Rate models compared

Semiclassical KIE — no tunneling

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with tunneling

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The upper number is k_H/k_D from zero-point energy alone, at the X–H stretch frequency and temperature set below. The classical prefactor k_BT/h carries no isotope mass at all, so it is exp[(ZPE_H − ZPE_D)/k_BT] — independent of the barrier height, its width and the asymmetry. The lower number adds tunneling through the Eckart barrier.

Preset

Barrier

Modes and temperature

At 298 K

Crossover temperature T_c = ħ|ω^‡|/2πk_B (set by the barrier frequency) is where the reaction stops behaving classically. Above it, passage over the barrier top dominates and a simple correction like Wigner's is a good approximation. Below it, the dominant path tunnels well beneath the top — Wigner then badly underestimates the rate, and only the full Eckart transmission captures it. It only appears as the dashed red line when it falls inside the T range set below — the "Above the crossover" preset keeps T_c off the bottom of the plot on purpose, since every plotted temperature is meant to sit above it.

Rate expression

k = κ (T) \frac{k_{B} T}{h} \exp [- \frac{V_{eff}^{‡}}{k_{B} T}]

model	barrier	κ(T)
classical TST	$V^{‡}$	$1$
ZPE-corrected	$V^{‡} - \frac{1}{2} ℏ ω_{XH}$	$1$
Wigner	$V^{‡} - \frac{1}{2} ℏ ω_{XH}$	$1 + \frac{1}{24} {(\frac{ℏ \| ω^{‡} \|}{k_{B} T})}^{2}$
Eckart	$V^{‡} - \frac{1}{2} ℏ ω_{XH}$	thermal average of P(E)

V^‡ is the barrier height and ω_XH the X–H stretch, whose zero-point energy is lost at the transition state. P(E) is the exact quantum transmission probability through the Eckart barrier; a classical step at the barrier top would give κ = 1. The potential is the same for every isotope; the mass enters through ω_XH and |ω^‡|, both of which scale as 1/√m, and through P(E) itself.

Rate models compared Transition state theory and its tunneling corrections