Electron charge dynamics and charge separation: A response theory approach

A detailed account with a clear logic that makes the manuscript a useful reference for a reader who would like to develop the idea further, or to compare with other approaches/ideas

While the simplicity of the model makes the analysis of the approach clearer, it may also hide weaknesses that could emerge in more complex systems (see requested changes)

The manuscript meets the criteria for publication. It provides the community with an interesting proposal on how to calculate charge transfer, demonstrating its effectiveness for a model system and reporting both its advantages and limitations. I believe there is "clear potential for multi-pronged follow-up work".

I would like to see some discussion on the model used to test the idea. Simplifications are introduced with respect to real systems. While it may cover a class of systems/problems, are there situations/systems which are not considered? Are these simplifications limiting the assessment of the approach to a particular class of systems or situations?

Eq. 10-11, it is a bit confusing since (apparently) the same LHS leads to different RHS. While it is well-explained in the text, it may be a bit puzzling when simply comparing the two equations. I would recommend that the authors make the LHS explicitly different.
Similarly, for the second-order response, it would be clearer if, e.g a superscript/subscript is added to the Bs, to distinguish the density/current cases (eqs 16-17 vs 18-19).

In Eq. 15, there is a double minus: would that give a plus? I am not sure what conventions are adopted by the journal for repeating signs when breaking an equation into two lines, but I believe the resulting sign should match that of the unbroken equation. Note that in Eq. 17, the authors adopted a different convention for sign repetition when breaking an equation.

Eq. 23, though the model is well-known and the symbols may be self-evident, for completeness, all symbols must be introduced before discussing the parameter choice.

Figure 3: The left/middle plots are difficult to read. Would it be possible to present the plots differently so that the time-dependent plots are legible?

p.13, "The advantage of the response theory is that it is possible to separate oscillating linear response from a less oscillating second order". I do not understand this statement. Is that coming from some particular plot? What does it mean by " less oscillating? Does it refer to the amplitude or to the frequency of the oscillation?

p.17 "under a various weak perturbation"... is it 'various weak perturbations'?

Ask for minor revision

Response theory is a well-known alternative to solving the full time-dependent Schroedinger equation for interacting (or noninteracting) systems, and as such has been very widely used. Questions of the validity of different orders of response are subtle and can be strongly system-dependent. This paper makes a useful contribution to this problem by studying lattice model systems in time-dependent linear and quadratic response, compared to full time propagation.

The authors first carefully work out the response theory expressions up to second order and explain in detail, based on matrix elements, what kinds of processes can be expected to be well described to lowest order, and which processes require higher-order contributions. This is then confirmed by numerical examples. In most cases, quadratic response does an excellent job to describe charge dynamics (i.e., transfer processes), but the validity depends on intensity and frequency. The authors derive a very useful criterion to estimate the validity of quadratic response in relation to resonances. Effects of interactions are also studied and found to enhance the significance of the linear order.

The paper is well written and very clear, and the examples are instructive and make a good case for what the authors want to show. This work shows that there are situations where it may be preferable to use second-order response theory over full time propagation in real materials, which could lead to significant savings in computer time.

The paper can be published essentially as it is. I only have a few minor comments.

• In the introduction: “Linear response is widely used for calculating optical properties such as absorption and electron energy loss”. I would think that electron loss is not an optical phenomenon. Maybe rephrase that sentence.

• After Eq. (8), in the following sentence: “Since it depends only on the overlap between the ground state, an excited state and the perturbation, the linear response is not able to have a non-vanishing value beyond the extention of the ground state…” What do the authors precisely mean by the word “extention” (spelling… should be “extension” or better “extent”)? Is it the spatial range?

• It should be “spatial part” rather than “spacial part”.

• Equation (12), first line of the right-hand side: should the sum be to the left of the square bracket?

• I don’t quite understand the right panel in Figure 3. It shows the Fourier transform of the dipole response to a delta perturbation. Are the blue and dashed red lines calculated using the numerically exact dipole moments? In the text, it is said that this “illustrates how second-order effects dominate away from the absorber region.” How can one see this? I don’t see a comparison between exact dipole spectra and first- or second-order dipole spectra.

• Page 12, somewhere in the middle: a close-parenthesis “)” seems to be misplaced, which alters the meaning of the sentence. I think it should be “(or for the smallest DeltaIJ +- omega excluding resonance) it remains….”

• Page 16, I think it should be “Fig, 7 (right)” instead of “Fig. 7 (right bottom)”.

Publish (easily meets expectations and criteria for this Journal; among top 50%)