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.     

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.     

Semiclassical IVR approach to rotational excitation of non-polar diatomic molecules by non-resonant laser pulses

We apply the ‘classic’ semiclassical initial value representation (SC-IVR) approach to describe rotational excitation of non-polar diatomic molecules by intense short non-resonant laser pulses. We also investigate the applicability of the quantum mechanical sudden approximation. It is found that the SC-IVR approach gives accurate rotational excitation probabilities in regimes where the sudden approximation fails. Primitive semiclassical wavefunction propagation is used to illustrate the phenomenon of angular focussing of rotation states by strong pulses.     

Wavepacket propagation using time-sliced semiclassical initial value methods

A new semiclassical initial value representation (SC-IVR) propagator and a SC-IVR propagator originally introduced by Kay [J. Chem. Phys. 100, 4432 (1994)], are investigated for use in the split-operator method for solving the time-dependent Schrodinger equation. It is shown that the SC-IVR propagators can be derived from a procedure involving modified Filinov filtering of the Van Vleck expression for the semiclassical propagator. The two SC-IVR propagators have been selected for investigation because they avoid the need to perform a coherent state basis set expansion that is necessary in other time-slicing propagation schemes. An efficient scheme for solving the propagators is introduced and can be considered to be a semiclassical form of the effective propagators of Makri [Chem. Phys. Lett. 159, 489 (1989)]. Results from applications to a one-dimensional, two-dimensional, and three-dimensional Hamiltonian for a double-well potential are presented.     

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.     

Semiclassical quantum unimolecular reaction rate theory revisited

We describe a semiclassical quantum unimolecular reaction rate theory derived from the corresponding classical theory developed by Davis, Gray, Rice and Zhao (DGRZ). The analysis retains the intuitively useful mechanistic distinctions between intramolecular energy transfer and reaction, with the consequence that the semiclassical quantum theory version neglects some interference effects in the reaction dynamics. In the limiting case that intramolecular energy transfer is very fast compared to the rate of reaction we show that the DGRZ representation of the rate constant can be transformed, using the Weyl correspondence between quantum operators and classical variables, to the quantum flux–flux correlation function representation of the rate constant. In the more general case that the rate of intramolecular energy transfer influences the reaction dynamics, the semiclassical representation of the Wigner function for a classical system with both quasiperiodic and chaotic motion is used to obtain the reaction rate constant. Our analysis identifies the quantum analogue of the classical bottleneck to intramolecular energy transfer with the scars of unstable periodic orbits; it leads to a flux–flux correlation function representation of the rate constant for intramolecular energy transfer.     

Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation

We propose to measure nonadiabaticity of molecular quantum dynamics rigorously with the quantum fidelity between the Born-Oppenheimer and fully nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies in situations where other criteria, such as the energy gap criterion or the extent of population transfer, fail. We further propose to estimate this quantum fidelity efficiently with a generalization of the dephasing representation to multiple surfaces. Two variants of the multiple-surface dephasing representation (MSDR) are introduced, in which the nuclei are propagated either with the fewest-switches surface hopping or with the locally mean field dynamics (LMFD). The LMFD can be interpreted as the Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used previously in the nonadiabatic semiclassical initial value representation. In addition to propagating an ensemble of classical trajectories, the MSDR requires evaluating nonadiabatic couplings and solving the Schro?dinger (or more generally, the quantum Liouville-von Neumann) equation for a single discrete degree of freedom. The MSDR can be also used in the diabatic basis to measure the importance of the diabatic couplings. The method is tested on three model problems introduced by Tully and on a two-surface model of dissociation of NaI.     

Trajectory-based solution of the nonadiabatic quantum dynamics equations: an on-the-fly approach for molecular dynamics simulations

The non-relativistic quantum dynamics of nuclei and electrons is solved within the framework of quantum hydrodynamics using the adiabatic representation of the electronic states. An on-the-fly trajectory-based nonadiabatic molecular dynamics algorithm is derived, which is also able to capture nuclear quantum effects that are missing in the traditional trajectory surface hopping approach based on the independent trajectory approximation. The use of correlated trajectories produces quantum dynamics, which is in principle exact and computationally very efficient. The method is first tested on a series of model potentials and then applied to study the molecular collision of H with H(2) using on-the-fly TDDFT potential energy surfaces and nonadiabatic coupling vectors.     

Quantized Hamilton Dynamics

The paper describes the quantized Hamilton dynamics (QHD) approach that extends classical Hamiltonian dynamics and captures quantum effects, such as zero point energy, tunneling, decoherence, branching, and state-specific dynamics. The approximations are made by closures of the hierarchy of Heisenberg equations for quantum observables with the higher order observables decomposed into products of the lower order ones. The technique is applied to the vibrational energy exchange in a water molecule, the tunneling escape from a metastable state, the double-slit interference, the population transfer, dephasing and vibrational coherence transfer in a two-level system coupled to a phonon, and the scattering of a light particle off a surface phonon, where QHD is coupled to quantum mechanics in the Schrödinger representation. Generation of thermal ensembles in the extended space of QHD variables is discussed. QHD reduces to classical mechanics at the first order, closely resembles classical mechanics at the higher orders, and requires little computational effort, providing an efficient tool for treatment of the quantum effects in large systems.     

A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces

The multiconfigurational time-dependent Hartree (MCTDH) approach facilitates multidimensional quantum dynamics calculations by representing the wavepacket in an optimal set of time-dependent basis functions, called single-particle functions. Choosing these single-particle functions to be themselves multidimensional wavefunctions which are represented using a MCTDH representation, a multilayer MCTDH scheme has been constructed and used for quantum dynamics calculations treating up to 1000 degrees of freedom rigorously [Wang and Thoss, J. Chem. Phys. 199, 1289 (2003)]. The present work gives a practical scheme which facilitates the application of the multilayer MCTDH approach, which previously has only been employed to study systems described by model-type Hamiltonians, to molecular systems described by more complicated Hamiltonians and general potential energy surfaces. A multilayer extension of the correlation discrete variable representation (CDVR) scheme employed in MCTDH calculations studying quantum dynamics on general potential energy surfaces is developed and tested in a simple numerical application. The resulting multilayer MCTDH/CDVR approach might offer a perspective to rigorously describe the quantum dynamics of larger polyatomic systems.     

Semiclassical initial value series solution of the spin boson problem

A numerical solution for the quantum dynamics of the spin boson problem is obtained using the semiclassical initial value series representation approach to the quantum dynamics. The zeroth order term of the series is computed using the new forward-backward representation for correlation functions presented in the preceding adjacent paper. This leads to a rapid convergence of the Monte Carlo sampling, as compared to previous attempts. The zeroth order results are already quite accurate. The first order term of the series is small, demonstrating the rapid convergence of the semiclassical initial value representation series. This is the first time that the first order term in the semiclassical initial value representation series has been converged for systems with the order of 50 degrees of freedom.