field-free adiabatic surfaces, W, and W2, are then calculated by coupling adiabatic surfaces through a constant spin-orbit interaction term.W, and W2 plotted against a reaction coordinate s are shown schematically in fig. 1. The transition dipole moment between the 'C and Ill states of HF as a function of interatomic distance has been calculated by Bender and Davidson.' The overall transition dipole coupling is taken to be proportional \ +-S FIG. 1 .--Scheme of the field-free adiabatic surfaces, W, and W,, including spin-orbit interaction, for the reactive F + H, system along a reaction coordinate s. The vertical line indicates the photon energy from a Nd-glass laser, which comes into resonance with W, and W2 along s.92 F + H, COLLISIONS I N LASER FIELDS to the moment they have calculated, so that the coupling is a function of just the H-F distance.The electronic-field surfaces, E- and E,, can then be constructed from W,, W, and the radiative coupling d12 according to the relation where E , = *(W1 + W2 + hm [ W2 - W1 - hw)’ + 4d12]1/2} (1) 4 2 = P12% (2) p12 is the overall transition dipole coupling, E is the field strength and co is the laser photon frequency. The reaction we have studied is hw (1.06 pm) F(2P3/2) + H ~ ( u = 0) - HF(u’) + H (3) in the presence of a Nd-glass laser, which involves no net absorption or emission of photons. We are especially interested in the change in the reaction probability and the final vibrational distribution of the HF molecule induced by the laser field. We have also calculated inelastic transition probabilities for the process (4) which does involve absorption of a photon.In the reactant asymptotic region, E,, on which we start all the trajectories, be- comes Wl + hm, and Wl itself becomes a Morse potential of a hydrogen molecule plus a free fluorine atom. To select initial conditions for a grid of trajectories, we transform the coordinates of the Morse oscillator of the reactant diatomic molecule into action-angle variables.’ The vibrational quantum number and the phase of the oscillator are the corresponding action and angle variables, respectively. All trajec- tories start in the asymptotic region with H2 in the ground vibrational state, and we have selected the initial vibrational phase with a uniform grid of one hundred values between 0 and 27t.Let rl be the H2 internuclear distance and r3 the distance between F and the closer H atom. Associated with rl and r3 are the translational coordinates R1 and &, where R1 is the separation between F and the centre of mass of H2, and, denoting H, (H,) as the H atom further from (closer to) F, R3 is the separation between H, and the centre of mass of H,F. To apply the Miller-George theory we need to locate complex branch points (ix., intersection points) between the electronic-field surfaces. For each field strength of the laser field we have a different set of electronic-field surfaces, and therefore a different set of branch points. The way we find the proper set of branch points is that for a fixed real rl we change r3 iteratively in the complex plane until the equation E, = E- holds to a high degree of accuracy a.u.).The curve formed by projecting the set of branch points onto the real plane will be called a “ seam ”. In the decoupling approximation one needs to integrate the trajec- tory perpendicularly to the “ seam ” into the complex space to find the transition probability. To separate effectively the nuclear kinetic energy term into two com- ponents, one containing the perpendicular momentum and the other the parallel one, we project the set of branch points onto the (r3, R3) plane. The seam lies approximately on a straight line, which defines an axis called RII with a unit vector $11. The axis perpendicular to the seam is labelled as RI, with unit vector ii;, PI and PI, are momen- tum components along RL and R;I, respectively.The separable approximation, namely, F(2P3/2) + H2(u = 0) + hw(1.06 pm) + F*(2P1,2) + H2(u’), u’ = 0, 1 We shall study the reaction only in a collinear configuration.P . L . DEVRIES, T . F . GEORGE A N D J . - M . YUAN 93 holds to a high degree of accuracy in the (r3, R3) coordinate system. In eqn ( 5 ) E is the total energy of the system, El is E- or E+, and MI and M ~ I are defined by MI' = M-l AI and Mi1 = All M-I ill, with the inverse mass tensor M-I given as where p is the reduced mass between H and HF and m is the reduced mass of HF. When a trajectory is propagated to a seam, the representative particle may either switch surfaces or stay on the original surface. The probability of switching surfaces is equal to the local nonadiabatic transition probability p t , which can be calculated by integrating PI, as defined in eqn (9, around the branch point and taking the expo- nential of the imaginary part of the action integral.The probability of staying on the original surface is then 1 - p t . For reaction (3), at all field intensities studied (109-1013 W cmM-2) only vibra- tional states u = 2 and 3 of HF are populated, which is also true for the field-free case. The population ratio of u = 3 to u = 2 for field intensities below 1 TW cm-, is almost a constant and is -0.75. However, as the field intensity increases to 10 TW cm-, the ratio increases to 1.64. The total reaction probability decreases slightly from 0.63 to 0.60 as the field intensity increases from 0 to 1 TW cm-2, which can be explained by the fact that as the separation of the electronic-field surfaces increases at the seam, pt becomes smaller.However, as the field intensity further increases to 10 TW cm-2, the total reaction probability jumps to 0.74. These results are interesting, because they suggest that laser emission from u = 3 to u = 2 is possible without con- sidering the rotational manifold, as is necessary for an ordinary HF laser. Our preliminary study then points out the possibility that laser characteristics can be affected by shining another laser through the laser cavity. The threshold field in- tensity (% 10 TW cm-2) predicted may be too high for any practical USC, but since this is the result of a collinear system, the threshold field intensity for a realistic three- dimensional system could be lower.Inelastic scattering probabilities for the u = 0 and 1 states in process (4), which is not energetically accessible in the corresponding field-free case, increase from the order of to the order of 0.01 as the field intensity increases from 1 GW cm-2 to 10 TW cm-2. It is worth noting that H, can be vi- brationally excited to u = 1 while the ground-state F(2P3/2) atom is a t the same time electronically excited to F(,P1,,), even though no vibrational transition moment has been included in our calculation. 3. NONREACTIVE PROCESSES In this section, we shall investigate the effect of intense radiation on nonreactive processes in the F + H, collision system. Specifically, we are interested in the quenching of fluorine in this situation. The potential surface correlating to the ground spin-orbit state of F leads to formation of HF, whereas the surface correlating to the excited spin-orbit state does not lead to reaction.Thus the quenching of the excited state can lead directly to an enhanced reaction cross section. This quenching does occur in the absence of a radiation field and, in fact, the asymptotic level of F* + H2(j = 0) is nearly resonant with F + H,(j = 2), leading to substantial quenching. (In this section H2 is treated as a rigid rotor.) The purpose of the present work is to determine to what extent the quenching can be enhanced by the presence of an intense radiation field. T!ic use of a nonreactive formalism to treat this quenching problem is justifiable only if the region of importance to the quenching process is far removed from the94 F + H2 COLLISIONS I N LASER FIELDS reaction region.This condition is satisfied in the absence of the field, where the " quenching region " occurs at an F-H2 separation of ~5 bohr. We have met this requirement in the presence of the radiation field by choosing a radiation field which is nearly resonant with the asymptotic state; the spin-orbit splitting of fluorine is z 404 cm-', and we chose the radiation field to be ~ 4 0 8 cm-'. This ensures that the region of significant interaction occurs far from the reaction region. However, the electric dipole moment of the system becomes vanishingly small for F-H, separations > ~3 bohr. Thus, unlike the usual physical situation, in the region of significant quenching the magnetic dipole is the dominant factor in the radiative process.The calculations were performed within a three-dimensional quantum mechanical close-coupled formalism in the body-fixed coordinate system, as described e1sewhere.l' Matrix elements of the hamiltonian were evaluated by the diatomics-in-molecules method." The presence of the radiative interaction introduces several complexities to the problem, including the fact that the total molecular angular momentum is not conserved (since photons have an intrinsic angular momentum) and that the total hamiltonian is not rotationally invariant." The first difficulty was minimized by truncating the basis-set expansion to include only those states coupled to the F* + H2 states by single photon transitions, and the second overcome by employing the orienta- tional average appr~ximation.'~*'~ In all, 48 channels were included in the calcu- lating, correlating to the states F + H2(j = 0, 2) + nhco, F* + H2(j = 0, 2) + nhco, and F + H2(j = 0) + (n + 1)hw.In the body-fixed system, the hamiltonian matrix is (approximately) block-diagonal so that a maximum of 14 states were considered simultaneously. The close-coupled equations were then solvedby the R-matrix method of Light and Walker." There are at least two very interesting intermediate results of this calculation. The first is that the transitions F* + H2(j = 0) + nhco -+ F + H2(j = 0, 2) + nhco are (within the accuracy of our calculation) independent of n (i.e., the intensity of the radiation field) for intensities as high as lot2 W cm-2.The effect of the radiation field in transitions of this sort is to distort the shape of the potential surfaces, and hence to effect the process indirectly, e.g., without net absorption or emission of photons. The fact that the cross-sections are not altered indicates that this distortion is small, which is easily confirmed. An order of magnitude estimate of the distortion can be obtained by considering two crossing surfaces W, + hw and W,, and computing the amount of splitting in the corresponding surfaces E, and E-. For our situation, in which only the magnetic dipole component of the radiation field interacts with the molecular system, it is found that the splitting at the avoided crossing is only z 2.5 cm-' for a field of 10l2 W ern-,.(This should be compared with the situation of the previ- ous section, in which the electric dipole interaction induced a splitting of z 260 cm-' in a field of lo1, W cm-2.) Thus it is understandable that the intensity of the radia- tion field plays an insignificant role in those processes not involving net absorption or emission of photons. The second result of some interest is the behaviour of the cross-section for the resonant quenching transition F* + H2(j = 0) + nhco -+ F + H2(j = 2) + nhco as a function of J , the initial total molecular angular momentum. (As discussed in the preceding paragraph, these results are found to be independent of the intensity of the radiation field.) Since states of different parity do not couple, the cross-sections for the even and odd parity states are computed separately from one another.These J- dependent cross-sections (for a collision energy of 0.03 eV) are exhibited in fig. 2, the even parity results indicated by the solid line and the odd parity results by the dashed line. As seen from the figure, the even and odd parity cross-sections alternate in magnitude as a function of J. This behaviour has been observed in previous (field-P . L . DEVRIES, T . F . GEORGE AND J . - M . YUAN 0.01 0.02 0.03 0.04 95 4.15 4.21 4.80 10.6 3.41 3.44 3.75 6.79 3.14 3.17 3.37 5.43 2.83 2.85 3.01 4.56 I i J I J FIG. 2.-Cross sections as a function of the total (initial) molecular angular momentum J for the process F* + Hz ( j = 0) + nho -+ F + Hz ( j = 2 ) + nho at a collision energy of 0.03 eV ( h o z 408 cm-I).The even parity results are indicated by the solid line and the odd parity results by the dashed line. free) calculations, for C + + Hz by Chu and Dalgarno,16 and F + H2 by Wyatt and Walker," who explained it in terms of the interweaving of avoided crossings. An- other feature of this figure is the presence of two distinct regions of J values where the cross-section is significant. Calculations at other collision energies indicate that as the energy increases, the second region moves to higher J values, and the contribution to the total cross section decreases. Note that our calculation differs from that of Wyatt and Walker in several respects, not the least of which is the use of different electronic surfaces; and it is well known that cross sections at low collision energies can be extremely sensitive to the shapes of the electronic surfaces.The major results of this investigation are presented in table 1. The cross sections TABLE TOTAL CROSS SECTIONS (A') FOR QUENCHING OF F* BY &(j = 0) IN PRESENCE OF A LASER FIELD (ko Z 408 Cm-') reported are the total degeneracy-averaged cross-sections for the quenching of F* by H2(j = 0), summed over final H, rotational states and over the final states of the radiation field. As indicated in table 1, the presence of the radiation field can sub- stantially alter the quenching cross section for a radiation field intensity as low as 10" W cm-'. Furthermore, the effect of the field is stronger at the lower collision energies. This is easily understandable since, at the lower energies, the system remains in the interaction region for a longer amount of time.(Similar behaviour has been observed in an atom-atom collision system in which the radiative process was dominated by the Liectric dipole interaction.)" These results clearly indicate that the quenching process can be considerably enhanced by the radiation field, even though the transition96 F + Hz COLLISIONS IN LASER FIELDS must proceed through the magnetic dipole interaction rather than the (typically larger) electric dipole interaction. 4. CONCLUSION The calculations reported in sections 2 and 3 by no means represent a complete description of the F + H2 collision system in the presence of intense laser radiation. In fact, these calculations mark the first time, for an atom-diatom collision system in a laser field, that reactive channels and rotational degrees of freedom have explicitly been treated.Nevertheless, the results should help maintain the interest among experimentalists to continue to explore the effects of intense laser radiation on the dynamics of molecular rate processes. Due to the various approximations employed, we do not want to state absolutely that field intensities must be in a range as high as 1011-1013 W cm-2 in order to observe effects as suggested by our calculations. We hope that interesting effects might be observable with lower field intensities. One of us (P. L. D.) thanks Prof. C. Moser and CECAM for their hospitality at the Workshop on Selective Excitation of Atoms and Molecules (University of Paris at Orsay, Summer, 1978), where part of this work was carried out.The research was financed by U.S. Government Agencies, including NASA, the U.S. Air Force and the National Science Foundation. * T. F. George, I. H. Zimmerman, P. L. DeVries, J.-M. Yuan, K. S. Lam, J. C. Bellum, H. W. Lee, M. S. Slutsky and J. T. Lin, in Chemical and Biochemical Applications of Lasers, ed. C. B. Moore (Academic Press, New York, 1979), vol. IV, pp. 253-354. P. L. DeVries and T. F. George, J. Chem. Phys., in press. A. Komornicki, T. F. George and K. Morokuma, J . Chem. Phys., 1976, 65, 48. W. H. Miller and T. F. George, J . Chem. Phys., 1972, 56, 5637. ' J. T. Muckerman, J. Chern. Phys., 1972, 56, 2997, and personal communication. ' R. L. Jaffe, K. Morokuma and T. F. George, J . Chem. Phys., 1975,63, 3417. * C. F. Bender and E. R. Davidson, J . Chem. Phys., 1968, 49,4989. C. C. 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