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Manifold curvature and Ehrenfest forces with a moving basis

by Jessica F. K. Halliday, Emilio Artacho

Submission summary

As Contributors: Emilio Artacho · Jessica Halliday
Preprint link: scipost_202110_00031v1
Date accepted: 2021-11-15
Date submitted: 2021-10-19 17:22
Submitted by: Artacho, Emilio
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approach: Theoretical


Known force terms arising in the Ehrenfest dynamics of quantum electrons and classical nuclei, due to a moving basis set for the former, can be understood in terms of the curvature of the manifold hosting the quantum states of the electronic subsystem. Namely, the velocity-dependent terms appearing in the Ehrenfest forces on the nuclei acquire a geometrical meaning in terms of the intrinsic curvature of the manifold, while Pulay terms relate to its extrinsic curvature.

Published as SciPost Phys. 12, 020 (2022)

Author comments upon resubmission

Dear Editor

Please find here the revised paper. We have addressed the points raised by the referees and yourself. As stated in our earlier replies to the referees, we amend the paper to include the clarifications requested, plus the ones that make clear in the paper what the referees were asking. We hope it is now acceptable for publication and will be happy to send the source material for processing.


Emilio Artacho

List of changes

The uploaded paper contains all changes marked in red. Namely:
- page 2 after eq 1: Clarification on N_n referring to number of nuclei, while the R_j refer to all their coordinates (j=1 .. 3N_n)
- page 2 after eq 2: the time-dependent Omega space is always a subspace of the infinite dimensional ambient Hilbert space.
- page 3 1st par: clarification it could be done for orthonormalised bases, but numerically inconvenient, hence leaving general formalism.
- page 3 before eq 5: better definition of f(r) and of vector r. And of vector v as an instantaneous nuclear velocity.
- Section 2.3 divided into two subsections, 2.3.1 and 2.3.2, given that the first one had grown, and so becoming clearer.
- Subsection 2.3.1. (page 4) now includes connection with geometric phases (and cites, added to references).
- page 5 before eq. 9: reminder and citation for covariant derivative of Hamiltonian tensor, towards clarifying eq. 9
- page 5 above Pulay forces: clearer definition of adiabatic states chi^a_mu
- page 5 after eq 10: Clarifying relation between Eq. 10 and the following one
- page 5 after last eq: reminder of Omega in ambient Hilbert space
- Page 5 end (last red bit): adding for clarification on why for a flat Omega space Pulay terms don't appear
- page 6 before conclusions; clarification on extrinsic curvature sufficient for non-zero Pulay even if not intrinsic.
- page 6 end of par 1: add brief mention of gauge field interpretation analogous to geometric-phase theory.
- page 6 last par of conclusions: remark on the question raised by the editor. Reference added.

More equations were numbered in this version, for handles on new clarifying text (new eq numbers)

Reports on this Submission

Anonymous Report 1 on 2021-11-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202110_00031v1, delivered 2021-11-01, doi: 10.21468/SciPost.Report.3775


The Authors of the Manuscript titled "Manifold curvature and Ehrenfest forces with a moving basis" have addressed all the points suggested in my initial report.
The similarities and differences between their formalism and the "gauge theory of molecular physics" of Koizumi, Hiroyasu, et al. have been discussed and Section 2.4 clarified.
I particularly appreciate the Authors' comment about the possible orthonormalisation of the basis set. That comment and the new conclusions further clarify the scope of their work, especially in view of numerical applications.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
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Claudio Attaccalite  on 2021-11-02  [id 1901]

suggestion for further work

In order to make the Referee/Editorial Process more transparent I put in this comment the additional questions I asked to the authors:

1) Does this new formulation will help practical calculations with moving basis? In the sense, I saw different calculations in the past that were obligated to add additional fictitious orbitals to achieve convergence.

2) Does this new formulation will help this convergence? and second will this new formulation allow to device better integrators for the the real-time dynamics?

the reply to these question is included in the new submission and I'm satisfied of the authors response.