Article pubs.acs.org/JPCA
Transport Properties for Systems with Deep Potential Wells: H + O2 Paul J. Dagdigian*,† and Millard H. Alexander‡ †
Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218-2685, United States Department of Chemistry and Biochemistry and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742-2021, United States
‡
ABSTRACT: Transport properties for collisions of the oxygen molecule with hydrogen atoms are computed by means of quantum scattering calculations. Because two potential energy surfaces (PESs) arise from the interaction of H(2S) with O2(X3Σ+g ), namely 2A″ and 4A″, collision integrals were computed using both PESs and then averaged with weighting by their respective spin multiplicities. A PES for the 4A″ state was computed for the interaction of O2, frozen at its equilibrium internuclear separation, with a hydrogen atom, using a coupled-cluster method that includes all single and double excitations as well as perturbative contributions of connected triple excitation. A PES of similar quality was taken from Kłos et al. [J. Chem. Phys. 2008, 129, 064306] for the 2A″ state. Because the 2A″ state correlates with the deep HO2(X̃ 2A″) well, statistical capture boundary conditions [Rackham et al., J. Chem. Phys. 2003, 119, 12895] were applied to compute the S matrix, and then the transport properties, for this state.
I. INTRODUCTION Chemical kinetic models for spatially inhomogeneous systems, such as flames and internal combustion engines, require knowledge of the relevant transport properties for an accurate description of the temporal and spatial dependence of the species concentrations, as well as for the calculation of flame velocities and the production of pollutants in combustion media. A recent article by Brown et al.1 has reviewed the principal methods currently employed for the estimation of transport properties. Typically, an isotropic Lennard-Jones (LJ) 12−6 potential with the length and depth parameters obtained from combination rules2 is assumed. For a few systems, the spherical average of a computed ab initio potential has been employed in quantum scattering calculations of the transport properties.3−6 There have also been a number of quantum scattering calculations of transport properties using accurate anisotropic atom−molecule ab initio potentials.7−14 Jasper and Miller15 have investigated methods to derive spherically averaged potentials for larger, floppy molecules. Quantum chemistry methods and resources have advanced to the point that accurate potential energy surfaces (PESs) for interactions involving both stable and reactive polyatomic species can be computed and employed in scattering calculations.16−18 We have carried out a series of quantum scattering calculations on exemplary systems, including OH− He,19 CH2(X̃ ,ã)−He,20 and H2O−H,21 for which state-of-the-art potential energy energy surfaces are available. Quantum scattering calculation of transport properties for anisotropic PESs is much more involved than for isotropic systems because of the necessity to compute state-to-state transport cross sections and then carry out a Boltzmann average and a sum over the initial and final rotational levels, respectively. © XXXX American Chemical Society
For the systems we studied, we considered whether transport properties could be computed with reasonable accuracy using the spherical average of the PES. We found for OH−He and H2O−H only slight differences ( 21 levels is
Figure 3. Dependence on collision energy of the state-specific transport (2) cross sections Q(1) ni [panels a and b] and Qni [panels c and d] for 2 collisions on the A″ PES of the ni = 1 and 15 rotational levels of O2 with hydrogen atoms . The elastic contribution to the transport cross section is indicated in each panel.
involving collisions on the 2A″ PES. The elastic contribution to the cross sections is also shown in the plots. As we have observed previously for other systems,19−21 the Q(1) ni transport cross sections decrease monotonically with increasing collision energy. At very low collision energies, the Q(1) ni transport cross sections are dominated by the elastic contribution, while at higher collision energies the inelastic contribution dominates. The relative unimportance of the elastic contribution in the transport cross section is in stark contrast with the integral cross section, for which the elastic contribution dominates. This difference results from the weighting factor Φn(E) in eq 6 for the transport cross section.20 The state-specific transport cross section Q(2) ni for the lowest rotational level ni = 1 displays a collision energy dependence similar to that for the corresponding Q(1) ni cross section. We see, D
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Figure 7. Collision integrals Ω(1,1) and Ω(2,2) as a function of temperature for the H−O2 system, computed from quantum scattering calculations (labeled as 2A″, 4A″ PESs) and with the LJ 12−6 potential.
Figure 5. State-specific collision integrals Q(1,1) as a function of the ni rotational angular momentum n for collisions on the 2A″ and 4A″ PESs at temperatures of (a) 300 and (b) 1500 K.
computed with the LJ potential are larger than those from the quantum scattering calculation, except for Ω(2,2) at low temperatures. Moreover, the LJ estimate has a temperature dependence that is much steeper than that of the quantum scattering calculation. We observed a temperature dependence of the LJ collision integral that was similarly steeper than that in the quantum scattering calculation in our previous studies of the OH−He, CH2(X̃ ,ã), and H2O−H systems.19−21 This was ascribed to a repulsive wall for the LJ potential that was much steeper than that in the ab initio PESs. We have advocated the use of the LJ 9−6 potential as a more realistic parametrized potential for estimating transport properties. Tables of collision integrals are available for this potential.61 We have employed the collision integrals Ω(1,1) from the quantum scattering calculations and the LJ estimate to calculate the diffusion coefficient Dab for the H−O2 system. Figure 8
equal to that for ni = 21. It should be noted that the most probable ni for O2 is 9 and 19 for T = 300 and 1500 K, respectively. Figure 6
Figure 6. Collision integrals Ω(1,1) and Ω(2,2) as a function of temperature for the H−O2 system. These quantities are presented for collisions on the 2A″ and 4A″ PESs separately and for their degeneracyweighted average.
presents these collision integrals computed for collisions on the A″ and 4A″ PESs separately and their spin-multiplicity weighted average. It is the latter that we will use to compute transport properties for collisions of H atoms with O2. As with the Q(n) ni transport cross sections, the n = 2 collision integrals are larger than the corresponding n = 1 quantities. 2
VI. DISCUSSION We have employed quantum scattering calculations to calculate collision integrals for the H−O2 system. It is interesting to compare the present results with a conventional transport calculation using a LJ 12−6 potential. The parameters specifying this potential for H−O2 were estimated with conventional combining rules: The well depth ε and length parameter σ for H−O2 were taken as the geometric and arithmetic means, respectively, of the corresponding parameters for the like systems (ε = 145 K and σ = 2.05 Å for H−H and ε = 107.4 K and σ = 3.458 Å for O2−O2).60 This yields the following LJ parameters for H− O2: ε = 87.0 cm−1 and σ = 5.20 bohr. We present in Figure 7 a comparison of the collision integrals for H−O2 computed from our quantum scattering calculations and the LJ 12−6 estimate. For the former, the spin-multiplicity weighted average of the collision integrals for collisions on the 2 A″ and 4A″ PESs was taken. We see that the collision integrals
Figure 8. Diffusion coefficient for the H−O2 system, computed through quantum scattering calculations (labeled as 2A″, 4A″ PESs) and with the LJ 12−6 potential.
presents a comparison of these calculations. The diffusion coefficient computed with the LJ 12−6 potential is seen to be smaller than that from the quantum scattering calculation. At 300 K, the LJ diffusion coefficient is 7% smaller, while it is 25% smaller at 1500 K. These differences are consistent with the differences in the Ω(1,1) collision integrals plotted in Figure 7. In this study, we have developed a procedure, involving the use of coupled-channel statistical theory,25,26 for the calculation of collision integrals, and hence transport properties, for systems having a PES that possesses a deep well. We have applied this E
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total angular momentum J. This suggests that the centrifugal barrier, which grows with increasing J, also inhibits access to the deep HO2 well. Because the capture probability declines as the O2 rotational excitation increases, the effect of formation and decay of the HO2 complex should be less at higher temperatures. The rotational state distribution extends to higher rotational levels at higher temperatures. In principle, one could extend the coupled-channel statistical theory to include collision events involving formation and decay of a long-lived complex. This is beyond the scope of the present work and would require calculation of differential cross sections for decay of the complex back to the collision partners, so that the transport cross sections (eq 6) could be computed. In addition, it should be noted that there is evidence39 for nonstatistical behavior in reaction 1. A more exact treatment might lead to small corrections to the predictions of the coupled-channel statistical treatment used here. We have nonetheless shown that it is possible to carry out quantum scattering calculations of transport properties for systems that possess a PES with a deep well. We have found significant differences in the values of transport properties computed with quantum scattering theory as compared to estimates using a parametrized LJ potential. It will be interesting to extend our theory to other important combustion collision partners for which there is a deep well.
theory to the calculation of transport cross sections, and hence collision integrals, for H−O2 collisions that take place on the 2A″ PES, which includes the deep HO2(X̃ 2A″) well. Included in the transport cross sections are contributions from only those collisions that do not access the deep well. Our calculation neglects the contribution from collisions that access the deep well and subsequently decay back to H + O2. For high collision energies, the HO2 complex can, in principle, decay to O + OH reaction products; such collisions will not contribute to the H− O2 transport cross sections. However, statistical factors dictate that the complex will decay predominantly to the lower-energy H + O2 channel, rather than O + OH. At high pressures, the transient HO2 complex can be collisionally stabilized; thus, under these conditions the decay of the complex back to H + O2 can be ignored. In the coupled-channel statistical theory,25,26 the probability for capture into the deep HO2 well can be computed for a given channel at total angular momentum J from the S matrix: pnJ l (E) = 1 − ii
∑ |SnJ l ,n l (E)|2 f f
nf l f
ii
(12)
where the initial and final channels are specified by the rotational and orbital angular momenta n and l, respectively. We present in Figure 9 capture probabilities for the O2 rotational levels ni for
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This material is based upon work supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Award DESC0002323. The authors gratefully acknowledge the encouragement of Al Wagner and Nancy Brown.
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Figure 9. Capture probabilities in collisions on the 2A″ PES for rotational levels ni, averaged over the allowed values of the orbital angular momentum li, for several values of the total angular momentum J for (a) 300 and (b) 1000 cm−1 collision energy.
REFERENCES
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selected values of the total angular momentum J at two different collision energies in collisions on the 2A″ PES. To minimize congestion in the plots, the capture probabilities are presented as averages over the allowed orbital angular momenta li for each rotational level. We see in Figure 9 that for a given collision energy and total angular momentum J, the capture probability generally decreases with increasing rotational angular momentum ni. This results from the restricted angular access to the deep HO2(X̃ 2A′) well (see panel a of Figure 1). For high rotational angular momentum, the molecule is rotating too fast for the forces due to the PES to steer the molecule into the well before the collision is over. A similar effect of molecular rotation on the rate constant is observed in the electronic quenching of OH(X2Π) by Kr.62 In this case, the quenching is mediated by a crossing of PESs emanating from the ground and excited electronic states at a restricted orientation near linear geometry. We also observe in Figure 9 that for a given rotational level and collision energy the capture probability declines with increasing F
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H
dx.doi.org/10.1021/jp505769h | J. Phys. Chem. A XXXX, XXX, XXX−XXX
Transport Properties for Systems with Deep Potential Wells: H + O2 Paul J. Dagdigian*,† and Millard H. Alexander‡ †
Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218-2685, United States Department of Chemistry and Biochemistry and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742-2021, United States
‡
ABSTRACT: Transport properties for collisions of the oxygen molecule with hydrogen atoms are computed by means of quantum scattering calculations. Because two potential energy surfaces (PESs) arise from the interaction of H(2S) with O2(X3Σ+g ), namely 2A″ and 4A″, collision integrals were computed using both PESs and then averaged with weighting by their respective spin multiplicities. A PES for the 4A″ state was computed for the interaction of O2, frozen at its equilibrium internuclear separation, with a hydrogen atom, using a coupled-cluster method that includes all single and double excitations as well as perturbative contributions of connected triple excitation. A PES of similar quality was taken from Kłos et al. [J. Chem. Phys. 2008, 129, 064306] for the 2A″ state. Because the 2A″ state correlates with the deep HO2(X̃ 2A″) well, statistical capture boundary conditions [Rackham et al., J. Chem. Phys. 2003, 119, 12895] were applied to compute the S matrix, and then the transport properties, for this state.
I. INTRODUCTION Chemical kinetic models for spatially inhomogeneous systems, such as flames and internal combustion engines, require knowledge of the relevant transport properties for an accurate description of the temporal and spatial dependence of the species concentrations, as well as for the calculation of flame velocities and the production of pollutants in combustion media. A recent article by Brown et al.1 has reviewed the principal methods currently employed for the estimation of transport properties. Typically, an isotropic Lennard-Jones (LJ) 12−6 potential with the length and depth parameters obtained from combination rules2 is assumed. For a few systems, the spherical average of a computed ab initio potential has been employed in quantum scattering calculations of the transport properties.3−6 There have also been a number of quantum scattering calculations of transport properties using accurate anisotropic atom−molecule ab initio potentials.7−14 Jasper and Miller15 have investigated methods to derive spherically averaged potentials for larger, floppy molecules. Quantum chemistry methods and resources have advanced to the point that accurate potential energy surfaces (PESs) for interactions involving both stable and reactive polyatomic species can be computed and employed in scattering calculations.16−18 We have carried out a series of quantum scattering calculations on exemplary systems, including OH− He,19 CH2(X̃ ,ã)−He,20 and H2O−H,21 for which state-of-the-art potential energy energy surfaces are available. Quantum scattering calculation of transport properties for anisotropic PESs is much more involved than for isotropic systems because of the necessity to compute state-to-state transport cross sections and then carry out a Boltzmann average and a sum over the initial and final rotational levels, respectively. © XXXX American Chemical Society
For the systems we studied, we considered whether transport properties could be computed with reasonable accuracy using the spherical average of the PES. We found for OH−He and H2O−H only slight differences ( 21 levels is
Figure 3. Dependence on collision energy of the state-specific transport (2) cross sections Q(1) ni [panels a and b] and Qni [panels c and d] for 2 collisions on the A″ PES of the ni = 1 and 15 rotational levels of O2 with hydrogen atoms . The elastic contribution to the transport cross section is indicated in each panel.
involving collisions on the 2A″ PES. The elastic contribution to the cross sections is also shown in the plots. As we have observed previously for other systems,19−21 the Q(1) ni transport cross sections decrease monotonically with increasing collision energy. At very low collision energies, the Q(1) ni transport cross sections are dominated by the elastic contribution, while at higher collision energies the inelastic contribution dominates. The relative unimportance of the elastic contribution in the transport cross section is in stark contrast with the integral cross section, for which the elastic contribution dominates. This difference results from the weighting factor Φn(E) in eq 6 for the transport cross section.20 The state-specific transport cross section Q(2) ni for the lowest rotational level ni = 1 displays a collision energy dependence similar to that for the corresponding Q(1) ni cross section. We see, D
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Figure 7. Collision integrals Ω(1,1) and Ω(2,2) as a function of temperature for the H−O2 system, computed from quantum scattering calculations (labeled as 2A″, 4A″ PESs) and with the LJ 12−6 potential.
Figure 5. State-specific collision integrals Q(1,1) as a function of the ni rotational angular momentum n for collisions on the 2A″ and 4A″ PESs at temperatures of (a) 300 and (b) 1500 K.
computed with the LJ potential are larger than those from the quantum scattering calculation, except for Ω(2,2) at low temperatures. Moreover, the LJ estimate has a temperature dependence that is much steeper than that of the quantum scattering calculation. We observed a temperature dependence of the LJ collision integral that was similarly steeper than that in the quantum scattering calculation in our previous studies of the OH−He, CH2(X̃ ,ã), and H2O−H systems.19−21 This was ascribed to a repulsive wall for the LJ potential that was much steeper than that in the ab initio PESs. We have advocated the use of the LJ 9−6 potential as a more realistic parametrized potential for estimating transport properties. Tables of collision integrals are available for this potential.61 We have employed the collision integrals Ω(1,1) from the quantum scattering calculations and the LJ estimate to calculate the diffusion coefficient Dab for the H−O2 system. Figure 8
equal to that for ni = 21. It should be noted that the most probable ni for O2 is 9 and 19 for T = 300 and 1500 K, respectively. Figure 6
Figure 6. Collision integrals Ω(1,1) and Ω(2,2) as a function of temperature for the H−O2 system. These quantities are presented for collisions on the 2A″ and 4A″ PESs separately and for their degeneracyweighted average.
presents these collision integrals computed for collisions on the A″ and 4A″ PESs separately and their spin-multiplicity weighted average. It is the latter that we will use to compute transport properties for collisions of H atoms with O2. As with the Q(n) ni transport cross sections, the n = 2 collision integrals are larger than the corresponding n = 1 quantities. 2
VI. DISCUSSION We have employed quantum scattering calculations to calculate collision integrals for the H−O2 system. It is interesting to compare the present results with a conventional transport calculation using a LJ 12−6 potential. The parameters specifying this potential for H−O2 were estimated with conventional combining rules: The well depth ε and length parameter σ for H−O2 were taken as the geometric and arithmetic means, respectively, of the corresponding parameters for the like systems (ε = 145 K and σ = 2.05 Å for H−H and ε = 107.4 K and σ = 3.458 Å for O2−O2).60 This yields the following LJ parameters for H− O2: ε = 87.0 cm−1 and σ = 5.20 bohr. We present in Figure 7 a comparison of the collision integrals for H−O2 computed from our quantum scattering calculations and the LJ 12−6 estimate. For the former, the spin-multiplicity weighted average of the collision integrals for collisions on the 2 A″ and 4A″ PESs was taken. We see that the collision integrals
Figure 8. Diffusion coefficient for the H−O2 system, computed through quantum scattering calculations (labeled as 2A″, 4A″ PESs) and with the LJ 12−6 potential.
presents a comparison of these calculations. The diffusion coefficient computed with the LJ 12−6 potential is seen to be smaller than that from the quantum scattering calculation. At 300 K, the LJ diffusion coefficient is 7% smaller, while it is 25% smaller at 1500 K. These differences are consistent with the differences in the Ω(1,1) collision integrals plotted in Figure 7. In this study, we have developed a procedure, involving the use of coupled-channel statistical theory,25,26 for the calculation of collision integrals, and hence transport properties, for systems having a PES that possesses a deep well. We have applied this E
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total angular momentum J. This suggests that the centrifugal barrier, which grows with increasing J, also inhibits access to the deep HO2 well. Because the capture probability declines as the O2 rotational excitation increases, the effect of formation and decay of the HO2 complex should be less at higher temperatures. The rotational state distribution extends to higher rotational levels at higher temperatures. In principle, one could extend the coupled-channel statistical theory to include collision events involving formation and decay of a long-lived complex. This is beyond the scope of the present work and would require calculation of differential cross sections for decay of the complex back to the collision partners, so that the transport cross sections (eq 6) could be computed. In addition, it should be noted that there is evidence39 for nonstatistical behavior in reaction 1. A more exact treatment might lead to small corrections to the predictions of the coupled-channel statistical treatment used here. We have nonetheless shown that it is possible to carry out quantum scattering calculations of transport properties for systems that possess a PES with a deep well. We have found significant differences in the values of transport properties computed with quantum scattering theory as compared to estimates using a parametrized LJ potential. It will be interesting to extend our theory to other important combustion collision partners for which there is a deep well.
theory to the calculation of transport cross sections, and hence collision integrals, for H−O2 collisions that take place on the 2A″ PES, which includes the deep HO2(X̃ 2A″) well. Included in the transport cross sections are contributions from only those collisions that do not access the deep well. Our calculation neglects the contribution from collisions that access the deep well and subsequently decay back to H + O2. For high collision energies, the HO2 complex can, in principle, decay to O + OH reaction products; such collisions will not contribute to the H− O2 transport cross sections. However, statistical factors dictate that the complex will decay predominantly to the lower-energy H + O2 channel, rather than O + OH. At high pressures, the transient HO2 complex can be collisionally stabilized; thus, under these conditions the decay of the complex back to H + O2 can be ignored. In the coupled-channel statistical theory,25,26 the probability for capture into the deep HO2 well can be computed for a given channel at total angular momentum J from the S matrix: pnJ l (E) = 1 − ii
∑ |SnJ l ,n l (E)|2 f f
nf l f
ii
(12)
where the initial and final channels are specified by the rotational and orbital angular momenta n and l, respectively. We present in Figure 9 capture probabilities for the O2 rotational levels ni for
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This material is based upon work supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Award DESC0002323. The authors gratefully acknowledge the encouragement of Al Wagner and Nancy Brown.
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Figure 9. Capture probabilities in collisions on the 2A″ PES for rotational levels ni, averaged over the allowed values of the orbital angular momentum li, for several values of the total angular momentum J for (a) 300 and (b) 1000 cm−1 collision energy.
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H
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