Two-dimensional Model Potential Research Articles

Reconstructing free-energy landscapes for cyclic molecular motors using full multidimensional or partial one-dimensional dynamic information.

Diffusion on a free-energy landscape is a fundamental framework for describing molecular motors. In the landscape framework, energy conversion between different forms of energy, e.g., chemical and mechanical, is explicitly described using multidimensional nonseparable potential landscapes. We present a k-space method for reconstructing multidimensional free-energy landscapes from stochastic single-molecule trajectories. For a variety of two-dimensional model potential landscapes, we demonstrate the robustness of the method by reconstructing the landscapes using full dynamic information, i.e., simulated two-dimensional stochastic trajectories. We then consider the case where the stochastic trajectory is known only along one dimension. With this partial dynamic information, the reconstruction of the full two-dimensional landscape is severely limited in the majority of cases. However, we reconstruct effective one-dimensional landscapes for the two-dimensional model potentials. We discuss the interpretation of the one-dimensional landscapes and identify signatures of energy conversion. Finally, we consider the implications of these results for biological molecular motors experiments.

Time Recovery for a Complex Process Using Accelerated Dynamics.

The hyperdynamics method (HD) developed by Voter (J. Chem. Phys. 1996, 106, 4665) sets the theoretical basis to construct an accelerated simulation scheme that holds the time scale information. Since HD is based on transition state theory, pseudoequilibrium conditions (PEC) must be satisfied before any system in a trapped state may be accelerated. As the system evolves, many trapped states may appear, and the PEC must be assumed in each one to accelerate the escape. However, since the system evolution is a priori unknown, the PEC cannot be permanently assumed to be true. Furthermore, the different parameters of the bias function used may need drastic recalibration during this evolution. To overcome these problems, we present a general scheme to switch between HD and conventional molecular dynamics (MD) in an automatic fashion during the simulation. To decide when HD should start and finish, criteria based on the energetic properties of the system are introduced. On the other hand, a very simple bias function is proposed, leading to a straightforward on-the-fly set up of the required parameters. A way to measure the quality of the simulation is suggested. The efficiency of the present hybrid HD-MD method is tested for a two-dimensional model potential and for the coalescence process of two nanoparticles. In spite of the important complexity of the latter system (165 degrees of freedoms), some relevant mechanistic properties were recovered within the present method.

Theoretical Studies of Dynamic Interactions in Excited States of Hydrogen-Bonded Systems

Theoretical model for vibrational interactions in the hydrogen-bonded benzoic acid dimer is presented. The model takes into account anharmonic-type couplings between the high-frequency O–H and the low-frequency O⋯O stretching vibrations in two hydrogen bonds, resonance interactions between two hydrogen bonds in the dimer, and Fermi resonance between the O–H stretching fundamental and the first overtone of the O–H in-plane bending vibrations. The model is used for theoretical simulation of the O–H stretching IR absorption bands of benzoic acid dimers in the gas phase in the first excited singlet state. Ab initio CIS and CIS(D)/CIS/6-311++G(d,p) calculations have been carried out in the à state of tropolone. The grids of potential energy surfaces along the coordinates of the tunneling vibration and the low-frequency coupled vibration have been calculated. Two-dimensional model potentials have been fitted to the calculated potential energy surfaces. The tunneling splittings for vibrationally excited states have been calculated and compared with the available experimental data. The model potential energy surfaces give good estimation of the tunneling splittings in the vibrationally ground and excited states of tropolone, and explain monotonic decrease in tunneling splittings with the excitation of low-frequency out-of-plane modes and increase of the tunneling splittings with the excitation of low-frequency planar modes.

Efficient Computation, Sensitivity, and Error Analysis of Committor Probabilities for Complex Dynamical Processes

In many fields of physics, chemistry, and biology, the characterization of rates and pathways between certain states or species is of fundamental interest. The central mathematical object in such situations is the committor probability—a generalized reaction coordinate that measures the progress of the process as the probability of proceeding to the target state rather than relapsing to the source state. Here, we conduct a numerical analysis of the committor. First, it is shown that committors can be expressed by the stationary eigenfunctions of a modified dynamical operator, thus relating the committors to the dominant eigenfunctions of the original operator. Based on this reformulation, committors can be efficiently computed for systems with large state spaces. Moreover, a sensitivity analysis of the committor is conducted, which allows its statistical uncertainty from estimation to be quantified within a Bayesian framework. The methods are illustrated on two examples of diffusive dynamics: a two-dimensional model potential with three minima, and a three-dimensional model representing protein-ligand binding.

Rovibrational quenching of BH in ultracold 3He collisions

The interaction potential of a He–BH complex is investigated by the coupled-cluster single-double plus perturbative triples (CCSD (T)) method and an augmented correlation consistent polarized valence (aug-cc-pV)5Z basis set extended with a set of (3s3p2d1f1g) midbond functions. Using the five two-dimensional model potentials, the first three-dimensional interaction potential energy surface is constructed by interpolating along (r–re) by using a fourth-order polynomial. The cross sections for the rovibrational relaxation of BH in cold and ultracold collisions with 3He atom are calculated based on the three-dimensional potential. The results show that the Δν = −1 transition is more efficient than the Δν = −2 transition, and that the process of relaxation takes place mainly between rotational energy levels with the same vibration state and the Δj = −1 transition is the most efficient. The zero temperature quenching rate coefficient is finite as predicted by Wigner's law. The resonance is found to take place around 0.1–1 cm−1 translational energy, which gives rise to a step in the rate coefficients for temperatures around 0.1–1 K. The final rotational distributions in the state ν = 0 resulting from the quenching of state (ν = 1, j = 0) at three energies corresponding to the three different regimes are also given.

Theoretical study of proton tunneling in the excited state of tropolone

Ab initio CIS/6-311++G(d,p) calculations of geometry and vibrational frequencies have been carried out in the A state of tropolone. The grids of potential energy surfaces along the coordinates of high frequency tunneling vibration and the low-frequency coupled vibration have been calculated. Two-dimensional model potentials, formed from symmetric mode coupling potential and squeezed double well potential, have been fitted to the calculated potential energy surfaces and used to analyze proton dynamics. The tunneling splittings for different vibrationally excited states have been calculated and compared with the available experimental data. The model potential energy surfaces, based on the CIS/6-311++G(d,p) calculations, give good estimation of the tunneling energy splittings in the vibrationally ground and excited states of tropolone and explain monotonic decrease in tunneling splittings with the excitation of low-frequency out-of-plane modes and increase in the tunneling splittings with the excitation of low-frequency planar modes.

Proton Dynamics in the Strong Chelate Hydrogen Bond of Crystalline Picolinic Acid N-Oxide. A New Computational Approach and Infrared, Raman and INS Study

Infrared, Raman and INS spectra of picolinic acid N-oxide (PANO) were recorded and examined for the location of the hydronic modes, particularly O-H stretching and COH bending. PANO is representative of strong chelate hydrogen bonds (H-bonds) with its short O...O distance (2.425 A). H-bonding is possibly well-characterized by diffraction, NMR and NQR data and calculated potential energy functions. The analysis of the spectra is assisted by DFT frequency calculations both in the gas phase and in the solid state. The Car-Parrinello quantum mechanical solid-state method is also used for the proton dynamics simulation; it shows the hydron to be located about 99% of time in the energy minimum near the carboxylic oxygen; jumps to the N-O acceptor are rare. The infrared spectrum excels by an extended absorption (Zundel's continuum) interrupted by numerous Evans transmissions. The model proton potential functions on which the theories of continuum formation are based do not correspond to the experimental and computed characteristics of the hydrogen bond in PANO, therefore a novel approach has been developed; it is based on crystal dynamics driven hydronium potential fluctuation. The envelope of one hundred 0 --> 1 OH stretching transitions generated by molecular dynamics simulation exhibits a maximum at 1400 cm-1 and a minor hump at approximately 1600 cm-1. These positions square well with ones predicted for the COH bending and OH stretching frequencies derived from various one- and two-dimensional model potentials. The coincidences with experimental features have to be considered with caution because the CPMD transition envelope is based solely on the OH stretching coordinate while the observed infrared bands correspond to heavily mixed modes as was previously shown by the normal coordinate analysis of the IR spectrum of argon matrix isolated PANO, the present CPMD frequency calculation and the empirical analysis of spectra. The experimental infrared spectra show some unusual characteristics such as large temperature effects on the intensity of some bands, thus presenting a challenge for theoretical band shape treatments. Our calculations clearly show that the present system is characterized by an asymmetric single well potential with no large amplitudes in the hydronium motion, which extends the existence of Zundel-type spectra beyond the established set of hydrogen bonds with large hydronic vibrational amplitudes.

Semiclassical tunneling splittings from short time dynamics: Herman-Kluk-propagation and harmonic inversion.

We investigate a recently proposed method [J. Chem. Phys. 108, 9206 (1998)] to obtain tunneling splittings from short time cross-correlation matrices that were propagated according to the semiclassical propagator of Herman and Kluk. The energy levels were extracted by harmonic inversion of the cross-correlation matrix using the filter diagonalization technique. The aim of this study is twofold: First, the short time behavior of the Herman-Kluk-propagator and the meaning of using cross-correlation matrices rather than autocorrelation functions is addressed. Numerical examples are given for one- and two-dimensional model potentials. Second, the performance of the method is investigated for a system with considerable anharmonicity and coupling. Here the proton transfer in 3,7-dichlorotropolone is considered using an ab initio reaction surface Hamiltonian approach. For this example also the extension to more dimensions is critically discussed.

Theoretical Study of Multidimensional Proton Tunnelling in Benzoic Acid Dimer

Ab initio B3LYP/6-311++G** calculations have been carried out for the benzoic acid dimer for the stable and saddle point structures. The energy barrier for the proton tunneling amounts to 6.5 kcal/mol. The normal mode frequencies have been computed including modes coupled to the proton tunneling mode. Two-dimensional model potentials, formed from symmetric mode coupling potential and squeezed double well potential, have been fitted to the calculated energy barrier, geometries and frequencies, and used to analyze proton dynamics. The calculated proton tunneling energy splitting in the vibrationally ground states of the low-frequency modes is ~230 cm-1. The two-dimensional model PES predict monotonic increase of the tunneling splitting with the excitation of the planar modes. Depending of the sign of the coupling parameter out-of-plane modes can either suppress or promote the splittings.

Tunneling splittings. A classical trajectory approach

We propose an extension of the well-known Makri–Miller model [J. Chem. Phys. 91 (1989) 4026] for the calculation of tunneling splittings of ground and excited states of symmetric double well potentials. Our approach is based on generalized trajectories which can penetrate into the classically forbidden region. The performance of the new method is demonstrated for a two-dimensional model potential. We compare our results with data from exact quantum mechanical calculations as well as from various semiclassical methods. The numerical effort is only slightly increased as compared to that needed for the Makri–Miller model.

Theoretical study of multidimensional proton tunneling in the hydrogen carbonate dimer ion [(HCO 3) 2] 2−

Ab initio HF/6-31G ∗∗, MP2/6-31G ∗∗ and Becke3LYP/6-311++G ∗∗ calculations have been carried out for the hydrogen carbonate dimer ion [(HCO 3) 2] 2− for the stable and saddle point structures. The energy barrier for the proton tunneling amounts to 9.3 kcal/mol. The normal mode frequencies have been computed including modes coupled to the proton tunneling mode. Two-dimensional model potentials, formed from symmetric mode coupling potential (SMC) and squeezed double well potential (SQZ), have been fitted to the calculated energy barrier, geometries and frequencies, and used to analyze proton dynamics. The two-dimensional model PES predicts monotonic increase of the tunneling splitting with the excitation of the planar modes and decrease in tunneling splittings with the excitation of the out-of-plane modes.

Simulations of adiabatic bond-breaking electron transfer reactions on metal electrodes

Adiabatic electron transfer at metal electrodes accompanied by bond breaking is studied by molecular dynamic simulations in a two-dimensional model potential. In the simulations the redox system is coupled to a thermal bath that ensures proper statistical mechanics and provides the energy fluctuations required for the system to cross the activation barrier. The rate constants and transmission factors obtained from the simulations are analysed for various system parameters such as the applied overpotential and the strength of the interaction between the reactants and the electrode. A dependency of the rate constant on the friction parameter is also tested considering both isotropy and anisotropy effects. The Kramers turnover region is found in both cases, but the rate constant is much lower than predicted by the Kramers theory. In all simulations a strong saddle-point avoidance was observed.

Hydrogen bonding in picolinic acid N-oxide. Part II: A proposal for dissipative laser driven proton transfer dynamics

Laser control of proton dynamics in the medium-strong intramolecular hydrogen bond of picolinic acid N-oxide (PANO) is investigated. This work is an extension of our recent article dealing with the “statical” effects of hydrogen bonding in PANO [J. Mol. Struct. (Theochem) 500 (2000) 429–440]. A two-dimensional model potential is extracted from DFT calculations that include the proton transfer motion and the heavy atom mode. The effects of the environmental degrees of freedom were treated by means of their spectral density within the density matrix formalism. The proton dynamics is monitored over time by calculating the nonlinear optical response nonperturbatively in the driving field.

Inverse isotope effects in the surface diffusion of atoms

The inverse isotope effect in hydrogen diffusion at metal surfaces has been investigated with the Quantum Transition State Theory using a two-dimensional model potential with confinement orthogonal to the diffusion path. It is clarified by quantum Monte Carlo simulation that a decrease in the hopping rate by confinement, competing with an increase by tunneling, induces the inverse isotope effect within a certain temperature range above a classical-quantum crossover temperature. We also show that, at high temperature, the confinement effect can be included in the potential barrier for diffusion and that the inverse isotope effect can be described with a reduced one-dimensional model within Feynman's variational method.

Numerical evaluation of FC factors for highly anharmonic multidimensional potentials

For calculation of quantum-statistical transition probabilities for chemical conversion processes in condensed systems in the Franck-Condon formalism, overlap integrals for nuclear wavefunctions in multidimensional, anharmonic, nonseparable potentials with large spatial displacement of potential surface minima are needed. We calculate them by expansion of the coupled wavefunction in terms of eigenfunctions of the uncoupled problem. Thereby, different quantum-mechanical approximations were tested (adiabatic, SCF and two-dimensional x- y-product basis method). Results are given for a two-dimensional model potential.

Quantum Monte Carlo method for some model and realistic coupled anharmonic oscillators

A new quantum Monte Carlo (QMC) method of evaluating low lying vibrational levels for coupled modes is presented. We use a modified fixed-node (FN) approach in which an extremum principle for energy levels is invoked. In this way, the nodal hypersurfaces of the nuclear wave function are parametrized and then optimized for each excited state. The method is tested on the fundamental excitations of some two-dimensional model potentials and is applied to the case of realistic coupled modes of the CO molecule adsorbed on a palladium cluster. The effect of an external electric field is also examined. The quantum Monte Carlo results are compared with those obtained in the conventional variational treatment of the nuclear Schrödinger equation for coupled vibrations. The QMC results give the exact values with an error which is in general less than 1 cm−1 . In all cases (even in the case of strong coupling) the use of our procedure leads to ‘‘optimal’’ nodal lines (in the sense of the extremum principle used in this work) which are practically undistorted. A salient feature of the Monte Carlo method presented here is that it readily permits the evaluation of the fundamental excitations of an arbitrary number of coupled vibrations. Furthermore, the potential energy surface may be represented by any analytical form without practical difficulties.

Poincare map for scattering states

For scattering states in classical mechanics a map is constructed, which is an analogue to the Poincare map for bound state trajectories. Iteration of this map indicates which parts of the phase space are filled by unstable trajectories that are sensitive to small changes in the initial conditions, and which regions are filled by stable trajectories. Examples for this map are given by numerical calculations for scattering off a simple two-dimensional model potential.

Structure in excimer line shape

Recently monomer-like structured emission has been observed in the spectra of excimers both in solids and in solution. In a two-dimensional model potential an intramolecular mode is considered in addition to the relative degree of freedom. It is shown that under certain conditions structured line shapes are obtained.