Exact Quantum Mechanical Results Research Articles

Mean field theory of thermal energy transport in molecular junctions.

Mean field theory is applied to nonequilibrium thermal energy transport in a model molecular junction. An approximation to the total time-dependent heat current in the junction is constructed using an ensemble of Ehrenfest trajectories, and the average heat current in the steady state is obtained. The accuracy of this treatment is verified through benchmark comparisons with exact quantum mechanical results and various approximate quantum transport theories for the nonequilibrium spin-boson model. The performance of the multitrajectory Ehrenfest approach is found to be quite robust, displaying good accuracy in intermediate cases that remain elusive to many perturbative approximations and in the strong coupling limit where many methods break down. Thus, mean field theory and related trajectory-based approximate quantum dynamics methods emerge as a promising toolkit for the study of transport properties in nanoscale systems.

Floquet theory of radical pairs in radiofrequency magnetic fields.

We present a new method for calculating the product yield of a radical pair recombination reaction in the presence of a weak time-dependent magnetic field. This method successfully circumvents the computational difficulties presented by a direct solution of the Liouville-von Neumann equation for a long-lived radical pair containing many hyperfine-coupled nuclear spins. Using a modified formulation of Floquet theory, treating the time-dependent magnetic field as a perturbation, and exploiting the slow radical pair recombination, we show that one can obtain a good approximation to the product yield by considering only nearly degenerate sub-spaces of the Floquet space. Within a significant parameter range, the resulting method is found to give product yields in good agreement with exact quantum mechanical results for a variety of simple model radical pairs. Moreover it is considerably more efficient than the exact calculation, and it can be applied to radical pairs containing significantly more nuclear spins. This promises to open the door to realistic theoretical investigations of the effect of radiofrequency electromagnetic radiation on the photochemically induced radical pair recombination reactions in the avian retina which are believed to be responsible for the magnetic compass sense of migratory birds.

Non-Condon equilibrium Fermi's golden rule electronic transition rate constants via the linearized semiclassical method.

In this paper, we test the accuracy of the linearized semiclassical (LSC) expression for the equilibrium Fermi's golden rule rate constant for electronic transitions in the presence of non-Condon effects. We do so by performing a comparison with the exact quantum-mechanical result for a model where the donor and acceptor potential energy surfaces are parabolic and identical except for shifts in the equilibrium energy and geometry, and the coupling between them is linear in the nuclear coordinates. Since non-Condon effects may or may not give rise to conical intersections, both possibilities are examined by considering: (1) A modified Garg-Onuchic-Ambegaokar model for charge transfer in the condensed phase, where the donor-acceptor coupling is linear in the primary mode coordinate, and for which non-Condon effects do not give rise to a conical intersection; (2) the linear vibronic coupling model for electronic transitions in gas phase molecules, where non-Condon effects give rise to conical intersections. We also present a comprehensive comparison between the linearized semiclassical expression and a progression of more approximate expressions. The comparison is performed over a wide range of frictions and temperatures for model (1) and over a wide range of temperatures for model (2). The linearized semiclassical method is found to reproduce the exact quantum-mechanical result remarkably well for both models over the entire range of parameters under consideration. In contrast, more approximate expressions are observed to deviate considerably from the exact result in some regions of parameter space.

Semiclassical Monte Carlo: a first principles approach to non-adiabatic molecular dynamics.

Modeling the dynamics of photophysical and (photo)chemical reactions in extended molecular systems is a new frontier for quantum chemistry. Many dynamical phenomena, such as intersystem crossing, non-radiative relaxation, and charge and energy transfer, require a non-adiabatic description which incorporate transitions between electronic states. Additionally, these dynamics are often highly sensitive to quantum coherences and interference effects. Several methods exist to simulate non-adiabatic dynamics; however, they are typically either too expensive to be applied to large molecular systems (10's-100's of atoms), or they are based on ad hoc schemes which may include severe approximations due to inconsistencies in classical and quantum mechanics. We present, in detail, an algorithm based on Monte Carlo sampling of the semiclassical time-dependent wavefunction that involves running simple surface hopping dynamics, followed by a post-processing step which adds little cost. The method requires only a few quantities from quantum chemistry calculations, can systematically be improved, and provides excellent agreement with exact quantum mechanical results. Here we show excellent agreement with exact solutions for scattering results of standard test problems. Additionally, we find that convergence of the wavefunction is controlled by complex valued phase factors, the size of the non-adiabatic coupling region, and the choice of sampling function. These results help in determining the range of applicability of the method, and provide a starting point for further improvement.

Asymmetric recombination and electron spin relaxation in the semiclassical theory of radical pair reactions.

We describe how the semiclassical theory of radical pair recombination reactions recently introduced by two of us [D. E. Manolopoulos and P. J. Hore, J. Chem. Phys. 139, 124106 (2013)] can be generalised to allow for different singlet and triplet recombination rates. This is a non-trivial generalisation because when the recombination rates are different the recombination process is dynamically coupled to the coherent electron spin dynamics of the radical pair. Furthermore, because the recombination operator is a two-electron operator, it is no longer sufficient simply to consider the two electrons as classical vectors: one has to consider the complete set of 16 two-electron spin operators as independent classical variables. The resulting semiclassical theory is first validated by comparison with exact quantum mechanical results for a model radical pair containing 12 nuclear spins. It is then used to shed light on the spin dynamics of a carotenoid-porphyrin-fullerene triad containing considerably more nuclear spins which has recently been used to establish a "proof of principle" for the operation of a chemical compass [K. Maeda, K. B. Henbest, F. Cintolesi, I. Kuprov, C. T. Rodgers, P. A. Liddell, D. Gust, C. R. Timmel, and P. J. Hore, Nature (London) 453, 387 (2008)]. We find in particular that the intriguing biphasic behaviour that has been observed in the effect of an Earth-strength magnetic field on the time-dependent survival probability of the photo-excited C(·+)PF(·-) radical pair arises from a delicate balance between its asymmetric recombination and the relaxation of the electron spin in the carotenoid radical.

Decoherence-induced surface hopping

A simple surface hopping method for nonadiabatic molecular dynamics is developed. The method derives from a stochastic modeling of the time-dependent Schrödinger and master equations for open systems and accounts simultaneously for quantum mechanical branching in the otherwise classical (nuclear) degrees of freedom and loss of coherence within the quantum (electronic) subsystem due to coupling to nuclei. Electronic dynamics in the Hilbert space takes the form of a unitary evolution, intermittent with stochastic decoherence events that are manifested as a localization toward (adiabatic) basis states. Classical particles evolve along a single potential energy surface and can switch surfaces only at the decoherence events. Thus, decoherence provides physical justification of surface hopping, obviating the need for ad hoc surface hopping rules. The method is tested with model problems, showing good agreement with the exact quantum mechanical results and providing an improvement over the most popular surface hopping technique. The method is implemented within real-time time-dependent density functional theory formulated in the Kohn-Sham representation and is applied to carbon nanotubes and graphene nanoribbons. The calculated time scales of non-radiative quenching of luminescence in these systems agree with the experimental data and earlier calculations.

Linear dependence and energy conservation in Gaussian wavepacket basis sets

We propose a method for dealing with the problem of linear dependence in quantum dynamics simulations employing over-complete Gaussian wavepacket (GWP) basis sets. In particular, by periodically projecting out redundant basis functions using the matching pursuit algorithm whilst simultaneously introducing GWPs which avoid linear dependence with the current basis set, we find that numerical conditioning of the equations-of-motion can be readily controlled. In applications to particle tunnelling in one- and two-dimensional potentials, this method allows us to reproduce the exact quantum-mechanical results with fewer GWP basis functions than similar calculations with non-adaptive basis sets, a result which we trace back to the improved energy conservation of our adaptive approach.

Accelerating the convergence of path integral dynamics with a generalized Langevin equation

The quantum nature of nuclei plays an important role in the accurate modelling of light atoms such as hydrogen, but it is often neglected in simulations due to the high computational overhead involved. It has recently been shown that zero-point energy effects can be included comparatively cheaply in simulations of harmonic and quasiharmonic systems by augmenting classical molecular dynamics with a generalized Langevin equation (GLE). Here we describe how a similar approach can be used to accelerate the convergence of path integral (PI) molecular dynamics to the exact quantum mechanical result in more strongly anharmonic systems exhibiting both zero point energy and tunnelling effects. The resulting PI-GLE method is illustrated with applications to a double-well tunnelling problem and to liquid water.

Numerical testing of the Fowler–Nordheim equation for the electronic field emission from a flat metal and proposition for an improved equation

The author uses a transfer-matrix technique to simulate field electronic emission from a flat metal. He compares, in particular, the results provided by this numerical scheme with those predicted by the standard Fowler–Nordheim equation. He considers for this study electric fields between 1 and 10 V/nm as well as work functions between 1.5 and 5 eV. The results demonstrate that the Fowler–Nordheim theory and the transfer-matrix calculations are globally in good agreement. With the Fermi energy of 10 eV considered in this work, the results provided by the standard Fowler–Nordheim equation are, however, systematically larger than the quantum-mechanical result, especially for low values of the work function and for high electric fields. This is essentially due to the fact the standard Fowler–Nordheim theory relies on the simple Jeffreys–Wentzel–Kramers–Brillouin approximation for evaluating the electronic transmission through the surface barrier of the emitter. A correction factor is thus established that enables the temperature-dependent version of the standard Fowler–Nordheim equation to match the exact quantum-mechanical result.

A comparative study of the electron transmission through one-dimensional barriers relevantto field-emission problems

We study the transmission coefficient of one-dimensional barriers that are relevant tofield-emission problems. We compare, in particular, the results provided by the simpleJeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation, the continued-fractiontechnique and the transfer-matrix methodology for the electronic transmission throughsquare, triangular and Schottky–Nordheim barriers (the Schottky–Nordheim barrier isoften used in models of field emission from flat metals). For conditions that are typical offield emission (Fermi energy of 10 eV, work function of 4.5 eV and field strength of5 V nm − 1), it is shown that the simple JWKB approximation must be completed by an effective prefactorPeff inorder to match the exact quantum-mechanical result. This prefactor takes typical values around 3.4for square barriers, 1.8 for triangular barriers and 0.84 for the Schottky–Nordheim barrier. For fieldsF between 1 and10 V nm − 1 and for work functionsϕ between 1 and5 eV, the prefactor Peff to consider in the case of the Schottky–Nordheim barrier actually ranges between 0.28 and0.98. This study hence demonstrates that the Fowler–Nordheim equation (in its standardform that accounts for the image interaction and that actually relies on the simple JWKBapproximation) overestimates the current emitted from a flat metal by a factor that may beof the order of 2–3 for the conditions considered in this work. The study thus confirmsForbes’s opinion that this prefactor should be reintegrated in field-emission theories.

Wave-packet dynamics in energy space of a chaotic trimeric Bose-Hubbard system

We study the energy redistribution of interacting bosons in a ring-shaped quantum trimer as the coupling strength between neighboring sites of the corresponding Bose-Hubbard Hamiltonian undergoes a sudden change $\ensuremath{\delta}k$. Our analysis is based on a threefold approach combining linear response theory calculations as well as semiclassical and random matrix theory considerations. The $\ensuremath{\delta}k$ borders of applicability of each of these methods are identified by direct comparison with the exact quantum-mechanical results. We find that while the variance of the evolving quantum distribution shows a remarkable quantum-classical correspondence (QCC) for all $\ensuremath{\delta}k$ values, other moments exhibit this QCC only in the nonperturbative $\ensuremath{\delta}k$ regime.

Algebraic description of the inelastic collision between an atom and a Morse oscillator in one dimension

The 1D collision between an atom and a diatomic molecule is investigated using an algebraic approach. Closed expressions for the quantum mechanical excitation transition probabilities are obtained. A Morse potential for the diatomic molecule is used. The classical trajectories are obtained in the united atom limit by taking an average over a finite number of states using the matrix density formalism, which allows the problem to be reformulated in terms of a time-dependent Hamiltonian. The system is studied in the interaction picture, which permits us to carry out an algebraic description through the approximation of the interaction potential in terms of a linear combination of the generators of the su(2) algebra. Comparisons with exact quantum mechanical results for transition probabilities in different systems are presented.

Complex trajectory method in time-dependent WKB

We present a significant improvement to a complex time-dependent WKB (CWKB) formulation developed by Boiron and Lombardi [J. Chem. Phys. 108, 3431 (1998)] in which the time-dependent WKB equations are solved along classical trajectories that propagate in complex space. Boiron and Lombardi showed that the method gives very good agreement with the exact quantum mechanical result as long as the wavefunction does not exhibit interference effects such as oscillations and nodes. In this paper, we show that this limitation can be overcome by superposing the contributions of crossing trajectories. Secondly, we demonstrate that the approximation improves when incorporating higher order terms in the expansion. Thirdly, equations of motion for caustics and Stokes lines are implemented to help overcome Stokes discontinuities. These improvements could make the CWKB formulation a competitive alternative to current time-dependent semiclassical methods.

Quantum corrections to classical evaluation of nonadiabatic transition rates

A recently developed quantum correction approach is applied to evaluating the nonadiabatic quantum-mechanical transition rate between Born-Oppenheimer states of a subsystem embedded in a thermal bath of harmonic oscillators. In the first-order perturbation theory, the nonadiabatic rate can be expressed in terms of a quantum-mechanical correlation function, which can be estimated directly from classical data. Application to a popular spin-boson model shows that our results are in excellent agreement with the exact quantum-mechanical results.

Correlated non-perturbative electron dynamics with quantum trajectories

An approach to electron correlation effects in atoms that uses quantum trajectories is presented. A comparison with the exact quantum mechanical results for 1D Helium atom shows that the major features of the correlated ground state distribution and of the strong field ionization dynamics are reproduced with quantum trajectories. The intra-atomic resonant transitions are described accurately by a trajectory ensemble. The present approach reduces significantly the computational time and it can be used for both bound and ionizing electrons.

A confined noninteracting many electron system: Accurate corrections to a statistical model

We derive a correction term to the Fermi–Dirac energy for the system of N el noninteracting electrons confined to a cube with sides of infinite potential. This correction term is compared to one derived by Pauncz [R. Pauncz, Acta Physica Academiae Scientarium Hungaricae 1 (1952) 277]. Both models are compared to the exact quantum mechanical result with systems of 1 to 5000 electrons being considered. Agreement is seen to increase with particle number.

An initial value representation semiclassical approach for the study of molecular systems with geometric constraints

We present an approach for the inclusion of geometric constraints in quantum dynamics calculations based on the semiclassical initial value representation. An important feature of the method is that a Cartesian coordinate system is used throughout, resulting in a general approach that does not require the definition of new coordinates. The Herman–Kluk [M. F. Herman and E. Kluk, Chem. Phys. 91, 27 (1984)] coherent state formulation is used. The required (constrained) classical trajectories are calculated using the standard techniques of molecular dynamics and initial conditions are sampled from a distribution that obeys the constraints. An approximate form of the Herman–Kluk prefactor is used and its evaluation requires the construction of a projected Hessian matrix. In its present form, the approach allows the calculation of energy levels from the Fourier transform of the autocorrelation function. The approach is tested on a model problem consisting of two particles in a harmonic trap and constrained to remain at a fixed distance from one another. The approach yields exact results for this simplified case when compared to the exact quantum mechanical formulation. The method is then applied to a real molecular system consisting of a water molecule with fixed OH bonds and yields accurate results when compared to exact quantum mechanical results. The accuracy of the method is comparable to that of the usual semiclassical implementation where the problem is written in a new set of coordinates. The approach can be extended to more complex cases in a straightforward manner.

Semiclassical quantization of complex Henon–Heiles systems

Abstract Semiclassical eigenenergies and power spectra of PT -symmetric Henon–Heiles systems are obtained by using classical trajectories in complex phase space. It was found that some of the semiclassical methods developed for real multidimensional systems are equally valid when systems are complex as well. Semiclassical results are compared with exact quantum mechanical results.

Long-range effects in the density of loosely bound states of ion–dipole systems: the gas-phase SN2 complex Cl−⋯CH3Cl

We present a methodology for the calculation of the number and density of bound vibrational states close to the dissociation threshold of large polyatomic molecules. It is shown that an evaluation based on the classical phase-space volume dramatically underestimates the number and density of the loosely bound states. The reason for this behaviour is the combination of two effects: the long-range interactions between the fragments and the quantum nature of the intramolecular modes of these fragments in the vicinity of their zero point energy level. Our approach is applied to the Cl−⋯CH3Cl complex, for which multidimensional potential energy surfaces describing the long-range interactions between the Cl− ion and the CH3Cl molecule are available. The proposed adiabatic method, that combines a quantum treatment of the intramolecular modes and a semi-classical description of the intermolecular modes, is first applied to the two C–Cl stretching degrees of freedom and compared with the semi-classical approach and the exact quantum mechanical results. Then, we extend the methodology to a full-dimensional, but approximate, 12D potential energy surface and compare it with various semi-classical approaches. Finally, the quantum nature of the molecular dynamics in the vicinity of the dissociation threshold of a polyatomic molecule is discussed, in particular with respect to the importance of the density of states when deriving dissociation rates using statistical theories, e.g. RRKM.

Magnetoconductance fluctuations and weak localization in quantum dots: Reliability of the semiclassical approach

We utilize a semiclassical (SC) approach to calculate the conductance and weak-localization (WL) corrections in a triangular billiard of a given shape in the presence of nonzero magnetic field. The semiclassical conductance is given as a sum of all classical trajectories between the leads, each of them carrying the quantum-mechanical phase. The present SC approach is numerically exact (i.e., free from any approximations), explicitly includes diffractive effects in the leads, and is valid for arbitrary (low) mode numbers in the leads. We find however that the symmetry of the SC conductance/reflectance with respect to the direction of magnetic field or direction of the current is not satisfied, as well as that the WL corrections for the conductance and reflectance are inconsistent with each other; the SC approach does not satisfy the current conservation requirements and does not reproduce the corresponding exact quantum-mechanical results. The reason for that is traced to the topological difference in the sets of classical trajectories between the leads for different current or magnetic field directions which determine the conductance in the SC approximation. Our findings raise a question as to what extent one can rely on numerous predictions for statistical properties of the conductance oscillations of ballistic cavities including the WL line shapes and fractal conductance which were essentially based on the standard SC approach.