The Entropic Barrier around the Conical Intersection Seam

TL;DR

This work demonstrates that, under classical nuclear dynamics, exact degeneracy sampling at conical intersections is statistically forbidden due to an entropic barrier in the free energy as the adiabatic energy gap vanishes, $F(\Delta E_{IJ}) \to \infty$ as $\Delta E_{IJ} \to 0$. Using a linear vibronic coupling (LVC) model, the authors derive a closed-form expression showing the marginal density $\rho(\Delta E)$ vanishes at zero gap, while molecular-dynamics simulations on CH2NH2+ in the S1 state confirm trajectories approach but do not reach the CI seam. This reconciles why mixed quantum-classical methods reproduce nonadiabatic transitions without sampling exact degeneracies and clarifies that the CI seam acts as an entropic barrier in the branching-plane of nuclear motion. The findings have implications for interpreting nonadiabatic dynamics, guiding near-CI sampling strategies, and highlighting potential quantum-nuclear effects that could partially lift the barrier in more complete treatments.

Abstract

Conical intersections (CIs) are seen as the main mediators of nonadiabatic transitions; yet, mixed quantum-classical (MQC) simulations rarely, if ever, sample geometries with exactly degenerate electronic energies. Here we show that this behavior arises from a fundamental statistical-mechanical constraint. Using a linear vibronic coupling model, we derive the free energy along the adiabatic energy gap and demonstrate analytically that as the gap approaches zero, an infinite free-energy barrier arises around the CI seam. Molecular dynamics simulations of the methaniminium cation on the S$_1$ surface confirm this prediction: trajectories can approach regions with small adiabatic gaps, but never reach the CI seam, even if the CI corresponds to a region of lowest potential energy. These results clarify why MQC methods successfully capture nonadiabatic behavior without sampling exact degeneracies and agree with recent findings that classical trajectories can sense the presence of CIs without visiting them.

Figures (5)

• Figure 1: Conical intersection between the ground and first excited singlet states of the methaniminium cation (CH$_2$NH$_2^+$). (a) Optimized geometry of the minimum-energy conical intersection (MECI) corresponding to the S$_1$/S$_0$ degeneracy point. (b) Three dimensional rendering of the adiabatic potential-energy surfaces in the vicinity of the MECI, shown as cut along the branching-plane coordinates $g$ and $h$ (gradient-difference and nonadiabatic-coupling directions) to illustrate the characteristic double-cone topography of the CI.
• Figure 2: Histogram of the adiabatic energy gap $\Delta E$ between the ground and first excited singlet states from simulations of CH$_2$NH$_2^+$. In the log–log representation, the low-$\Delta E$ region exhibits a power-law behavior; the fitted slope $m$ corresponds to the polynomial exponent governing the dependence of the density on $\Delta E$.
• Figure 3: Free energy $F$, internal energy $U$, and entropy $T S$ profiles from the S$_1$ dynamics of CH$_2$NH$_2^+$ as a function of the adiabatic energy gap $\Delta E$. The right panel shows a magnified view of the small-gap region. The minimum of the free-energy curve as well as $U(0)$ were shifted to zero for clarity. All energies are given in electron volts. The shaded regions indicate the standard deviation between curves obtained by splitting the 196 trajectories into batches of 28 each. The solid line represents the average of the seven curves computed this way.
• Figure 4: Free-energy profile $F(z)$ as a function of (left) the minimal RMSD from the MECI geometry and (right) the difference between the S$_1$ energy and the S$_1$ energy at the MECI.
• Figure 5: Two-dimensional potential of mean force (PMF) along the projections onto the gradient-difference coordinate $g$ and the nonadiabatic-coupling coordinate $h$. The dashed gray lines mark $g = 0$ and $h = 0$, whose intersection corresponds to the MECI geometry.