The exact molecular Kohn-Sham density functional theory of Fromager and Lasorne (Electronic Structure 6, 025002, 2024) maps the fully quantum electron-nuclear molecular problem onto an electronically non-interacting Kohn-Sham molecule whose Born-Huang expansion yields self-consistent "Kohn-Sham beyond Born-Oppenheimer" (KS-bBO) electronic and nuclear equations, together with an exact adiabatic-connection formula for the molecular Hartree-exchange-correlation energy EHxc[Γ,n]\mathcal{E}_{\mathrm{Hxc}}[\Gamma,n] and a simple adiabatic approximation EHxc[Γ,n]≈∫dR Γ(R)EHxc[nR]\mathcal{E}_{\mathrm{Hxc}}[\Gamma,n]\approx\int dR\,\Gamma(R)E_{\mathrm{Hxc}}[n_R] that recycles any standard electronic density functional. The theory is in-principle exact but unexercised: the adiabatic approximation has never been implemented or quantitatively assessed on any continuous-coordinate molecular system, no explicitly nonadiabatic approximation to EHxc[Γ,n]\mathcal{E}_{\mathrm{Hxc}}[\Gamma,n] exists, and the time-dependent extension — in which potentials that divide by the nuclear density Γ(R)\Gamma(R) may become unstable — has not been formulated. The problem posed here is to turn this exact framework into a practically validated computational method: construct and quantitatively validate nonadiabatic density-functional approximations against exact nonadiabatic references on continuous-coordinate molecular model systems, and formulate the time-dependent molecular KS theory with a definite characterization (proof or reproducible demonstration) of the Γ(R)\Gamma(R)-division stability. A 2026 follow-up by the same group has derived the exactly-factorized reformulation and tested a first-order truncation on lattice models, but explicitly leaves all of these components open.

Background

Nearly all routine electronic-structure theory rests on the Born-Oppenheimer (BO) approximation: the electronic Schrödinger equation is solved at fixed nuclear geometry RR, and nuclear motion is treated afterwards. The approximation fails when electronic and nuclear motion genuinely couple, as in proton-coupled electron transfer, passage through conical intersections, and electron transfer, where the state of the electrons cannot be described at any single fixed geometry.

Kohn-Sham density functional theory (KS-DFT) maps the interacting electronic problem onto non-interacting electrons moving in an effective one-body potential; its usefulness hinges on approximations to the exchange-correlation energy functional. Extending this logic to the fully quantum electron-nuclear problem is an old goal. In multicomponent DFT (Kreibich and Gross, Physical Review Letters 86, 2984, 2001), both the electronic density and the nuclear density Γ(R)\Gamma(R) (the probability density of the nuclear geometry RR) are the basic variables. In exact-factorization (EF) DFT (Abedi, Maitra, and Gross, Physical Review Letters 105, 123002, 2010; Requist and Gross, Physical Review Letters 117, 193001, 2016; Li, Requist, and Gross, Journal of Chemical Physics 148, 084110, 2018), the molecular wavefunction is decomposed as Ψ(R,r)=χ(R)ΦR(r)\Psi(R,r)=\chi(R)\Phi_R(r) into a marginal nuclear wavefunction χ\chi and a conditional electronic wavefunction ΦR\Phi_R, and the theory works with the nuclear density and the conditional electronic density nR(r)n_R(r).

Fromager and Lasorne (Electronic Structure 6, 025002, 2024) derived an alternative, in-principle exact "molecular KS-DFT": both densities are mapped onto a fictitious, electronically non-interacting KS molecule in which the electrons still interact with the nuclei but not among themselves. Expanding the KS molecular wavefunction in the Born-Huang expansion — the standard representation of a molecular wavefunction as Ψ(R,r)=∑νχν(R)Φν(r;R)\Psi(R,r)=\sum_\nu\chi_\nu(R)\Phi_\nu(r;R), a sum over electronic eigenfunctions at each geometry weighted by nuclear wavefunction components — yields a self-consistent set of "KS beyond Born-Oppenheimer" (KS-bBO) electronic equations coupled to nuclear equations, from which nonadiabatic couplings — the matrix elements that mix electronic states through nuclear motion and drive transitions between them — follow directly, without a separate response calculation. Two structural features of the exact theory matter here. First, the molecular electronic density n(R,r)n(R,r) that the KS-bBO orbitals must reproduce contains off-diagonal transition-density terms nμνKS(R,r)=⟨ΦμKS(R)∣n^(r)∣ΦνKS(R)⟩n^{\mathrm{KS}}_{\mu\nu}(R,r)=\langle\Phi^{\mathrm{KS}}_\mu(R)|\hat n(r)|\Phi^{\mathrm{KS}}_\nu(R)\rangle weighted by products of different nuclear wavefunction components — genuinely nonadiabatic content. Second, the exact Hartree-exchange-correlation (Hxc) functional EHxc[Γ,n]\mathcal{E}_{\mathrm{Hxc}}[\Gamma,n], which describes the electron-electron repulsion within the molecule as a functional of both densities, enters nuclear-electron and nuclear-nuclear potentials that contain an explicit division by Γ(R)\Gamma(R).

By adapting the adiabatic connection (AC) — the coupling-constant integration technique of electronic DFT, here integrating over the strength λ\lambda of the electron-electron repulsion in the full molecular problem — Fromager and Lasorne derived an exact expression for EHxc[Γ,n]\mathcal{E}_{\mathrm{Hxc}}[\Gamma,n] and, from it, the practical adiabatic density-functional approximation (DFA) EHxc[Γ,n]≈∫dR Γ(R)EHxc[nR]\mathcal{E}_{\mathrm{Hxc}}[\Gamma,n]\approx\int dR\,\Gamma(R)E_{\mathrm{Hxc}}[n_R], where any standard local or semi-local electronic functional may be inserted. The exact AC integrand, however, involves a partially-interacting effective electronic wavefunction that is a linear combination of ground and excited electronic states at each geometry, so there is a priori no reason it should match its adiabatic substitute; the accuracy of the approximation, including what the off-diagonal transition-density content costs, is unknown. In the time-dependent regime, prior numerical experience with the EF formalism (Gossel, Lacombe, and Maitra, Journal of Chemical Physics 150, 154112, 2019) suggests that division by Γ(R)\Gamma(R) may induce instabilities, which the authors explicitly flag as deserving detailed study.

Problem Statement

The problem is to turn the in-principle-exact molecular KS-DFT of Fromager and Lasorne into a practically validated computational method for nonadiabatic molecular quantum dynamics. All components below serve this single objective within the KS-bBO framework and are complete only jointly.

  1. Implement the adiabatic DFA EHxc[Γ,n]≈∫dR Γ(R)EHxc[nR]\mathcal{E}_{\mathrm{Hxc}}[\Gamma,n]\approx\int dR\,\Gamma(R)E_{\mathrm{Hxc}}[n_R], with its associated nuclear-electron and nuclear-nuclear potentials, inside the self-consistent coupled KS-bBO electronic and nuclear equations, and quantitatively assess it on molecular systems with continuous electronic and nuclear coordinates (not only lattice models) against exact nonadiabatic references obtained by direct solution of the full molecular Schrödinger equation. The assessment must quantify errors in nonadiabatic observables — nuclear density distributions, conditional electronic densities, and electron-nuclear correlation energies — relative to both the exact reference and the plain Born-Oppenheimer KS-DFT baseline, and establish what the off-diagonal transition-density content that the adiabatic ansatz omits costs in accuracy.

  2. Design explicitly nonadiabatic approximations to EHxc[Γ,n]\mathcal{E}_{\mathrm{Hxc}}[\Gamma,n] that go beyond the adiabatic ansatz and capture the correlation induced by the nuclear kinetic-energy operator T^n\hat T_{\mathrm{n}} (for example along the exact molecular adiabatic-connection formula), and demonstrate their accuracy on the same class of benchmarks.

  3. Formulate the time-dependent extension of molecular KS-DFT and characterize the stability of the resulting equations: determine whether the potentials that divide by the nuclear density Γ(R)\Gamma(R) remain regular during exact and self-consistent time propagation, or develop the singularities and instabilities suggested by exact-factorization experience.

An answer may deliver these components in one combined development or in sequence. Completion is judged by expert review against the overarching objective: a molecular KS-DFT-based method whose density-functional approximations (adiabatic baseline and explicitly nonadiabatic) are explicit, implemented, and quantitatively validated against exact nonadiabatic references on at least continuous-coordinate molecular model systems, together with a formulated time-dependent theory and a definite statement — proof or reproducible demonstration — on the Γ(R)\Gamma(R)-division stability.

References

  1. E. Fromager and B. Lasorne, "Density functional theory beyond the Born-Oppenheimer approximation: exact mapping onto an electronically non-interacting Kohn-Sham molecule", Electronic Structure 6, 025002 (2024). DOI: 10.1088/2516-1075/ad45d5 (arXiv:2312.15080). https://doi.org/10.1088/2516-1075/ad45d5
  2. L. Dupuy, B. Lasorne, and E. Fromager, "Exactly Factorized Molecular Kohn-Sham Density Functional Theory", Journal of Computational Chemistry 47, e70459 (2026). DOI: 10.1002/jcc.70459 (arXiv:2601.03972). https://doi.org/10.1002/jcc.70459
  3. C. Li, R. Requist, and E. K. U. Gross, "Beyond Born-Oppenheimer Time-Dependent Density Functional Theory", arXiv:2511.09899 (2025). https://arxiv.org/abs/2511.09899
  4. Z. Wang, Y. Li, and C. Li, "Testing exact-factorization-based density functional approximation on a continuous density model", The Journal of Chemical Physics 162, 234104 (2025). DOI: 10.1063/5.0260749. https://doi.org/10.1063/5.0260749
  5. T. Kreibich and E. K. U. Gross, "Multicomponent Density-Functional Theory for Electrons and Nuclei", Physical Review Letters 86, 2984 (2001). DOI: 10.1103/PhysRevLett.86.2984. https://doi.org/10.1103/PhysRevLett.86.2984
  6. A. Abedi, N. T. Maitra, and E. K. U. Gross, "Exact Factorization of the Time-Dependent Electron-Nuclear Wave Function", Physical Review Letters 105, 123002 (2010). DOI: 10.1103/PhysRevLett.105.123002. https://doi.org/10.1103/PhysRevLett.105.123002
  7. R. Requist and E. K. U. Gross, "Exact Factorization-Based Density Functional Theory of Electrons and Nuclei", Physical Review Letters 117, 193001 (2016). DOI: 10.1103/PhysRevLett.117.193001. https://doi.org/10.1103/PhysRevLett.117.193001
  8. C. Li, R. Requist, and E. K. U. Gross, "Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF", The Journal of Chemical Physics 148, 084110 (2018). DOI: 10.1063/1.5011663. https://doi.org/10.1063/1.5011663
  9. G. H. Gossel, L. Lacombe, and N. T. Maitra, "On the numerical solution of the exact factorization equations", The Journal of Chemical Physics 150, 154112 (2019). DOI: 10.1063/1.5090802. https://doi.org/10.1063/1.5090802