A second approach to simulating non-adiabatic electron dynamics is to propagate the electronic structure in real time in the presence of the perturbing potential. Rather than casting the non-adiabatic problem as a set of electronic transitions between electronic states as in FSSH (i.e. changing occupancy of electronic states with a probability of “hopping” related to the coupling between states), real-time propagation evolves the electronic wavefunctions, and thus electronic excitations are captured via the changing character of the wavefunctions themselves. Real-time time-dependent density functional theory (RT-TDDFT) [67] has been used as an approach to describe the quantum dynamics of electrons for systems of various sizes and complexity. TDDFT is an extension of DFT that captures the electron dynamics of many-body systems. Here, we would like to clarify that the RT-TDDFT method that we discuss here is distinctly different from the more commonly recognized linear- response TDDFT [68], often referred to as TDDFT without any further distinction. Linear- response TDDFT uses the linear response of the electron density in response to a perturbation
to compute the excitation energies of a system [68]. This formalism can only be used when the perturbation is small and the changes in electron density do not dramatically change the overall electronic structure.
To make these many-body equations accessible, a proof similar to the Hohenberg-Kohn theorem that accounts for time dependence is necessary. It was the Runge-Gross theorem [67] in 1984 that served as proof that there indeed exists a one-to-one correspondence between densities and potentials for a given fixed initial state. Taking a similar form as the time-dependent Schrödinger equation, the time-dependent Kohn-Sham (TDKS) equations are
𝑖ℏ 𝜕 𝜕𝑡𝜓! r,𝑡 = − ℏ! 2𝑚∇!+𝑣ext r,𝑡 + 𝑑!r 𝑛 r r-r' +𝑣XC 𝑛 𝑡 r 𝜓! r,𝑡 Eq. 2.19
As was true for the time-independent case, the relationship between electron density and TDKS orbital is 𝑛 r,𝑡 = !! 𝜓! r,𝑡 !
!!! . Here, the XC potential is a functional of the time- dependent density. (Further discussion regarding the XC potential in RT-TDDFT is provided below.) Provided an initial set of KS orbitals 𝜓! , RT-TDDFT simulates the dynamics of the system by propagating these 𝑁! single-particle, nonlinear TDKS equations by applying the time-evolution operator 𝜓! r,𝑇 = 𝒯exp −𝑖 𝑑𝜏 ! ! 𝐻!" 𝜏 𝜓 ! r,𝑡 =0 Eq. 2.20
where 𝒯 is the time-ordering operator, and 𝜓! r,𝑡 =0 represents the initial set of KS orbitals. At this point, it is worthwhile to reiterate that these coupled equations are non-linear as the KS Hamiltonian is a functional of the electron density as determined by the TDKS orbitals. The solution to Eq. 2.20 becomes essentially intractable. In practice, this propagation is performed using numerical approximations. The propagation scheme used to evolve a set of TDKS equations depends on a variety of factors including the basis set representation and the time-step taken throughout the simulation. For example, in the Octopus code [69], which is calculated on a real-space mesh, the Crank-Nicholson (Cayley) scheme [70] and Magnus expansions [71] are both used to propagate the TDKS equations. In codes that use a planewave basis such as the Qbox/Qb@ll code[72,73], the 4th-order Runge-Kutta propagation scheme was found to be most stable and scalable to systems of thousands of electrons [74].
Exchange-Correlation Functionals in RT-TDDFT
An additional assumption made in these RT-TDDFT calculations is in the expression used to express the exchange correlation potential. Throughout much of this work, the PBE- GGA XC functional [30] is used because of its computational efficiency and its accuracy. Due to the increased computational cost associated with calculating real-time dynamics, it is necessary to carefully select the XC functional such that simulations remain computationally accessible, especially for large system sizes and phenomena that require an ensemble of simulations to accurately compute. GGA XC functionals have known shortcomings, however, such as accurately capturing long-range interactions, more specifically the non- Coulomb contributions to the total energy. Advanced XC functionals such as the long-range corrected Becke-Lee-Yang-Parr (LC-BLYP) [29,75] have been developed such that the
Hartree-Fock exchange integral is gradually turned on for long-range distances using a standard error function. As can be expected with the computation of the double integral, the computational cost of Hartree-Fock limits the use of these class of XC functionals.
An additional approximation within applications of RT-TDDFT is the use of the adiabatic exchange-correlation approximation [76,77]. Eq. 2.19 indicates that 𝑣XC 𝑛 𝑡 r is a functional of the time-dependent density, and more specifically the instantaneous electron density. The development of XC potential functionals has been formulated based on ground- state DFT and as was mentioned above, uses the homogeneous electron gas as the reference to describe electron correlation. In strong fields that drive large excitations, however, it is less clear whether or not these ground-state functionals are reasonably constructed for non- equilibrium, excited-state electron densities. In particular, this critique comes from work showing that the KS potential depends on the history of the density (i.e. how the excited electron density was generated) [77]. Work has continued to incorporate history effects
(𝑡 < 𝑡’) into the XC-potential functions themselves as well as testing the contexts in which the adiabatic approximation performs with reasonable accuracy [78-82].
Computational Resources and Code Parallelization
It cannot go unstated that the feasibility of these real-time electron dynamic simulations go beyond just the theory and require an intentional consideration of the computational resources and the parallelization of the electronic structure code. The work contained within Chapter 4 (‘Electronic Stopping in Water’) makes use of the Qbox/Qb@ll code [72,73], which has been optimized to perform large-scale simulations on massively-parallel, multi- core supercomputers [83]. As a plane-wave based code (𝜓! r,𝑡 = !
! G𝐶! G,𝑡 𝑒𝑥𝑝 𝑖G∙r ,
Blue Gene/Q architecture by distributing the expansion coefficients , 𝐶! G,𝑡 , on a process
grid with electronic states distributed over columns and planewaves over rows[83]. As a result, the Qb@ll code has been shown to scale to hundreds of thousands of computing cores for a system of 27,000 electrons. This level of parallelization is necessary for systems of large sizes in order to reduce communication cost within codes. Future development of real- time dynamic codes must also take advantage of and develop new algorithms that can be parallelized over similar numbers of cores in order to avoid introducing computational bottlenecks into the simulations.
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CHAPTER 3 : EXCITED ELECTRON RELAXATION - THE ROLE OF SURFACE