Mixed Quantum Classical Reaction Rates based on the Phase Space Formulation of the Hierarchical Equations of Motion

In a previous work [J. Chem. Phys. 140 , 174105 (2014)], we have shown that a mixed quantum classical (MQC) rate theory can be derived to investigate the quantum tunneling effects in the proton transfer reactions. However, the method is based on the high temperature approximation of the hierarchical equation of motion (HEOM) with the Debye-Drude spectral density, and results in a multi-state Zusman type of equation. We now extend this theory to include quantum effects of the bath degrees of freedom. By writing the full HEOM into a multidimensional partial differential equation in phase space, we can define a new reaction coordinate, and the previous method can be generalized to the full quantum regime. The validity of the new method is demonstrated by using numerical examples, including the spin-Boson model, and the double well model for proton transfer reaction. The new method is found to resolve some key problems of the previous theory based on high temperature approximation, including possible numerical instability in long time simulation and wrong rate constant at low temperatures.     

Quantum generalized Langevin equation approach to gas/solid collisions

The classical generalized Langevin equation (GLE) approach to gas/solid collisions is generalized to quantum scattering. Using Feynman's method of partial path integration, the full gas/solid propagator is reduced to a form in which only the dynamics of the incident atom and the surface oscillator(s) directly struck appear explicitly. Solving this effective dynamical problem in the semiclassical limit yields a stationary phase equation of motion identical in form to the classical GLE. The noise, however, is distributed according to quantum rather than classical statistics. From the GLE a quantum phase can be constructed and an S-matrix computed. The resulting theory is capable of describing inelastic-diffractive scattering which has been seen experimentally by Williams.     

Time propagation of constrained coupled Gaussian wave packets

The dynamics of quantum systems can be approximated by the time propagation of Gaussian wave packets. Applying a time dependent variational principle, the time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations. Unfortunately, the set of equations is ill behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. We present a general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well behaved. The power of the method is demonstrated for a nonintegrable system with two degrees of freedom.     

Molecular structure calculations without clamping the nuclei

A number of recent papers have considered ways in which molecular structure may be calculated when both the electrons and the nuclei are treated from the outset as quantum particles. This is in contrast to the conventional approach in which the nuclei initially have their positions fixed and so merely provide a potential for electronic motion. The usual approach is generally assumed to be justified by the 1927 work of Born and Oppenheimer. In this paper we discuss what precisely might be anticipated in the way of molecular structure from a mathematical consideration of the spectral properties of the full Coulomb Hamiltonian, to what extent the very idea of molecular structure might be dependent upon treating the nuclei simply as providing a potential and the extent to which the work of Born and Oppenheimer can be used to support this position.     

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.     

Intermolecular torsional motion of a π-aggregated dimer probed by two-dimensional electronic spectroscopy

The energetic splitting of the two exciton states of a molecular dimer depends strongly on the relative orientation of the monomers with respect to each other. The curvature of the corresponding adiabatic potential energy surfaces can lead to torsional motion of the monomers. It has been suggested recently that this torsional motion could provide a possible relaxation mechanism for the upper state which proceeds via a crossing of the two singly excited state potentials. Another, competing, relaxation mechanism is provided by coupling to the environment, leading to direct exciton relaxation. Here we examine theoretically the combined dynamics of torsional motion and excitonic relaxation for a π-aggregated dimer. Using two-dimensional (2D) spectroscopy, it is shown how torsional motion through a crossing of the adiabatic excitonic potential surfaces could be distinguished from direct relaxation. For the calculations a mixed quantum/classical approach is used, where the torsional motion is treated by an Ehrenfest type of equation, while the excitonic dynamics including dephasing and direct relaxation is described by a quantum master equation.     

Simulation of tunneling in enzyme catalysis by combining a biased propagation approach and the quantum classical path method: application to lipoxygenase

The ability of using wave function propagation approaches to simulate isotope effects in enzymes is explored, focusing on the large H/D kinetic isotope effect of soybean lipoxygenase-1 (SLO-1). The H/D kinetic isotope effect (KIE) is calculated as the ratio of the rate constants for hydrogen and deuterium transfer. The rate constants are calculated from the time course of the H and D nuclear wave functions. The propagations are done using one-dimensional proton potentials generated as sections from the full multidimensional surface of the reacting system in the protein. The sections are obtained during a classical empirical valence bond (EVB) molecular dynamics simulation of SLO-1. Since the propagations require an extremely long time for treating realistic activation barriers, it is essential to use an effective biasing approach. Thus, we develop here an approach that uses the classical quantum path (QCP) method to evaluate the quantum free energy change associated with the biasing potential. This approach provides an interesting alternative to full QCP simulations and to other current approaches for simulating isotope effects in proteins. In particular, this approach can be used to evaluate the quantum mechanical transmission factor or other dynamical effects, while still obtaining reliable quantized activation free energies due to the QCP correction.     

Does ℏ play a role in multidimensional spectroscopy? Reduced hierarchy equations of motion approach to molecular vibrations

To investigate the role of quantum effects in vibrational spectroscopies, we have carried out numerically exact calculations of linear and nonlinear response functions for an anharmonic potential system nonlinearly coupled to a harmonic oscillator bath. Although one cannot carry out the quantum calculations of the response functions with full molecular dynamics (MD) simulations for a realistic system which consists of many molecules, it is possible to grasp the essence of the quantum effects on the vibrational spectra by employing a model Hamiltonian that describes an intra- or intermolecular vibrational motion in a condensed phase. The present model fully includes vibrational relaxation, while the stochastic model often used to simulate infrared spectra does not. We have employed the reduced quantum hierarchy equations of motion approach in the Wigner space representation to deal with nonperturbative, non-Markovian, and nonsecular system-bath interactions. Taking the classical limit of the hierarchy equations of motion, we have obtained the classical equations of motion that describe the classical dynamics under the same physical conditions as in the quantum case. By comparing the classical and quantum mechanically calculated linear and multidimensional spectra, we found that the profiles of spectra for a fast modulation case were similar, but different for a slow modulation case. In both the classical and quantum cases, we identified the resonant oscillation peak in the spectra, but the quantum peak shifted to the red compared with the classical one if the potential is anharmonic. The prominent quantum effect is the 1-2 transition peak, which appears only in the quantum mechanically calculated spectra as a result of anharmonicity in the potential or nonlinearity of the system-bath coupling. While the contribution of the 1-2 transition is negligible in the fast modulation case, it becomes important in the slow modulation case as long as the amplitude of the frequency fluctuation is small. Thus, we observed a distinct difference between the classical and quantum mechanically calculated multidimensional spectra in the slow modulation case where spectral diffusion plays a role. This fact indicates that one may not reproduce the experimentally obtained multidimensional spectrum for high-frequency vibrational modes based on classical molecular dynamics simulations if the modulation that arises from surrounding molecules is weak and slow. A practical way to overcome the difference between the classical and quantum simulations was discussed.     

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.     

Mixed quantum-classical molecular dynamics simulation of vibrational relaxation of ions in an electrostatic field

The vibrational relaxation of ions in low-density gases under the action of an electrostatic field is reproduced through a molecular dynamics simulation method. The vibration is treated though quantum mechanics and the remaining degrees of freedom are considered classical. The procedure is tested through comparison against analytic results for a two-dimensional quantum model and by studying energy exchange during binary ion-atom collisions. Finally, the method has been applied successfully to the calculation of the mobility and the vibrational relaxation rate of O2+ in Kr as a function of the mean collision energy using a model interaction potential that reproduces the potential minimum of a previously known ab initio potential surface. The calculation of the steady mean vibrational motion of the ions in (flow) drift tubes seems straightforward, though at the expense of large amounts of computer time.     

Incorporation of quantum effects for selected degrees of freedom into the trajectory-based dynamics using spatial domains

The approach of defining quantum corrections on nuclear dynamics of molecular systems incorporated approximately into selected degrees of freedom, is described. The approach is based on the Madelung-de-Broglie-Bohm formulation of time-dependent quantum mechanics which represents a wavefunction in terms of an ensemble of trajectories. The trajectories follow classical laws of motion except that the quantum potential, dependent on the wavefunction amplitude and its derivatives, is added to the external, classical potential. In this framework the quantum potential, determined approximately for practical reasons, is included only into the "quantum" degrees of freedom describing light particles such as protons, while neglecting with the quantum force for the heavy, nearly classical nuclei. The entire system comprised of light and heavy particles is described by a single wavefunction of full dimensionality. The coordinate space of heavy particles is divided into spatial domains or subspaces. The quantum force acting on the light particles is determined for each domain of similar configurations of the heavy nuclei. This approach effectively introduces parametric dependence of the reduced dimensionality quantum force, on classical degrees of freedom. This strategy improves accuracy of the quantum force and does not restrict interaction between the domains. The concept is illustrated for two-dimensional scattering systems, where the quantum force is required to reproduce vibrational energy of the quantum degree of freedom.     

Wigner function approach to the quantum Brownian motion of a particle in a potential

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.     

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.     

An improved Hellmann-type pseudopotential for atoms and molecules

The concept of pseudopotentials offers much attractiveness for the quantum mechanical evaluation of the physical properties of atoms and molecules. The ideas of Hellmann, in which the repulsive and fermion character of inner electrons can be mimicked by an experimentally fitted, exponentially damped potential term, are especially attractive. Unfortunately, it is found that such a simple expression can only be used in a very limited number of cases, such as for the alkali metals, and even then fails for the simple case of lithium. The present study shows that the Hellmann idea can readily be extended by including a second “shielded potential” term evaluated from tabulated previous Hartree-Fock calculations. The new expression for the model pseudopotential is both simple and effective. With it, the inner potential of any of the alkali metal atoms, including lithium, can be represented so that calculation of the molecular properties of the metal dimers can be accomplished. Calculations for Li₂, Na₂, and K₂ show the binding energies and equilibrium interatomic distances to be quite well given, in agreement with both chemical experience and spectroscopic evidence.     

Effect of solvation on the ionization of guanine nucleotide: A hybrid QM/EFP study

Ionization of nucleobases is affected by their biological environment, which includes both the effect of adjacent nucleotides as well as the presence of water around it. Guanine and its nucleotide have the lowest ionization potentials among the various DNA bases. Therefore, the threshold of ionization is dependent on that of guanine and its characterization is crucial to the prediction of interaction of light with DNA. We investigate the effect of solvation on the vertical ionization energies (VIEs) of guanine and its nucleotide. In this work, we have used hybrid quantum mechanics/molecular mechanics (QM/MM) approach with effective fragment potential as the MM method of choice and equation‐of‐motion coupled‐cluster for ionization potential with singles and doubles (EOM‐IP‐CCSD) as the QM method. The performance of the hybrid scheme with respect to the full QM method shows an accuracy of 0.02–0.04 eV. The lowest few ionizations of the nucleotide are found to be from different parts of the moiety, that is, the nucleic acid base, phosphate, or sugar, and these ionization energies are very closely spaced giving rise to a very complicated spectrum. Furthermore, microsolvation has large effects on these ionizations and can lead to red or blue shift depending on the position of the water molecule. Even a single water molecule can change the order of ionized states in the nucleotide. The VIEs of the bulk solvated chromophores are predicted and compared to existing experimental spectra. The predominant role of polarization in the solvatochromic shift is noticed. © 2017 Wiley Periodicals, Inc.     

Electronic Energy Spectrum of Two-Dimensional Solids and a Chain of C Atoms from a Quantum Network Model

The aim of this study is to explain how a quantum network can be used as simple model to calculate complex band structures. The paper contains an introduction, a mathematical exposure of the method, and applications to graphene, boron nitride, and polyacetylene chains. Using a quantum network is a simple, intuitive, and, yet, rather accurate way to obtain a band structure for a complex material. One focus here is to invoke physical and chemical intuition to construct the effective one-body potential along the wires of a quantum network.