A Temperature-Dependent Hartree Approach for Excess Proton Transport in Hydrogen-Bonded Chains

We develop a temperature-dependent Hartree (TeDH) approach to solving the N-dimensional Schrödinger equation, based on the time dependent Hartree (TDH) approximation, where the Ndimensional Schrödinger equation is approximated as N one-dimensional equations with a time dependent Hamiltonian arising from the mean field potentials. The TeDH method uses the temperature-dependent analog of the TDH method to efficiently obtain the approximate ground state wavefunction. The TeDH is formulated with the goal of describing the dynamics of an excess proton in a hydrogen-bonded chain, as found in biological systems. A quantum description of all the protons involved in making and breaking hydrogen bonds of the chain must be used; hence the utility of a SCF approximation whose computational cost scales linearly with the number of quantum degrees of freedom. We show by comparison with numerically exact results that the TeDH scheme gives accurate ground state wavefunctions for chains with up to three protons. (Comparison with longer chains was not possible due to the computational requirements of numerically exact solutions). Sharing of a proton between flanking heavy atoms, as well as “split” protons, respectively corresponding to proton potential surfaces with one and two, symmetric wells are properly described by the TeDH method.