On the validity range of the Born-Oppenheimer approximation: A semiclassical study for all-particle quantization of three-body Coulomb systems

Geometrical decomposition of nonadiabatic interactions to collective coordinates in many-dimensional and many-state mixed fast-slow dynamics

In general, for many-dimensional and many-state nonadiabatic dynamics composed of slow and fast modes, we geometrically decompose the nonadiabatic interactions by means of the method of singular value decomposition. Each pair of the left and right singular vectors connecting the slow (nuclear) and fast (electronic) modes gives rise to a one-dimensional collective coordinate, and the sum of them amounts to the total nonadiabatic interaction. The analysis identifies how efficiently the slow modes, thus decomposed, can induce a transition in their fast counterparts. We discuss the notions of nonadiabatic resonance and nonadiabatic chaos in terms of the decomposition.

Quantum Chaos in the Dynamics of Molecules

Quantum chaos is reviewed from the viewpoint of "what is molecule?", particularly placing emphasis on their dynamics. Molecules are composed of heavy nuclei and light electrons, and thereby the very basic molecular theory due to Born and Oppenheimer gives a view that quantum electronic states provide potential functions working on nuclei, which in turn are often treated classically or semiclassically. Therefore, the classic study of chaos in molecular science began with those nuclear dynamics particularly about the vibrational energy randomization within a molecule. Statistical laws in probabilities and rates of chemical reactions even for small molecules of several atoms are among the chemical phenomena requiring the notion of chaos. Particularly the dynamics behind unimolecular decomposition are referred to as Intra-molecular Vibrational energy Redistribution (IVR). Semiclassical mechanics is also one of the main research fields of quantum chaos. We herein demonstrate chaos that appears only in semiclassical and full quantum dynamics. A fundamental phenomenon possibly giving birth to quantum chaos is "bifurcation and merging" of quantum wavepackets, rather than "stretching and folding" of the baker's transformation and the horseshoe map as a geometrical foundation of classical chaos. Such wavepacket bifurcation and merging are indeed experimentally measurable as we showed before in the series of studies on real-time probing of nonadiabatic chemical reactions. After tracking these aspects of molecular chaos, we will explore quantum chaos found in nonadiabatic electron wavepacket dynamics, which emerges in the realm far beyond the Born-Oppenheimer paradigm. In this class of chaos, we propose a notion of Intra-molecular Nonadiabatic Electronic Energy Redistribution (INEER), which is a consequence of the chaotic fluxes of electrons and energy within a molecule.

Nature of chemical bond and potential barrier in an invariant energy-orbital picture

Physical nature of the chemical bond and potential barrier is studied in terms of energy natural orbitals (ENOs), which are extracted from highly correlated electronic wavefunctions. ENO provides an objective one-electron picture about energy distribution in a molecule, just as the natural orbitals (NOs) represent one electron view about electronic charge distribution. ENO is invariant in the same sense as NO is, that is, ENOs converge to the exact ones as a series of approximate wavefunctions approach the exact one, no matter how the methods of approximation are adopted. Energy distribution analysis based on ENO can give novel insights about the nature of chemical bonding and formation of potential barriers, besides information based on the charge distribution alone. With ENOs extracted from full configuration interaction wavefunctions in a finite yet large enough basis set, we analyze the nature of chemical bonding of three low-lying electronic states of a hydrogen molecule, all being in different classes of the so-called covalent bond. The mechanism of energy lowering in bond formation, which gives a binding energy, is important, yet not the only concern for this small molecule. Another key notion in chemical bonding is whether a potential basin is well generated stiff enough to support a vibrational state(s) on it. Based on the virial theorem in the adiabatic approximation and in terms of the energy and force analyses with ENOs, we study the roles of the electronic kinetic energy and its nuclear derivative(s) on how they determine the curvature (or the force constant) of the potential basins. The same idea is applied to the potential barrier and, thereby, the transition states. The rate constant within the transition-state theory is formally shown to be described in terms of the electronic kinetic energy and the nuclear derivatives only. Thus, the chemical bonding and rate process are interconnected behind the scenes. Besides this aspect, we pay attention to (1) the effects of electron correlation that manifests itself not only in the orbital energy but also in the population of ENOs and (2) the role of nonadiabaticity (diabatic state mixing), resulting in double basins and a barrier on a single potential curve in bond formation. These factors differentiate a covalent bond into subclasses.

An orbital picture extracted from correlated electronic wavefunctions and application to forbidden reactions: 70 years of the frontier orbital theory

The one-electron picture in molecular electronic state theory, particularly the molecular orbital (MO) theory with the Hartree-Fock approximation, has set a foundation to develop chemical science. Frontier orbital theory, or the theory of HOMO (highest occupied MO)-LUMO (lowest unoccupied MO) interaction, and the conservation rule of orbital symmetry are among the brightest achievements in a molecular orbital picture. After 70 years from the birth of frontier orbital theory, however, electronic wavefunctions treated in current quantum chemistry are often highly correlated and consist of extensive scales of electronic configurations to be more accurate and to cope with far more complicated reactions than concerted reactions. Under such circumstances, the MO approximation itself readily loses its validity, let alone the utter dominance of the HOMO-LUMO interaction. Recently, we have proposed an invariant method to extract general orbitals from such correlated electronic wavefunctions, which we refer to as Energy Natural Orbitals (ENOs) [K. Takatsuka and Y. Arasaki, J. Chem. Phys. 154, 094103 (2021)]. The energies of ENOs are summed exactly to the total electronic energy. The topological (symmetry) properties of a total wavefunction are represented by the relative phases of ENOs along with the continuity and crossing (avoided and conical intersection) among them. Only a small number of ENOs often dominate and characterize chemical reactions. With these properties of ENO, we explore a couple of simple and typical symmetry forbidden reactions, illustrating the effects of electron correlation and degeneracy in relevant ENOs. We propose the notion of "internal conical intersection" among ENOs, which leads to Jahn-Teller effect, pseudo-Jahn-Teller effect, and so on. We dare to explain the primary origin of elementary conical intersections and multidimensional avoided crossing in chemical reactions with the use of the notion of orbital crossing between those of HOMO-HOMO and LUMO-LUMO interactions and so on.

Nuclear-electronic all-particle density matrix renormalization group

We introduce the Nuclear-Electronic All-Particle Density Matrix Renormalization Group (NEAP-DMRG) method for solving the time-independent Schrödinger equation simultaneously for electrons and other quantum species. In contrast to the already existing multicomponent approaches, in this work, we construct from the outset a multi-reference trial wave function with stochastically optimized non-orthogonal Gaussian orbitals. By iterative refining of the Gaussians' positions and widths, we obtain a compact multi-reference expansion for the multicomponent wave function. We extend the DMRG algorithm to multicomponent wave functions to take into account inter- and intra-species correlation effects. The efficient parameterization of the total wave function as a matrix product state allows NEAP-DMRG to accurately approximate the full configuration interaction energies of molecular systems with more than three nuclei and 12 particles in total, which is currently a major challenge for other multicomponent approaches. We present the NEAP-DMRG results for two few-body systems, i.e., H₂ and H₃ ⁺, and one larger system, namely, BH₃.

Electron correlation via frozen Gaussian dynamics

We investigate the accuracy and efficiency of the semiclassical frozen Gaussian method in describing electron dynamics in real time. Model systems of two soft-Coulomb-interacting electrons are used to study correlated dynamics under non-perturbative electric fields, as well as the excitation spectrum. The results show that a recently proposed method that combines exact-exchange with semiclassical correlation to propagate the one-body density-matrix holds promise for electron dynamics in many situations that either wavefunction or density-functional methods have difficulty describing. The results also however point out challenges in such a method that need to be addressed before it can become widely applicable.

Exploring dynamical electron theory beyond the Born-Oppenheimer framework: from chemical reactivity to non-adiabatically coupled electronic and nuclear wavepackets on-the-fly under laser field

Chemical theory and its application to dynamical electrons in molecules under intense electromagnetic fields is explored, in which we take an explicit account of nuclear nonadiabatic (kinematic) interactions along with simultaneous coupling with intense optical interactions. All the electronic wavefunctions studied here are necessarily time-dependent, and thereby beyond stationary state quantum chemistry based on the Born-Oppenheimer framework. As a general and tractable alternative framework with which to track the electronic and nuclear simultaneous dynamics, we propose an on-the-fly method to calculate the electron and nuclear wavepackets coupled along the branching non-Born-Oppenheimer paths, through which their bifurcations, strong quantum entanglement between nuclear electronic motions, and coherence and decoherence among the phases associated with them are properly represented. Some illustrative numerical examples are also reported, which are aimed at our final goals; real time tracking of nonadiabatic electronic states, chemical dynamics in densely degenerate electronic states coupled with nuclear motions and manipulation and/or creation of new electronic states in terms of intense lasers, and so on. Other examples are also presented as to how the electron wavepacket dynamics can be used to analyze chemical reactions, shedding a new light on some typical and conventional chemical reactions such as proton transfer followed by tautomerization.

Semiclassical initial value calculations of the collinear helium atom

Semiclassical calculations using the Herman-Kluk initial value treatment are performed to determine energy eigenvalues of bound and resonance states of the collinear helium atom. Both the eZe configuration (where the classical motion is fully chaotic) and the Zee configuration (where the classical dynamics is nearly integrable) are treated. The classical motion is regularized to remove singularities that occur when the electrons collide with the nucleus. Very good agreement is obtained with quantum energies for bound and resonance states calculated by the complex rotation method.

Phase quantization of chaos in the semiclassical regime

Since the early stage of the study of Hamilton chaos, semiclassical quantization based on the low-order Wentzel-Kramers-Brillouin theory, the primitive semiclassical approximation to the Feynman path integrals (or the so-called Van Vleck propagator), and their variants have been suffering from difficulties such as divergence in the correlation function, nonconvergence in the trace formula, and so on. These difficulties have been hampering the progress of quantum chaos, and it is widely recognized that the essential drawback of these semiclassical theories commonly originates from the erroneous feature of the amplitude factors in their applications to classically chaotic systems. This forms a clear contrast to the success of the Einstein-Brillouin-Keller quantization condition for regular (integrable) systems. We show here that energy quantization of chaos in semiclassical regime is, in principle, possible in terms of constructive and destructive interference of phases alone, and the role of the semiclassical amplitude factor is indeed negligibly small, as long as it is not highly oscillatory. To do so, we first sketch the mechanism of semiclassical quantization of energy spectrum with the Fourier analysis of phase interference in a time correlation function, from which the amplitude factor is practically factored out due to its slowly varying nature. In this argument there is no distinction between integrability and nonintegrability of classical dynamics. Then we present numerical evidence that chaos can be indeed quantized by means of amplitude-free quasicorrelation functions and Heller's frozen Gaussian method. This is called phase quantization. Finally, we revisit the work of Yamashita and Takatsuka [Prog. Theor. Phys. Suppl. 161, 56 (2007)] who have shown explicitly that the semiclassical spectrum is quite insensitive to smooth modification (rescaling) of the amplitude factor. At the same time, we note that the phase quantization naturally breaks down when the oscillatory nature of the amplitude factor is comparable to that of the phases. Such a case generally appears when the Planck constant of a large magnitude pushes the dynamics out of the semiclassical regime.

Generalization of classical mechanics for nuclear motions on nonadiabatically coupled potential energy surfaces in chemical reactions

Classical trajectory study of nuclear motion on the Born-Oppenheimer potential energy surfaces is now one of the standard methods of chemical dynamics. In particular, this approach is inevitable in the studies of large molecular systems. However, as soon as more than a single potential energy surface is involved due to nonadiabatic coupling, such a naive application of classical mechanics loses its theoretical foundation. This is a classic and fundamental issue in the foundation of chemistry. To cope with this problem, we propose a generalization of classical mechanics that provides a path even in cases where multiple potential energy surfaces are involved in a single event and the Born-Oppenheimer approximation breaks down. This generalization is made by diagonalization of the matrix representation of nuclear forces in nonadiabatic dynamics, which is derived from a mixed quantum-classical representation of the electron-nucleus entangled Hamiltonian [Takatsuka, K. J. Chem. Phys. 2006, 124, 064111]. A manifestation of quantum fluctuation on a classical subsystem that directly contacts with a quantum subsystem is discussed. We also show that the Hamiltonian thus represented gives a theoretical foundation to examine the validity of the so-called semiclassical Ehrenfest theory (or mean-field theory) for electron quantum wavepacket dynamics, and indeed, it is pointed out that the electronic Hamiltonian to be used in this theory should be slightly modified.

Energy quantization of chaos with the semiclassical phases alone

The mechanism of energy quantization is studied for classical dynamics on a highly anharmonic potential, ranging from integrable, mixed, and chaotic motions. The quantum eigenstates (standing waves) are created by the phase factors (the action integrals and the Maslov index) irrespective of the integrability, when the amplitude factors are relatively slowly varying. Indeed we show numerically that the time Fourier transform of an approximate semiclassical correlation function in which the amplitude factors are totally removed reproduces the spectral positions (energy eigenvalues) accurately in chaotic regime. Quantization with the phase information alone brings about dramatic simplification to molecular science, since the amplitude factors in the lowest order semiclassical approximation diverge exponentially in a chaotic domain.

Adiabatic Jacobi corrections for H2+-like systems

The Coulomb three-body problem in Jacobi coordinates was solved by treating the distance of the particles having equal charge as a parameter. This method allows computation of electronic energies with finite nuclear masses while maintaining the notion of a potential energy curve. The rotationless ground-state electronic and the so-called adiabatic Jacobi correction (AJC) energies are presented for H2+, D2+, and HD+ at fixed internuclear separations. The AJCs are defined as the difference between the results obtained from calculations using proper finite and infinite nuclear masses. Except at the united atom limit, the AJCs are smaller than the traditional first-order diagonal Born-Oppenheimer corrections. Expectation values of proton-electron, p-e, and deuteron-electron, d-e, distances for HD+ have been computed as a function of internuclear separation. Similarly to the fully nonadiabatic approach, the present method is able to follow the symmetry breaking in HD+. Exact and approximate analytical and numerical results are given for counterfactual systems as well. In these cases changes are allowed for the values of the electron rest mass or the elementary charge, as well as for the mass or charge of the unique particle (electron).