The Hackfort index Φ_H: construction, first applications and tools

Many chemical and enzymatic rate constants contain a contribution from quantum-mechanical tunnelling through the energy barrier. This paper presents the Hackfort index Φ_H, a dimensionless number that combines the tunnelling contributions of the competing channels of one system, weighted by their shares of the total rate, with two further factors for functional impairment and coherence dynamics. Its core is the tunnelling fraction η = 1 − 1/κ, already in use as the degree of tunnelling of a reaction, where κ is the ratio of the rate constant with and without the tunnelling correction in the same model. When all channels are intact and the coherence term vanishes, Φ_H equals the tunnelling fraction of the total rate. η can be negative; for the one-dimensional Eckart barrier exactly computed values show where, and every application here is checked to lie outside that region. For the oxidation of melatonin by CYP1A1, published DFT barriers and imaginary frequencies give a strongly tunnelling methoxy hydrogen abstraction (η = 0.65 and 0.81 for the two binding orientations, 0.62 and 0.74 with the actual reverse barriers) and a weakly tunnelling aromatic hydroxylation (η = 0.08–0.16). Weighted with the product shares quoted for CYP1A1, with each channel taken from the orientation that produces it, Φ_H = 0.076 if the aromatic hydroxylation is treated as classical, as the analysis protocol prescribes, and Φ_H = 0.22 with its computed tunnelling fraction (0.19–0.24 for an O-demethylation share of 5–15 %); in the second case more than half of the tunnelling-attributable rate comes from the major, weakly tunnelling channel. Reconciling the quoted shares with the DFT barriers requires the minor binding orientation to be populated at about one in a million. The temperature dependence of the tunnelling part of the methoxy isotope effect is predicted; a measurement would need high precision and could be decisive for at most one of the two orientation models. A comparison with QM/MM tunnelling factors for formate dehydrogenase agrees for protium within the spread reported by the source, though not within the standard error of its ten-path average; it cannot distinguish the Eckart kernel from Wigner's correction, and the mass-scaled one-dimensional kernel overstates the tritium isotope effect. Only two of the four factors of Φ_H are evaluated. All numbers are reproducible from the attached code. The upload contains the paper (English; German reading version with the same content) and the reproduction package: Python scripts with self-tests, result files, the analysis protocol written down before the calculation with its SHA-256 digest, the LaTeX sources and checksums of all files. It also contains the OpenTimestamps proof of the original manuscript of 29 March 2026, anchored in Bitcoin block 942854, with a script that checks it.