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Proton transfer and hydrogen tunneling play key roles in many processes of chemical and biological importance. The generalized nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method was developed in order to capture hydrogen tunneling effects in systems involving the transfer and tunneling of one or more protons. The generalized NEO-MSDFT method treats the transferring protons quantum mechanically on the same level as the electrons and obtains the delocalized vibronic states associated with hydrogen tunneling by mixing localized NEO-DFT states in a nonorthogonal configuration interaction scheme. Herein, we present the derivation and implementation of analytical gradients for the generalized NEO-MSDFT vibronic state energies and the nonadiabatic coupling vectors between these vibronic states. We use this methodology to perform adiabatic and nonadiabatic dynamics simulations of the double proton transfer reactions in the formic acid dimer and the heterodimer of formamidine and formic acid. The generalized NEO-MSDFT method is shown to capture the strongly coupled synchronous or asynchronous tunneling of the two protons in these processes. Inclusion of vibronically nonadiabatic effects is found to significantly impact the double proton transfer dynamics. This work lays the foundation for a variety of nonadiabatic dynamics simulations of multiple proton transfer systems, such as proton relays and hydrogen-bonding networks. 

Single-molecule and related experiments yield time series of an observable as it fluctuates due to thermal motion. In such data, it can be difficult to distinguish fluctuating signal from fluctuating noise. We present a method of separating signal from noise using nonlinear-correlation functions. The method is fully nonparametric: No a priori model for the system is required, no knowledge of whether the system is continuous or discrete is needed, the number of states is not fixed, and the system can be Markovian or not. The noise-corrected, nonlinear-correlation functions can be converted to the system’s Green’s function; the noise-corrected moments yield the system’s equilibrium-probability distribution. As a demonstration, we analyze synthetic data from a three-state system. The correlation method is compared to another fully nonparametric approach—time binning to remove noise, and histogramming to obtain the distribution. The correlation method has substantially better resolution in time and in state space. We develop formulas for the limits on data quality needed for signal recovery from time series and test them on datasets of varying size and signal-to-noise ratio. The formulas show that the signal-to-noise ratio needs to be on the order of or greater than one-half before convergence scales at a practical rate. With experimental benchmark data, the positions and populations of the states and their exchange rates are recovered with an accuracy similar to parametric methods. The methods demonstrated here are essential components in building a complete analysis of time series using only high-order correlation functions.