Revisiting nuclear tunnelling in the aqueous ferrous–ferric electron transfer

Abstract

The aqueous ferrous-ferric system provides a classic example of an electron-transfer process in solution. There has been a long standing argument spanning more than three decades around the importance of nuclear tunnelling in this system, with estimates based on Wolynes theory suggesting a quantum correction factor of 65, while estimates based on a related spin-boson model suggest a smaller factor of 7-36. Recently, we have shown that Wolynes theory can break down for systems with multiple transition states leading to an overestimation of the rate, and we suggest that a liquid system such as the one investigated here may be particularly prone to this. We re-investigate this old yet interesting system with the first application of the recently developed golden-rule quantum transition-state theory (GR-QTST). We find that GR-QTST can be applied to this complex system without apparent difficulties and that it gives a prediction for the quantum rate 6 times smaller than that from Wolynes theory. The fact that these theories give different results suggests that although it is well known that the system can be treated using linear response and therefore resembles a spin-boson model in the classical limit, this approximation is questionable in the quantum case. It also intriguingly suggests the possibility that the previous predictions were overestimating the rate due to a break down of Wolynes theory.Semiclassical instanton rate theory 22-24 predicts the tunnelling rate and mechanism via locating the optimal tunnelling pathway (the instanton) defined by a stationary-action principle. Based on a similar first-principles derivation as in the normal regime, 25,26 instanton theory has been extended to treat electron-transfer reactions 27-29 in both the normal and inverted regimes. 30 It has the most rigorous derivation of the methods discussed in this paper, shows excellent agreement with exact methods on model systems and is well suited for gas-phase electron-transfer reactions. How-J o u r n a l N a me , [ y e a r ] , [ v o l . ] , 1-12 | 1 arXiv:1912.09811v2 [physics.chem-ph] 3 Feb 2020 2 | 1-12 J o u r n a l N a me , [ y e a r ] , [ v o l . ] , aqueous ferrous-ferric electron transfer. AbstractIn this supporting information we provide further details of the simulations reported in the manuscript and additional discussions in support of the conclusions reached. * These authors contributed equally †

An improved path-integral method for golden-rule rates

“…The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp ferrous-ferric electron transfer exhibits a range of qualitatively different tunneling pathways, leading to a breakdown of Wolynes theory and also the assumptions of linear response inherent in the Marcus picture of electron transfer. 62 The conclusions of this study will turn out to be entirely consistent with what we have found here.…”

Confirming the role of nuclear tunneling in aqueous ferrous–ferric electron transfer

“…Section IV analyzes how a lack of size consistency in GR-QTST affects its prediction of the ferrous-ferric rate and investigates the recent suggestion that Wolynes theory breaks down for this reaction. 80 Section V reassesses the validity of the spinboson mapping in the light of the preceding sections, and Section VI concludes the paper.…”

“…79 On applying GR-QTST to a fully atomistic model of ferrous-ferric electron transfer, they found a tunneling enhancement around 6 times smaller than that of Wolynes theory and used this to argue that Wolynes theory may be overestimating the rate due to the presence of multiple tunneling pathways. 80 This was then used to argue that even the linear response assumption may break down in the presence of tunneling for ferrous-ferric electron transfer, casting doubt not only on the quantitative accuracy but also the qualitative validity of mapping to the spin-boson model. 80 However, as was acknowledged by Richardson and co-workers, they were unable to conclusively prove that it was Wolynes theory that was breaking down for this system, and hence, they could not rule out an error on the part of GR-QTST.…”

“…80 This was then used to argue that even the linear response assumption may break down in the presence of tunneling for ferrous-ferric electron transfer, casting doubt not only on the quantitative accuracy but also the qualitative validity of mapping to the spin-boson model. 80 However, as was acknowledged by Richardson and co-workers, they were unable to conclusively prove that it was Wolynes theory that was breaking down for this system, and hence, they could not rule out an error on the part of GR-QTST.…”

A general non-adiabatic quantum instanton approximation

“…Imaginary time path integral techniques, 5 which can accurately capture zero point energy and tunneling effects and yet scale only linearly with system size, have therefore become popular for studying more complex reactions. While there now exist several such methods that are routinely used to study electronically adiabatic reactions, [6][7][8][9][10][11][12] as well as theories that can be applied to electronically non-adiabatic reactions in the golden rule limit, [13][14][15][16][17][18][19][20] the development of accurate path integral techniques for more general electronically non-adiabatic reactions with intermediate electronic coupling strengths is still a very active area of research. [21][22][23][24][25][26][27][28][29][30][31][32][33] When developing new path integral methods for studying nonadiabatic systems, a common approach has been to generalize a pre-existing adiabatic method.…”