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 †

Self-consistent-field (SCF) approximations formulated using Hartree−Fock (HF) or Kohn−Sham density-functional theory (KS-DFT) have the potential to yield multiple solutions. However, the formal relationship between multiple solutions identified using HF or KS-DFT remains generally unknown. We investigate the connection between multiple SCF solutions for HF or KS-DFT by introducing a parameterized functional that scales between the two representations. Using the hydrogen molecule and a model of electron transfer, we continuously map multiple solutions from the HF potential to a KS-DFT description. We discover that multiple solutions can coalesce and vanish as the functional changes, forming a direct analogy with the disappearance of real HF solutions along a change in molecular structure. To overcome this disappearance of solutions, we develop a complex-analytic extension of DFTthe "holomorphic DFT" approachthat allows every SCF stationary state to be analytically continued across all molecular structures and exchange−correlation functionals.

We recently introduced a method called conveyor belt (CB) thermodynamic integration (TI) for the calculation of alchemical free-energy differences based on molecular dynamics simulations. In the present work, the CBTI approach is generalized to conformational free-energy changes, i.e., to the determination of the potential of mean force (PMF) along a conformational coordinate ξ of interest. The proposed conveyor belt umbrella sampling (CBUS) scheme relies on the parallel simulation of K replicas k = 0,1, ..., K – 1 of the system, with K even. For each replica k, the instantaneous value of ξ is restrained to an anchor value λ k . The latter anchor points are equally spaced along a forward-turn-backward-turn path (i.e., a CB) between two extreme values defining the ξ-range of interest. The rotation of the CB is controlled by a variable Λ (range from 0 to 2π) which evolves dynamically along the simulation. The evolution of Λ results from the forces exerted by the restraining potentials on the anchor points, taken equal and opposite to those they exert on the replicas. Because these forces tend to cancel out along the CB, the dynamics of Λ is essentially diffusive, and the continuous distribution of ξ-values sampled by the replica system is automatically close to homogeneous. The latter feature represents an advantage over direct counting (DCNT) and traditional umbrella sampling (TRUS), shared to some extent with replica-exchange umbrella sampling (REUS). In this work, the CBUS scheme is introduced and compared to the three latter schemes in the calculation of 45 standard absolute binding free energies. These correspond to the binding of five alkali cations to three crown ethers in three solvents. Different free-energy estimators are considered for the PMF calculation, and the calculated values are also compared to those of a previous study relying on an alchemical path, as well as to experimental data.