Analysis of kinetic isotope effects for nonadiabatic reactions

Factors influencing the rates of quantum mechanical particle transfer reactions in many-body systems are discussed. The investigations are carried out on a simple model for a proton transfer reaction that captures generic features seen in more realistic models of condensed phase systems. The model involves a bistable quantum oscillator coupled to a one-dimensional double-well reaction coordinate, which is in turn coupled to a bath of harmonic oscillators. Reactive-flux correlation functions that involve quantum-classical Liouville dynamics for chemical species operators and quantum equilibrium sampling are used to estimate the reaction rates. Approximate analytical expressions for the quantum equilibrium structure are derived. Reaction rates are shown to be influenced significantly by both the quantum equilibrium structure and nonadiabatic dynamics. Nonadiabatic dynamical effects are found to play the major role in determining the magnitude of the kinetic isotope effect for the model transfer reaction.     

Few-body quantum and many-body classical hyperspherical approaches to reactions and to cluster dynamics

The hyperspherical method is a widely used and successful approach for the quantum treatment of elementary chemical processes. It has been mostly applied to three-atomic systems, and current progress is here outlined concerning the basic theoretical framework for the extension to four-body bound state and reactive scattering problems. Although most applications only exploit the advantages of the hyperspherical coordinate systems for the formulation of the few-body problem, the full power of the technique implies representations explicitly involving quantum hyperangular momentum operators as dynamical quantities and hyperspherical harmonics as basis functions. In terms of discrete analogues of these harmonics one has a universal representation for the kinetic energy and a diagonal representation for the potential (hyperquantization algorithm). Very recently, advances have been made on the use of the approach in classical dynamics, provided that a hyperspherical formulation is given based on “classical” definitions of the hyperangular momenta and related quantities. The aim of the present paper is to offer a retrospective and prospective view of the hyperspherical methods both in quantum and classical dynamics. Specifically, regarding the general quantum hyperspherical approaches for three- and four-body systems, we first focus on the basis set issue, and then we present developments on the classical formulation that has led to applications involving the implementations of hyperspherical techniques for classical molecular dynamics simulations of simple nanoaggregates.     

Exact quantum mechanical vibration-rotation kinetic energy operator of four-particle system in various coordinates

The vibration-rotational kinetic energy operators of four-particle system in various coordinates are derived using a new and simple angular momentum method. The operators are respectively suitable for studying the systems described by scattering coordinate, valence coordinate, Radau coordinate, Radau/Jacobi and Jacobi/valence hybrid coordinates and so on. Certain properties of these operators and their possible applications are discussed.     

Analytical treatment of Coriolis coupling for three-body systems

In a previous article [J. Chem. Phys. 108 (1998) 5216], an efficient method was presented for performing “exact” quantum calculations for the three-body rovibrational Hamiltonian, within the helicity-conserving approximation. This approach makes use of a certain three-body “effective potential,” enabling the same bend angle basis set to be employed for all values of the rotational quantum numbers, J, K and M. In the present work, the method is extended to incorporate Coriolis coupling, for which the relevant matrix elements are derived exactly. These can be used to solve the full three-body rovibrational problem, in the standard Jacobi coordinate vector embedding. Generalization of the method for coupled kinetic energy operators arising from other coordinate systems, embeddings, and/or system sizes, is also discussed.     

Thermal Gaussian molecular dynamics for quantum dynamics simulations of many-body systems: application to liquid para-hydrogen

A new method, here called thermal Gaussian molecular dynamics (TGMD), for simulating the dynamics of quantum many-body systems has recently been introduced [I. Georgescu and V. A. Mandelshtam, Phys. Rev. B 82, 094305 (2010)]. As in the centroid molecular dynamics (CMD), in TGMD the N-body quantum system is mapped to an N-body classical system. The associated both effective Hamiltonian and effective force are computed within the variational Gaussian wave-packet approximation. The TGMD is exact for the high-temperature limit, accurate for short times, and preserves the quantum canonical distribution. For a harmonic potential and any form of operator A?, it provides exact time correlation functions C(AB)(t) at least for the case of B, a linear combination of the position, x, and momentum, p, operators. While conceptually similar to CMD and other quantum molecular dynamics approaches, the great advantage of TGMD is its computational efficiency. We introduce the many-body implementation and demonstrate it on the benchmark problem of calculating the velocity time auto-correlation function for liquid para-hydrogen, using a system of up to N = 2592 particles.     

A canonical averaging in the second-order quantized Hamilton dynamics

Quantized Hamilton dynamics (QHD) is a simple and elegant extension of classical Hamilton dynamics that accurately includes zero-point energy, tunneling, dephasing, and other quantum effects. Formulated as a hierarchy of approximations to exact quantum dynamics in the Heisenberg formulation, QHD has been used to study evolution of observables subject to a single initial condition. In present, we develop a practical solution for generating canonical ensembles in the second-order QHD for position and momentum operators, which can be mapped onto classical phase space in doubled dimensionality and which in certain limits is equivalent to thawed Gaussian. We define a thermal distribution in the space of the QHD-2 variables and show that the standard beta=1/kT relationship becomes beta'=2/kT in the high temperature limit due to an overcounting of states in the extended phase space, and a more complicated function at low temperatures. The QHD thermal distribution is used to compute total energy, kinetic energy, heat capacity, and other canonical averages for a series of quartic potentials, showing good agreement with the quantum results.     

Simulating quantum dynamics in classical environments

Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum–classical systems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approximation. The dynamics of these composite systems are governed by a quantum–classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of trajectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many-body environments.     

The main beam correction term in kinetic energy release from metastable peaks

The correction term for the precursor ion signal width in determination of kinetic energy release is reviewed, and the correction term is formally derived. The derived correction term differs from the traditionally applied term. An experimental finding substantiates the inaccuracy in the latter. The application of the “T ‐value” to study kinetic energy release is found preferable to kinetic energy release distributions when the metastable peaks are slim and simple Gaussians. For electronically predissociated systems, a “borderline zero” kinetic energy release can be directly interpreted in reaction dynamics with strong curvature in the reaction coordinate.     

A computational study of the intramolecular deprotonation of a carbon acid in aqueous solution

Proton transfer reactions are the rate-limiting steps in many biological and synthetic chemical processes, often requiring complex cofactors or catalysts to overcome the generally unfavourable thermodynamic process of carbanion intermediate formation. It has been suggested that quantum tunnelling processes enhance the kinetics of some of these reactions, which when coupled to protein motions may be an important consideration for enzyme catalysis. To obtain a better fundamental and quantitative understanding of these proton transfer mechanisms, a computational analysis of the intramolecular proton transfer from a carbon acid in the small molecule, 4-nitropentanoic acid, in aqueous solution is presented. Potential-energy surfaces from gas-phase, implicit and QM/MM (quantum mechanical/molecular mechanical) explicit solvation quantum chemistry models are compared, and the potential of mean force, for the full reaction coordinate, using umbrella-sampling molecular dynamics is analysed. Semi-classical multidimensional tunnelling corrections are also used to estimate the quantum tunnelling contributions and to understand the origin of the primary deuterium kinetic isotope effects (KIEs). The computational results are found to be in excellent agreement with the KIEs and the energetics obtained experimentally.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II. Thermal correlation functions

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.     

Atom-atom partitioning of total (super)molecular energy: the hidden terms of classical force fields

Classical force fields describe the interaction between atoms that are bonded or nonbonded via simple potential energy expressions. Their parameters are often determined by fitting to ab initio energies and electrostatic potentials. A direct quantum chemical guide to constructing a force field would be the atom-atom partitioning of the energy of molecules and van der Waals complexes relevant to the force field. The authors used the theory of quantum chemical topology to partition the energy of five systems [H2, CO, H2O, (H2O)2, and (HF)2] in terms of kinetic, Coulomb, and exchange intra-atomic and interatomic contributions. The authors monitored the variation of these contributions with changing bond length or angle. Current force fields focus only on interatomic interaction energies and assume that these purely potential energy terms are the only ones that govern structure and dynamics in atomistic simulations. Here the authors highlight the importance of self-energy terms (kinetic and intra-atomic Coulomb and exchange).     

State-to-state reactive scattering in six dimensions using reactant-product decoupling: OH + H2 → H2O + H (J = 0)

We extend to full dimensionality a recently developed wave packet method [M. T. Cvitas? and S. C. Althorpe, J. Phys. Chem. A 113, 4557 (2009)] for computing the state-to-state quantum dynamics of AB + CD → ABC + D reactions and also increase the computational efficiency of the method. This is done by introducing a new set of product coordinates, by applying the Crank-Nicholson approximation to the angular kinetic energy part of the split-operator propagator and by using a symmetry-adapted basis-to-grid transformation to evaluate integrals over the potential energy surface. The newly extended method is tested on the benchmark OH + H(2) → H(2)O + H reaction, where it allows us to obtain accurately converged state-to-state reaction probabilities (on the Wu-Schatz-Fang-Lendvay-Harding potential energy surface) with modest computational effort. These methodological advances will make possible efficient calculations of state-to-state differential cross sections on this system in the near future.     

Non-adiabatic molecular dynamics with quantum solvent effects

Three novel approaches extending quantum-classical non-adiabatic (NA) molecular dynamics (MD) to include quantum effects of solvent environments are described. In a standard NA-MD the solute subsystem is treated quantum mechanically, while the larger solvent part of a system is treated classically. The three novel approaches presented here are based on the Bohmian formulation of quantum mechanics, the stochastic Schrödinger equation for the evolution of open quantum systems and the quantized Hamilton dynamics generalization of classical mechanics. The approaches extend the standard NA-MD to incorporate the following quantum effects of the solvent. (1) Branching, i.e. the ability of solvent quantum wave packets to split and follow asymptotically diverging trajectories correlated with different quantum states of the solute. (2) Decoherence, i.e. loss of quantum interference within the solute subsystem induced by the diverging solvent trajectories. (3) Zero point energy that contributes to NA coupling and must be preserved during the energy exchange between solvent and solute degrees of freedom. The Bohmian quantum-classical mechanics, stochastic mean-field and quantized mean-field approximations incorporate the quantum solvent effects into the standard quantum-classical NA-MD in a straightforward and efficient way that can be easily applied to quantum dynamics of condensed phase chemical systems.     

The non-Markovian quantum master equation in the collective-mode representation: application to barrier crossing in the intermediate friction regime

The calculation of chemical reaction rates in the condensed phase is a central preoccupation of theoretical chemistry. At low temperatures, quantum-mechanical effects can be significant and even dominant; yet quantum calculations of rate constants are extremely challenging, requiring theories and methods capable of describing quantum evolution in the presence of dissipation. In this paper we present a new approach based on the use of a non-Markovian quantum master equation (NM-QME). As opposed to other approximate quantum methods, the quantum dynamics of the system coordinate is treated exactly; hence there is no loss of accuracy at low temperatures. However, because of the perturbative nature of the NM-QME it breaks down for dimensionless frictions larger than about 0.1. We show that by augmenting the system coordinate with a collective mode of the bath, the regime of validity of the non-Markovian master equation can be extended significantly, up to dimensionless frictions of 0.5 over the entire temperature range. In the energy representation, the scaling goes as the number of levels in the relevant energy range to the third power. This scaling is not prohibitive even for chemical systems with many levels; hence we believe that the current method will find a useful place alongside the existing techniques for calculating quantum condensed-phase rate constants.     

A theoretical study on excited state double proton transfer reaction of a 7-azaindole dimer: an ab initio potential energy surface and its empirical valence bond model

Double proton transfer (DPT) reaction of a 7-azaindole dimer in the first ππ* electronically excited state was studied theoretically. We investigated the reaction mechanism through constructing a full dimensional empirical valence bond potential energy function (PEF) based on potential energies evaluated by ab initio molecular orbital methods, and carrying out quantum dynamics calculations with the PEF. Potential energy surfaces of the DPT obtained at the multi-reference perturbation level of theory favors a concerted DPT mechanism, although a stepwise channel is suggested to open for an excited initial vibrational state. Reduced two dimensional quantum dynamics calculations for a reaction surface Hamiltonian of DPT coordinates were performed. Time constants of the reaction were evaluated to be on the order of picoseconds, which is consistent with experiments. On the other hand, the computed kinetic isotope effect deviates from experimental evidence, suggesting the importance of intermolecular stretching motion, which is not explicit in the present calculations for the quantum effect.     

Kinetic Energy Partition Method for Competing Modes

We present a new basis set expansion method for quantum dynamics systems with two competing modes where the interaction potentials are equally dominant. The new idea introduced here is a kinetic energy partition scheme instead of the usual division of the potential energy. The partition results in two kinetic energy terms with their effective masses. By distributing each partial kinetic energy to the respective potential, the full Hamiltonian can be expressed as the sum of the two competing modes. The solution procedure is illustrated by using a system consisting of a particle under the action of two harmonic potentials with different equilibrium distances and force constants. Next we apply this method to obtain the potential energy curves for the prototype hydrogen molecule ion. This new expansion converges very fast to the exact solutions for both eigenvalues and eigenfunctions.     

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.