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Exactly factorized molecular Kohn-Sham density functional theory

Lucien Dupuy, Benjamin Lasorne, Emmanuel Fromager

Abstract

Fromager and Lasorne [Electron. Struct. 6 025002 (2024)] have recently derived an in-principle exact Kohn-Sham density functional theory (KS-DFT) of electrons and nuclei, where the nuclear density and the (so-called conditional) electronic density are mapped onto a fictitious electronically non-interacting KS molecule. In this work, we apply the exact factorization formalism to the molecular KS wavefunction, thus leading to disentangled (but coupled) marginal and conditional KS equations. We show that, while being equivalent to the original theory, these equations open new perspectives in the practical extension of regular (electronic) KS-DFT beyond the Born-Oppenheimer approximation. The importance and treatment of correlations induced in this context by second-order geometrical derivatives is also discussed.

Exactly factorized molecular Kohn-Sham density functional theory

Abstract

Fromager and Lasorne [Electron. Struct. 6 025002 (2024)] have recently derived an in-principle exact Kohn-Sham density functional theory (KS-DFT) of electrons and nuclei, where the nuclear density and the (so-called conditional) electronic density are mapped onto a fictitious electronically non-interacting KS molecule. In this work, we apply the exact factorization formalism to the molecular KS wavefunction, thus leading to disentangled (but coupled) marginal and conditional KS equations. We show that, while being equivalent to the original theory, these equations open new perspectives in the practical extension of regular (electronic) KS-DFT beyond the Born-Oppenheimer approximation. The importance and treatment of correlations induced in this context by second-order geometrical derivatives is also discussed.
Paper Structure (13 sections, 111 equations, 2 figures)

This paper contains 13 sections, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Brief review of the exact factorization and molecular KS-DFT
  3. Exact factorization of molecular wavefunctions
  4. Molecular KS-DFT
  5. Applying the exact factorization to molecular KS-DFT
  6. Exact formulation
  7. Approximate KS-like one-electron conditional equation
  8. Model system study
  9. Outlook: Correlating the conditional KS wavefunction
  10. Conclusions
  11. Key steps in the derivation of the conditional and marginal equations
  12. Perturbative solutions to the "first-order" approximation
  13. Computation of KS coefficients for the model system study

Figures (2)

  • Figure 1: Top panel - BO ground (deep blue) and first excited (light blue) state PESs as a function of interatomic distance. The beyond-BO exact ground state nuclear density is shown in red. Bottom panel - BO (deep blue) and exact (red) electronic density difference between site 1 and 2 as a function of $R$.
  • Figure 2: Comparison of the "first-order" approximation (dashed lines) and full resolution (full lines) of the conditional KS electronic coefficients' geometrical derivative. The corresponding key equations are Eq. (\ref{['eq:KS_cond_eq_no_second_order_deriv']}), which involves first-order geometrical derivatives only and is equivalent to the one-electron conditional KS Eq. (\ref{['eq:one_elec_XF-based_KS_eq']}), and the exact Eq. \ref{['eq:exact_cond_KS_eq']}, respectively.