Adiabatic dynamics

The simplest approach to simulating non-adiabatic dynamics is by surface hopping [175. 176]. In its simplest fomi, the approach is as follows. One carries out classical simulations of the nuclear motion on a specific adiabatic electronic state (ground or excited) and at any given instant checks whether the diabatic potential associated with that electronic state is mtersectmg the diabatic potential on another electronic state. If it is, then a decision is made as to whedier a jump to the other adiabatic electronic state should be perfomied. 

Finally, semi-classical approaches to non-adiabatic dynamics have also been fomuilated and siiccessfLilly applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This mediod is also quite accurate, except drat the number of required trajectories grows with time, as in any semi-classical approach. 

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

The standard semiclassical methods are surface hopping and Ehrenfest dynamics (also known as the classical path (CP) method [197]), and they will be outlined below. More details and comparisons can be found in [30-32]. The multiple spawning method, based on Gaussian wavepacket propagation, is also outlined below. See [1] for further infomiation on both quantum and semiclassical non-adiabatic dynamics methods. 

The first study in which a full CASSCE treatment was used for the non-adiabatic dynamics of a polyatomic system was a study on a model of the retinal chromophore [86]. The cis-trans photoisomerization of retinal is the primary event in vision, but despite much study the mechanism for this process is still unclear. The minimal model for retinal is l-cis-CjH NHj, which had been studied in an earlier quantum chemisti7 study [230]. There, it had been established that a conical intersection exists between the Si and So states with the cis-trans defining torsion angle at approximately a = 80° (cis is at 0°). Two... 

The first role of a reservoir is to impose on the system a gradient that makes the subsystem structure nonzero. The adiabatic flux that consequently develops continually decreases this structure, but the second role of the reservoir is to cancel this decrement by exchange of variables conjugate to the gradient. This does not affect the adiabatic dynamics. Hence provided that the flux is maximal in the above sense, then this procedure ensures that both the structure and the dynamics of the subsystem are steady and unchanging in time. (See also the discussion of Fig. 9.) A corollary of this is that the first entropy of the reservoirs increases at the greatest possible rate for any unconstrained flux. 

Mitchell PJ, Fincham D (1993) Shell-model simulations by adiabatic dynamics. J Phys Conden Matter 5(8) 1031-1038... 

At the most fundamental level one follows the time development of the system in detail. The reactants are started in a specific initial (quantum) state and the equation of motion are propagated to give the final state. The equation of motion of the system is the time dependent Schroinger equation, or, if the atoms involved are heavy enough (not H or Li) Newtons equation. The starting point is the adiabatic potential energy surface on which the process takes place. For some reactions electronic excitations during the reaction are important and must be included in addition to the electronically adiabatic dynamics. 

Figure 3.47 A schematic diagram of the evolution of the population of each instantaneous eigenstate of a reference Hamiltonian for the dynamics driven by counter-diabatic Hamiltonian and the fast-forward driving potential. The sizes of the green circles represent the populations of the levels. The red curves correspond to adiabatic dynamics, which is accelerated by the fast-forward driving potential.

S. Masuda and K. Nakamura. Fast-forward of adiabatic dynamics in quantum mechanics. Proc. R. Soc.A, 466 1135-1154(2009). 

Including 7]xx and Fx in Langevin-type classical dynamics has been termed molecular dynamics with electronic frictions (MDEF) [70], and has now been used in several simulations of non-adiabatic dynamics. Of course, the key unknown is the magnitude of the electronic frictions (since they also determine Fx). 

Fortunately, the same limiting conditions that validate the friction approximation can also be used with time-dependent density functional theory to give a theoretical description of rjxx. This expression was originally derived to describe vibrational damping of molecules adsorbed on surfaces [71]. It was later shown to also be applicable to any molecular or external coordinate and at any location on the PES, and thus more generally applicable to non-adiabatic dynamics at surfaces [68,72]. The expression is... 

Atomic/molecular scattering is the fraction of incident particles that do not trap or stick irreversibly on impact with the surface. However, in adiabatic dynamic theory, the scattering results from the same PES as that which causes bond making/breaking, so that a study of those that got away also gives complimentary information about the bond making/ breaking dynamics. 

Figure 3.25. Probability of a given energy loss into e-h pairs of magnitude eh vs. A/icM occurring in associative desorption of a diatomic from a metal surface from 3D non-adiabatic dynamics, (a) is for H2 associative desorption from Cu(lll), with ( ) 0.02 eV and (b) is N2 associative desorption from Ru(0001), with (A/i ch) 0.5 eV. From Ref. [68].

Figure 3.28. N2 vibrational state distribution in associative desorption from Ru(0001). (a) Observed in experiment. From Ref. [126]. (b) From 3D (Z, R, q) first principles quasi-classical dynamics, with the solid triangles pointing upward being adiabatic dynamics and the squares from molecular dynamics with electronic frictions also from DFT. Based on the PES and frictions of Ref. [68]. The open triangles pointing downward are the results of 6D first principles adiabatic quasi-classical dynamics from Ref. [253].

A significant limitation in both the adiabatic and non-adiabatic dynamics discussed above is that they are only 3D. Recently quasi-classical adiabatic dynamics calculations of S on a 6D DFT PES have shown that the predominant reason that 5 <<1... 

S. Mahapatra, Quantum non-adiabatic dynamics through conical intersections Spectroscopy to reactive scattering, Int. Rev. Phys. Chem. 23 (2006) 483. 

---