Fast calculation of retarded potentials in multi-domain TDDFT

TL;DR

This work introduces a scalable, multi-domain RT-TDDFT framework that computes retarded electromagnetic potentials in the Lorenz gauge for systems with spatially separable densities. By partitioning the domain into non-overlapping sub-domains and leveraging space-time FFTs, the method achieves significant computational speedups while preserving retardation effects, demonstrated on dimers of cumulene molecules and N$_2$. The approach combines a discrete space-time formulation, predictor-corrector time propagation, and parallelizable inter-domain coupling, with detailed complexity analyses showing favorable scaling relative to a single-domain implementation. The results validate retardation physics at nano-to-meso scales and highlight practical advantages for large molecular assemblies and nano-antenna-like configurations, including potential multipole-augmented acceleration future work. Overall, the paper provides a concrete, implementable pathway to include retarded potentials in real-time TDDFT for spatially extended, multi-domain systems.

Abstract

A formulation for the efficient calculation of the electromagnetic retarded potential generated by time-dependent electron density in the context of real-time time dependent density functional theory (RT-TDDFT) is presented. The electron density is considered to be spatially separable, which is suitable for systems that include several molecules or nano-particles. The formulation is based on splitting the domain of interest into sub-domains and calculating the time dependent retarded potentials from each sub-domain separately. The computations are accelerated by using the fast Fourier transform and parallelization. We demonstrate this formulation by solving the orbitals dynamics in systems of two molecules at varied distances. We first show that for small distances we get exactly the results that are expected from non-retarded potentials, we then show that for large distances between sub-domains we observe substantial retardation effects.

Figures (13)

• Figure 1: A graphic representation of a system with two sub-domains. Each sub-domain induces a delayed scalar and vector potential on the other sub-domain.
• Figure 2: A setup of two elongated sub-domains of a size ${W}\times{W}\times{L}$, with a dominant axis ($L>>W$). The sub-domains are set in parallel to each other with a distance of $d$ with respect to their horizontal axis. The green shaped box represents the size of a single large domain that contains both sub-domains (it is slightly larger only for illustration purposes).
• Figure 3: Computational complexities ratio between the single domain and the multi-domain i.e $speedup$. The domains' parameters are $W=10$ a.u., $L$ varies from $50$ a.u. to $1000$ a.u. and $d$ according to the y axis.
• Figure 4: Propagation scheme modified to integrate multi-domain calculations. Each domain produces the current time induced potentials on all the domains and obtains the retarded potentials from other domains (orange rectangle processes). Then these potentials are accumulated for a total effective potential, and propagated forward in time (red rectangle processes). The predictor-corrector scheme is used as in mundt2009realmundt2007photoelectron, where the induced retarded potentials are injected into the total effective potential in each step.
• Figure 5: The setup used to demonstrate the model. Two sub-domains of C$_{12}$H$_4$ with parallel axes, located at various distances. An external field is applied to Domain 1. Domain 2 is affected by the induced potential of Domain 1.