Energetics and partition function of H$_3^+$ molecular ion

Full $NVT$ quantum statistics of the H$_3^+$ ion is simulated at low densities using the path integral Monte Carlo approach. For the first time, the molecular total energy, partition function, free energy, entropy and heat capacity are evaluated in temperatures relevant for planetary atmospheric physics. Temperature and density dependent dissociation recombination reaction balance of the molecule and its fragments above 4000 K is described, and also, the density dependence of thermal ionization above $10 000$ K is demonstrated. We introduce a new well-behaving analytical model for the molecular partition function of the H$_3^+$ ion for the temperature range below dissociation and fit the parameters to the energetics from our simulations. The approach presented here can be regarded as an extension of the traditional {\it ab initio} quantum chemistry beyond the Born–Oppenheimer approximation to description of nonadiabatic phenomena, and even further, account of nuclear quantum dynamics.

💡 Research Summary

This paper presents a comprehensive quantum‑statistical study of the H₃⁺ molecular ion using the path‑integral Monte Carlo (PIMC) method. Simulations were performed at low number densities (≈10⁻⁶–10⁻⁴ g cm⁻³) over a wide temperature range from 160 K to 15 000 K, thereby covering regimes where the ion is stable, partially dissociated, and thermally ionized. By treating all five particles (three protons and two electrons) fully quantum mechanically, the authors avoid the Born–Oppenheimer approximation and explicitly include non‑adiabatic and nuclear‑quantum effects.

Methodology
A single H₃⁺ ion was placed in a cubic periodic cell with three different volumes to emulate three densities. The Coulomb interaction was handled with the pair‑approximation action, and the Metropolis algorithm with bisection moves sampled the configuration space. The imaginary‑time step was fixed at τ = 0.03 E_H⁻¹, and the Trotter number M was adjusted to satisfy β = Mτ/k_B for each temperature. Energies were obtained via the virial estimator, while thermodynamic quantities were derived from the partition function Z = Tr e^{‑βĤ} using standard relations (F = ‑k_B T ln Z, S = ‑∂F/∂T, C_V = ∂⟨E⟩/∂T).

Key Results

1. Total Energy and Dissociation Balance

   • Below ~4 000 K the ion remains essentially non‑dissociated; the total internal energy (excluding the center‑of‑mass kinetic term) matches the non‑adiabatic zero‑K ground‑state energy (≈‑1.169 E_H).
   • Between 4 000 K and 10 000 K the system progressively dissociates into fragments H₂ + H⁺, H₂⁺ + H, and 2H + H⁺. Energy histograms reveal distinct peaks corresponding to each fragment, and the relative heights depend strongly on density. Higher densities suppress dissociation, while lower densities favor it.
   • Above ~10 000 K, thermal ionization of hydrogen atoms becomes evident; the onset temperature shifts upward with increasing density (≈15 000 K for the highest density considered).

2. Analytical Partition‑Function Model

   • For the low‑temperature regime (0–3 900 K) the authors fitted the internal‑energy data to the functional form
     ⟨E⟩ = k_B T²