Figure 1. Upon photoexcitation, charge density is transferred from the Na atom to the F atom, thus strengthening the NaF bond, weaking the HF bond, and enhancing the likelihood of the system fragmenting to the NaF + H products.

Excited electronic states play an important role in many areas of chemistry from combustion to light harvesting via a variety of elementary chemical mechanisms, including intersystem crossing, photodissociation, internal conversion, and charge transfer. The term "non-BornOppenheimer (NBO) may be generally applied to these processes to emphasize the idea that the BornOppenheimer separation of the nuclear (slow) and electronic (fast) time scales breaks down and that electronic surfaces other than the ground surface play a key role in the dynamics. An accurate treatment of the chemistry of NBO systems poses significant conceptual and computational challenges to current theoretical models.

CRF researcher Ahren Jasper, in collaboration with University of Minnesota professor Donald Truhlar, is developing and validating methods for performing quantitative simulations of NBO systems using reactive molecular dynamics (MD), which has traditionally been limited to single surface simulations. NBO MD simulations retain the classical, trajectory-based description of the nuclear motion, while incorporating transitions between the various electronic states based on a quantum mechanical treatment of the electronic degrees of freedom.

Because there is no unique way to couple quantum and classical subsystems, practical mixed quantum/classical treatments of quantum mechanical events are necessarily somewhat ad hoc, and several NBO MD methods have been proposed. An important aspect of current research is therefore the quantification and reduction of the errors associated with the additional assumptions and approximations used to model electronic transitions in MD simulations.

A recent target for theoretical treatment was the photodissociation of the NaFH van der Waals complex. Thermal excitation tends to break the weak van der Waals bond, producing the Na and HF products exclusively. Upon electronic excitation with visible light, however, the complex is promoted to a metastable complex called an exciplex. The exciplex is proposed to exhibit enhanced reactivity to form NaF + H via the "harpooning" mechanism shown in Figure 1.

Figure 2. Ground (blue) and first excited (red) potential energy surfaces of the NaFH system.

Analytic representations of the ground and first-excited potential energy surfaces were developed based on high-level quantum chemistry calculations and are shown in Figure 2. The two surfaces are most strongly coupled along a line of avoided crossings occurring at slightly extended HF distances. Avoided crossings typically represent the "shoulder" of a nearby but energetically inaccessible conical intersection.

NBO MD simulations were carried out using the fewest switches (FS) surface hopping algorithm. Each trajectory was initially propagated in the excited electronic state with its starting coordinates and momenta selected from a distribution designed to simulate the result of laser excitation. As the trajectory vibrated in the exciplex, a time-dependent electronic density matrix was calculated, representing the nonradiative flow of electronic density between the two states. This flow of electronic density depends on terms in the Hamiltonian that are neglected in the BO approximation but that become significant when the character of the electronic states changes rapidly as a function of nuclear geometry. At each time step along the trajectory, a probability of "hopping" between the states was computed based on the local rate of change of the electronic density matrix, and hops were carried out stochastically. If a hop was called for, the nuclear momentum was adjusted such that total energy was conserved. The NBO simulation consisted of thousands of trajectories, and the results were averaged to obtain product branching probabilities and exciplex lifetimes.

Figure 3. Ground (blue) and first excited (red) potential energy surfaces. The initial hops down are shown as red triangles. Subsequent successful hops up are shown as blue triangles, and frustrated hops up are shown as black dots. The solid black line is the line of avoided crossings.

The NBO MD simulations confirmed the enhanced reactivity of the harpooning mechanism, and NaF + H was predicted to be the dominant photodissociated bimolecular product. Quantitatively, however, the FS method was found to overpredict the formation of Na + HF by a factor of four when compared with accurate quantum mechanical results. The lifetime of the exciplex was likewise overestimated by a factor of two.

An analysis of the NBO MD trajectories revealed that an important dynamical feature was missing from the FS method. Figure 3 shows contour plots of the excited and ground electronic states, as well as hopping information for a subset of trajectories. More than three-fourths of the trajectories attempted to hop back into the exciplex after their first hop down, but many did so at geometries where the excited state is energetically forbidden. The majority of these so-called "frustrated" hops occur near the line of avoided crossings, a region where the two electronic surfaces have very different shapes. Frustrated hops have previously been identified as a significant source of error in NBO MD surface hopping simulations.

In the quantum mechanical calculation, the nuclear motions associated with divergent parts of the wave function cause the electronic subsystem to decohere, which reduces the likelihood of electronic transitions. Decoherence due to wave packet divergence may be expected to be significant near the line of avoided crossings, as shown schematically in Figure 4.

Figure 4. The stochastic decoherence method accurately treats both coherent motion (the solid arrow) and decoherent motion due to wave packet divergence (the dashed arrows).

In contrast, the FS method incorrectly treats the electronic motion in this region as fully coherent, independent of the shapes of the potential energy surfaces. This failure of the FS method is due to its artificial separation of the classical nuclear and quantum mechanical electronic motions. A new model for decoherence, called "stochastic decoherence," was developed and introduced into the FS method, resulting in reduced errors associated with frustrated hopping and near quantitative agreement with the quantum mechanical results.

This study represents a step toward a more complete understanding of the complex chemistry of coupled electronic states, including the important role coherence plays in determining the products of laser-induced chemistry. The development of simple models for coherence and multistate dynamics that are readily applicable to large systems may aid in the understanding and design of technologies that seek to exploit these quantum effects for microscopic control.