2.2 Quantum dynamics
2.2.8 Quantum dynamics using independent trajectories
Many systems of real-word interest, especially those involving interactions with light or the electromagnetic field in general, involve multiple electronic states. Fur- thermore, breakdown of the Born-Oppenheimer approximation is rather common, that is the dynamics of the system occur around geometries where electronic states can rapidly change with respect to the nuclear coordinates. Such non-adiabatic
dynamics often involve high frequency oscillation of the wavefunction between elec- tronic states, which presents a challenge for quantum dynamics methods. There are a number of quantum dynamics approaches which are rely on independent trajectories to account for dynamics across multiple electronic states, and given the ultimate goal of the work presented herein being light-matter interactions, it is worthwhile briefly discussing some common approaches. Given that the work
2.2. QUANTUM DYNAMICS 1 1.25 1.5 1.75 2 −5 0 5 (a) S1 S0 R E ( R )
Pure electronic states EEhr(r) moreS0character EEhr(r) moreS1character
1 1.25 1.5 1.75 2 −5 0 5 (b) S1 S0 R “Parent” trajectory “Child” trajectories 1 1.25 1.5 1.75 2 −5 0 5 (c) S1 S0 R TSH trajectory
Figure 2.3: Multi-state quantum dynamics methods: (a) MCE trajectories, evolve on a state averaged PES, (b) AIMS trajectories evolve classically, spawning copies of themselves to account for non-adiabatic transitions and (c) TSH trajectories “hop” from state to state with a probability proportional to the non-adiabatic coupling.
presented herein is only loosely related to these methods, in depth discussion is forgone in the interest of brevity here, especially since the scope of methods oc- cupying this area of quantum dynamics is relatively regularly surveyed by far more experienced authors.75
The Ehrenfest approximation, which is discussed in more detail in Chapter 3, has been used to derive a set of equations of motion for a “frozen” Gaussian basis set, resulting in the multi-configuration Ehrenfest (MCE) method.45 While initially developed and applied as a way of treating systems with many environ- mental modes, the extension to multiple electronic states was quickly achieved,84 and very successfully applied to the pyrazine benchmark discussed above. One of the key benefits of this approach is its simplicity: the trajectories propagating the basis set evolve on a state-averaged, mean-field PES, the extent of each function on the states determined by a set of time-dependent coefficients, the equations of motion are determined variationally. Figure 2.3 (a) illustrates the nature of the MCE trajectories. The MCE approach has recently been interfaced with ab
initio electronic structure routines (AI-MCE), thus allowing dynamics to be calcu-
lated on the fly, the application of AI-MCE to excited state dynamics of ethylene yielding encouraging results.85
Another approach to multi-state dynamics relies on classical trajectories to propagate its basis functions, the equations of motion for the expansion coefficients being determined variationally as usual. In order to account for non-adiabatic transitions, the effective non-adiabatic coupling is calculated and if this value reaches a certain threshold, each trajectory has a chance to spawn identical copies of itself on the state it is coupling with. This approach is referred to as the multiple spawning (MS) method.43,86The direct dynamics version of this approach is known
asab initio multiple spawning56(AIMS), which has been applied very successfully to a number of challenging systems.27,28,87,88 The MS strategy is visualised in Figure 2.3(b).
Recently a hybrid method, combining the state-averaged potentials of the MCE method and the basis set expansion strategy from AIMS has been introduced, referred to asab initiomultiple cloning46,89(AIMC). More specifically, this method addresses the shortcomings of MCE in regions of low non-adiabatic coupling where the wavepacket is spread significantly across multiple states. In such a case, the state-averaged potential does not reproduce the branching of the wavepacket that should occur due to the different shapes of the PESs. The multiple spawning like cloning spawning algorithm allows for this to be corrected however: In AIMC, after the wavepacket has passed through a region of strong non-adiabatic coupling, the wavepacket bifurcates into two initially identical copies, each predominantly on one of the two electronic states involved in the coupling event. Recently, this approach has been applied to the photodissociation of pyrrole with some encouraging initial results.89
As an alternative to spawning or cloning in order to account for non-adiabatic transitions, the wavepacket may be expanded in a set of trajectories moving on a single surface, which in regions of strong non-adiabatic coupling have the capability to “hop” and thereby change the electronic state they evolve on. This constitutes one of the earliest approaches to non-adiabatic dynamics and has come to be re- ferred to as trajectory surface hopping (TSH).42,90–93 Again, the propagation of the time-dependent basis functions occurs via classical mechanics, although in the case of TSH the basis functions are purely electronic, while the equations of mo- tion for the coefficients are derived variationally. At any point, the probability of a “hop” to any given surface may be calculated as a function of the non-adiabatic coupling. Given its relatively long developmental history, a large number of modi- fications and improvements have been made to TSH,94 it having been applied to numerous systems. Recently, some rather complex photodynamics problems have been tackled with a direct dynamics version of TSH with encouraging results.95,96 In conclusion then, MCE,AIMS and AIMC all share a similar wavefunction
ansatz,
Ψ(r;t) =X j
Aj(t)φj(r;t),
where the basis functions φ(r;t) are multi-dimensional “frozen” Gaussians, the time dependence of which is achieved by evolving the centre with classical, or in the case of MCE, semi-classical equations of motion. TSH employs a similar expansion, however the basis is made up of orthonormal electronic basis functions. Time evolution of the coefficients, Aj(t), for all methods occurs via variational