Marcus Electron Transfer & Inverted Region Simulator

Potential Energy Surfaces G(q)

Reactant G_R

Product G_P

Barrier ΔG^‡

Marcus Rate vs. Driving Force (-ΔG°)

k_ET operating point

Model Parameters

Driving Force (ΔG°) -0.80 eV

Reorganization (λ) 0.80 eV

Coupling (H_DA) 0.020 eV

Temperature (T) 298.15 K

Experimental Presets

Kinetic Telemetry

Activation Barrier ΔG^‡

0.0000 eV

0.00 kJ/mol

Rate Constant k_ET

7.51e12 s⁻¹

Half-life (t_1/2)

0.092 ps

Crossing Coordinate q^‡

0.000

• Classical Barrier: ΔG^‡ = (λ + ΔG°)² / (4λ)

• Golden Rule: k_ET = (2π/ħ) |H_DA|² (4πλk_BT)^-1/2 exp(-ΔG^‡ / k_BT)

• Inverted Region: When |-ΔG°| > λ, rate drops as exothermicity increases.

Marcus electron transfer: why the inverted region exists

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Read the explanation

The classical nonadiabatic Marcus lab uses two free energy curves on a collective nuclear coordinate. The donor is lambda q squared. The acceptor is lambda times q minus one squared, plus reaction free energy. Fix lambda at point eight electron volts and start with reaction free energy zero. Their crossing is q point five, and its barrier is point two electron volts above the donor minimum. Hold that initial configuration on the shared axes. Now lower reaction free energy. The gold acceptor curve moves down. Watch the green crossing move left along the blue donor curve while the vertical green barrier line shrinks. At the endpoint, reaction free energy is minus point four electron volts. The crossing is now q point two five and the barrier is point zero five electron volts. The crossing formula is one half times one plus reaction free energy divided by lambda. The barrier is the square of lambda plus reaction free energy, divided by four lambda. These axes stay fixed throughout. This coordinate represents collective nuclear reorganization, not an electron travelling down a literal track. Fix lambda at point eight electron volts. Begin with reaction free energy minus point four electron volts. The barrier is point zero five electron volts: this horizontal bar is three hundred pixels long at six thousand pixels per electron volt. Now increase the driving force toward the activationless configuration. Watch the bar shrink as the computed barrier falls. At reaction free energy minus point eight electron volts the exact mathematical barrier is zero. Hold the zero-length bar. This is where the driving force equals the reorganization energy. Now increase the driving force beyond that point. The computed barrier rises again, and the bar grows. At reaction free energy minus one point two electron volts the barrier returns to point zero five electron volts, the same three hundred pixel length as the initial case. This is the Marcus inverted region. More driving force no longer means a lower barrier. The source badge has a small tolerance for its activationless label, but nearby mathematical barriers are not identically zero. At two hundred ninety-eight point one five kelvin, thermal energy k B T is about point zero two five seven electron volts. Keep temperature, lambda and weak coupling fixed. Start with a point zero five electron volt barrier. The rate normalized to the same-parameter barrierless reference is point one four two eight three. At six hundred pixels per unit normalized rate the blue bar is about eighty-five point seven pixels long; the faint reference remains six hundred pixels. Now lower the barrier toward zero. The exponential suppression weakens, so the predicted rate rises and the blue bar extends. At zero barrier the normalized rate is one. Hold the blue bar at the full reference length. Now raise the barrier back to point zero five electron volts. Exponential suppression increases and the blue bar contracts. The endpoint normalized rate is again point one four two eight three. Equal barriers at fixed temperature and coupling give equal predicted rates. The full expression also contains coupling squared and the inverse square root of lambda k B T. The illustrative presets are not measured rates or a guarantee that the nonadiabatic approximation remains valid at strong coupling. Use the original controls and export the actual JSON kinetics and CSV rate curve to inspect those assumptions.