Free Energies of Quantum Particles: The Coupled-Perturbed Quantum Umbrella Sampling Method

Abstract

We introduce a new simulation method called Coupled-Perturbed Quantum Umbrella Sampling that extends the classical umbrella sampling approach to reaction coordinates involving quantum mechanical degrees of freedom. The central idea in our method is to solve coupled-perturbed equations to find the response of the quantum system's wave function along a reaction coordinate of interest. This allows for propagation of the system's dynamics under the influence of a quantum biasing umbrella potential and provides a method to rigorously undo the effects of the bias to compute equilibrium ensemble averages. In this way, one can drag electrons into regions of high free energy where they would otherwise not go, thus enabling chemistry by fiat. We demonstrate the applicability of our method for two condensed-phase systems of interest. First, we consider the interaction of a hydrated electron with an aqueous sodium cation, and we calculate a potential of mean force that shows that an e − :Na + contact pair is the thermodynamically favored product starting from either a neutral sodium atom or the separate cation and electron species. Second, we present the first determination of a hydrated electron's free-energy profile relative to an air/ water interface. For the particular model parameters used, we find that the hydrated electron is more thermodynamically stable in the bulk rather than at the interface. Our analysis suggests that the primary driving force keeping the electron away from the interface is the long-range electron−solvent polarization interaction rather than the short-range details of the chosen pseudopotential.

Temperature dependence of the hydrated electron’s excited-state relaxation. I. Simulation predictions of resonance Raman and pump-probe transient absorption spectra of cavity and non-cavity models

“…The equilibrium ground-state MQC MD simulations in this work were performed in the canonical (NVT) ensemble using in-house developed codes; the methods are essentially the same as those detailed in our previous published work on this topic. 14,22,23,40,41 Briefly, 499 water molecules were included in a cubic simulation box of length 24.64 Å with one excess electron. The excess electron was treated quantum mechanically, the wave function of which computed using a plane-wave basis set, while the water molecules were treated classically by the flexible simple point charge (SPC-flex) 42 model.…”

“…15 We have chosen to taper the interactions for the simulations discussed here, however, both because Ewald summation is known to give a stronger finite size effect for this system, as we have discussed in Ref. 28, and because this choice is consistent with our previously published work, 14,22,23,41 allowing for direct comparison to the data presented below.…”

Flexible boundary layer using exchange for embedding theories. I. Theory and implementation

“…52, we periodically shifted particle positions by an integer number of grid spacings in order to recenter the electron's wavefunction, and avoid FG boundary artefacts. Furthermore, a harmonic restraint with a force constant of k e-COM = 10 eV/ Å2 was placed on the electron centroid using quantum-biased MD 49,53 in order to tether the electron to the center of the droplet, and prevent it from diffusing to the droplet surface. Coulomb and Lennard-Jones pair interactions were not truncated.…”

Calcium Chloride (CaCl2) : Characteristics and Molecular Interaction in Solution

“…Kation (188)(189)(190)(191)(192) bereaksi dengan percepatan menuju elektroda negative dan anion (179-181; 193-199) bereaksi dengan percepatan menuju elektroda (237)(238) positif. Akan tetapi, saat ion bergerak melalui pelarut maka ion (200) akan mengalami gaya gesekan memperlambat Ie' yang sebanding dengan kecepatannya.dan jika diasumsikan bahwa hukum Stokes untuk bola radius a dan s berlaku pada skala mikroskopis , maka kita dapat menuliskan gaya perlambatan ini sebagai:…”