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Auxiliary many-body wavefunctions for TDDFRT electronic excited states: Consequences for the representation of molecular electronic transitions

Jérémy Morere, Enzo Monino, Thibaud Etienne

TL;DR

This paper tackles the lack of an intrinsic excited-state ansatz in TDDFRT by analyzing three post hoc auxiliary many-body wavefunctions (AMBW) and their impact on the natural-orbital description of electronic transitions. It develops and compares ζ-type CIS-like AMBW, the Luzanov–Zhikol schemes, and the Subotnik constructions, focusing on how 1-TDM and 1-DDM matrices decompose into detachment/attachment and NDO/NTO representations. A key result is that detachment/attachment matrices in the EOM-TDDFRT framework can be obtained without diagonalizing the 1-DDM, and that NTO-based pictures are not universally applicable outside CIS/TDA, necessitating careful interpretation. The work highlights how different AMBW choices influence the preservation or loss of de-excitation content, affecting transition-picture fidelity and orthogonality of auxiliary states. Overall, the paper clarifies how to coherently connect TDDFRT transitions with CIS-like and other auxiliary pictures, facilitating more reliable analyses of excited-state dynamics and non-adiabatic couplings.

Abstract

This contribution reports the study of a set of molecular electronic-structure reorganization representations related to light-induced electronic transitions, modeled in the framework of time-dependent density-functional response theory. More precisely, the work related in this paper deals with the consequences, for the electronic transitions natural-orbital characterization, that are inherent to the use of auxiliary many-body wavefunctions constructed a posteriori and assigned to excited states - since time-dependent density-functional response theory does not provide excited state ansatze in its native formulation. Three types of such auxiliary many-body wavefunctions are studied, and the structure and spectral properties of the relevant matrices (the one-electron reduced difference and transition density matrices) is discussed and compared with the native equation-of-motion time-dependent density functional response theory picture of an electronic transition - we see for instance that within this framework the detachment and attachment density matrices can be derived without diagonalizing the one-body reduced difference density matrix. The common ''departure/arrival'' wavefunction-based representations of electronic transitions are also extensively discussed.

Auxiliary many-body wavefunctions for TDDFRT electronic excited states: Consequences for the representation of molecular electronic transitions

TL;DR

This paper tackles the lack of an intrinsic excited-state ansatz in TDDFRT by analyzing three post hoc auxiliary many-body wavefunctions (AMBW) and their impact on the natural-orbital description of electronic transitions. It develops and compares ζ-type CIS-like AMBW, the Luzanov–Zhikol schemes, and the Subotnik constructions, focusing on how 1-TDM and 1-DDM matrices decompose into detachment/attachment and NDO/NTO representations. A key result is that detachment/attachment matrices in the EOM-TDDFRT framework can be obtained without diagonalizing the 1-DDM, and that NTO-based pictures are not universally applicable outside CIS/TDA, necessitating careful interpretation. The work highlights how different AMBW choices influence the preservation or loss of de-excitation content, affecting transition-picture fidelity and orthogonality of auxiliary states. Overall, the paper clarifies how to coherently connect TDDFRT transitions with CIS-like and other auxiliary pictures, facilitating more reliable analyses of excited-state dynamics and non-adiabatic couplings.

Abstract

This contribution reports the study of a set of molecular electronic-structure reorganization representations related to light-induced electronic transitions, modeled in the framework of time-dependent density-functional response theory. More precisely, the work related in this paper deals with the consequences, for the electronic transitions natural-orbital characterization, that are inherent to the use of auxiliary many-body wavefunctions constructed a posteriori and assigned to excited states - since time-dependent density-functional response theory does not provide excited state ansatze in its native formulation. Three types of such auxiliary many-body wavefunctions are studied, and the structure and spectral properties of the relevant matrices (the one-electron reduced difference and transition density matrices) is discussed and compared with the native equation-of-motion time-dependent density functional response theory picture of an electronic transition - we see for instance that within this framework the detachment and attachment density matrices can be derived without diagonalizing the one-body reduced difference density matrix. The common ''departure/arrival'' wavefunction-based representations of electronic transitions are also extensively discussed.

Paper Structure

This paper contains 40 sections, 23 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Hypotheses and notations
  3. Preliminary considerations
  4. Basic results from linear algebra
  5. Auxiliary functions
  6. Natural-orbital representation of molecular electronic transitions
  7. One-body reduced difference density matrices
  8. One-body reduced transition density matrices
  9. A quick review of CIS(--like) many-body wavefunctions features
  10. CIS(--like) wavefunctions
  11. Properties of $\zeta$--type ansatzes
  12. $\zeta$-type one-body reduced transition density matrix
  13. V.2.1.A. Structure
  14. V.2.1.B. Singular value decomposition
  15. $\zeta$-type one-body reduced difference density matrix
  16. ...and 25 more sections

Key Result

Lemma III.1

Let ${\normalfont \textbf{A}}$ be an $m\times n$ complex matrix and ${\normalfont \textbf{A}}^\dag$ its adjoint. Let $q$ be the strictly positive integer equal to $\mathrm{min}\left(m,n\right)$. Then, ${\normalfont \textbf{AA}^\dag}$ and ${\normalfont\textbf{A}^\dag\textbf{A}}$ share $q$ non-negativ

Theorems & Definitions (52)

  • Definition III.1
  • Lemma III.1
  • Corollary III.1
  • Lemma III.2
  • Definition III.2: System-$\mathbb{I}$
  • Definition III.3: System-$\mathbb{J}$
  • Definition III.4: $\mathbb{I}$--auxiliary function
  • Example III.1
  • Example III.2
  • Definition IV.1
  • ...and 42 more