Semiclassical nonadiabatic dynamics using a mixed wave-function representation

Nonadiabatic effects in quantum dynamics are described using a mixed polar/coordinate space representation of the wave function. The polar part evolves on dynamically determined potential surfaces that have diabatic and adiabatic potentials as limiting cases of weak localized and strong extended diabatic couplings. The coordinate space part, generalized to a matrix form, describes transitions between the surfaces. Choice of the effective potentials for the polar part and partitioning of the wave function enables one to represent the total wave function in terms of smooth components that can be accurately propagated semiclassically using the approximate quantum potential and small basis sets. Examples are given for two-state one-dimensional problems that model chemical reactions that demonstrate the capabilities of the method for various regimes of nonadiabatic dynamics.     

Semiclassical nonadiabatic dynamics based on quantum trajectories for the O(3P,1D) + H2 system

The O(3P,1D) + H2 --> OH + H reaction is studied using trajectory dynamics within the approximate quantum potential approach. Calculations of the wave-packet reaction probabilities are performed for four coupled electronic states for total angular momentum J = 0 using a mixed coordinate/polar representation of the wave function. Semiclassical dynamics is based on a single set of trajectories evolving on an effective potential-energy surface and in the presence of the approximate quantum potential. Population functions associated with each trajectory are computed for each electronic state. The effective surface is a linear combination of the electronic states with the contributions of individual components defined by their time-dependent average populations. The wave-packet reaction probabilities are in good agreement with the quantum-mechanical results. Intersystem crossing is found to have negligible effect on reaction probabilities summed over final electronic states.     

Semiclassical analysis of Hénon-Heiles coupled oscillators: quasi-periodic and chaotic quantum behavior and the resonance model of unimolecular decay

The quantum mechanics of the Hénon-Heiles potential is analyzed using an adiabatic representation in polar coordinates and exploiting the asymptotic separability of the radius. The procedure allows us to establish a correlation between quasiperiodic and chaotic classical behavior, and regular or irregular quantum modes: It is found that irregularity can be attributed to nonadiabatic effects at the potential ridge. The resonance widths for this prototypic system of coupled oscillators are studied with reference to the lifetime in the quantum theory of unimolecular decay. The near separability of the radius of the polar coordinate representation is exploited for discussing energy dependence and symmetry effects on the widths. The relevance of this analysis for the characterization of quantum mechanical behavior near an elliptic umbilic catastrophe point is also briefly considered.     

Quantum theory of chemical reactions in the presence of electromagnetic fields

We present a theory for rigorous quantum scattering calculations of probabilities for chemical reactions of atoms with diatomic molecules in the presence of an external electric field. The approach is based on the fully uncoupled basis set representation of the total wave function in the space-fixed coordinate frame, the Fock-Delves hyperspherical coordinates, and the adiabatic partitioning of the total Hamiltonian of the reactive system. The adiabatic channel wave functions are expanded in basis sets of hyperangular functions corresponding to different reaction arrangements, and the interactions with external fields are included in each chemical arrangement separately. We apply the theory to examine the effects of electric fields on the chemical reactions of LiF molecules with H atoms and HF molecules with Li atoms at low temperatures and show that electric fields may enhance the probability of chemical reactions and modify reactive scattering resonances by coupling the rotational states of the reactants. Our preliminary results suggest that chemical reactions of polar molecules at temperatures below 1 K can be selectively manipulated with dc electric fields and microwave laser radiation.     

An analysis of the accuracy of an initial value representation surface hopping wave function in the interaction and asymptotic regions

The behavior of an initial value representation surface hopping wave function is examined. Since this method is an initial value representation for the semiclassical solution of the time independent Schrodinger equation for nonadiabatic problems, it has computational advantages over the primitive surface hopping wave function. The primitive wave function has been shown to provide transition probabilities that accurately compare with quantum results for model problems. The analysis presented in this work shows that the multistate initial value representation surface hopping wave function should approach the primitive result in asymptotic regions and provide transition probabilities with the same level of accuracy for scattering problems as the primitive method.     

Solution reaction space Hamiltonian based on an electrostatic potential representation of solvent dynamics

Quantum chemical solvation models usually rely on the equilibrium solvation condition and is thus not immediately applicable to the study of nonequilibrium solvation dynamics, particularly those associated with chemical reactions. Here we address this problem by considering an effective Hamiltonian for solution-phase reactions based on an electrostatic potential (ESP) representation of solvent dynamics. In this approach a general ESP field of solvent is employed as collective solvent coordinate, and an effective Hamiltonian is constructed by treating both solute geometry and solvent ESP as dynamical variables. A harmonic bath is then attached onto the ESP variables in order to account for the stochastic nature of solvent dynamics. As an illustration we apply the above method to the proton transfer of a substituted phenol-amine complex in a polar solvent. The effective Hamiltonian is constructed by means of the reference interaction site model self-consistent field method (i.e., a type of quantum chemical solvation model), and a mixed quantum/classical simulation is performed in the space of solute geometry and solvent ESP. The results suggest that important dynamical features of proton transfer in solution can be captured by the present approach, including spontaneous fluctuations of solvent ESP that drives the proton from reactant to product potential wells.     

Momentum representation of the solute/bath interaction in the dynamic theory of chemical processes in condensed phase

For the system consisting of the chemically reactive solute immersed in the oscillator bath, we consider an approach based on the solute/medium interaction expressed in terms of momenta rather than coordinates. In the adiabatic representation the medium reorganization effects are suppressed, being hidden in the solute renormalized potential and new spectral density function. The advantage proposed by the bilinear interaction in momentum representation is its spatial uniformity important for approximate dynamical treatments. The procedure of explicit transforming a standard spectral density (coordinate representation of interaction) into the spectral density in adiabatic representation (momentum representation of interaction) is the main new result of the present study. Illustrative calculations for several types of spectral functions are performed. Special discussion is devoted to clarifying the nature of the slow diffusion coordinate, to which the present approach is mainly addressed.     

Photo-induced desorption of NO from NiO(100): calculation of the four-dimensional potential energy surfaces and systematic wave packet studies

The velocity distributions of the laser-induced desorption of NO molecules from an epitaxially grown film of NiO(100) on Ni(100) have been studied [Mull et al., J. Chem. Phys., 1992, 96, 7108]. A pronounced bimodality of velocity distributions has been found, where the NO molecules desorbing with higher velocities exhibit a coupling to the rotational quantum states J. In this article we present simulations of state resolved velocity distributions on a full ab initio level. As a basis for this quantum mechanical treatment a 4D potential energy surface (PES) was constructed for the electronic ground and a representative excited state, using a NiO5Mg(18+)13 cluster. The PESs of the electronic ground and an excited state were calculated at the CASPT2 and the configuration interaction (CI) level of theory, respectively. Multi-dimensional quantum wave packet simulations on these two surfaces were performed for different sets of degrees of freedom. Our key finding is that at least a 3D wave packet simulation, in which the desorption coordinate Z, polar angle theta and lateral coordinate X are included, is necessary to allow the simulation of experimental velocity distributions. Analysis of the wave packet dynamics demonstrates that essentially the lateral coordinate, which was neglected in previous studies [Klüner et al., Phys. Rev. Lett. 1998, 80, 5208], is responsible for the experimentally observed bimodality. An extensive analysis shows that the bimodality is due to a bifurcation of the wave packet on the excited state PES, where the motion of the molecule parallel to the surface plays a decisive role.     

Spectroscopic interpretation: the high vibrations of CDBrClF

We extract the dynamics implicit in an algebraic fitted model Hamiltonian for the deuterium chromophore's vibrational motion in the molecule CDBrClF. The original model has four degrees of freedom, three positions and one representing interbond couplings. A conserved polyad allows in a semiclassical approach the reduction to three degrees of freedom. For most quantum states we can identify the underlying motion that when quantized gives the said state. Most of the classifications, identifications, and assignments are done by visual inspection of the already available wave function semiclassically transformed from the number representation to a representation on the reduced dimension toroidal configuration space corresponding to the classical action and angle variables. The concentration of the wave function density to lower dimensional subsets centered on idealized simple lower dimensional organizing structures and the behavior of the phase along such organizing centers already reveals the atomic motion. Extremely little computational work is needed.     

On the relationship between the one‐electron and Bohm's quantum potential

The one‐electron potential, derived from the electron density, is a three‐dimensional function, whereas Bohm's quantum potential depends on the spatial coordinates of all involved electrons. To analyze the relationship between the two potentials in the many‐electron case, first the dimensionality of the quantum potential needs to be reduced to match that of the one‐electron potential. A possible approach for such dimensionality reduction by calculating the expectation value of the quantum potential over all but one electron, using the factorization of the real part of the wave function into a marginal and conditional function is presented, and a relation between such three‐dimensional local quantum potential and the one‐electron potential is given. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Quantum wave packet ab initio molecular dynamics: an approach to study quantum dynamics in large systems

A methodology to efficiently conduct simultaneous dynamics of electrons and nuclei is presented. The approach involves quantum wave packet dynamics using an accurate banded, sparse and Toeplitz representation for the discrete free propagator, in conjunction with ab initio molecular dynamics treatment of the electronic and classical nuclear degree of freedom. The latter may be achieved either by using atom-centered density-matrix propagation or by using Born-Oppenheimer dynamics. The two components of the methodology, namely, quantum dynamics and ab initio molecular dynamics, are harnessed together using a time-dependent self-consistent field-like coupling procedure. The quantum wave packet dynamics is made computationally robust by using adaptive grids to achieve optimized sampling. One notable feature of the approach is that important quantum dynamical effects including zero-point effects, tunneling, as well as over-barrier reflections are treated accurately. The electronic degrees of freedom are simultaneously handled at accurate levels of density functional theory, including hybrid or gradient corrected approximations. Benchmark calculations are provided for proton transfer systems and the dynamics results are compared with exact calculations to determine the accuracy of the approach.     

Compact multipolar representation of the electrostatic potential for flexible molecules

A new method for generating a compact multipolar representation of the electrostatic potential (EP) for flexible molecules is presented. The method is based on a constrained minimization of the difference between the quantum mechanical and the classical EP. The fitting procedure used adopts the least absolute shrinkage and selection operator technique [R. Tibshirani, J. Roy. Stat. Soc. B 58, 267 (1996)] which can be seen as penalized ordinary least squares. The penalty function optimized for the particular molecule of interest effectively removes redundant multipoles. It is shown that the use of multiple conformations is crucial for the predictive ability of the EP model for flexible molecules. The multipole local coordinate systems are chosen in a way that best reflects the key conformational changes. It was demonstrated that such an approach improves the predictive ability of EP models. It also allows to exploit equivalence of atoms in the calculation of multipoles components. In the case of polar flexible molecules, the augmentation of the EP model based on charges by higher multipoles decreases the relative root mean square error by a factor of 1.5-5. The corresponding effect of enlargement of the set of multipoles was significantly reduced.     

Scattered wave packet formalism for the energy-resolved reaction probability

The scattered wave packet formalism developed for a quantum subsystem interacting with reservoirs through open boundaries is utilized to calculate the energy-resolved transmission probability. The total wave function is split into incident and scattered components. Markovian outgoing wave boundary conditions are imposed on the scattered or total wave function by the polynomial method. The wave packet correlation function approach is employed to compute the energy-resolved transmission probability for a one-dimensional potential barrier and a one-dimensional model chemical reaction exhibiting a quantum resonance. Accurate results demonstrate that this formalism can significantly reduce the number of grid points required in a dynamical calculation for the reaction probability.     

Coping with the node problem in quantum hydrodynamics: the covering function method

A conceptually simple approach, the covering function method (CFM), is developed to cope with the node problem in the hydrodynamic formulation of quantum mechanics. As nodes begin to form in a scattering wave packet (detected by a monitor function), a nodeless covering wave function is added to it yielding a total function that is also nodeless. Both local and global choices for the covering function are described. The total and covering functions are then propagated separately in the hydrodynamic picture. At a later time, the actual wave function is recovered from the two propagated functions. The results obtained for Eckart barrier scattering in one dimension are in excellent agreement with exact results, even for very long propagation times t=1.2 ps. The capability of the CFM is also demonstrated for multidimensional propagation of a vibrationally excited wave packet.     

Representations of Shavitt Graphs Within the Graphical Unitary Group Approach

The Shavitt graph is a visual representation of a distinct row table (DRT) within the graphical unitary group approach. The DRT is a compact representation of the entire configuration state function expansion space within a molecular electronic structure calculation. Each node of the graph is associated with an integer triple (a _k,b _k,c _k). These integers may be mapped to other quantum numbers, including the number of orbitals, number of electrons, and spin quantum number, and used to display Shavitt graphs in various ways that emphasize different aspects of the expansion space or that reveal different aspects of computed wave functions. The features of several graph density plots are discussed, including electron–hole symmetries and the bonding–antibonding wave function character. © 2019 Wiley Periodicals, Inc.     

Energy conserving approximations to the quantum potential: dynamics with linearized quantum force

Solution of the Schrodinger equation within the de Broglie-Bohm formulation is based on propagation of trajectories in the presence of a nonlocal quantum potential. We present a new strategy for defining approximate quantum potentials within a restricted trial function by performing the optimal fit to the log-derivatives of the wave function density. This procedure results in the energy-conserving dynamics for a closed system. For one particular form of the trial function leading to the linear quantum force, the optimization problem is solved analytically in terms of the first and second moments of the weighted trajectory distribution. This approach gives exact time-evolution of a correlated Gaussian wave function in a locally quadratic potential. The method is computationally cheap in many dimensions, conserves total energy and satisfies the criterion on the average quantum force. Expectation values are readily found by summing over trajectory weights. Efficient extraction of the phase-dependent quantities is discussed. We illustrate the efficiency and accuracy of the linear quantum force approximation by examining a one-dimensional scattering problem and by computing the wavepacket reaction probability for the hydrogen exchange reaction and the photodissociation spectrum of ICN in two dimensions.     

Computation of correlation functions and wave function projections in the context of quantum trajectory dynamics

The de Broglie-Bohm formulation of the Schrodinger equation implies conservation of the wave function probability density associated with each quantum trajectory in closed systems. This conservation property greatly simplifies numerical implementations of the quantum trajectory dynamics and increases its accuracy. The reconstruction of a wave function, however, becomes expensive or inaccurate as it requires fitting or interpolation procedures. In this paper we present a method of computing wave packet correlation functions and wave function projections, which typically contain all the desired information about dynamics, without the full knowledge of the wave function by making quadratic expansions of the wave function phase and amplitude near each trajectory similar to expansions used in semiclassical methods. Computation of the quantities of interest in this procedure is linear with respect to the number of trajectories. The introduced approximations are consistent with approximate quantum potential dynamics method. The projection technique is applied to model chemical systems and to the H+H(2) exchange reaction in three dimensions.     

A parallel Green's operator for multidimensional quantum scattering calculations

A parallel algorithm for computing multidimensional scattering wave functions is introduced. The inhomogeneous scattering (Lippmann–Schwinger) equation is solved within the discrete variable representation with absorbing boundary conditions, using iterative (Krylov) methods. A parallel Green's operator enables one to distribute the wave function to orthogonal subspaces in which it is processed in parallel. Application to a model problem of electron scattering in a three-dimensional rectangular quantum wire is given. Speedup is demonstrated with an increasing number of processors and with increasing dimensions and/or sampling density. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 167–173, 1998