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A practical quantum theory for unifying electronic and nuclear dynamics, which were separated by the Born–Oppenheimer approximation, is proposed. The theory consists of two processes. Nonadiabatic (quantum) electron wavepacket dynamics on branching (non-Born–Oppenheimer) nuclear paths are first constructed. Since these paths are not the classical trajectories, most of the existing semiclassical theories to generate quantum wavepacket do not work. Therefore, we apply our own developed semiclassical wavepacket theory to these generated non-Born–Oppenheimer paths. This wavepacket is generated based on what we call the action decomposed function, which does not require the information of the so-called stability matrix. Thus, the motion of nuclei is also quantized, and consequently the total wavefunction is represented as a series of entanglement between the electronic and nuclear wavepackets. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

We extend our formerly proposed theory for non-Born-Oppenheimer electronic and nuclear wavepacket dynamics within on-the-fly scheme [T. Yonehara, S. Takahashi, and K. Takatsuka, J. Chem. Phys. 130, 214113 (2009)] to a case of nonadiabatic dynamics under an intense laser field: electron wavepacket in a molecule is propagated in attosecond time-scale along non-Born-Oppenheimer nuclear paths that smoothly branch due to nonadiabatic coupling and/or optical interactions. Such branching paths are determined consistently with the motion of the electron wavepackets. Furthermore, these nuclear paths are quantized in terms of Gaussian wavepackets (action decomposed function), which can be applied to nonclassical paths. Both electronic wavepacket dynamics and quantization of non-Born-Oppenheimer paths are generalized so as to include the direct effects of the classical vector potential of electromagnetic fields. In the second half of this paper, we perform numerical studies to explore nonadiabatic dynamics in a laser field by examining two cases: one is a two-state model system having an avoided crossing, and the other is two-state dynamics in HF molecule on the two low lying ab initio potential curves. Both are placed in laser fields. With the former system, we survey some basic properties of the coupling of nonadiabatic dynamics and laser interaction varying the relevant coupling parameters such as the laser timing with respect to the incident of nonadiabatic transition. This investigation will set a foundation for the future studies of control of electronic states in realistic multidimensional molecular systems. Application to the latter system shows that non-Born-Oppenheimer quantum chemistry in laser fields is indeed useful in the study of dynamics in ab initio level. Through the comparison with full quantum data, we verify that the formalism and methodology developed here work accurately. Furthermore, we attain some basic insight about the characteristics of molecules in laser fields.

A practical quantum theory for unifying electronic and nuclear dynamics, which were separated by the Born–Oppenheimer approximation, is proposed. The theory consists of two processes. Nonadiabatic (quantum) electron wavepacket dynamics on branching (non-Born–Oppenheimer) nuclear paths are first constructed. Since these paths are not the classical trajectories, most of the existing semiclassical theories to generate quantum wavepacket do not work. Therefore, we apply our own developed semiclassical wavepacket theory to these generated non-Born–Oppenheimer paths. This wavepacket is generated based on what we call the action decomposed function, which does not require the information of the so-called stability matrix. Thus, the motion of nuclei is also quantized, and consequently the total wave function is represented as a series of entanglement between the electronic and nuclear wavepackets. In the last half of the article, we show the practice to demonstrate how these independent theories can be unified to give electron-nuclear wavepackets in a two-state model. The wavepackets up to the phases and resultant transition probabilities are compared to the full quantum-mechanical counterparts. It turns out that the lowest level approximation to the wavepacket approach already shows a good agreement with the full quantum quantities. Thus, the present theoretical framework gives a basic method with which to study non-Born–Oppenheimer electronic and nuclear wavepacket states relevant to ultrafast chemical events.

A serious difficulty in the semiclassical Ehrenfest theory for nonadiabatic transitions is that a path passing across the avoided crossing is forced to run on a potential averaged over comprising adiabatic potential surfaces that commit the avoided crossing. Therefore once a path passes through the crossing region, it immediately becomes incompatible with the standard view of "classical trajectory" running on an adiabatic surface. This casts a fundamental question to the theoretical structure of chemical dynamics. In this paper, we propose a non-Born-Oppenheimer path that is generated by an anti-Hermitian Hamiltonian, whose complex-valued eigenenergies can cross in their real parts and avoid crossing in the imaginary parts in the vicinity of the nonadiabatic transition region. We discuss the properties of this non-Born-Oppenheimer path and thereby show its compatibility with the Born-Oppenheimer classical trajectories. This theory not only allows the geometrical branching of the paths but gives the nonadiabatic transition amplitudes and quantum phases along the generated paths.

We develop the path-branching representation for nonadiabatic electron wavepacket dynamics [T. Yonehara and K. Takatsuka, J. Chem. Phys. 132, 244102 (2010)] so as to treat dynamics in an energy range comparable to the barrier height of adiabatic potential energy curves. With this representation two characteristic chemical reaction dynamics are studied, in which an incident nuclear wavepacket encounters a potential barrier, on top of which lies another nonadiabatically coupled adiabatic potential curve: (1) Dynamics of initial paths coming into the nonadiabatic interaction region with energy lower than the barrier height. They branch into two pieces (and repeat branching subsequently), the upper counterparts of which can penetrate into a classically inaccessible high energy region and eventually branch back to the product region on the ground state curve. This is so to say surmounting the potential barrier via nonadiabatically coupled excited state, and phenomenologically looks like the so-called deep tunneling. (2) Dynamics of classical paths whose initial energies are a little higher than the barrier but may be lower than the bottom of the excited state. They can undergo branching and some of those components are trapped on top of the potential barrier, being followed by the population decay down to the lower state flowing both to product and reactant sites. Such expectations arising from the path-branching representation are numerically confirmed with full quantum mechanical wavepacket dynamics. This phenomenon may be experimentally observed as time-delayed pulses of wavepacket trains.

Chaos in the multimode nonadiabatic system constructed by Heller [J. Chem. Phys. 92, 1718 (1990)], which consists of two diabatic two-dimensional harmonic potentials with the Condon coupling, is studied. A thorough investigation is carried out by scanning the magnitudes of the Condon coupling and the Duschinsky angle. To elucidate mechanisms that can cause chaos in this quantum system, the statistical properties of the energy levels and eigenfunctions of the system are investigated. We find an evidence in terms of the nearest-neighbor spacing distribution of energy levels and other measures that a certain class of chaos is purely induced by the nonadiabatic interaction due to breakdown of the Born-Oppenheimer approximation. Since the nonadiabatic transition can induce repeated bifurcation and merging of a wave packet around the region of quasicrossing between two potential surfaces, and since this interaction does not have a counterpart in the lower adiabatic system, the present chaos deserves being called "nonadiabatic chaos." Another type of chaos in a nonadiabatic system was previously identified [D. M. Leitner et al., J. Chem. Phys. 104, 434 (1996)] that reflects the inherent chaos of a corresponding adiabatic potential. We present a comparative study to establish the similarity and difference between these kinds of chaos.

The validity range of the Born-Oppenheimer (BO) approximation is studied with respect to the variation of the mass (m) of negatively charged particle by substituting an electron (e) with muon (mu) and antiproton (p) in hydrogen molecule cation. With the use of semiclassical quantization applied to these (ppe), (ppmu), and (ppp) under a constrained geometry, we estimate the energy difference of the non-BO vibronic ground state from the BO counterpart. It is found that the error in the BO approximation scales to the power of 3/2 to the mass of negative particles, that is, m(1.5). The origin of this clear-cut relation is analyzed based on the original perturbation theory due to Born and Oppenheimer, with which we show that the fifth order term proportional to m(5/4) is zero and thereby the first correction to the BO approximation should arise from the sixth order term that is proportional to m(6/4). Therefore, the validity range of the Born-Oppenheimer approximation is wider than that often mistakenly claimed to be proportional to m(1/4).