OPUS-Rota5: A highly accurate protein side-chain modeling method with 3D-Unet and RotaFormer.

Accurate protein side-chain modeling is crucial for protein folding and design. This is particularly true for molecular docking as ligands primarily interact with side chains. In this study, we introduce a two-stage side-chain modeling approach called OPUS-Rota5. It leverages a modified 3D-Unet to capture the local environmental features, including ligand information of each residue, and then employs the RotaFormer module to aggregate various types of features. Evaluation on three test sets, including recently released targets from CAMEO and CASP15, shows that OPUS-Rota5 significantly outperforms some other leading side-chain modeling methods. We also employ OPUS-Rota5 to refine the side chains of 25 G protein-coupled receptor targets predicted by AlphaFold2 and achieve a significantly improved success rate in a subsequent "back" docking of their natural ligands. Therefore, OPUS-Rota5 is a useful and effective tool for molecular docking, particularly for targets with relatively accurate predicted backbones but not side chains such as high-homology targets.

Is there a finite mobility for the one vibrational mode Holstein model? Implications from real time simulations.

The question of whether there exists a finite mobility in the standard Holstein model with one vibrational mode on each site remains unclear. In this Communication, we approach this problem by employing the hierarchical equation of motion method to simulate model systems where the vibrational modes are dissipative. It is found that, as the friction becomes smaller, the charge carrier mobility increases significantly and a friction-free limit cannot be obtained. The current autocorrelation functions are also calculated for the friction-free Holstein model, and converged results cannot be obtained with an increase in the number of sites. Based on these observations, we conclude that a finite mobility cannot be defined for the standard Holstein model in the parameter regime explored in this work.

Revealing Intermolecular Electronic and Vibronic Coherence with Polarization-Dependent Two-Dimensional Beating Maps.

Two-dimensional electronic spectroscopy (2DES) has been widely employed as an efficient tool to reveal the impact of intermolecular electronic and/or vibronic quantum coherence on excitation energy transfer in light-harvesting complexes. However, intramolecular vibrational coherence would also contribute to oscillating signals in 2D spectra, along with the intermolecular coherence signals that are directly related to energy transfer. In this work, the possibility of screening the vibrational coherence signals is explored through polarization-dependent 2DES. The all-parallel (AP) and double-crossed (DC) polarization-dependent two-dimensional rephasing spectra (2DRS) are simulated for a minimalist heterodimer model with vibrational coupling. By combining the DC-2DRS and the 2D beating maps, we demonstrate that the population and vibrational coherence signals can be largely suppressed, resulting in highlighted intermolecular electronic and vibronic coherence signals. Moreover, the AP- and DC-2DBMs show rather different patterns at the vibrational frequency, indicating a possible way to identify pure vibrational coherence.

Taming Quantum Noise for Efficient Low Temperature Simulations of Open Quantum Systems.

The hierarchical equations of motion (HEOM), derived from the exact Feynman-Vernon path integral, is one of the most powerful numerical methods to simulate the dynamics of open quantum systems. Its applicability has so far been limited to specific forms of spectral reservoir distributions and relatively elevated temperatures. Here we solve this problem and introduce an effective treatment of quantum noise in frequency space by systematically clustering higher order Matsubara poles, equivalent to an optimized rational decomposition. This leads to an elegant extension of the HEOM to arbitrary temperatures and very general reservoirs in combination with efficiency, high accuracy, and long-time stability. Moreover, the technique can directly be implemented in other approaches such as Green's function, stochastic, and pseudomode formulations. As one highly nontrivial application, for the subohmic spin-boson model at vanishing temperature the Shiba relation is quantitatively verified which predicts the long-time decay of correlation functions.

Application of the imaginary time hierarchical equations of motion method to calculate real time correlation functions.

We investigate the application of the imaginary time hierarchical equations of motion method to calculate real time quantum correlation functions. By starting from the path integral expression for the correlated system-bath equilibrium state, we first derive a new set of equations that decouple the imaginary time propagation and the calculation of auxiliary density operators. The new equations, thus, greatly simplify the calculation of the equilibrium correlated initial state that is subsequently used in the real time propagation to obtain the quantum correlation functions. It is also shown that a periodic decomposition of the bath imaginary time correlation function is no longer necessary in the new equations such that different decomposition schemes can be explored. The applicability of the new method is demonstrated in several numerical examples, including the spin-Boson model, the Holstein model, and the double-well model for proton transfer reaction.

Generalized master equation for charge transport in a molecular junction: Exact memory kernels and their high order expansion.

We derive a set of generalized master equations (GMEs) to study charge transport dynamics in molecular junctions using the Nakajima-Zwanzig-Mori projection operator approach. In the new GME, time derivatives of population on each quantum state of the molecule, as well as the tunneling current, are calculated as the convolution of time non-local memory kernels with populations on all system states. The non-Markovian memory kernels are obtained by combining the hierarchical equations of motion (HEOM) method and a previous derived Dyson relation for the exact kernel. A perturbative expansion of these memory kernels is then calculated using the extended HEOM developed in our previous work [M. Xu et al., J. Chem. Phys. 146, 064102 (2017)]. By using the resonant level model and the Anderson impurity model, we study properties of the exact memory kernels and analyze convergence properties of their perturbative expansions with respect to the system-bath coupling strength and the electron-electron repulsive energy. It is found that exact memory kernels calculated from HEOM exhibit short memory times and decay faster than the population and current dynamics. The high order perturbation expansion of the memory kernels can give converged results in certain parameter regimes. The Padé and Landau-Zener resummation schemes are also found to give improved results over low order perturbation theory.

A low-temperature quantum Fokker-Planck equation that improves the numerical stability of the hierarchical equations of motion for the Brownian oscillator spectral density.

We investigate the numerical stability of the hierarchical equations of motion (HEOM) method applied to systems with the Brownian oscillator (BO) and multimode BO (MBO) spectral densities. It is shown that, with the increase in the system-bath coupling strength, the standard HEOM may become unstable, and a simple increase in the truncation depth of the HEOM cannot remove the instability at long times. To solve this problem, we first show that the high-temperature approximation of the HEOM with the BO spectral density is equivalent to the celebrated quantum Fokker-Planck equation (QFPE). By starting from the HEOM, we then derive a new multidimensional phase space differential equation that generalizes the QFPE to arbitrary temperature. It is further shown that the numerical instability can be removed if the new low-temperature QFPE is expanded in a basis set different than the one that leads to the conventional HEOM. The matrix product state method is also employed to propagate the new equation based on the low-temperature QFPE and to resolve the numerical instability problem for an electron transfer model with the MBO spectral density presented in the recent literature.

Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors.

We develop new methods to efficiently propagate the hierarchical equations of motion (HEOM) by using the Tucker and hierarchical Tucker (HT) tensors to represent the reduced density operator and auxiliary density operators. We first show that by employing the split operator method, the specific structure of the HEOM allows a simple propagation scheme using the Tucker tensor. When the number of effective modes in the HEOM increases and the Tucker representation becomes intractable, the split operator method is extended to the binary tree structure of the HT representation. It is found that to update the binary tree nodes related to a specific effective mode, we only need to propagate a short matrix product state constructed from these nodes. Numerical results show that by further employing the mode combination technique commonly used in the multi-configuration time-dependent Hartree approaches, the binary tree representation can be applied to study excitation energy transfer dynamics in a fairly large system including over 104 effective modes. The new methods may thus provide a promising tool in simulating quantum dynamics in condensed phases.

Understanding the Large Kinetic Isotope Effect of Hydrogen Tunneling in Condensed Phases by Using Double-Well Model Systems.

In recent years, many experiments have shown large kinetic isotope effects (KIEs) for hydrogen transfer reactions in condensed phases as evidence of strong quantum tunneling effects. Since accurate calculation of the tunneling dynamics in such systems still present significant challenges, previous studies have employed different types of approximations to estimate the tunneling effects and KIEs. In this work, by employing model systems consisting of a double-well coupled to a harmonic bath, we calculate the tunneling effects and KIEs using the numerically exact hierarchical equations of motion (HEOM) method. It is found that hydrogen and deuterium transfer reactions in the same system may show rather different behaviors, where hydrogen transfer is dominated by tunneling between the two lowest vibrational states and deuterium transfer is controlled by excited vibrational states close to the barrier top. The simulation results are also used to test the validity of various approximate methods. It is shown that the Wolynes theory of dissipative tunneling gives a good estimation of rate constants in the over-the-barrier regime, while the nonadiabatic reaction rate theory based on the Landau-Zener formula is more suitable for deep tunneling reactions.

A new method to improve the numerical stability of the hierarchical equations of motion for discrete harmonic oscillator modes.

The hierarchical equations of motion (HEOMs) have developed into an important tool in simulating quantum dynamics in condensed phases. Yet, it has recently been found that the HEOM may become numerically unstable in simulations using discrete harmonic oscillator modes [I. S. Dunn, et al., J. Chem. Phys. 150, 184109 (2019)]. In this paper, a new set of equations of motion are obtained based on the equivalence between the HEOM for discrete harmonic oscillator modes and the mixed quantum-classical Liouville equation. The new set of equations can thus be regarded as the expansion of the same phase space partial differential equation using different basis sets. It is shown that they have similar structures as the original HEOM but are free from the problem of numerical instability. The new set of equations are also incorporated into the matrix product state method, where it is found that the trace of the reduced density operator is not well conserved during the propagation. A modified time-dependent variational principle is then proposed to achieve better trace conservation.

Theoretical study on resonance Raman spectra of meso-tetrakis(3,5-di-tertiarybutylphenyl)-porphyrin due to the second-order Herzberg-Teller mechanism.

Using quantum chemistry computations TDDFT//B3LYP/6-31G(d), we study resonance Raman spectra of the Q band for meso-tetrakis(3,5-di-tertiarybutylphenyl)-porphyrin (H2TBPP) molecule due to the Franck-Condon and non-Condon mechanisms including the Herzberg-Teller and second-order Herzberg-Teller terms. Generally, the Herzberg-Teller terms are large. However, for some vibrational modes, the second-order Herzberg-Teller terms are the strongest and dominate resonance Raman spectra, which may also impact on fluorescence and absorption spectra. Hence, the Taylor expansion of the electric dipole transition moment with respect to the normal coordinates at the equilibrium structure of the ground electronic state may not converge for H2TBPP. A method to solve this problem is suggested.

Theoretical study of charge carrier transport in organic molecular crystals using the Nakajima-Zwanzig-Mori generalized master equation.

There has been a long history of applying the generalized master equation (GME) to study charge carrier and exciton transport in molecular systems. Yet exact memory kernels in the GME are generally difficult to obtain. In this work, exact memory kernels of the Nakajima-Zwanzig-Mori GME for a one dimensional Holstein type of model are calculated by employing the Dyson relation for the exact memory kernel, combined with the hierarchical equations of motion method. Characteristics of the exact memory kernels, as well as the transition rate constants within the Markovian approximation, are then analyzed for different sets of parameters ranging from the hopping to bandlike transport regimes. It is shown that, despite the memory effect of the exact kernels, the Markovian approximation to the exact GME can reproduce the diffusion constants accurately. We also investigate the validity of the second and fourth order perturbation theories with respect to the electronic coupling constant in calculating the rate constants and the diffusion constant. It is found that, due to the cancellation of errors, the second order diffusion constant gives a reasonable estimate of the exact one within a wide range of electronic coupling constants.

Efficient propagation of the hierarchical equations of motion using the matrix product state method.

We apply the matrix product state (MPS) method to propagate the hierarchical equations of motion (HEOM). It is shown that the MPS approximation works well in different type of problems, including boson and fermion baths. The MPS method based on the time-dependent variational principle is also found to be applicable to HEOM with over one thousand effective modes. Combining the flexibility of the HEOM in defining the effective modes and the efficiency of the MPS method thus may provide a promising tool in simulating quantum dynamics in condensed phases.

Convergence of high order memory kernels in the Nakajima-Zwanzig generalized master equation and rate constants: Case study of the spin-boson model.

The Nakajima-Zwanzig generalized master equation provides a formally exact framework to simulate quantum dynamics in condensed phases. Yet, the exact memory kernel is hard to obtain and calculations based on perturbative expansions are often employed. By using the spin-boson model as an example, we assess the convergence of high order memory kernels in the Nakajima-Zwanzig generalized master equation. The exact memory kernels are calculated by combining the hierarchical equation of motion approach and the Dyson expansion of the exact memory kernel. High order expansions of the memory kernels are obtained by extending our previous work to calculate perturbative expansions of open system quantum dynamics [M. Xu et al., J. Chem. Phys. 146, 064102 (2017)]. It is found that the high order expansions do not necessarily converge in certain parameter regimes where the exact kernel show a long memory time, especially in cases of slow bath, weak system-bath coupling, and low temperature. Effectiveness of the Padé and Landau-Zener resummation approaches is tested, and the convergence of higher order rate constants beyond Fermi's golden rule is investigated.

Simulation of the Two-Dimensional Electronic Spectroscopy and Energy Transfer Dynamics of Light-Harvesting Complex II at Ambient Temperature.

We simulate the two-dimensional electronic spectra (2DES) of the light-harvesting complex II (LHCII) at room temperature by combining the hierarchical equations of motion method and the equation-of-motion phase-matching approach. The laser-excited population dynamics of LHCII is also calculated to help understanding the 2DES. Three different excitation schemes are studied, including (1) only the chlorophyll (Chl) b Q y states of LHCII are excited, (2) only the Chl a Q y states are excited, and (3) both the Chl b and Chl a states are excited. The energy transfer pathways and time scales revealed from the 2DES in schemes (1) and (2) agree with the recent experimental studies for the Chl b to Chl a energy transfer and the excitation energy relaxation process within the Chl a manifold. We also propose a different way to better present the signals of bottleneck states by investigating the diagonal peaks of the 2DES in scheme (3).

Understanding the free energy barrier and multiple timescale dynamics of charge separation in organic photovoltaic cells.

By employing several lattice model systems, we investigate the free energy barrier and real-time dynamics of charge separation in organic photovoltaic (OPV) cells. It is found that the combined effects of the external electric field, entropy, and charge delocalization reduce the free energy barrier significantly. The dynamic disorder reduces charge carrier delocalization and results in the increased charge separation barrier, while the effect of static disorder is more complicated. Simulation of the real-time dynamics indicates that the free charge generation process involves multiple time scales, including an ultrafast component within hundreds of femtoseconds, an intermediate component related to the relaxation of the hot charge transfer (CT) state, and a slow component on the time scale of tens of picoseconds from the thermally equilibrated CT state. Effects of hot exciton dissociation as well as its dependence on the energy offset between the Frenkel exciton and the CT state are also analyzed. The current results indicate that only a small energy offset between the band gap and the lowest energy CT state is needed to achieve efficient free charge generation in OPV devices, which agrees with recent experimental findings.

In Situ Visualization of Assembly and Photonic Signal Processing in a Triplet Light-Harvesting Nanosystem.

Real-time visualization of assembly processes and sophisticated signal processing at the nanoscale are two challenging topics in photonic nanomaterials. Here, high-quality light-harvesting crystalline nanorods were developed by the coassembly of two polypyridyl Ir(III) and Ru(II) metallophosphors, behaving as the antenna chromophore and energy acceptor, respectively. By using a one-pot or stepwise growth condition, homogeneous and multiblock heterojunction nanorods were prepared, respectively. These nanostructures display multicolor phosphorescence from green to red due to the efficient triplet energy transfer and light-harvesting capability at low acceptor doping ratios. Heterojunction nanorods show gradient emission-color switches during different growth stages, in which the real-time stepwise assembly can be vividly visualized using fluorescence microscopy techniques. Triplet excitons were successfully manipulated in both homogeneous and heterojunction nanorods to realize waveguided green, orange, and red emissions and advanced photonic signal logics and encoding/decoding on single multiblock heterojunction nanorod.