Semiclassical Initial Value Representation Method Research Articles

Semiclassical reaction rate constant of non-adiabatic system by means of initial value representation

We applied the semiclassical initial value representation (SC-IVR) method with the classical electron analog to deal with electronically non-adiabatic reactive barriers and calculated the thermal reaction rate constant. The symmetric form of the flux–flux correlation function is used, and the matrix element of the Boltzmannized flux operator is derived using imaginary-time path integral techniques. This combined use of non-adiabatic SC-IVR and non-adiabatic path-integral sampling (NA-IVR&PI) is beyond the treatment of mean-field, and we analyze its performance in comparison with some existing mean-field methods for different scenarios of potential energy surface. We found a systematically better performance. One exception is the deep tunneling regime, where the mean-field ring-polymer instanton is superior. In the golden rule limit, where all these mean-field methods break down, the significant advantage of NA-IVR&PI is shown by using the electronic states for description of the dividing surface.

Nonadiabatic simulations of photoisomerization and dissociation in ethylene using abinitio classical trajectories.

We simulate the nonadiabatic dynamics of photo-induced isomerization and dissociation in ethylene using abinitio classical trajectories in an extended phase space of nuclear and electronic variables. This is achieved by employing the linearized semiclassical initial value representation method for nonadiabatic dynamics, where discrete electronic states are mapped to continuous classical variables using either the Meyer-Miller-Stock-Thoss representation or a more recently introduced spin mapping approach. Trajectory initial conditions are sampled by constraining electronic state variables to a single initial excited state and by drawing nuclear phase space configurations from a Wigner distribution at a finite temperature. An ensemble of classical abinitio trajectories is then generated to compute thermal population correlation functions and analyze the mechanisms of isomerization and dissociation. Our results serve as a demonstration that this parameter-free semiclassical approach is computationally efficient and accurate, identifying mechanistic pathways in agreement with previous theoretical studies and also uncovering dissociation pathways observed experimentally.

Machine learning for vibrational spectroscopy via divide-and-conquer semiclassical initial value representation molecular dynamics with application to N-methylacetamide.

A machine learning algorithm for partitioning the nuclear vibrational space into subspaces is introduced. The subdivision criterion is based on Liouville's theorem, i.e., the best preservation of the unitary of the reduced dimensionality Jacobian determinant within each subspace along a probe full-dimensional classical trajectory. The algorithm is based on the idea of evolutionary selection, and it is implemented through a probability graph representation of the vibrational space partitioning. We interface this customized version of genetic algorithms with our divide-and-conquer semiclassical initial value representation method for the calculation of molecular power spectra. First, we benchmark the algorithm by calculating the vibrational power spectra of two model systems, for which the exact subspace division is known. Then, we apply it to the calculation of the power spectrum of methane. Exact calculations and full-dimensional semiclassical spectra of this small molecule are available and provide an additional test of the accuracy of the new approach. Finally, the algorithm is applied to the divide-and-conquer semiclassical calculation of the power spectrum of 12-atom trans-N-methylacetamide.

Simplified approach to the mixed time-averaging semiclassical initial value representation for the calculation of dense vibrational spectra.

We present and test an approximate method for the semiclassical calculation of vibrational spectra. The approach is based on the mixed time-averaging semiclassical initial value representation method, which is simplified to a form that contains a filter to remove contributions from approximately harmonic environmental degrees of freedom. This filter comes at no additional numerical cost, and it has no negative effect on the accuracy of peaks from the anharmonic system of interest. The method is successfully tested for a model Hamiltonian and then applied to the study of the frequency shift of iodine in a krypton matrix. Using a hierarchic model with up to 108 normal modes included in the calculation, we show how the dynamical interaction between iodine and krypton yields results for the lowest excited iodine peaks that reproduce experimental findings to a high degree of accuracy.

Application of the mixed time-averaging semiclassical initial value representation method to complex molecular spectra.

The recently introduced mixed time-averaging semiclassical initial value representation of the molecular dynamics method for spectroscopic calculations [M. Buchholz, F. Grossmann, and M. Ceotto, J. Chem. Phys. 144, 094102 (2016)] is applied to systems with up to 61 dimensions, ruled by a condensed phase Caldeira-Leggett model potential. By calculating the ground state as well as the first few excited states of the system Morse oscillator, changes of both the harmonic frequency and the anharmonicity are determined. The method faithfully reproduces blueshift and redshift effects and the importance of the counter term, as previously suggested by other methods. Different from previous methods, the present semiclassical method does not take advantage of the specific form of the potential and it can represent a practical tool that opens the route to direct ab initio semiclassical simulation of condensed phase systems.

On the accuracy of the LSC-IVR approach for excitation energy transfer in molecular aggregates.

We investigate the applicability of the linearized semiclassical initial value representation (LSC-IVR) method to excitation energy transfer (EET) problems in molecular aggregates by simulating the EET dynamics of a dimer model in a wide range of parameter regime and comparing the results to those obtained from a numerically exact method. It is found that the LSC-IVR approach yields accurate population relaxation rates and decoherence rates in a broad parameter regime. However, the classical approximation imposed by the LSC-IVR method does not satisfy the detailed balance condition, generally leading to incorrect equilibrium populations. Based on this observation, we propose a post-processing algorithm to solve the long time equilibrium problem and demonstrate that this long-time correction method successfully removed the deviations from exact results for the LSC-IVR method in all of the regimes studied in this work. Finally, we apply the LSC-IVR method to simulate EET dynamics in the photosynthetic Fenna-Matthews-Olson complex system, demonstrating that the LSC-IVR method with long-time correction provides excellent description of coherent EET dynamics in this typical photosynthetic pigment-protein complex.

Semiclassical initial value representation for the quantum propagator in the Heisenberg interaction representation

One of the challenges facing on-the-fly ab initio semiclassical time evolution is the large expense needed to converge the computation. In this paper, we suggest that a significant saving in computational effort may be achieved by employing a semiclassical initial value representation (SCIVR) of the quantum propagator based on the Heisenberg interaction representation. We formulate and test numerically a modification and simplification of the previous semiclassical interaction representation of Shao and Makri [J. Chem. Phys. 113, 3681 (2000)]. The formulation is based on the wavefunction form of the semiclassical propagation instead of the operator form, and so is simpler and cheaper to implement. The semiclassical interaction representation has the advantage that the phase and prefactor vary relatively slowly as compared to the "standard" SCIVR methods. This improves its convergence properties significantly. Using a one-dimensional model system, the approximation is compared with Herman-Kluk's frozen Gaussian and Heller's thawed Gaussian approximations. The convergence properties of the interaction representation approach are shown to be favorable and indicate that the interaction representation is a viable way of incorporating on-the-fly force field information within a semiclassical framework.

Representation of the Boltzmann operator via the chain of imaginary time coherent states and its application to calculation of real time correlation functions

Coherent states in imaginary time are used to represent the Boltzmann operator in terms of a classical chain of N coupled beads. The resulting representation is similar to the path integral molecular dynamics (PIMD) approach, except for additional nearest-neighbor coupling terms in the effective Hamiltonian. Each time step of the molecular dynamics evolution of the effective classical system provides appropriately thermalized Gaussians that can be used as distribution functions to sample initial conditions for real-time semiclassical initial value representation (IVR) methods. The approach is illustrated with numerical calculations of thermal expectation values and canonical time correlation functions for several one-dimensional quantum systems.

Determination of molecular vibrational state energies using the ab initio semiclassical initial value representation: Application to formaldehyde

We have demonstrated the use of ab initio molecular dynamics (AIMD) trajectories to compute the vibrational energy levels of molecular systems in the context of the semiclassical initial value representation (SC-IVR). A relatively low level of electronic structure theory (HF/3-21G) was used in this proof-of-principle study. Formaldehyde was used as a test case for the determination of accurate excited vibrational states. The AIMD-SC-IVR vibrational energies have been compared to those from curvilinear and rectilinear vibrational self-consistent field/vibrational configuration interaction with perturbation selected interactions-second-order perturbation theory (VSCF/VCIPSI-PT2) and correlation-corrected vibrational self-consistent field (cc-VSCF) methods. The survival amplitudes were obtained from selecting different reference wavefunctions using only a single set of molecular dynamics trajectories. We conclude that our approach is a further step in making the SC-IVR method a practical tool for first-principles quantum dynamics simulations.

Zero point energy leakage in condensed phase dynamics: An assessment of quantum simulation methods for liquid water

The approximate quantum mechanical ring polymer molecular dynamics (RPMD) and linearized semiclassical initial value representation (LSC-IVR) methods are compared and contrasted in a study of the dynamics of the flexible q-TIP4P/F water model at room temperature. For this water model, a RPMD simulation gives a diffusion coefficient that is only a few percent larger than the classical diffusion coefficient, whereas a LSC-IVR simulation gives a diffusion coefficient that is three times larger. We attribute this discrepancy to the unphysical leakage of initially quantized zero point energy (ZPE) from the intramolecular to the intermolecular modes of the liquid as the LSC-IVR simulation progresses. In spite of this problem, which is avoided by construction in RPMD, the LSC-IVR may still provide a useful approximation to certain short-time dynamical properties which are not so strongly affected by the ZPE leakage. We illustrate this with an application to the liquid water dipole absorption spectrum, for which the RPMD approximation breaks down at frequencies in the O-H stretching region owing to contamination from the internal modes of the ring polymer. The LSC-IVR does not suffer from this difficulty and it appears to provide quite a promising way to calculate condensed phase vibrational spectra.

First-principles semiclassical initial value representation molecular dynamics

In this work, we explore the use of the semiclassical initial value representation (SC-IVR) method with first-principles electronic structure approaches to carry out classical molecular dynamics. The proposed approach can extract the vibrational power spectrum of carbon dioxide from a single trajectory providing numerical results that agree with experiment and quantum calculations. The computational demands of the method are comparable to those of classical single-trajectory calculations, while describing uniquely quantum features such as the zero-point energy and Fermi resonances. The method can also be used to identify symmetry properties of given vibrational peaks and investigate vibrational couplings by selected classical trajectories. The accuracy of the method degrades for the reproduction of anharmonic shifts for high-energy vibrational levels.

Simulations of the emission spectra of fac-tris(2-phenylpyridine) iridium and Duschinsky rotation effects using the Herman–Kluk semiclassical initial value representation method

The phosphorescent emission spectra of fac-tris(2-phenylpyridine) iridium [fac-Ir(ppy)(3)] due to the lowest triplet T(1) and T(2) states are simulated using the harmonic oscillator approximation for the S(0), T(1), and T(2) potential energy surfaces (PESs) and taking the Duschinsky rotation into account. The simulations involve the propagation of 177-dimensional wave packets on the coupled PES according to the Herman-Kluk (HK) semiclassical (SC) initial value representation (IVR) method. The HK SC-IVR method is employed because of its accuracy for the PES with mode mixing and its efficiency in dealing with coupled degrees of freedom for large systems. The simulated emission spectrum due to T(1) reproduces the structures of the emission spectra observed experimentally, while T(2) is found very unlikely to participate in the phosphorescent emission. Although the effect of the Duschinsky mode mixing is small for the T(1) state, neglecting it blueshifts the spectrum due to the T(2) state by 800 cm(-1) and changes the relative intensities, indicating that the importance of the Duschinsky rotation is rather unpredictable and should not be overlooked. The present simulations demonstrate that the simple harmonic oscillator approximation combined with the Duschinsky rotation can adequately describe the photophysics of fac-Ir(ppy)(3) and that the HK SC-IVR method is a powerful tool in studies of this kind.

Semiclassical wave packet study of anomalous isotope effect in ozone formation

We applied the semiclassical initial value representation method to calculate energies, lifetimes, and wave functions of scattering resonances in a two-dimensional potential for O+O2 collision. Such scattering states represent the metastable O3* species and play a central role in the process of ozone formation. Autocorrelation functions for scattering states were computed and then analyzed using the Prony method, which permits one to extract accurate energies and widths of the resonances. We found that the results of the semiclassical wave packet propagation agree well with fully quantum results. The focus was on the 16O16O18O isotopomer and the anomalous isotope effect associated with formation of this molecule, either through the 16O16O+18O or the 16O+16O18O channels. An interesting correlation between the local vibration mode character of the metastable states and their lifetimes was observed and explained. New insight is obtained into the mechanism by which the long-lived resonances in the delta zero-point energy part of spectrum produce the anomalously large isotope effect.

A justification for a nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation of the time evolution operator

A justification is given for the validity of a nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method. The method is based on a propagator that combines the single surface HK SC-IVR method [J. Chem. Phys. 84, 326 (1986)] and Herman's nonadiabatic semiclassical surface hopping theory [J. Chem. Phys. 103, 8081 (1995)], which was originally developed using the primitive semiclassical Van Vleck propagator. We show that the nonadiabatic HK SC-IVR propagator satisfies the time-dependent Schrodinger equation to the first order of variant Planck's over 2pi and the error is O(variant Planck's over 2pi(2)). As a required lemma, we show that the stationary phase approximation, under current assumptions, has an error term variant Planck's over 2pi(1) order higher than the leading term. Our derivation suggests some changes to the previous development, and it is shown that the numerical accuracy in applications to Tully's three model systems in low energies is improved.

Matching-pursuit split-operator Fourier-transform simulations of excited-state intramolecular proton transfer in 2-(2′-hydroxyphenyl)-oxazole

The excited-state intramolecular proton-transfer dynamics associated with the keto-enolic tautomerization reaction in 2-(2(')-hydroxyphenyl)-oxazole is simulated according to a numerically exact quantum-dynamics propagation method and a full-dimensional excited-state potential energy surface, based on an ab initio reaction surface Hamiltonian. The reported simulations involve the propagation of 35-dimensional wave packets according to the recently developed matching-pursuit/split-operator-Fourier-transform (MP/SOFT) method by Wu and Batista. The underlying propagation scheme recursively applies the time-evolution operator as defined by the Trotter expansion to second order accuracy in dynamically adaptive coherent-state expansions. Computations of time-dependent survival amplitudes, photoabsorption cross sections, and time-dependent reactant(product) populations are compared to the corresponding calculations based on semiclassical approaches, including the Herman-Kluk semiclassical initial value representation method. The reported results demonstrate the capabilities of the MP/SOFT method as a valuble computational tool to study ultrafast reaction dynamics in polyatomic systems as well as to validate semiclassical simulations of complex (nonintegrable) quantum dynamics in multidimensional model systems.

Nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation method revisited: Applications to Tully’s three model systems

The nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method for nonadiabatic problems is reformulated. The method has the same spirit as Tully's surface hopping technique [J. Chem. Phys. 93, 1061 (1990)] and almost keeps the same structure as the original single-surface HK SC-IVR method except that trajectories can hop to other surfaces according to the hopping probabilities and phases, which can be easily integrated along the paths. The method is based on a rather general nonadiabatic semiclassical surface hopping theory developed by Herman [J. Chem. Phys. 103, 8081 (1995)], which has been shown to be accurate to the first order in h and through all the orders of the nonadiabatic coupling amplitude. Our simulation studies on the three model systems suggested by Tully demonstrate that this method is practical and capable of describing nonadiabatic quantum dynamics for various coupling situations in very good agreement with benchmark calculations.

Is the Filinov integral conditioning technique useful in semiclassical initial value representation methods?

The utility of the Filinov integral conditioning technique, as implemented in semiclassical initial value representation (SC-IVR) methods, is analyzed for a number of regular and chaotic systems. For nonchaotic systems of low dimensionality, the Filinov technique is found to be quite ineffective at accelerating convergence of semiclassical calculations since, contrary to the conventional wisdom, the semiclassical integrands usually do not exhibit significant phase oscillations in regions of large integrand amplitude. In the case of chaotic dynamics, it is found that the regular component is accurately represented by the SC-IVR, even when using the Filinov integral conditioning technique, but that quantum manifestations of chaotic behavior was easily overdamped by the filtering technique. Finally, it is shown that the level of approximation introduced by the Filinov filter is, in general, comparable to the simpler ad hoc truncation procedure introduced by Kay.

Strong field tunnel ionization by real-valued classical trajectories.

Strong field tunnel ionization of an atom is considered from the point of view of semiclassical initial value representation methods which are based on real-valued classical trajectories alone. While the straightforward application of such propagators fails to give an accurate description of tunnel ionization in one dimension, incorporating the semiclassical propagator into S-matrix techniques standard in strong field physics leads to a more accurate method which recovers the tunneling dynamics. From the point of view of strong field physics, this procedure offers a method of incorporating core effects into the standard strong field approximation. In two dimensions, both the standard and the new semiclassical propagators are shown to give equally accurate results at sufficiently short times, but the new method exhibits much better scaling of the convergence rate with increasing dimensionality.

Semiclassical Molecular Dynamics Simulations of the Excited State Photodissociation Dynamics of H2O in the AB1 Band

The photodissociation dynamics of H2O in the A1B1 band is investigated by implementing a recently developed time-sliced semiclassical initial value representation method (Burant, J. C.; Batista, V. S. J. Chem. Phys. 2002, 116, 2748). The capabilities of the computational method are explored as applied to calculations of partial and total photodissociation cross sections associated with highly excited vibrational states of H2O. The direct comparison of our semiclassical results with full quantum mechanical calculations shows that the semiclassical approach is able to describe the H2O spectroscopy and the overall photodissociation dynamics in excellent agreement with full quantum mechanical results.

Ultrafast non-adiabatic dynamics of systems with multiple surface crossings: a test of the Meyer–Miller Hamiltonian with semiclassical initial value representation methods

Three three-state models have been studied to test the capabilities of the Meyer–Miller Hamiltonian studying non-adiabatic photodissociation dynamics. within the framework of the semiclassical initial value representation (SC-IVR). The absorption spectra and the evolution of the electronic populations were calculated using the SC-IVR methodology, and they both show excellent agreement with accurate quantum results. A linearized version (LSC-IVR) was also tested, and it describes the dynamics well for short times but becomes less accurate at longer time.