Including Tunneling in Non-Born–Oppenheimer Simulations

Letter pubs.acs.org/JPCL

Including Tunneling in Non-Born−Oppenheimer Simulations Jingjing Zheng, Rubén Meana-Pañeda, and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States S Supporting Information *

ABSTRACT: For electronically nonadiabatic processes in all but the simplest systems, the most practical multidimensional simulation method is a semiclassical approximation in which a trajectory or the center of a wave packet follows a classical path governed by an effective potential energy function. Here, we show how such simulations can be made more realistic by including tunneling by the army ants tunneling method. We illustrate the theory by calculations with model potential energy surfaces; one model study is in the adiabatic limit, and the other one has nonadiabatic transitions between two electronic states during the tunneling event. The army ants tunneling algorithm is used to efficiently sample tunneling events in the trajectories in both cases. This work makes it possible to simulate complex nonadiabatic chemical processes by efficiently including the important quantum effect of tunneling. SECTION: Spectroscopy, Photochemistry, and Excited States

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modeling of electronically nonadiabatic processes by other methods as well, for example, quantal wave packet methods where the wave packet is not restricted to follow a classical trajectory,18 path integral dynamics,19−21 and the initial value representation,22 but these methods will require more development of efficient computational strategies for their practical application to complex systems, whereas the method proposed here has a cost comparable to non-Born−Oppenheimer (NBO) trajectory calculations that do not include tunneling. Two types of methods are often used to calculate NBO trajectories, in particular, mean-field methods and surface hopping methods. In the present study, we use a mean-field method and assume that the electronic wave functions are fully coherent in the tunneling region, which is a reasonable approximation for a short-time process like tunneling. In practice, the assumed coherence means that the evolution of the electronic wave function follows the Schrödinger equation, which results in its propagation by the semiclassical Ehrenfest23−26 (SE) equations. In a real process, the electronic system’s evolution is governed not by the Schrödinger equation but rather by a nonunitary Liouville−von Neumann equation because the electrons are a subsystem evolving in the presence of the nuclear degrees of freedom; the nuclear wave packet components associated with the different electronic states get out of phase and out of overlap, and this decoheres the coherent superposition of electronic states into the measured probabilistic mixture.27 More accurate mean-field methods, for example, the coherent switching with decay-of-mixing (CSDM)

n recent work, we demonstrated how to efficiently include quantum mechanical tunneling in classical simulations by the army ants quantum mechanical rare event sampling algorithm combined with a semiclassical treatment of tunneling in internal coordinates.1 Tunneling is also very important in electronically nonadiabatic chemical processes, for example, in some electrontransfer processes,2 coupled electron−proton transfer,3 many photoinduced processes,4,5 some spin-forbidden reactions,6,7 and reactions involving a narrowly avoided crossing at a saddle point.8 A prominent example is the photodissociation of phenol and substituted phenols.9−11 Another example where tunneling could be intimately coupled with nonadiabatic processes is the isomerization rate of the retinal chromophore of visual pigments.5 To study the reaction mechanisms of nonadiabatic chemical processes in which tunneling plays an important role, dynamic simulations should not only cover the energies higher than that of the conical intersection or avoided crossing along the reaction path but also cover the lower energies that constitute the so-called nonadiabatic tunneling regime. Nakamura and his co-workers12−14 have developed complete analytic solutions for the one-dimensional Landau−Zener− Stueckelberg15−17 problem including the nonadiabatic tunneling case. Their work provides an analytical model for the physics of nonadiabatic tunneling using a one-dimensional framework. However, to simulate real nonadiabatic chemical processes in the general case, we have to deal more fully with the multidimensional aspects of the problem. In this Letter, we will show how to include tunneling in multidimensional nonadiabatic trajectories (sometimes called molecular dynamics simulations) and demonstrate the method for multidimensional model problems (full-dimensional calculations on triatomic systems). We note that tunneling may be included in the © 2014 American Chemical Society

Received: April 3, 2014 Accepted: May 15, 2014 Published: May 15, 2014 2039

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⎛ 2 P = exp⎜ − ⎝ ℏ

method,28,29 reduces to the SE method if one neglects decoherence, and the SE method is used here to demonstrate how to include nonadiabatic tunneling in NBO trajectories. Therefore, in the work presented here, we assume that the electronic wave functions are fully coherent in the tunneling region. A very significant advantage of the SE method, as we shall see below, is that the dynamics is independent of representationadiabatic or diabaticin the SE approximation;30 thus, one does not have to choose a representation for treating the tunneling process; one can use either representation. We emphasize that the examples in this Letter are presented using only the SE method because the SE method is simpler but has all of the elements needed to allow one to understand the method with a minimum of extraneous additional background knowledge. Although for simplicity we assume the Ehrenfest approximation for the entire trajectory in the present demonstration of the method, for practical applications to real molecules, the Ehrenfest method does not yield physical final states (due to lack of decoherence), and we recommend including decoherence by the CSDM algorithm in the classically allowed regions; one just turns off decoherence in the tunneling region. One could also include decoherence during the tunneling event, but that is beyond our scope and, as argued above, often not necessary due to the short time scale for tunneling. (In principle, the tunneling method presented in this Letter should be also extendable to surface hopping methods, but one would have to devise a protocol for hopping in the middle of a tunneling event. Because CSDM is both more natural and more accurate than surface hopping, we recommend employing CSDM instead.) Army Ants Tunneling. The army ants tunneling algorithm for a trajectory governed by a single potential energy surface is presented in detail elsewhere;1 here, we summarize the relevant details. One selects trajectories from an initial ensemble. One selects an internal coordinate or a combination of two internal coordinates for the tunneling path and calculates a tunneling probability P whenever one reaches a turning point of that coordinate (P is zero if the tunneling path never emerges into a classically allowed region of phase space). One then picks a random number between 0 and 1 and compares it to P. The uniform sampling method of including tunneling is to follow the tunneling path if the random number is less than P and classically reflect without tunneling if it is greater than P. For small P, this algorithm seldom follows the tunneling path; therefore, the statistical error is large due to sampling only a few tunneling events. Army ants sampling, in contrast, follows the tunneling path more often; therefore, the final result is based on a statistically significant number of tunneling events without having to run an excessive number of trajectories, but each tunneling event and some nontunneling events have a weight less than unity to produce an unbiased stochastic result (that is, the result would be the same as that for the uniform method if the ensemble of initial conditions were the same and there were no other sources of error, such as numerical integration step size, a finite number of trajectories, and roundoff). In single-surface trajectories, one propagates the coordinates and momenta under the influence of a potential energy surface V, which is a function of nuclear position q; V usually corresponds to the ground adiabatic electronic state. The potential V is also used to calculate P, which results from exponentiation of an imaginary action integral along the tunneling path31

∫0

ξmax

⎞ 2μ[V (q) − V (q 0)] dξ⎟ ⎠

(1)

where ξ is the distance (in isoinertial coordinates scaled to a reduced mass μ) at position q along the tunneling path relative to the starting point q0 of the tunneling path and ξmax is the length of the whole tunneling path in isoinertial coordinates. The tunneling path is determined by the choice of tunneling coordinate, as explained previously.1,31 The integration in eq 1 was carried out by Gaussian−Legendre quadrature in our previous work on adiabatic trajectories. However, here, we calculate the integral by the trapezoidal rule because small step sizes are required to propagate the electronic coefficients in nonadiabatic trajectories. Extension to the Electronically Nonadiabatic Case. In meanfield and decay-of-mixing methods (presented and reviewed in detail elsewhere29,32), the nuclear motion is governed by an effective potential V̅ , which is a function not only of nuclear position q but also of the coefficients ci in the expansion of the multiconfigurational electronic wave function in terms of either adiabatic or diabatic states. One could equally well propagate the electronic density matrix instead of the electronic wave function, but our computer program propagates the wave functions.33 The coefficients, nuclear coordinates, and momenta are propagated together along the trajectory. When we combine a mean-field method with the tunneling algorithm, it is straightforward to substitute V̅ for V in eq 1, but the new issue that arises is that we need to convert the time derivatives ċi of the SE electronic wave function coefficients to their rate of change along the tunneling path, which requires assigning a time to each point on the tunneling path. The problem of defining a time in the tunneling region presents conceptual difficulties and has been extensively discussed in the literature.34−37 The difficulty with determining the transmission time for a tunneling particle is seen most clearly if we try to derive the tunneling time by the time it takes a wave packet to tunnel through a barrier. A wave packet inevitably contains a distribution of momenta and coordinates, and the tunneling is not dominated by the components with the average momentum or by the momentum corresponding to the maximum value of the wave packet.34 Furthermore, in the regime of most interest, most of the wave packet is reflected, and yet a portion of the wave packet has enough energy to go over the barrier; furthermore, the wave packet continues to spread throughout the tunneling event. Sokolovski and Connor35 note that “the question ‘How long does it take for a quantum particle to tunnel through a barrier’ is of the same type as, for instance, the question ‘What is the position of a tunneling particle at a given time during the collision?’” However, we also note that although defining momentum as a function of position violates the uncertainty principle, that is precisely what we do when we make the WKB (Wentzel−Kramers−Brillouin) approximation. We therefore use the WKB approximation to resolve the question of traversal time for a tunneling particle. Büttiker and Landauer36 have shown the WKB traversal time for tunneling of a particle with mass μ is

Δt =

Δξ 2(V − V0)/μ

(2)

where ξ is again the distance along the tunneling path in isoinertial coordinates scaled to mass μ, V is the potential energy at Δξ in the tunneling path, and V0 is the energy of the 2040

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turning point. Büttiker and Landauer showed that this is the appropriate time for a particle that interacts with a time-varying field while it tunnels, and thus, we conclude that this is appropriate for the interaction of the nuclear coordinates with the time-varying mean potential V̅ . (V̅ varies with time not only because the adiabatic and diabatic potentials depend on the coordinates but also because of the additional time dependence in the mean field due to the dependence on time of the coefficients ci.) Note that the simple physical interpretation of eq 2 is that it is the distance divided by the magnitude of the imaginary velocity. This result was further confirmed by Sokolovski and Connor,37 who showed that the Büttiker− Landauer formula provides a natural time scale for the region of time that contributes the most to the transmission amplitude integral. [An interesting aspect of eq 1 is that the transit time for a constant-width barrier is smaller for deep tunneling (low energy) than that for shallow tunneling (energies only slightly below the barrier top). This is consistent with the result12 that electronically nonadiabatic processes become more diabatic (less adiabatic) both when the energy is increased to much greater than the potential energies at the avoided crossing and also when the energy is much lower than the potential energies at the avoid crossing.] Applying the Büttiker−Landauer result to our problem yields dci = dξ

Figure 1. Potentials and couplings along the H−X1 dissociation coordinate with other coordinates having their equilibrium structure values; Vk is the potential for adiabatic state k; Ukk is an element of the diabatic potential matrix; and dH,x is the hydrogen atom x component of the Cartesian nonadiabatic coupling vector d in isoinertial coordinates (when calculating d, the H−X1 internuclear axis is parallel to the x axis, and the center of mass of HX2 is at the origin).

ci̇ 2(V̅ (q) − V̅ (q 0))/μ

The trajectories are stopped when reaction has occurred; we define this as the time when the H−X1 distance is larger than 8 Å. We calculate the probability of reaction as a function of time, Pr(t), as the sum of the weights of trajectories that have dissociated divided by the sum of all of the weights at that time, and we plot the natural logarithm of [1 − Pr(t)] versus t; the negative value of the slope of the linear portion of the plot is the rate constant k. Note that our method is independent of whether the overall angular momentum is removed or not; in the case of nonzero overall angular momentum, more complicated derivations will be required to conserve total angular momentum before and after each tunneling events. To illustrate model N, we started with the equilibrium geometry, fixed-EJ ensemble on the ground-state surface, and we carried out a vertical excitation (in the adiabatic or diabatic representation), as explained in Supporting Information. The tunneling coordinate is the H−X1 bond stretching internal coordinate. Results: Model A. Trajectories were calculated for model A in both the adiabatic and diabatic representations. The trajectories start on surface 1 of the given representation. The total energy is taken as 3.04 eV, which is 0.21 eV above the classical barrier. We also ran Born−Oppenheimer trajectories with the same total energy on the adiabatic ground state for comparison. The plots of ln (1− Pr) versus time and the resulting rate constants for dissociation are in Figure 2a. Comparing the results is a difficult test of the method because obtaining electronically adiabatic behavior in the diabatic representation is only achieved by treating the very large coupling consistently so that trajectories transfer completely from the U11 surface to the U22 surface. However, the method passes the test in that, within statistical accuracy, all three runs (adiabatic representation, diabatic representation, and single-surface) give the same rate constants. This set of runs proves that we obtain the correct electronic evolution in the tunneling region. Results: Model N. Model N was also run with both representations, and for this model, we ran calculations with and without tunneling. The initial electronic state is vertically

(3)

Systems Studied. Two models, defined by their potential energy surfaces and couplings and their masses, are used to demonstrate nonadiabatic tunneling in NBO trajectories. Both models are of the chemical type HX2 with two electronic states, where H and X are model atoms; we set the mass of H to that of hydrogen and the mass of X to that of nitrogen. We label the X atoms as X1 and X2, with X1 initially closer to H. The potential energy surfaces and couplings are defined in the diabatic representation (surfaces U11 and U22, coupling U12), and the adiabatic surfaces (V1 and V2) and nonadiabatic coupling vector (d12, which equals ⟨ψ1|∇k ψ2⟩, where ψk is the adiabatic electronic wave function of state k) are obtained by diagonalization in the standard way.32 The two models are called model A and model N, and their surfaces and couplings are shown in both representations in Figure 1 (only the component of d along the H−X1 bond dissociation direction r̂ is shown). In model A, we set U12 to very large values, so that the resulting two adiabatic states are very well separated and the nonadiabatic coupling vector is small enough that the dynamics should have the behavior of the electronically adiabatic limit, that is, the two-state dynamics with initial population on the ground state should be the same as the single-state dynamics on the ground adiabatic state. In model N, the diabatic coupling is reasonably small so that nonadiabatic transitions can be observed during the tunneling event. Computational Details. To illustrate model A, we use a simple ensemble that we call the equilibrium geometry, fixed-EJ ensemble, as used previously.1 In this ensemble, the initial geometry is the classical equilibrium geometry (we use the minimum-energy geometry on the ground-state adiabatic surface), and, except for a fixed total internal energy E and zero total angular momentum J, the momenta are selected to be random (see the Supporting Information for details). For each calculation, we sample 20000 trajectories from this ensemble. 2041

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Figure 3. A typical tunneling path in the trajectories for model N. This figure shows the mean potential V̅ , adiabatic ground- and excited-state potential V1 and V2, and the adiabatic ground-state density ρ(1,1). This path yields a tunneling probability of 2.54 × 10−5. The total energy is E = 3.20 eV, and V̅ (q0) is 3.17 eV.

ries; calculations in three kinds of runs agree with each other. The nonadiabatic case shows how one can calculate a tunneling probability for the case where the states are strongly mixed in both the adiabatic and the diabatic representations in the tunneling region. Electronic nonadiabaticity and nuclear tunneling are two aspects of molecular dynamics that require an extension beyond strictly classical molecular dynamics. We and others have previously studied these extensions separately. Here, we combine our approaches to enable the treatment of dynamical events that involve both electronic nonadiabaticity and nuclear tunneling at the same time. The new method, because it may easily be incorporated in standard molecular dynamics codes, provides new capabilities for understanding and simulating complex molecular dynamics.

Figure 2. Curves of the natural logarithm of nonreactive probability 1 − Pr versus time. The negative value of the slope of the linear part of each curve is the first-order reaction rate constant. (a) Model A; (b) model N.

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ASSOCIATED CONTENT

S Supporting Information *

excited to the V2 surface (adiabatic representation) from the ground-state minimum. When the diabatic representation is used, the populations of diabatic states are calculated based on the population of the adiabatic states (details of the vertical excitation algorithm are in the Supporting Information). The total energy is 3.2 eV. The decay curves and calculated rate constants are in Figure 2b. Calculations in the two representations agree well both when tunneling is included and when it is not. Tunneling increases the reaction rate by almost one order of magnitude for this case. Figure 3 shows the effective potential and 1,1 element of the electronic density matrix for a typical tunneling path from the adiabatic representation simulation for model N. Because both states have finite populations, the mean potential is between two adiabatic states. Therefore, tunneling through the mean potential has larger probability than tunneling through the adiabatic excited-state surface. This shows how the electronic nonadiabaticity is coupled to the tunneling. In this work, we showed how to include tunneling in nonadiabatic trajectories by using the army ants tunneling algorithm for efficient rare quantum event sampling. Two model potentials were used to demonstrate the method. The adiabatic limit case was used to validate the method by comparing the nonadiabatic trajectories and adiabatic trajecto-

Fortran subroutines and analytic forms of the potential energy functions and couplings and details of the initial conditions and numerical integration. This material is available free of charge via the Internet at http://pubs.acs.org.

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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 work was supported in part by the U.S. Department of Energy under SciDAC Grant No. DE-SC0008666 (J.Z. and D.T.) and the Air Force Office of Scientific Research under Grant No. FA9550-12-1-0486 (R.M.-P. and D.T.).

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