Conditions for the equivalence of the “surface hopping” and complex-valued time methods to affect nonadiabatic transitions

In this paper we examine two semiclassical theories used to describe collision induced electronic transitions, namely the surface hopping method of Tully and Preston and the “exact” semiclassical method of Miller and George, and give conditions such that the two methods are equivalent. An example using a DIM potential energy for H⁺₃ is discussed.     

The three-dimensional nonadiabatic dynamics calculation of DH(2)(+) and HD(2)(+) systems by using the trajectory surface hopping method based on the Zhu-Nakamura theory

A theoretical investigation on the nonadiabatic processes of the full three-dimensional D(+)+H(2) and H(+)+D(2) reaction systems has been performed by using trajectory surface hopping (TSH) method based on the Zhu-Nakamura (ZN) theory. This ZN-TSH method refers to not only classically allowed hops but also classically forbidden hops. The potential energy surface constructed by Kamisaka et al. is employed in the calculation. A new iterative method is proposed to yield the two-dimensional seam surface from the topography of the adiabatic potential surfaces, in which the inconvenience of directly solving the first-order partial differential equation is avoided. The cross sections of these two systems are calculated for three competing channels of the reactive charge transfer, the nonreactive charge transfer, and the reactive noncharge transfer, for ground rovibrational state of H(2) or D(2). Also, this study provides reaction probabilities of these three processes for the total angular momentum J=0 and ground initial vibrational state of H(2) or D(2). The calculated results from ZN-TSH method are in good agreement with the exact quantum calculations and the experimental measurements.     

On the properties of a primitive semiclassical surface hopping propagator for nonadiabatic quantum dynamics

A previously developed nonadiabatic semiclassical surface hopping propagator [M. F. Herman J. Chem. Phys. 103, 8081 (1995)] is further studied. The propagator has been shown to satisfy the time-dependent Schrodinger equation (TDSE) through order h, and the O(h2) terms are treated as small errors, consistent with standard semiclassical analysis. Energy is conserved at each hopping point and the change in momentum accompanying each hop is parallel to the direction of the nonadiabatic coupling vector resulting in both transmission and reflection types of hops. Quantum mechanical analysis and numerical calculations presented in this paper show that the h2 terms involving the interstate coupling functions have significant effects on the quantum transition probabilities. Motivated by these data, the h2 terms are analyzed for the nonadiabatic semiclassical propagator. It is shown that the propagator can satisfy the TDSE for multidimensional systems by including another type of nonclassical trajectories that reflect on the same surfaces. This h2 analysis gives three conditions for these three types of trajectories so that their coefficients are uniquely determined. Besides the nonadiabatic semiclassical propagator, a numerically useful quantum propagator in the adiabatic representation is developed to describe nonadiabatic transitions.     

Directionally negative friction: a method for enhanced sampling of rare event kinetics

A method exploiting the properties of an artificial (nonphysical) Langevin dynamics with a negative frictional coefficient along a suitable manifold and positive friction in the perpendicular directions is presented for the enhanced calculation of time-correlation functions for rare event problems. Exact time-correlation functions that describe the kinetics of the transitions for the all-positive, physical system can be calculated by reweighting the generated trajectories according to stochastic path integral treatment involving a functional weight based on an Onsager-Machlup action functional. The method is tested on a prototypical multidimensional model system featuring the main elements of conformational space characteristic of complex condensed matter systems. Using the present method, accurate estimates of rate constants require at least three order of magnitudes fewer trajectories than regular Langevin dynamics. The method is particularly useful in calculating kinetic properties in the context of multidimensional energy landscapes that are characteristic of complex systems such as proteins and nucleic acids.     

A justification for a nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation of the time evolution operator

A justification is given for the validity of a nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method. The method is based on a propagator that combines the single surface HK SC-IVR method [J. Chem. Phys. 84, 326 (1986)] and Herman's nonadiabatic semiclassical surface hopping theory [J. Chem. Phys. 103, 8081 (1995)], which was originally developed using the primitive semiclassical Van Vleck propagator. We show that the nonadiabatic HK SC-IVR propagator satisfies the time-dependent Schrodinger equation to the first order of variant Planck's over 2pi and the error is O(variant Planck's over 2pi(2)). As a required lemma, we show that the stationary phase approximation, under current assumptions, has an error term variant Planck's over 2pi(1) order higher than the leading term. Our derivation suggests some changes to the previous development, and it is shown that the numerical accuracy in applications to Tully's three model systems in low energies is improved.     

Regarding the validity of the time-dependent Kohn-Sham approach for electron-nuclear dynamics via trajectory surface hopping

The implementation of fewest-switches surface-hopping (FSSH) within time-dependent Kohn-Sham (TDKS) theory [Phys. Rev. Lett. 95, 163001 (2005)] has allowed us to study successfully excited state dynamics involving many electronic states in a variety of molecular and nanoscale systems, including chromophore-semiconductor interfaces, semiconductor and metallic quantum dots, carbon nanotubes and graphene nanoribbons, etc. At the same time, a concern has been raised that the KS orbital basis used in the calculation provides only approximate potential energy surfaces [J. Chem. Phys. 125, 014110 (2006)]. While this approximation does exist in our method, we show here that FSSH-TDKS is a viable option for computationally efficient calculations in large systems with straightforward excited state dynamics. We demonstrate that the potential energy surfaces and nonadiabatic transition probabilities obtained within the TDKS and linear response (LR) time-dependent density functional theories (TDDFT) agree semiquantitatively for three different systems, including an organic chromophore ligating a transition metal, a quantum dot, and a small molecule. Further, in the latter case the FSSH-TDKS procedure generates results that are in line with FSSH implemented within LR-TDDFT. The FSSH-TDKS approach is successful for several reasons. First, single-particle KS excitations often give a good representation of LR excitations. In this regard, DFT compares favorably with the Hartree-Fock theory, for which LR excitations are typically combinations of multiple single-particle excitations. Second, the majority of the FSSH-TDKS applications have been performed with large systems involving simple excitations types. Excitation of a single electron in such systems creates a relatively small perturbation to the total electron density summed over all electrons, and it has a small effect on the nuclear dynamics compared, for instance, with thermal nuclear fluctuations. In such cases an additional, classical-path approximation can be made. Third, typical observables measured in time-resolved experiments involve averaging over many initial conditions. Such averaging tends to cancel out random errors that may be encountered in individual simulated trajectories. Finally, if the flow of energy between electronic and nuclear subsystems is insignificant, the ad hoc FSSH procedure is not required, and a straightforward mean-field, Ehrenfest approach is sufficient. Then, the KS representation provides rigorously a convenient and efficient basis for numerically solving the TDDFT equations of motion.     

Excited state dynamics with the direct trajectory surface hopping method: azobenzene and its derivatives as a case study

We review the recent studies of the photoisomerization dynamics of azobenzene and its derivatives by surface hopping simulations based on semiempirical potential energy surfaces. We examine the ability of semiclassical methods to predict the excited state dynamics and to reproduce transient spectroscopic signals that constitute the most direct experimental evidence in this field. We show that the available simulation methods yield a deep insight into the mechanism of photochemical reactions and excited state decay, and a fairly good quantitative agreement with experimental findings. Probably the most important technical improvements we can envisage concern the surface hopping algorithm and the usage of ab initio data in the simulation of transient spectra. Concerning azobenzene, our results show that the isomerization mechanism is torsion of the N=N double bond, both by n → π* and by π → π* excitation. The influence of the solvent and the findings of some recent femtochemistry experiments deserve further work to be fully interpreted.     

Thermodynamic verification of the gradient method for surface area estimation

We present thermodynamic verification of Nonaka's gradient method for surface area estimation. On the basis of calorimetric measurements and integral molar entropy calculations it is shown that the main assumption of this method, when used at around room temperature, is an approximation. The specific surface areas of Sterling MT graphitized carbon black, obtained on the basis of three methods are compared. It is shown that the gradient method may be used, for its operational convenience, in cases where an uncertainty of several per cent may be tolerated.     

Optimizing the sampling and staging for simulations of rare events via forward flux sampling schemes

In this work, we present an adaptive algorithm to optimize the phase space sampling for simulations of rare events in complex systems via forward flux sampling (FFS) schemes. In FFS, interfaces are used to partition the phase space along an order parameter lambda connecting the initial and final regions of interest. Since the kinetic "bottleneck" regions along the order parameter are not usually known beforehand, an adaptive procedure is used that first finds these regions by estimating the rate constants associated with reaching subsequent interfaces; thereafter, the FFS simulation is reset to concentrate the sampling on those bottlenecks. The approach can optimize for either the number and position of the interfaces (i.e., optimized lambda phase staging) or the number M of fired trial runs per interface (i.e., the {M(i)} set) to minimize the statistical error in the rate constant estimation per simulation period. For example, the optimization of the lambda staging leads to a net constant flux of partial trajectories between interfaces and hence a constant flux of connected paths throughout the region between the two end states. The method is demonstrated for several test systems, including the folding of a lattice protein. It is shown that the proposed approach leads to an optimized lambda staging and {M(i)} set which increase the computational efficiency of the sampling algorithm.     

A surface hopping method for chemical reaction dynamics in solution described by diabatic representation: an analysis of tunneling and thermal activation

We present a surface hopping method for chemical reaction in solution based on diabatic representation, where quantum mechanical time evolution of the vibrational state of the reacting nuclei as well as the reaction-related electronic state of the system are traced simultaneously together with the classical motion of the solvent. The method is effective in describing the system where decoherence between reactant and product states is rapid. The diabatic representation can also give a clear picture for the reaction mechanism, e.g., thermal activation mechanism and a tunneling one. An idea of molecular orbital theory has been applied to evaluate the solvent contribution to the electronic coupling which determines the rate of reactive transition between the reactant and product potential surfaces. We applied the method to a model system which can describe complex chemical reaction of the real system. Two numerical examples are presented in order to demonstrate the applicability of the present method, where the first example traces a chemical reaction proceeded by thermal activation mechanism and the second examines tunneling mechanism mimicking a proton transfer reaction.     

A surface hopping quasiclassical trajectory study of the H2+ + H2 and (H2 + D2)+ systems

In this work we use a complete surface hopping quasiclassical trajectory method to determine cross sections for the reactions H₂⁺ + H₂ → H₃⁺ + H and the isotopic variants (H₂⁺ + D₂ and D₂⁺ + H₂). Initial translational energies ranged between 0.5 and 6 eV. The vibrational quantum number (v⁺) of the charged diatom is either 0 or 3. Comparing these results with our previous results with a partial treatment of surface hopping, we find essentially no change for v⁺ = 0 and reductions in cross sections of up to 30% for v⁺ = 3 trajectories.     

Minima hopping: an efficient search method for the global minimum of the potential energy surface of complex molecular systems

A method is presented that can find the global minimum of very complex condensed matter systems. It is based on the simple principle of exploring the configurational space as fast as possible and of avoiding revisiting known parts of this space. Even though it is not a genetic algorithm, it is not based on thermodynamics. The efficiency of the method depends strongly on the type of moves that are used to hop into new local minima. Moves that find low-barrier escape-paths out of the present minimum generally lead into low energy minima.     

Spectrographic a.c. spark method for the estimation of yttrium in rare earth mixtures

Summary The spectrographic estimation of yttrium in rare earth mixtures was achieved within a concentration range of 1–90% by means of the a.c. spark method using a large quartz spectrograph. The sample was mixed with ceria and zinc oxide in the ratio 212. The Y lines 2414.68 Å and 2856.36 Å were used as analytical lines for the ranges of 1–15% and 10–90%, respectively. The Ce line at 2603.65 Å served as internal standard line. The standard deviation was 2–5%.

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Spektrographische Bestimmung von Yttrium in Gemischen Seltener Erden mit Hilfe der Wechselstrom-Funken-methode

Zusammenfassung Yttrium konnte im Konzentrationsbereich von 1–90% mit einem großen Quarzspektrographen bestimmt werden. Die Probe wird mit Cerdioxid und Zinkoxid im Verhältnis 212 gemischt. Als Analysenlinie dient die Y-Linie 2414,68 Å (1–15%) bzw. 2856,36 Å (10–90%). Als innere Standardlinie wird die Ce-Linie 2603,65 Å verwendet. Die Standardabweichung betrug 2–5%.

     

Collision-induced nonadiabatic transitions in the second-tier ion-pair states of iodine molecule: experimental and theoretical study of the I2(f0g+) collisions with rare gas atoms

Nonadiabatic transitions induced by collisions with He, Ar, Kr, and Xe atoms in the I(2) molecule excited to the f0(g)(+) second-tier ion-pair state are investigated by means of the optical-optical double resonance spectroscopy. Fluorescence spectra reveal that the transition to the F0(u)(+) state is a dominant nonradiative decay channel for f state in He, Ar, and Kr, whereas the reactive quenching is more efficient for collisions with Xe atom. Total rate constants and vibrational product state distributions for the f-->F electronic energy transfer are determined and analyzed in terms of energy gaps and Franck-Condon factors for the combining vibronic levels at initial vibrational excitations v(f)=8, 10, 14, and 17. Quantum scattering calculations are performed for collisions with He and Ar atoms, implementing a combination of the diatomics-in-molecule and long-range perturbation theories to evaluate diabatic PESs and coupling matrix elements. Calculated rate constants and vibrational product state distributions agree well with the measured ones, especially in case of Ar. Qualitative comparison is made with the previous results for the second-tier f0(g)(+)-->F0(u)(+) transition in collisions with I(2)(X) molecule and the first-tier E0(g)(+)-->D0(u)(+) transition induced by collisions with the rare gas atoms.     

A revised method, and errors in the determination of thiosulphate by the Wollak method

The Wollak method for the determination of thiosulfate has been investigated. The experimental procedure has been revised in order to eliminate several problems associated with the method.     

Numerical study of the accuracy and efficiency of various approaches for Monte Carlo surface hopping calculations

A one-dimensional, two-state model problem with two well-separated avoided crossing points is employed to test the efficiency and accuracy of a semiclassical surface hopping technique. The use of a one-dimensional model allows for the accurate numerical evaluation of both fully quantum-mechanical and semiclassical transition probabilities. The calculations demonstrate that the surface hopping procedure employed accounts for the interference between different hopping trajectories very well and provides highly accurate transition probabilities. It is, in general, not computationally feasible to completely sum over all hopping trajectories in the semiclassical calculations for multidimensional problems. In this case, a Monte Carlo procedure for selecting important trajectories can be employed. However, the cancellation due to the different phases associated with different trajectories limits the accuracy and efficiency of the Monte Carlo procedure. Various approaches for improving the accuracy and efficiency of Monte Carlo surface hopping procedures are investigated. These methods are found to significantly reduce the statistical sampling errors in the calculations, thereby increasing the accuracy of the transition probabilities obtained with a fixed number of trajectories sampled.     

A kernel‐based method to determine optimal sampling times for the simultaneous estimation of the parameters of rival mathematical models

When several models are proposed for one and the same process, experimental design techniques are available to design optimal discriminatory experiments. However, because the experimental design techniques are model‐based, it is important that the required model predictions are not too uncertain. This uncertainty is determined by the quality of the already available data, since low‐quality data will result in poorly estimated parameters, which on their turn result in uncertain model predictions. Therefore, model discrimination may become more efficient and effective if this uncertainty is reduced first. This can be achieved by performing dedicated experiments, designed to increase the accuracy of the parameter estimates. However, performing such an additional experiment for each rival model may undermine the overall goal of optimal experimental design, which is to minimize the experimental effort. In this article, a kernel‐based method is presented to determine optimal sampling times to simultaneously estimate the parameters of rival models in a single experiment. The method is applied in a case study where nine rival models are defined to describe the kinetics of an enzymatic reaction (glucokinase). The results clearly show that the presented method performs well, and that a compromise experiment is found which is sufficiently informative to improve the overall accuracy of the parameters of all rival models, thus allowing subsequent design of an optimal discriminatory experiment. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

An algorithm for the uniform sampling of iso-energy surfaces and for the calculation of microcanonical averages

In this article an algorithm is proposed to efficiently perform the uniform sampling of an iso-energy surface corresponding to a fixed potential energy U of a molecular system, and for calculating averages of certain quantities over microstates having this energy (microcanonical averages). The developed sampling technique is based upon the combination of a recently proposed method for performing constant potential energy molecular dynamics simulations [Rapallo, A. J Chem Phys 2004, 121, 4033] with well-established thermostatting techniques used in the framework of standard molecular dynamics simulations, such as the Andersen thermostat, and the Nose-Hoover chain thermostat. The proposed strategy leads to very accurate and drift-free potential energy conservation during the whole sampling process, and, very important, specially when dealing with high-dimensional or complicated potential functions, it does not require the calculation of the potential energy function hessian. The technique proved to be very reliable for sampling both low- and high-dimensional surfaces.