A combined explicit-implicit method for high accuracy reaction path integration

We present the use of an optimal combined explicit-implicit method for following the reaction path to high accuracy. This is in contrast to most purely implicit reaction path integration algorithms, which are only efficient on stiff ordinary differential equations. The defining equation for the reaction path is considered to be stiff, however, we show here that the reaction path is not uniformly stiff and instead is only stiff near stationary points. The optimal algorithm developed in this work is a combination of explicit and implicit methods with a simple criterion to switch between the two. Using three different chemical reactions, we combine and compare three different integration methods: the implicit trapezoidal method, an explicit stabilized third order algorithm implemented in the code DUMKA3 and the traditional explicit fourth order Runge-Kutta method written in the code RKSUITE. The results for high accuracy show that when the implicit trapezoidal method is combined with either explicit method the number of energy and gradient calculations can potentially be reduced by almost a half compared with integrating either method alone. Finally, to explain the improvements of the combined method we expand on the concepts of stability and stiffness and relate them to the efficiency of integration methods.     

Exponentially fitted symmetric and symplectic DIRK methods for oscillatory Hamiltonian systems

The order conditions for modified Runge–Kutta methods are derived via the rooted trees. Symmetry and symplecticity conditions and exponential fitting conditions for modified diagonally implicit Runge–Kutta (DIRK) are considered. Three new exponentially fitted symmetric and symplectic diagonally implicit Runge–Kutta (EFSSDIRK) methods of respective second order and fourth order are constructed. Phase properties of the new methods are analyzed. The new EFSSDIRK methods are applied to several Hamiltonian problems and compared to the results obtained by the existing symplectic DIRK methods in the literature.     

基于直接关系图方法的丁酸甲酯燃烧反应机理的框架简化

由于直接关系图(DRG)方法的概念简单和计算量较小,使得DRG方法目前已经成为详细燃烧反应机理框架简化的主流方法。DRG方法中评价物种之间依赖关系的相互作用系数和关系图连接权重的计算方法控制着DRG方法的简化效果。采用四种不同形式的相互作用系数的定义方法,分别与标准的DRG搜索算法和基于误差传播的搜索算法结合,构建了丁酸甲酯的框架燃烧反应机理。通过系统的误差分析比较了不同简化方法构建的框架机理的模拟可靠性。重点采用基于元素流量分析的反应路径分析方法研究了框架机理的化学动力学。最后,通过交集的思想构建了一个只包含96个物种的丁酸甲酯框架燃烧反应机理,并且反应路径分析结果表明丙烯的燃烧化学动力学在丁酸甲酯燃烧过程中占有重要地位。本文通过对不同DRG方法的系统比较研究表明了反应路径分析在框架机理可靠性验证中的重要性,对进一步发展更为有效的框架简化方法提供重要依据。     

Some embedded modified Runge-Kutta methods for the numerical solution of some specific Schrödinger equations

Some embedded Runge-Kutta methods for the numerical solution of the eigenvalue Schrödinger equation are developed. More specifically, a new embedded modified Runge-Kutta 4(6) Fehlberg method with minimal phase-lag and a block embedded Runge-Kutta-Fenlberg method are developed. For the numerical solution of the eigenvalue Schrödinger equation we investigate two cases. (i) The specific case, in which the potential V _x is an even function with respect to x. It is assumed, also, that the wavefunctions tend to zero for x → ± ∞. (ii) The general case for the well-known cases of the Morse potential and Woods-Saxon or Optical potential. Numerical and theoretical results show that the new approaches are more efficient compared with the well-known Runge-Kutta-Fehlberg 4(5) method.     

Improvement of methods for molecular dynamics integration

An application of symplectic implicit Runge–Kutta (RK ) integration schemes, the s-stage Gauss–Legendre Runge–Kutta (GLRK ) methods of order 2s, for the numerical solution of molecular dynamics (MD ) equation is described. The two-stage fourth-order GLRK method, the implicit midpoint rule, and the three-stage diagonally implicit RK method of order four are studied. The fixed-point iteraction was used for solving the resulting nonlinear system of equations. The algorithms were applied to a complex system of N particles interacting through a Lennard-Jones potential. The proposed symplectic methods for MD integration permit a wide range of time steps, are highly accurate and stable, and are thus suitable for the MD integration. © 1994 John Wiley & Sons, Inc.     

Synthesis of Pd@graphene oxide framework nanocatalyst with enhanced activity in Heck-Mizoroki cross-coupling reaction

A new method was developed for producing a catalyst involving a Pd nanoparticle (NP) embedded in a graphene oxide framework (Pd@GOF) with ordered macro- and mesoporous structures. First, 5,5′-diamino-2,2′-bipyridine was selected as cross-linking for covalent modification of GO nanosheets to prepare a three-dimensional (3D) framework with interlayer spaces in which well-dispersed and ultra-small Pd NPs in situ grew and embedded the framework. The synthesized nanopores 3D Pd@GOF can act as nanoreactors to help the reaction substrates thoroughly come into contact with the surface of Pd NPs, thereby exhibiting high activity toward the Heck reaction, rarely reported concerning Pd NPs supported on one-side functionalized graphene. The Pd@GOF catalyst can be used 10 times without any significant loss in the catalytic activity, confirming the long-term stability of this catalyst. Therefore, the covalently assembled GOF was proposed as a universal platform for hosting noble metal NPs to construct the desired metal@GOF nanocatalyst with improved activity and stability that can be used in a broad range of practical applications.     

Single‐ended transition state finding with the growing string method

Reaction path finding and transition state (TS) searching are important tasks in computational chemistry. Methods that seek to optimize an evenly distributed set of structures to represent a chemical reaction path are known as double‐ended string methods. Such methods can be highly reliable because the endpoints of the string are fixed, which effectively lowers the dimensionality of the reaction path search. String methods, however, require that the reactant and product structures are known beforehand, which limits their ability for systematic exploration of reactive steps. In this article, a single‐ended growing string method (GSM) is introduced which allows for reaction path searches starting from a single structure. The method works by sequentially adding nodes along coordinates that drive bonds, angles, and/or torsions to a desired reactive outcome. After the string is grown and an approximate reaction path through the TS is found, string optimization commences and the exact TS is located along with the reaction path. Fast convergence of the string is achieved through use of internal coordinates and eigenvector optimization schemes combined with Hessian estimates. Comparison to the double‐ended GSM shows that single‐ended method can be even more computationally efficient than the already rapid double‐ended method. Examples, including transition metal reactivity and a systematic, automated search for unknown reactivity, demonstrate the efficacy of the new method. This automated reaction search is able to find 165 reaction paths from 333 searches for the reaction of NH₃BH₃ and (LiH)₄, all without guidance from user intuition. © 2015 Wiley Periodicals, Inc.     

Proton Transfer Studied Using a Combined Ab Initio Reactive Potential Energy Surface with Quantum Path Integral Methodology

The rates of intramolecular proton transfer are calculated on a full-dimensional reactive electronic potential energy surface that incorporates high level ab initio calculations along the reaction path and by using classical Transition State theory, Path-Integral Quantum Transition State Theory, and the Quantum Instanton approach. The specific example problem studied is malonaldehyde. Estimates of the kinetic isotope effect using the latter two methods are found to be in reasonable agreement with each other. Improvements and extensions of this practical, yet chemically accurate framework for the calculations of quantized, reactive dynamics are also discussed.     

Rapid one-pot synthesis of riboflavin isotopomers

Flavocoenzymes labeled with stable isotopes are important reagents for the study of flavoproteins using isotope-sensitive methods such as NMR, ENDOR, infrared, and Raman spectroscopy. We describe highly versatile one-pot methods for the preparation of riboflavin isotopomers labeled with (13)C in every desired position of the xylene moiety. The starting materials are commercially available (13)C-labeled glucose samples, which are converted into riboflavin using enzymes of the oxidative pentose phosphate pathway in combination with recombinant enzymes of the riboflavin biosynthetic pathway. The overall reaction comprises six enzyme-catalyzed reaction steps for the synthesis of the vitamin and two auxiliary enzymes for in situ recycling of cofactors. The overall yields of riboflavin based on isotope-labeled glucose are 35-50%.     

New optimized explicit modified RKN methods for the numerical solution of the Schrödinger equation

This paper investigates a family of modified Runge-Kutta-Nyström (RKN) methods for the integration of second-order ordinary differential equations with oscillatory solutions. The order conditions for up to order five are presented. Two new optimized explicit four-stage modified RKN methods are derived by nullifying their dispersions and the dissipations in two different ways, respectively. These methods are checked to be of algebraic order five and both are dispersive of order six and dissipative of order five. The stability is examined and the error formulas are analyzed to show that advantages of the new methods compared with some highly efficient integrators from the recent literature. The high accuracy of the second new method is explained by its comparatively small dispersion and dissipation constants. In the integration of the resonance problem and the bound-states problem of the radial Schrödinger equation with the Woods-Saxon potential, the numerical results show the effectiveness and robustness of the new methods.     

An algorithm to find minimum free-energy paths using umbrella integration

The calculation of free-energy barriers by umbrella sampling and many other methods is hampered by the necessity for an a priori choice of the reaction coordinate along which to sample. We avoid this problem by providing a method to search for saddle points on the free-energy surface in many coordinates. The necessary gradients and Hessians of the free energy are obtained by multidimensional umbrella integration. We construct the minimum free-energy path by following the gradient down to minima on the free-energy surface. The change of free energy along the path is obtained by integrating out all coordinates orthogonal to the path. While we expect the method to be applicable to large systems, we test it on the alanine dipeptide in vacuum. The minima, transition states, and free-energy barriers agree well with those obtained previously with other methods.     

Implementation and performance of the artificial force induced reaction method in the GRRM17 program

This article reports implementation and performance of the artificial force induced reaction (AFIR) method in the upcoming 2017 version of GRRM program (GRRM17). The AFIR method, which is one of automated reaction path search methods, induces geometrical deformations in a system by pushing or pulling fragments defined in the system by an artificial force. In GRRM17, three different algorithms, that is, multicomponent algorithm (MC‐AFIR), single‐component algorithm (SC‐AFIR), and double‐sphere algorithm (DS‐AFIR), are available, where the MC‐AFIR was the only algorithm which has been available in the previous 2014 version. The MC‐AFIR does automated sampling of reaction pathways between two or more reactant molecules. The SC‐AFIR performs automated generation of global or semiglobal reaction path network. The DS‐AFIR finds a single path between given two structures. Exploration of minimum energy structures within the hypersurface in which two different electronic states degenerate, and an interface with the quantum mechanics/molecular mechanics method, are also described. A code termed SAFIRE will also be available, as a visualization software for complicated reaction path networks. © 2017 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Surfaces affect ion pairing

In water, positive ions attract negative ions. That attraction can be modulated if a hydrophobic surface is present near the two ions in water. Using computer simulations with explicit and implicit water, we study how an ion embedded on a hydrophobic surface interacts with another nearby ion in water. Using hydrophobic surfaces with different curvatures, we find that the contact interaction between a positive and negative ion is strongly affected by the curvature of an adjacent surface, either stabilizing or destabilizing the ion pair. We also find that the solvent-separated ion pair (SSIP) can be made more stable than the contacting ion pair by the presence of a surface. This may account for why bridging waters are often found in protein crystal structures. We also note that implicit solvent models do not account for SSIPs. Finally, we find that there are charge asymmetries: an embedded positive charge attracting a negative ion is different than an embedded negative charge attracting a positive ion. Such asymmetries are also not predicted by implicit solvent models. These results may be useful for improving computational models of solvation in biology and chemistry.     

Comparison of charge models for fixed-charge force fields: small-molecule hydration free energies in explicit solvent

In molecular simulations with fixed-charge force fields, the choice of partial atomic charges influences numerous computed physical properties, including binding free energies. Many molecular mechanics force fields specify how nonbonded parameters should be determined, but various choices are often available for how these charges are to be determined for arbitrary small molecules. Here, we compute hydration free energies for a set of 44 small, neutral molecules in two different explicit water models (TIP3P and TIP4P-Ew) to examine the influence of charge model on agreement with experiment. Using the AMBER GAFF force field for nonbonded parameters, we test several different methods for obtaining partial atomic charges, including two fast methods exploiting semiempirical quantum calculations and methods deriving charges from the electrostatic potentials computed with several different levels of ab initio quantum calculations with and without a continuum reaction field treatment of solvent. We find that the best charge sets give a root-mean-square error from experiment of roughly 1 kcal/mol. Surprisingly, agreement with experimental hydration free energies does not increase substantially with increasing level of quantum theory, even when the quantum calculations are performed with a reaction field treatment to better model the aqueous phase. We also find that the semiempirical AM1-BCC method for computing charges works almost as well as any of the more computationally expensive ab initio methods and that the root-mean-square error reported here is similar to that for implicit solvent models reported in the literature. Further, we find that the discrepancy with experimental hydration free energies grows substantially with the polarity of the compound, as does its variation across theory levels.     

Direct C-H bond arylation: selective palladium-catalyzed C2-arylation of N-substituted indoles

We present a new, practical method by which N-substituted indoles may be selectively arylated in the C2-position with good yields, low catalyst loadings, and a high degree of functional group tolerance. Our investigation found that two competitive processes, namely, the desired cross-coupling and biphenyl formation, were operative in this reaction. A simple kinetic model was formulated that proved to be instructive and provided useful guidelines for reaction optimization; the approach described within may prove to be useful in other catalytic cross-coupling processes.     

An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems

In this paper we present an optimized explicit Runge-Kutta method, which is based on a method of Fehlberg with six stages and fifth algebraic order and has improved characteristics of the phase-lag error. We measure the efficiency of the new method in comparison to other numerical methods, through the integration of the Schrödinger equation and three other initial value problems.     

The reaction path intrinsic reaction coordinate method and the Hamilton-Jacobi theory

The definition and location of an intrinsic reaction coordinate path is of crucial importance in many areas of theoretical chemistry. Differential equations used to define the path hitherto are complemented in this study with a variational principle of Fermat type, as Fukui [Int. J. Quantum Chem., Quantum Chem. Symp. 15, 633 (1981)] reported in a more general form some time ago. This definition is more suitable for problems where initial and final points are given. The variational definition can naturally be recast into a Hamilton-Jacobi equation. The character of the variational solution is studied via the Weierstrass necessary and sufficient conditions. The characterization of the local minima character of the intrinsic reaction coordinate is proved. Such result leads to a numerical algorithm to find intrinsic reaction coordinate paths based on the successive minimizations of the Weierstrass E-function evaluated on a guess curve connecting the initial and final points of the desired path.     

Automatic estimation of pressure-dependent rate coefficients

A general framework is presented for accurately and efficiently estimating the phenomenological pressure-dependent rate coefficients for reaction networks of arbitrary size and complexity using only high-pressure-limit information. Two aspects of this framework are discussed in detail. First, two methods of estimating the density of states of the species in the network are presented, including a new method based on characteristic functional group frequencies. Second, three methods of simplifying the full master equation model of the network to a single set of phenomenological rates are discussed, including a new method based on the reservoir state and pseudo-steady state approximations. Both sets of methods are evaluated in the context of the chemically-activated reaction of acetyl with oxygen. All three simplifications of the master equation are usually accurate, but each fails in certain situations, which are discussed. The new methods usually provide good accuracy at a computational cost appropriate for automated reaction mechanism generation.     

Reaction path determination for quantum mechanical/molecular mechanical modeling of enzyme reactions by combining first order and second order "chain-of-replicas" methods

A two-step procedure for the determination of reaction paths in enzyme systems is presented. This procedure combines two chain-of-states methods: a quantum mechanical/molecular mechanical (QM/MM) implementation of the nudged elastic band (NEB) method and a second order parallel path optimizer method both recently developed in our laboratory. In the first step, a reaction path determination is performed with the NEB method, along with a restrained minimization procedure for the MM environment to obtain a first approximation to the reaction path. In the second step, the calculated path is refined with the parallel path optimizer method. By combining these two methods the reaction paths are determined accurately, and in addition, the number of path optimization iterations are significantly reduced. This procedure is tested by calculating both steps of the isomerization of 2-oxo-4-hexenedioate by 4-oxalocrotonate tautomerase, which have been previously determined by our group. The calculated paths agree with the previously reported results and we obtain a reduction of 45%-55% in the number of path optimization cycles.     

A New Synthesis of α-Methylene Carbinols

The synthetic utility of α-methylene carbinols has prompted considerable recent preparative interest. However, many of the reported methods¹ lack scope and require reagents or reaction conditions which are incompatible with sensitive functional groups. We report below two closely related methods for the synthesis of α-methylene carbinols which are based on the reductive elimination of β,γ-epoxysulfones. Both methods give the desired products in moderate overall yield in a four-step sequence which is mild and experimentally facile.