Low-Lying Potential Energy Surfaces - American Chemical Society

Chapter 16

Model Studies of Intersystem Crossing Effects in theO+H Reaction 2

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Downloaded by UNIV MASSACHUSETTS AMHERST on September 14, 2012 | http://pubs.acs.org Publication Date: August 14, 2002 | doi: 10.1021/bk-2002-0828.ch016

Mark R. Hoffmann and George C. Schatz 1

Department of Chemistry, University of North Dakota, Box 9024, Grand Forks, ND 58202-9024 Theoretical Chemistry Group, Argonne National Laboratory, Argonne, IL 60439 Department of Chemistry, Northwestern University, Evanston, IL 60208-3113

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We use quantum scattering and trajectory surface hopping methods to examine the influence of intersystem crossing between the lowest energy triplet and singlet states on the O( P) + H reaction dynamics. Several two-state reaction path models of the potential energy surfaces and spin-orbit coupling are studied. In these models, the triplet state curve shows a barrier along the reaction path and the singlet state a well such that the two states intersect at a location near the barrier top. Eleven choices of the parameters in the Hamiltonian are examined in which the effect of the triplet-singlet crossing location, the singlet well depth, and the size and coordinate dependence of the spin-orbit coupling are varied. The quantum calculations show that if the crossing occurs on the reagent side of the triplet barrier, and the spin-orbit coupling at that point is similar to what exists in the reagent Ο atom, then the low energy reactivity is dominated by intersystem crossing. 3

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© 2002 American Chemical Society

In Low-Lying Potential Energy Surfaces; Hoffmann, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

329

330 This result is reasonably well described by surface hopping within the diabatic representation; the corresponding adiabatic representation results are less accurate below the adiabatic threshold, but more accurate above threshold. If the crossing occurs on the product side of the barrier, as actually occurs for the O + H reaction, the influence of intersystem crossing is much smaller, though not completely. The influence of Stuckelberg interference effects on the state-resolved reaction probabilities is also studied.


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Although spin-orbit induced intersystem crossing is an old subject that has long been studied in photochemistry and photophysics, its influence on small molecule bimoiecular reactions has received less attention. Such effects show up most commonly when the ground state of the reactants involves a triplet (or other high spin) potential surface (such as in the reaction of a triplet atom or molecule with a singlet molecule), while products correlate to both singlet and triplet states (such as is obtained from a doublet + doublet combination), thereby providing two pathways for reaction, a triplet spin-allowed path and a singlet spin-forbidden path. Often the singlet path involves the formation of a stable intermediate, while the triplet occurs over a barrier, so the reaction dynamics associated with these paths is quite different. In addition, the spin-forbidden path can in some cases take place over a lower barrier than the spin-allowed path, so the forbidden path could, in principle, dominate the dynamics. There have been several experimental and/or theoretical studies during the past 10 years involving intersystem crossing effects in bimoiecular reactions of 0(1-3), NH(4), CH (5-7), S(8) and CH(9,10). Important spin-forbidden effects have been observed for reactions involving iodine(i), however little is known about reactions involving lighter atoms where the allowed and forbidden pathways compete. Recently, Hoffmann and Schatz(ll) have developed a new level of treatment of spin-orbit effects in bimoiecular reactions which enables a more sophisticated treatment of intersystem crossing dynamics than in the past. In this treatment high quality electronic structure methods are used to determine global surfaces for the reaction and spin-orbit matrix elements, and then trajectory surface hopping (TSH) methods are used to determine properties of the bimoiecular collisions such as reactive cross sections and state distribution information. In an application of this theory to the Ο + H reaction, the spinorbit matrix elements were determined as a function of position, and then T S H calculations were done within a diabatic representation to determine cross sections. Intersystem crossing effects were found to be small for Ο + H due to 2

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the fact that the crossing of the singlet and triplet surfaces occurs in the product region, making it hard to access geometries where hopping can occur. In addition the spin-orbit coupling is small, which means that the hopping probability is small. However the small fraction of trajectories that do undergo surface hopping have significantly different product state distributions than those which do not. The Hoffmann-Sehatz (HS) work raises some important questions in the description of nonadiabatic dynamics that go beyond the original work. HS found that for Ο + H the triplet and singlet surfaces interact strongly in two places: where they cross near the barrier top, and where they become degenerate in the product region. The accuracy of T S H methods for this class of problems has not been described earlier, but HS found that unphysical results (excessively large cross sections) were obtained if T S H calculations were done in the adiabatic representation (a method that we denote TSH-A). Diabatic representation results (i.e., TSH-D) were more reasonable, but comparisons with quantum dynamics calculations were not performed, which means that the accuracy of the calculation is not known. In addition, the importance of interference in the coupled surface dynamics, something that would be imperfectly described using T S H - A or TSH-D, was not considered. In this paper, we compare quantum scattering, T S H - A and TSH-D results for several two-state reaction path models which describe the Ο + H reaction, and related reactions. Eleven model potentials have been considered, so as to determine the influence of triplet-singlet crossing location, the singlet well depth, and the size and coordinate dependence of the spin-orbit coupling. 2

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Theory In this section, we first describe the model Hamiltonians, and then the quantum and T S H nonadiabatic dynamics calculations that were performed.

Model Hamiltonians The model Hamiltonians are designed to mimic the lowest energy triplet and singlet potential surfaces for Ο + H along the minimum energy path of the triplet. As described by HS, the triplet state (1 A") has a collinear reaction path (1 Π symmetry) that correlates with 0( P)+H and ΟΗ( Π)+Η for reactant and product configurations, and with a linear O-H-H barrier (no wells) between these 2

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332 1

2

limits. The singlet state (1 *A') correlates with 0 ( D ) + H and ΟΗ( Π)+Η. It has a bent reaction path, and for C geometries there is a deep intermediate well (with no barrier). To develop a reaction path model that has this behavior we take the triplet potential to be an asymmetrical Eckart barrier function of the type: 2

2 v

"-τή^ — -— [2 cosh


ν

+

1 +

B

6

2

2

-ax]

where χ is the distance along the reaction path (not mass weighted) and the parameters A , Β and a are given below. The singlet surface is also written as an Eckart function:

V , = A£ +

C H 1

+

,.n

'

f

* [2 cosh

2

-i>(*

2

+

1

f)]

2

with parameters ΔΕ (the Ο atom singlet-triplet splitting), C, D, b and f, but here the parameters are chosen so that the potential exhibits a well rather than a barrier. The triplet-singlet coupling is assumed to be governed by a simple switching function of the type:

V =^[l + tanh £(* + /*)] 2l

4

(3)

where s, g and h are additional parameters. In order to specify the 11 parameters in terms of physically meaningful quantities, we use the formulas in E q . (4) below. With these definitions, the parameter C is no longer independent. Instead, the independent parameters are: A (exoergicity on the triplet surface), V (the triplet barrier), ΔΕ (defined above), a (triplet barrier width), b (singlet well width), g (determines how quickly the spin-orbit coupling is switched on as one moves from the reactants to the crossing point), w (singlet well depth), b


333

5 = -(-vWv -4Â ) 2 2

2

v = 2A-4V

b

C = A-AE

D = i(-vWv -4C ) Downloaded by UNIV MASSACHUSETTS AMHERST on September 14, 2012 | http://pubs.acs.org Publication Date: August 14, 2002 | doi: 10.1021/bk-2002-0828.ch016

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2

v'=2C-4(w-AE)

f (determines the singlet-triplet crossing location, with x=0 corresponding to the triplet barrier location), s (magnitude of spin-orbit splitting) and h (determines where the spin-orbit coupling is turned on relative to the barrier at x=0). To simplify the parameters, wefixthe first six parameters with the following values (modeled to mimic Ο + H for slightly bent structures (where the triplet-singlet crossing is more accessible than for linear geometries) with roughly the right V and ΔΕ): A = 0.003187, V = 0.03187, ΔΕ = 0.06374, a = 1.5, b=1.5, g = 1.0. These values are in atomic units, and we will use these units throughout the paper. The remaining four parameters are given in Table 1 for the eleven models that we have considered. Figure 1 plots V n , V and V as a function of x. Here is a summary of what each model does: 2

b

b

2 2

2 i

Model 1: Crossing is on the reagent side, coupling is turned on at the crossing point. Model 2: Same as Model 1 except that the coupling is turned on after the crossing. Model 3: Same as Model 1 except that the crossing point is moved to the product side of the barrier. Model 4: Same as Model 3 except that the coupling is turned on after the crossing. Model 5: Same as Model 2 except that the well depth is doubled. Model 6: Same as Model 1 except that the well depth is doubled. Model 7: Same as Model 1 except that the coupling is three times larger. Model 8: Same as Model 2 except that the coupling is three times larger. Model 9: Same as Model 2 except that the coupling is turned on further to the products. Model 10: Same as Model 1 except that the crossing point and coupling are shifted further to the reactants.

Model 11 : Same as 1 except that the crossing is right at the barrier top.

Quantum quantum Scattering Calculations Two-state scattering calculations were done using a time-


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336 Table 1. Parameters associated with the Model Hamiltonians Model

Well depth


M 1 2 3 4 5 6 7 8 9 10 11

-0.0319 -0.0319 -0.0319 -0.0319 -0.0637 -0.0637 -0.0319 -0.0319 -0.0319 -0.0319 -0.0319

Spin-orbit Crossing point shifiCf) splitting(s) 0.4 0. 0.4 0. 0.4 -2. 0.4 -2. 0.4 0. 0.4 0. 0. 1.2 1.2 0. 0.4 0. 1. 0.4 0.4 -1.

Coupling shift (h) 2.0 0.0 2.0 -2.0 0.0 2.0 2.0 0.0 -2.0 3.0 1.0

independent coupled channel method that is similar to a code that is described by Schatz (12). In this code the Schrôdinger equation for the two states is solved by sector propagation, integrating from χ = -10 to χ =6, using a step size of 0.01. The reduced mass in the calculation is 1 A M U (1732 atomic units) which is approximately the mass of hydrogen. B y propagating one set of independent solutions from negative χ to positive χ and then a second set from positive to negative, a complete set of linearly independent solutions is obtained, and then this is matched to proper asymptotic solutions to determine the scattering matrix and the reaction probabilities. Tests of convergence of these probabilities indicate that they are converged with respect to the integration parameters.

Trajectory Surface Hopping Calculations T S H calculations were done using the fewest switches method (13). Details are similar to work that was described earlier (11), except that here we have considered calculations in both the diabatic and adiabatic representations. A time step of 1 atomic unit was used for all calculations, and 2000 trajectories were used to determine the reaction probabilities at each energy. Variation of the results with respect to the numerical parameters was within statistical uncertainty. We found the diabatic results to be significantly less sensitive to time step than the adiabatic results.


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Results Figure 2 presents quantum reaction probabilities versus energy for each of the eleven models. These probabilities refer to the triplet (lower) initial state, and the two probabilities, labeled P and P , are for reaction to give the lower and upper final states, respectively. Note that the final states are defined in the adiabatic representation, and are roughly equal mixtures of singlet and triplet states that approximate the Π and Π ι states of O H . Linear and semilog plots are included for each model, so as to show both the below-barrier (E

Low-Lying Potential Energy Surfaces - American Chemical Society

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