Transmission Coefficients and Quantum Tunnelling

Introduction

Ideal transition-state theory counts thermal trajectories that cross a barrier surface and assumes they continue to products. Real trajectories may turn back, and quantum particles may pass through a barrier even when classical mechanics says they lack enough energy to go over it. These effects pull a rate prediction in different directions. A transmission coefficient makes the comparison explicit, but the word can cover different corrections in different texts, so its definition must accompany any numerical value.

Core explanation

Write k actual = κk TST for a stated TST baseline. For a classical trajectory calculation using the same dividing surface, κ recross can be interpreted as a successful-product flux divided by the forward-crossing flux. Recrossing makes this fraction no greater than one under that definition. If 100 trajectories are counted crossing forward and 80 ultimately commit to products, κ recross = 0.80. The value depends on the surface, temperature and surrounding dynamics; choosing a surface that minimises recrossing can improve a TST estimate. The IUPAC transition-state definition describes the fraction of assemblies reaching the saddle that pass through to products as a transmission coefficient.

Quantum tunnelling is different. A particle's quantum wavefunction can have nonzero transmission probability through a region where its energy is below the classical barrier. In chemistry, hydrogen, proton and hydride transfers are common cases where nuclear quantum effects can be important, though heavy-atom tunnelling can occur under suitable conditions. Tunnelling probability generally increases for a lighter effective mass and a narrower barrier at a given energy. Barrier height matters too, but height alone cannot predict tunnelling; a high, very narrow barrier and a lower, broad barrier may give different quantum transmission probabilities.

A simple one-dimensional barrier picture makes the mass and width dependence intuitive. Within a classically forbidden region, the wave amplitude decays roughly exponentially with an integral involving √[2m(V(x)−E)] divided by the reduced Planck constant. Increasing the distance across the barrier or increasing effective mass strengthens that exponential suppression. Real molecular reactions are multidimensional, so the effective mass changes with the coupled motions and the tunnelling path need not follow the classical minimum-energy path exactly. The formula is a guide to trends, not a substitute for a full calculation.

Authors may represent tunnelling with a separate factor κ tun greater than one relative to a classical over-barrier TST prediction, or combine recrossing and tunnelling into one overall κ. In the combined convention, κ need not be bounded above by one, because quantum enhancement can outweigh recrossing loss. Therefore the statement “the transmission coefficient is always less than one” is only valid for a specified classical recrossing fraction. The ACS account of ensemble-averaged variational TST discusses both recrossing and quantum reaction-coordinate motion in a combined coefficient.

Temperature dependence offers clues. Classical thermal crossing often falls steeply as temperature decreases. Tunnelling may sustain a larger low-temperature rate than a simple over-barrier Arrhenius extrapolation predicts. This can make ln k versus 1/T curve or flatten at sufficiently low temperature. However, curvature alone does not prove tunnelling: a changing rate-controlling step, activation heat capacity, conformer populations or diffusion effects can also produce it. Isotope substitution is a valuable additional probe because replacing H with D changes nuclear mass without directly changing electronic charge. Even then, isotope effects also arise from zero-point energy, so the next page treats them carefully. A primary RSC study of a low-temperature hydrogen-transfer system reports non-Arrhenius behaviour associated with tunnelling, while the wider lesson remains to use multiple lines of evidence.

Recrossing is especially important when a saddle-point geometry does not correspond to a clean commitment boundary. In solution, solvent motion can push a configuration back toward reactants after it passes a geometric saddle. An enzyme's active-site fluctuations can similarly affect the path through state space. Quantum tunnelling asks a different question: how much product flux can pass through classically inaccessible regions? A rate calculation may need both effects, plus a reliable potential-energy surface. Adjusting an arbitrary κ to fit one experimental number does not independently validate a proposed molecular mechanism.

Step-by-step reasoning

1. State the baseline TST prediction, including its dividing surface and whether it is classical. 2. Define the chosen κ: recrossing-only, tunnelling-only or combined. 3. For recrossing, compare successful product trajectories with counted forward crossings at identical conditions. 4. For tunnelling, inspect effective mass, barrier width and energy distribution, not just barrier height. 5. Compare temperature trends and isotope effects with alternative explanations such as multiple pathways. 6. Report how the correction was obtained instead of treating κ as an unexplained fitting constant.

Visual explanation

Draw a reaction-coordinate energy curve with reactants on the left and products on the right. A solid arrow climbs over the barrier; a dotted quantum path passes through its interior below the top. At the summit, show another classical arrow crossing toward products and curling back to reactants. Label the dotted path “tunnelling” and the returning path “recrossing.” Below, draw two ln k versus 1/T traces: a straight high-temperature extrapolation and a low-temperature curve above it. Annotate that the curve is a clue to investigate, not a unique fingerprint.

Real-world analogy

A security gate may count everyone who steps over a line, although some step back before entering: that resembles recrossing. A separate underground passage would let people arrive without crossing the visible gate: that resembles tunnelling only as a counting analogy. Quantum tunnelling is not a hidden classical tunnel cut through matter; it is a wave-mechanical probability for transmission through an energetic barrier.

Real-world example

Suppose a hydrogen-transfer reaction has a calculated classical TST rate that falls rapidly on cooling, but the measured low-temperature rate remains appreciable. Researchers could test deuterated reactants, calculate a multidimensional barrier and compare predicted product rates across temperatures. A larger-than-expected H/D contrast and matching quantum calculations would strengthen a tunnelling interpretation. They must still rule out an alternate mechanism or temperature-dependent reactant state population.

Why?

Why does lowering temperature sometimes make tunnelling more prominent relative to classical passage? Thermal populations with enough energy to climb over a barrier become rarer as temperature falls. A quantum pathway from lower-energy states may also slow or change, but it can decline less steeply. Its fraction of total product flux can therefore grow. This does not imply every tunnelling reaction becomes temperature-independent, nor that cooling always raises the absolute rate.

Common misconception

“A transmission coefficient larger than one violates probability.” If κ is defined as a classical recrossing fraction, it should not exceed one. But an overall correction relative to a classical TST baseline may include quantum tunnelling and can exceed one without any single-event probability exceeding unity. Another misconception is that curved Arrhenius data alone prove tunnelling. Several non-quantum mechanisms can cause curvature, so isotope, pressure, structural or computational evidence is needed.

Worked example

An ideal classical TST calculation gives k TST = 5.0 × 10³ s⁻¹ at a stated temperature. Trajectories show that 60% of counted forward crossings commit to products, so the recrossing-corrected classical rate is 0.60 × 5.0 × 10³ = 3.0 × 10³ s⁻¹. A separate quantum calculation predicts a tunnelling enhancement of 2.0 relative to that classical committed flux. Under a convention that multiplies these independent approximations, k ≈ 2.0 × 0.60 × 5.0 × 10³ = 6.0 × 10³ s⁻¹. The combined factor relative to ideal classical TST is 1.2; it is not a probability and does not mean 120% of trajectories crossed successfully.

Quick check

1. What is the classical recrossing fraction if 72 of 90 first forward crossings reach products? Answer: 72/90 = 0.80 for that chosen surface and condition.

Exam focus

Define κ before using it. Distinguish return of a counted classical crossing from quantum passage through a forbidden barrier. State that the classical recrossing fraction is at most one, while a combined correction to a classical baseline can exceed one. Describe barrier width and effective mass as well as height. Treat a non-Arrhenius curve as evidence requiring further testing.

Advanced insight

Multidimensional tunnelling paths may “cut a corner” relative to the classical minimum-energy path, reducing action even if the geometric distance looks longer in one coordinate. Variational TST changes the dividing surface to reduce recrossing, while semiclassical or quantum methods estimate transmission through the barrier. In enzyme reactions, conformational ensembles can give different barrier shapes and isotope effects, so a single static structure may not represent the measured rate. Modern modeling averages over these states and compares the predicted temperature and isotope trends, not only a single rate constant.

Summary

Recrossing reduces successful product formation after a classical dividing-surface crossing; tunnelling permits quantum transmission through a classically forbidden barrier. A recrossing-only κ is a fraction no greater than one, while a combined correction relative to classical TST may be larger. Tunnelling depends on mass, barrier width and energy, and may produce unusual low-temperature behaviour. Curved temperature plots motivate investigation but do not independently prove a quantum mechanism.

Practice questions

1. A classical TST rate is 100 s⁻¹ and κ recross = 0.75. Find the recrossing-corrected rate. Answer: 75 s⁻¹, because 0.75 × 100 = 75.

2. Why does a wider barrier usually suppress tunnelling at the same particle energy and height? Answer: The wave amplitude decays over a longer forbidden region, making the transmission probability smaller.

3. Can an overall tunnelling-corrected factor relative to classical TST exceed one? Answer: Yes. It is a ratio of rate predictions, not a single-event probability, and quantum enhancement can outweigh classical recrossing loss.

4. Name one non-tunnelling cause of a curved Arrhenius plot. Answer: A change in rate-controlling mechanism, temperature-dependent activation heat capacity or changing conformer populations can cause curvature.

5. Which is generally more tunnelling-prone at the same barrier shape, H or D transfer? Answer: H transfer, because its lower effective nuclear mass gives less exponential suppression, all else equal.