A quantum equation of motion for chemical reaction systems on an adiabatic double-well potential surface in solution based on the framework of mixed quantum-classical molecular dynamics

We present a quantum equation of motion for chemical reaction systems on an adiabatic double-well potential surface in solution in the framework of mixed quantum-classical molecular dynamics, where the reactant and product states are explicitly defined by dividing the double-well potential into the reactant and product wells. The equation can describe quantum reaction processes such as tunneling and thermal excitation and relaxation assisted by the solvent. Fluctuations of the zero-point energy level, the height of the barrier, and the curvature of the well are all included in the equation. Here, the equation was combined with the surface hopping technique in order to describe the motion of the classical solvent. Applying the present method to model systems, we show two numerical examples in order to demonstrate the potential power of the present method. The first example is a proton transfer by tunneling where the high-energy product state was stabilized very rapidly by solvation. The second example shows a thermal activation mechanism, i.e., the initial vibrational excitation in the reactant well followed by the reacting transition above the barrier and the final vibrational relaxation in the product well.     

Generalized electronic diabatic approach to structural similarity and the Hammond postulate

We revisit the notion of structural similarity along a reaction path within the context of a generalized electronic diabatic (GED) molecular model. In this approach, a reaction involving two closed‐shell stable species is described as the evolution of a quantum state that superimposes at least three diabatic electronic species (reactant, product, and an open‐shell transition state) coupled by an external electromagnetic field. Reactant and product amplitudes in this general state are also modulated by changing the geometry of a system of classical positive charges interacting with the electrons. By mapping these amplitudes over nuclear configurational space, we can follow the total quantum state along a reaction coordinate and establish its similarity to each of the diabatic species. As a result, chemical processes, and useful notions such as those of energy barriers and the Hammond postulate, emerge as consequence of Franck–Condon‐like transitions between quantum states. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

A quantum theory of chemical processes and reaction rates based on diabatic electronic functions coupled in an external field

The method discussed in this work provides a theoretical framework where simple chemical reactions resemble any other standard quantum process, i.e., a transition in quantum state mediated by the electromagnetic field. In our approach, quantum states are represented as a superposition of electronic diabatic basis functions, whose amplitudes can be modulated by the field and by the external control of nuclear configurations. Using a one-dimensional three-state model system, we show how chemical structure and dynamics can be represented in terms of these control parameters, and propose an algorithm to compute the reaction probabilities. Our analysis of effective energy barriers generalizes previous ideas on structural similarity between reactant, and product, and transition states using the geometry of conventional reaction paths. In the present context, exceptions to empirical rules such as the Hammond postulate appear as effects induced by the environment that supplies the external field acting on the quantum system.     

Generalized electronic diabatic approach to structural similarity in two‐dimensional potential energy surfaces of various topologies

Active site properties in some proteins can be affected by conformational fluctuations of neighbor residues, even when the latter are not involved directly in the binding process. A local environment thus appears to alter the relevant potential energy surface and its reaction paths. Here, some aspects of this phenomenon are simulated within a generalized electronic diabatic (GED) scheme to study the geometry and structural similarity for a class of two‐dimensional (2D) energy surfaces. The electronic quantum state is a linear superposition of diabatic basis functions, each of which is taken to represent a single (pure) electronic state for the isolated material system. Here, we describe a model reaction of isomerization by shifts in amplitudes for three diabatic species (reactant, product, and an open‐shell transition state) coupled in an external field. The “effective” 2D energy surface in the field is characterized in terms of critical points, and the amplitudes along the main reaction paths. A new feature is the introduction of a phase diagram where all possible potential‐energy‐surface topologies (consistent with three‐state systems in two linear coordinates) are matched with actual model parameters. By varying the coupling strengths between diabatic states, we classify regions of this phase diagram in terms of electronic and structural similarities; some regions comprise models whose reaction paths have geometries that belong to the catchment region of the reactant, yet are electronically akin to the diabatic transition state or product. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

A Three-State Model for Electronic Transitions Represented in a Generalized Diabatic Approach

We describe the chemical change between two diabatic closed-shell states as an electronic transition mediated by two factors: a bound diabatic transition state and the electromagnetic field. Using a three-state model for bond breaking, we compute the amplitudes of the total quantum state on the diabatic reactant, product, and transition states as a function of the external field. Changes in the total electronic state appear as sharp transitions between diabatic basis functions for particular configurations of the set of external positive charges. Depending on the diabatic states and the external field, the model predicts the possible occurrence of energy barriers for breaking or forming covalent bonds.     

Hydration effects on the reaction with an open-shell transition state: QM/MM-ER study for the dehydration reaction of alcohol in hot water

The reaction mechanism for the dehydration of 1,4-butanediol in hot water has been investigated by means of the hybrid quantum mechanical/molecular mechanical approach combined with the theory of energy representation (QM/MM-ER). We have assumed that the proton transfers along the hydrogen bonds of the water molecules catalyze the reaction, where the transition state (TS) forms a singlet biradical electronic structure. It has been revealed by the simulation that the biradical electronic state at the TS changes to zwitterionic structure in solution due to the hydration of the polar solvent. Such the electronic structure change gives rise to the substantial stabilization of the TS in hot water. As a result, the water-catalytic path becomes more favorable in aqueous solution than another possible path that proceeds without proton transfers as opposed to the reaction mechanism in the gas phase. Furthermore, the activation free energy computed by the present method is in excellent agreement with the experimental result.     

Franck-Condon theory of reactive scattering

A Franck-Condon theory of reactive scattering is introduced which generalizes previous works in a number of respects. We consider the collinear AB + C → A + BC reaction where A, B, and C may be polyatomic species and show how the multi-dimensional continuum-continuum Franck-Condon integral can exactly be reduced to a two-dimensional one involving the nonorthogonal reaction coordinates on the reactant and product diabatic surfaces. These integrals are then written in a rapidly convergent series of products of one-dimensional bound-continuum integrals of a form closely related to those studied in recent theories of photodissociation where accurate analytical and numerical methods are available for treating them. The theory is applied to the case of the D + Hl isotope exchange for a model surface to enable a comparison of exact quantum calculations with those of the Franck-Condon theory for the identical surface in both cases. The calculations are all performed at energies where the reaction is classically forbidden. The relative Dl vibrational distributions (as a function of initial Hl state) are accurately reproduced in a fashion that is fairly insensitive to the choice of the reactant and product diabatic surfaces, but the absolute probabilities are shown to be sensitively dependent on this choice.     

A diabatic representation including both valence nonadiabatic interactions and spin-orbit effects for reaction dynamics

A diabatic representation is convenient in the study of electronically nonadiabatic chemical reactions because the diabatic energies and couplings are smooth functions of the nuclear coordinates and the couplings are scalar quantities. A method called the fourfold way was devised in our group to generate diabatic representations for spin-free electronic states. One drawback of diabatic states computed from the spin-free Hamiltonian, called a valence diabatic representation, for systems in which spin-orbit coupling cannot be ignored is that the couplings between the states are not zero in asymptotic regions, leading to difficulties in the calculation of reaction probabilities and other properties by semiclassical dynamics methods. Here we report an extension of the fourfold way to construct diabatic representations suitable for spin-coupled systems. In this article we formulate the method for the case of even-electron systems that yield pairs of fragments with doublet spin multiplicity. For this type of system, we introduce the further simplification of calculating the triplet diabatic energies in terms of the singlet diabatic energies via Slater's rules and assuming constant ratios of Coulomb to exchange integrals. Furthermore, the valence diabatic couplings in the triplet manifold are taken equal to the singlet ones. An important feature of the method is the introduction of scaling functions, as they allow one to deal with multibond reactions without having to include high-energy diabatic states. The global transformation matrix to the new diabatic representation, called the spin-valence diabatic representation, is constructed as the product of channel-specific transformation matrices, each one taken as the product of an asymptotic transformation matrix and a scaling function that depends on ratios of the spin-orbit splitting and the valence splittings. Thus the underlying basis functions are recoupled into suitable diabatic basis functions in a manner that provides a multibond generalization of the switch between Hund's cases in diatomic spectroscopy. The spin-orbit matrix elements in this representation are taken equal to their atomic values times a scaling function that depends on the internuclear distances. The spin-valence diabatic potential energy matrix is suitable for semiclassical dynamics simulations. Diagonalization of this matrix produces the spin-coupled adiabatic energies. For the sake of illustration, diabatic potential energy matrices are constructed along bond-fission coordinates for the HBr and the BrCH(2)Cl molecules. Comparison of the spin-coupled adiabatic energies obtained from the spin-valence diabatics with those obtained by ab initio calculations with geometry-dependent spin-orbit matrix elements shows that the new method is sufficiently accurate for practical purposes. The method formulated here should be most useful for systems with a large number of atoms, especially heavy atoms, and/or a large number of spin-coupled electronic states.     

Theoretical study on solvation effects in chemical reactions: A vibrational coupling model

A vibrational coupling model to treat the solvation effects in chemical reaction rate calculations is proposed and applied to the intramolecular hydrogen transfer reaction CH₃O· → ·CH₂OH in the condensed phase. The effect of solvation is taken into account in two ways: (1) the solvent effect on the activation energy of the reaction is simulated by including 39 surrounding water molecules, represented by fractional charges at the assumed atomic positions, in the potential energy surface calculation; and (2) the vibrational couplings between the 10 nearest solvent molecules and the molecules constituting the reaction system are explicitly included in a vibrational frequency calculation. RRKM theory with Miller's tunneling correction included is employed to calculate the rate constants. The effect of solvation causes a significant change in the chemical reaction rate, mainly through a lowering of the activation energy. The vibrational coupling causes a slight increase of the rate constant in the tunneling region by perturbing the vibrational frequencies of the reactant and transition states, which appear in the rate-constant expression, but has little effect at higher temperatures.     

Characterizing the mechanism of the double proton transfer in the formamide dimer

The double proton transfer in the formamide dimer is characterized computationally by combining density functional theory and ab initio methods. The intrinsic reaction coordinate (IRC) is obtained at the B3LYP level of theory. Energies of several points along the IRC are treated by the more rigorous focal point method to test the validity of the B3LYP functional. The reaction mechanism is examined in terms of the energy profile, the reaction force, the chemical potential, and the reaction electronic flux. The energy profile for the activation process of the formamide dimer to the imino ether product obtained with the B3LYP functional is in agreement with the results of the focal point method. Together with the reaction force analysis and the reaction electronic flux a precise assignment of the structural and electronic contributions to the activation barrier becomes possible. The results show that the reaction starts with a structural rearrangement, where the two dimers approach each other, and is followed by electronic changes before the system reaches the transition state. This electronic contribution to the activation barrier steers the activation process. After the transition state is reached, deviations of the B3LYP functional from the more accurate focal point energies become apparent, where the errors may be rationalized in terms of the treatment of exchange. The inconsistency could be assigned to the incapacity of the functional to describe delocalization effects over the whole system.     

Infrared spectra of a model phenol-amine proton transfer complex in nanoconfined CH3Cl

The vibrational spectra of a model phenol-amine proton transfer complex dissolved in CH3Cl solvent confined in a 12 A radius spherical hydrophobic cavity were calculated using mixed quantum-classical molecular dynamics simulations. The reaction free energy of the proton transfer complex was varied in order to explore the contributions to the vibrational absorption band from product and reactant species. The vibrational spectra of the model proton transfer complex resulted in motionally narrowed spectral linewidths with two distinct peaks for products and reactants in cases where the system undergoes chemical exchange. It was found that the n=1 and n=2 vibrational excited states combine to form diabatic states such that the spectra have contributions from both n=0 --> n=1 and n=0 --> n=2 transitions. A strong relationship between the instantaneous vibrational frequency and a collective solvent coordinate was found that assists in understanding the origin of the spectral features.     

Kramers problem for nonequilibrium current-induced chemical reactions

We discuss the use of tunneling electron current to control and catalyze chemical reactions. Assuming the separation of time scales for electronic and nuclear dynamics we employ Langevin equation for a reaction coordinate. The Langevin equation contains nonconservative current-induced forces and gives nonequilibrium, effective potential energy surface for current-carrying molecular systems. The current-induced forces are computed via Keldysh nonequilibrium Green's functions. Once a nonequilibrium, current-depended potential energy surface is defined, the chemical reaction is modeled as an escape of a Brownian particle from the potential well. We demonstrate that the barrier between the reactant and the product states can be controlled by the bias voltage. When the molecule is asymmetrically coupled to the electrodes, the reaction can be catalyzed or stopped depending on the polarity of the tunneling current.     

Novel quantum mechanical/molecular mechanical method combined with the theory of energy representation: free energy calculation for the Beckmann rearrangement promoted by proton transfers in the supercritical water

The Beckmann rearrangement of acetone oxime promoted by proton transfers in the supercritical water has been investigated by means of the hybrid quantum mechanical/molecular mechanical approach combined with the theory of energy representation (QM/MM-ER) recently developed. The transition state (TS) structures have been explored by ab initio calculations for the reaction of hydrated acetone oxime on the assumption that the reaction is catalyzed by proton transfers along the hydrogen bonds connecting the solute and the solvent water molecules. Up to two water molecules have been considered as reactants that take part in the proton transfers. As a result of the density functional theory calculations with B3LYP functional and aug-cc-pVDZ basis set, it has been found that participation of two water molecules in the reaction reduces the activation free energy by -12.3 kcal/mol. Furthermore, the QM/MM-ER simulations have revealed that the TS is more stabilized than the reactant state in the supercritical water by 2.7 kcal/mol when two water molecules are involved in the reaction. Solvation free energies of the reactant and the TS have been decomposed into terms due to the electronic polarization of the solute, electron density fluctuation, and others to elucidate the origin of the stabilization of the TS as compared with the reactant. It has been revealed that the promotion of the chemical reaction due to the hydration mainly originates from the interaction between the nonpolarized solute and the solvent water molecules at the supercritical state.     

(4-溴甲基双环[4.4.1]-1,3,5,7,9-十一碳五烯基-3-)甲醇环氧化反应机理的理论研究

A reaction mechanism of epoxidation reaction of the 4-(bromomethyl)bicyclo4.4.1undeca-1,3,5,7,9-pentaen-3-ylmethanol has been studied by using the density functional theory(DFT) method at B3LYP level with 6 31G* basis set. The geometric structures of reactant, product and transition state have been optimized. The transition state is found by the QST2 method and characterized by the vibration frequency analysis. The intrinsic reaction coordinate(IRC) for this reaction is traced and confirms the reaction mechanism. The changes and the nature of related chemical bonds along the IRC path have been analyzed by the theory of electronic charge density. The result shows that the elimination reaction and ring closing reaction are synergistic, and the hydrogen in the elimination reaction derives from hydroxyl. In addition, the activation energy of the reaction is 139.2 kJ/mol.     

Free energy and dynamics of electron-transfer reactions in a room temperature ionic liquid

Reaction free energetics and dynamics of unimolecular electron-transfer processes in ionic liquid 1-ethyl-3-methylimidazolium hexafluorophosphate (EMI+PF6-) are investigated via molecular dynamics computer simulations employing a model diatomic solute and compared with those in aprotic acetonitrile. Using the free energy perturbation method, diabatic free energy curves relevant to charge separation and recombination processes are studied over a wide range of the reaction coordinate. The diabatic curves are found to vary with the solute charge distribution, especially in EMI+PF6-. Nevertheless, if the free energy of reaction is not that substantial, the Marcus free energy relationship holds reasonably well, provided that the reorganization free energy averaged between the reactant and product states is employed. The effective polarity, measured as solvation-induced stabilization of dipolar solutes, is higher for EMI+PF6- than for acetonitrile, consonant with many solvatochromic measurements. Thus, in the normal regime, activation barriers for charge separation and recombination reactions are, respectively, lower and higher in EMI+PF6- than in acetonitrile. The influence of solvent dynamics on reaction kinetics through modulations of activation, deactivation, and barrier crossing is analyzed. Even though overall solvent relaxation dynamics in EMI+PF6- are considerably slower than those in acetonitrile, the deviation of the rate constant from the transition state theory predictions is found to be small for both solvents. Implications of this finding for other reactions in ionic liquids are briefly discussed.     

Why and how do systems react in thermally fluctuating environments?

Many chemical reactions, including those of biological importance, take place in thermally fluctuating environments. Compared to isolated systems, there arise markedly different features due to the effects of energy dissipation through friction and stochastic driving by random forces reflecting the fluctuation of the environment. Investigation of how robustly the system reacts under the influence of thermal fluctuation, and elucidating the role of thermal fluctuation in the reaction are significant subjects in the study of chemical reactions. In this article, we start with overviewing the generalized Langevin equation (GLE), which has long been used and continues to be a powerful tool to describe a system surrounded by a thermal environment. It has been also generalized further to treat a nonstationary environment, in which the conventional fluctuation-dissipation theorem no longer holds. Then, within the framework of the Langevin equation we present a method recently developed to extract a new reaction coordinate that is decoupled from all the other coordinates in the region of a rank-one saddle linking the reactant and the product. The reaction coordinate is buried in nonlinear couplings among the original coordinates under the influence of stochastic random force. It was ensured that the sign of this new reaction coordinate (= a nonlinear functional of the original coordinates, velocities, friction, and random force) at any instant is sufficient to determine in which region, the reactant or the product, the system finally arrives. We also discuss how one can extend the method to extract such a coordinate from the GLE framework in stationary and nonstationary environments, where memory effects exist in dynamics of the reaction.     

Diabatic couplings for charge recombination via Boys localization and spin-flip configuration interaction singles

We describe a straightforward technique for obtaining diabatic couplings applicable to charge transfer from or charge recombination to the electronic ground state. Our method is nearly black box, requiring minimal chemical intuition from the user, and merges two well-established approaches in electronic structure theory: first, smooth and balanced adiabatic states are generated using spin-flip-configuration interaction singles (SF-CIS) based on a triplet HF state; second, Boys localization is applied to rotate all adiabatic states into charge-localized diabatic states. The method is computationally inexpensive, scaling only with the cost of CIS, and does not require a choice of active space, which is usually required for such intrinsically multiconfigurational problems. Molecular LiF in vacuum and LiF solvated by a single water molecule are examined as model systems. We find nearly smooth diabatic potential energy surfaces and couplings and we find that the Condon approximation is obeyed approximately for this model problem.     

Multipath variational transition state theory: rate constant of the 1,4-hydrogen shift isomerization of the 2-cyclohexylethyl radical

We propose a new formulation of variational transition state theory called multipath variational transition state theory (MP-VTST). We employ this new formulation to calculate the forward and reverse thermal rate constant of the 1,4-hydrogen shift isomerization of the 2-cyclohexylethyl radical in the gas phase. First, we find and optimize all the local-minimum-energy structures of the reaction, product, and transition state. Then, for the lowest-energy transition state structures, we calculate the reaction path by using multiconfiguration Shepard interpolation (MSCI) method to represent the potential energy surface, and, from this representation, we also calculate the ground-state vibrationally adiabatic potential energy curve, the reaction-path curvature vector, and the generalized free energy of activation profile. With this information, the path-averaged generalized transmission coefficients <γ> are evaluated. Then, thermal rate constant containing the multiple-structure anharmonicity and torsional anharmonicity effects is calculated using multistructural transition state theory (MS-TST). The final MP-VTST thermal rate constant is obtained by multiplying k(MS-T)(MS-TST) by <γ>. In these calculations, the M06 density functional is utilized to compute the energy, gradient, and Hessian at the Shepard points, and the M06-2X density functional is used to obtain the structures (conformers) of the reactant, product, and the saddle point for computing the multistructural anharmonicity factors.