Electron charge dynamics and charge separation: A response theory approach

This study applies response theory to investigate electron charge dynamics, with a particular focus on charge separation. We analytically assess the strengths and limitations of linear and quadratic response theories in describing charge density and current, illustrated by a model that simulates charge transfer systems. While linear response accurately captures optical properties, the quadratic response contains the minimal ingredients required to describe charge dynamics and separation. Notably, it closely matches exact time propagation results in some regime that we identify. We propose and test several approximations to the quadratic response and explore the influence of higher-order terms and the effect of an on-site interaction $U$.

💡 Research Summary

The paper investigates electron charge dynamics and charge separation using response theory, focusing on the capabilities and limits of linear (first‑order) and quadratic (second‑order) response functions. Starting from the exact time‑dependent Schrödinger equation, the authors expand the evolution operator in powers of an external scalar potential that couples to the electron density. This yields a hierarchy of response functions expressed as nested commutators, with the first‑order term involving only ground‑state–excited‑state matrix elements and the second‑order term coupling three states (ground, first excited, second excited).

The authors analytically show that linear response can correctly predict optical spectra but cannot generate net charge transfer beyond the spatial region where the ground‑state density is non‑zero. Because the linear density response depends solely on overlaps between the ground state, an excited state, and the perturbation, any induced charge density oscillates symmetrically around zero and the associated current has no DC component. Consequently, charge separation—defined as a persistent imbalance of electron and hole densities on opposite sides of the system—cannot be captured at first order.

In contrast, the quadratic response contains terms of the form O₀I n_IJ n_J0, which allow the observable (density or current) to be influenced by products of two transition densities. For an instantaneous perturbation (δ‑function) the second‑order density response involves sin Δt sin Δ′t terms, while for a sinusoidal drive it yields mixed sin Δt sin ωt and t sin ωt contributions. At resonance (Δ ≈ ω) the t sin ωt term leads to a non‑oscillatory growth of charge on one side of the system, i.e., a net charge separation, and the current acquires a finite time‑averaged (DC) component. Thus, the quadratic response provides the minimal ingredients required to describe charge migration and separation.

To benchmark these analytical insights, the authors introduce a minimal two‑site tight‑binding model that mimics an optoelectronic device. The model can be solved exactly by real‑time propagation (diagonalizing the Hamiltonian at each time step) and also evaluated using linear and quadratic response formulas. Numerical results confirm that for weak fields and short times the linear response matches the exact dynamics, but as the field strength or propagation time increases the linear approximation fails, while the quadratic response remains accurate up to a well‑defined crossover regime. The authors further explore three practical approximations to the full second‑order expression: (i) a simple product of two first‑order responses, (ii) a time‑separated factorization, and (iii) neglect of higher‑order time‑ordering terms. They find that the product approximation works well for low frequencies, the time‑separated form offers a good balance of accuracy and computational cost, and the neglect of higher‑order terms becomes acceptable only when the perturbation is far from resonance.

Finally, the effect of an on‑site Hubbard interaction U is examined. In the non‑interacting case (U = 0) the quadratic response reproduces the exact dynamics almost perfectly. As U increases, electron‑hole binding becomes stronger, reducing the amplitude of charge transfer and degrading the agreement of the quadratic approximation. This signals that for strongly correlated systems one may need to include third‑order (or higher) response contributions or adopt many‑body techniques such as the Bethe–Salpeter equation within the response framework.

In summary, the study demonstrates that second‑order response theory is the lowest order capable of describing charge separation and long‑range charge migration, identifies its regime of validity, proposes useful approximations for practical calculations, and highlights the role of electron‑electron interactions in limiting its accuracy. The work paves the way for applying higher‑order response methods to realistic materials and for integrating response‑based approaches with existing TDDFT or Green’s‑function techniques.