The non-adiabatic dynamics plays significant roles in many photochemical and pho-tophysical reactions, such as radiationless relaxation, charge and energy transfer. Thess processes pass through the region where non-adiabatic couplings are important. As a res-ult, going beyond Born-Oppenheimer approximation becomes an extraordinarily critical issue in the field of chemical physics.
Given the importance of non-adiabatic dynamics and its adversity in understanding and interpretation of experimental phenomena, a variety of theoretical methods used to simu-late these processes have been developed[47]. Here we will introduce some of the most widely used methods. The comparison between these methods and other methods can be found in Ref [48, 49]. For example, the full quantum methods such as split operator approach[50, 51], Gaussian wavepacket dynamics[52, 53].
The direct solution of time-dependent Schr¨odinger equation leads to wavepacket propaga-tion methods[54, 55]. In wavepacket dynamics, the system is represented in its initial state by a wavepacket, which is a nonstationary superposition of eigenstates. The time-evolution of this wavepacket followed the total Hamiltonian is then calculated numeric-ally. Not only can all the required information be extracted from the evolving wavepacket, but it provides a very pictorial description of the process of interest. Unfortunately, the standard wavepacket dynamics suffers from its need of tremendous computer resources, which scales exponentially with the number of degrees of freedom in the system. For systems with more than four to six degrees of freedom, the simulations are typically pre-cluded. This limitation on dimensionality forces us to search for approximate methods.
doi:10.6342/NTU201803678
2.2.1 Multi-Configuration Time-Dependent Hartree
Multi-configuration time-dependent Hartree (MCTDH) is one of the powerful wave-packet dynamics algorithms at present[53]. It was first introduced in 1990 by Meyer, Manthe, and Cederbaum[56, 57]. The idea of the MCTDH approach is to expand the MCTDH wavefunction in a set of basis, which is smaller than the basis set used in standard wavepacket method but time-dependent.The time-dependent weighted Hartree products with time-dependent single-particle functions is written as
Φ(Q1, ..., Qf, t) = By solving the time-dependent Schr¨odinger equation using a variational principle, one obtain the equations of motion for the expansion coefficients and the single-particle func-tions. The basis thus follows the evolving wavepacket. The high-accuracy potential en-ergy surfaces and non-adiabatic couplings are then obtained. Importantly, the result con-verges on the exact result as the basis is increased in size. MCTDH is a full quantum and numerically exct approach which can be applied in system with up to 24 nuclear degrees of freedom. More recently, multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) has been developed for large systems with hundreds of degrees of freedom.
However, those full quantum approaches require a significant amount of computational efforts and thus only applicable in small systems. Accordingly, many semi-classical meth-ods and mix quantum classical methmeth-ods have been developed.
One class of the mix quantum-classical methods is the nonadiabatic molecular dynamics which involves the separation of electron and nuclear subsystems by propagating nuclear trajectories on an electronic potential energy surface defined by electronic states.
2.2.2 Fewest Switch Surface Hopping
Surface hopping is one of the mixed quantum-classcal technique that includes quantum mechanical effects into molecular dynamics (MD) simulations[58]. It is developed to deal with the dynamics where Born-Oppenheimer approximation breads down. By including
doi:10.6342/NTU201803678 excited adiabatic surfaces in the calculation and allowing sudden changes (trajectory hop)
between these surfaces, surface hopping incorporates the non-adiabatic effects to the sim-ulations. For a large number of trajectories, the following procedure is repeated to have the nuclear wavepacket information. Along the propagation of one trajectory, for a fixed nuclear geometry, one solves the time-independent Schr¨odinger equation for electrons and evaluates the energy gradient. The energy gradient is then used to update the nuclear geometry classically according to the Newton’s equation. For the new step, nonadiabatic coupling is calculated and a hopping between electronic states is performed if necessary, while the transition probability is determined according to the coupling.
One of the most widely used surface hopping method was introduced by J. C. Tully, called Fewest Switch Surface Hopping (FSSH)[58]. In this approach, the non-adiabatic effect which treated by the trajectory hop is included. The popularity of this method cannot be understated. However, there exist many problems of surface hopping. The most import-ant among them is the unphysical way of determining the distribution of kinetic energy by adjusting the component of velocity in the direction of the nonadiabatic coupling after the trajectory hop.
2.2.3 Ehrenfest Mean Field
Another simulation method is Ehrenfest mean field (EMF) method[59] proposed in 1927. This method is not originally derived to study the non-adiabatic dynamics but has been proved to be an acceptable method in simulating non-adiabatic dynamics.
In Ehrenfest mean field method, the nuclear degrees of freedom is propagated along an average potential, hence the name mean field:
d2
dt2Rnu(t) = −∇nuhHel(Rnu)i. (2.24) Unlike FSSH, there is no hopping between different potential energy surface. This is because nuclear degrees of freedom is propagated along mean potential and thus non-adiabatic transistion is already included. EMF is a relatively cheap simulation method compare to FSSH and is useful in simulating dynamics over less important states.
How-doi:10.6342/NTU201803678 ever, if a simulation is performed using EMF, the result will not be accurate after nuclear
wavepacket leaves the region of strong non-adiabatic coupling. D. S. Sholl and J. C. Tully derived a combined method of FSSH and EMF[60]. In this method, states of interest are treated as adiabatic states as in FSSH while unimportant high energy states are treated as a single state with mean potential over these unimportant states.
2.2.4 Path Integral for Non-Adiabatic Dynamics
Path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum probab-ility amplitude.
Unlike FSSH and EMF, path integral formula for non-adiabatic dynamics is a numeric-ally exact formula. Exactness of this method implies that it is very time consuming and is not practically useful. However, both EMF and FSSH method mentioned above can be derived from this method. Path Integral thus provides a theoretical background for them.