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Partial Information Decomposition of Electronic Observables Along a Reaction Coordinate

Kyunghoon Han, Miguel Gallegos

Abstract

A reaction-coordinate--resolved information-theoretic analysis of chemical reactivity is developed using mutual information and partial information decomposition (PID). Along an intrinsic reaction coordinate (IRC), a local empirical distribution is constructed at each position $s$ that couples a coarse-grained geometric progress variable (target) to two electronic readouts (sources), and the joint mutual information $I(T;X,Y)$ is decomposed into redundant, unique, and synergistic contributions using the Williams--Beer PID formalism. In the numerical demonstrations, the target is a binned bond-asymmetry coordinate $ξ=d_{\mathrm{C}\!-\!\mathrm{nuc}}-d_{\mathrm{C}\!-\!\mathrm{LG}}$, while the sources are DDEC6 net atomic charges on the nucleophile and leaving-group centres. Application to three prototypical S$_\mathrm{N}$2 reactions (identity exchange $\mathrm{F^-+CH_3F}$, halide substitution $\mathrm{F^-+CH_3Br}$, and hydroxide substitution $\mathrm{OH^-+CH_3CH_2Br}$) yields compact, symmetry-sensitive signatures of bonding evolution: the identity reaction exhibits mirror-related information profiles with exchange of unique-information contributions between equivalent centres, whereas asymmetric reactions show shifted, centre-specific redistribution among redundancy and synergy as C--X cleavage couples to C--Nu formation. This Supplementary Information provides the formal construction, chemically motivated limiting toy models, a solvable analytic symmetric-transfer model, and the computational protocol used to obtain IRC-resolved PID curves.

Partial Information Decomposition of Electronic Observables Along a Reaction Coordinate

Abstract

A reaction-coordinate--resolved information-theoretic analysis of chemical reactivity is developed using mutual information and partial information decomposition (PID). Along an intrinsic reaction coordinate (IRC), a local empirical distribution is constructed at each position that couples a coarse-grained geometric progress variable (target) to two electronic readouts (sources), and the joint mutual information is decomposed into redundant, unique, and synergistic contributions using the Williams--Beer PID formalism. In the numerical demonstrations, the target is a binned bond-asymmetry coordinate , while the sources are DDEC6 net atomic charges on the nucleophile and leaving-group centres. Application to three prototypical S2 reactions (identity exchange , halide substitution , and hydroxide substitution ) yields compact, symmetry-sensitive signatures of bonding evolution: the identity reaction exhibits mirror-related information profiles with exchange of unique-information contributions between equivalent centres, whereas asymmetric reactions show shifted, centre-specific redistribution among redundancy and synergy as C--X cleavage couples to C--Nu formation. This Supplementary Information provides the formal construction, chemically motivated limiting toy models, a solvable analytic symmetric-transfer model, and the computational protocol used to obtain IRC-resolved PID curves.
Paper Structure (23 sections, 5 theorems, 34 equations, 8 figures, 1 algorithm)

This paper contains 23 sections, 5 theorems, 34 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theory
  3. Partial information decomposition
  4. PID along a reaction coordinate
  5. General structure of the local PID.
  6. Chemical interpretation.
  7. Chemically motivated limiting cases.
  8. Diagnostic value of the profile shapes.
  9. Numerical evaluation of the local PID
  10. Step 1: Arc-length parameterisation (line 1).
  11. Step 2: Discretisation (lines 2--3).
  12. Step 3: Adaptive kernel bandwidth (lines 5--17).
  13. Step 4: Renormalisation and PID evaluation (lines 18--22).
  14. Analytic model: symmetric atom transfer
  15. Valence-bond Hamiltonian
  16. ...and 8 more sections

Key Result

Proposition 1

Let $T$, $X$, $Y$ be discrete random variables. If $T$ is a deterministic function of $X$ (i.e. $H(T \mid X) = 0$) and also a deterministic function of $Y$ ($H(T \mid Y) = 0$), then A sufficient condition is that $X = f(T)$ and $Y = g(T)$ for injective $f,g$, since injectivity makes each map invertible on the support.

Figures (8)

  • Figure 1: Stacked PID decomposition of the joint mutual information (MI) $I(T;\,X,Y)$ along the IRC for the symmetric atom-transfer reaction $\mathrm{H} + \mathrm{H}_2 \to \mathrm{H}_2 + \mathrm{H}$. The blue region is the redundancy $\mathcal{R}$, the red region is the synergy $\mathcal{S}$, and the total joint MI $I(T;X,Y)$ (black curve). Dashed vertical lines mark the two MI peaks (A) and (B), where the charge magnitudes on the two terminals are equal ($|\bar{q}_X| = |\bar{q}_Y|$). Inset atom diagrams show the VB charge distributions at representative points. The unique PID contributions are zeroes along the IRC coordinates in this case.
  • Figure 2: Fractional PID composition $\mathcal{R}/I(T;X,Y)$ and $\mathcal{S}/I(T;X,Y)$ (in percent) along the IRC. At the transition state ($\xi = 0$, marked TS12) the redundancy fraction peaks at ${\sim}\,62\%$; deep in the basins (labels 1 and 2) both fractions converge to $50\%$, the low-SNR Gaussian limit \ref{['eq:gauss']}.
  • Figure 3: Mutual information and PI components along the IRC for the symmetric identity SN2 exchange $\mathrm{F^- + CH_3F \rightleftharpoons CH_3F + F^-}$. The black curve shows $I(T;X,Y)$, where the target $T=\mathrm{bin}(d_X-d_Y)$ is the binned bond-asymmetry coordinate and the sources are the DDEC6 charges $X=q({}^{A}\!F)$ and $Y=q({}^{B}\!F)$ on the two labelled fluorine centres. Coloured stacked areas show the PI components: redundancy $\mathcal{R}$ (blue), unique informations $U_X$ (orange) and $U_Y$ (green), and synergy $S$ (red), satisfying $I(T;X,Y)=\mathcal{R}+U_X+U_Y+S$ pointwise in $s$. Vertical dashed lines mark the symmetric maxima and the insets depict the corresponding mirror-related geometries.
  • Figure 4: Fractional PI components along the IRC for the symmetric identity SN2 exchange $\mathrm{F^- + CH_3F \rightleftharpoons CH_3F + F^-}$ with electronic sources $X=q({}^{A}\!F)$ and $Y=q({}^{B}\!F)$. The curves show the PI components normalised by the total joint mutual information, i.e. $\mathcal{R}/I(T;X,Y)$ (blue), $U_X/I(T;X,Y)$ (orange), $U_Y/I(T;X,Y)$ (green), and $S/I(T;X,Y)$ (red), expressed as percentages and satisfying $\mathcal{R}+U_X+U_Y+S=I(T;X,Y)$ pointwise in $s$. Representative geometries are shown at the reactant and product extremes (1 and 2), near the mutual-information maxima (1a and 2a), in the intermediate three-centre regime (1b and 2b), and at the transition state (TS12, $s=0$). The symmetry of the identity exchange is reflected in the exchange of unique-information fractions under $s\mapsto -s$ and in the near-even profiles of redundancy and synergy.
  • Figure 5: Mutual information and PI components along the IRC for the asymmetric SN2 reaction $\mathrm{F^- + CH_3Br \rightarrow CH_3F + Br^-}$. The black curve shows $I(T;X,Y)$ with $T=\mathrm{bin}(d_X-d_Y)$ and sources $X=q(\mathrm{Br})$ and $Y=q(\mathrm{F})$ (DDEC6 charges). Coloured stacked areas show redundancy $\mathcal{R}$ (blue), unique informations $U_X$ (orange) and $U_Y$ (green), and synergy $S$ (red), satisfying $I(T;X,Y)=\mathcal{R}+U_X+U_Y+S$ pointwise in $s$. Dashed vertical lines mark representative maxima on the reactant- and product-side branches; insets depict the corresponding geometries.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Proposition 1: Deterministic-source redundancy
  • proof
  • Remark 1: Determinism and coarse-graining
  • Proposition 2: Reaction symmetry
  • proof
  • Remark 2: Bin construction for symmetric reactions
  • Remark 3: Charge-sum constraint
  • Proposition 3: Equal unique information for signed target
  • proof
  • Proposition 4: Vanishing unique information for even target
  • ...and 3 more