Sonification of molecular electronic energy density and its dynamics.

A method is proposed for sonification of the molecular electronic energy density. The characteristic energetic structures of the individual complicated electronic wavefunctions are extracted in terms of the Energy Natural Orbitals (ENO), which are the eigenfunctions of the electronic energy density operator [K. Takatsuka and Y. Arasaki, J. Chem. Phys., 2021, 154, 094103]. Then, the frequency corresponding to each ENO energy is linearly transformed to the audible range. The time-variation of the population of the ENO serves as the volume (amplitude) of the sound. We demonstrate the sonification and associated voiceprints for a couple of very basic chemical bondings, from across an avoided crossing, and from the bond dissociation of a cluster.

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.

Spin current in the early stage of radical reactions and its mechanisms.

We study the electronic spin flux (atomic-scale flow of the spin density in molecules) by a perturbation analysis and ab initio nonadiabatic calculations. We derive a general perturbative expression of the charge and spin fluxes and identify the driving perturbation of the fluxes to be the time derivative of the electron-nucleus interaction term in the Hamiltonian. We then expand the expression in molecular orbitals so as to identify relevant components of the fluxes. Our perturbation theory describes the electronic fluxes in the early stage of reactions in an intuitively clear manner. The perturbation theory is then applied to an analysis of the spin flux obtained in ab initio calculations of the radical reaction of O2 and CH3· starting from three distinct spin configurations; (a) CH3· and triplet O2 with total spin of the system set Stot=1/2 (b) CH3· and singlet O2, Stot=1/2, and (c) CH3· and triplet O2, Stot=3/2. Further analysis of the time-dependent behaviors of the spin flux in these numerical simulations reveals (i) the spin flux induces rearrangement of the local spin structure, such as reduction of the spin polarization arising from the triplet O2 and (ii) the spin flux flows from O2 to CH3· in the reaction starting from spin configuration (a) and from CH3· to O2 in that starting from configuration (b), whereas no major intermolecular spin flux was observed in that starting from configuration (c). Our study thus establishes the mechanism of the spin flux that rearranges the local spin structures associated with chemical bonds.

Electronic-state chaos, intramolecular electronic energy redistribution, and chemical bonding in persisting multidimensional nonadiabatic systems.

We study the chaotic, huge fluctuation of electronic state, resultant intramolecular energy redistribution, and strong chemical bonding surviving the fluctuation with exceedingly long lifetimes of highly excited boron clusters. Those excited states constitute densely quasi-degenerate state manifolds. The huge fluctuation is induced by persisting multidimensional nonadiabatic transitions among the states in the manifold. We clarify the mechanism of their coexistence and its physical significance. In doing so, we concentrate on two theoretical aspects. One is quantum chaos and energy randomization, which are to be directly extracted from the properties of the total electronic wavefunctions. The present dynamical chaos takes place through frequent transitions from adiabatic states to others, thereby making it very rare for the system to find dissociation channels. This phenomenon leads to the concept of what we call intramolecular nonadiabatic electronic-energy redistribution, which is an electronic-state generaliztion of the notion of intramolecular vibrational energy redistribution. The other aspect is about the peculiar chemical bonding. We investigate it with the energy natural orbitals (ENOs) to see what kind of theoretical structures lie behind the huge fluctuation. The ENO energy levels representing the highly excited states under study appear to have four robust layers. We show that the energy layers responsible for chaotic dynamics and those for chemical bonding are widely separated from each other, and only when an event of what we call "inter-layer crossing" happens to burst can the destruction of these robust energy layers occur, resulting in molecular dissociation. This crossing event happens only rarely because of the large energy gaps between the ENO layers. It is shown that the layers of high energy composed of complex-valued ENOs induce the turbulent flow of electrons and electronic-energy in the cluster. In addition, the random and fast time-oscillations of those high energy ENOs serve as a random force on the nuclear dynamics, which can work to prevent a concentration of high nuclear kinetic energy in the dissociation channels.

Energy natural orbital characterization of nonadiabatic electron wavepackets in the densely quasi-degenerate electronic state manifold.

Dynamics and energetic structure of largely fluctuating nonadiabatic electron wavepackets are studied in terms of Energy Natural Orbitals (ENOs) [K. Takatsuka and Y. Arasaki, J. Chem. Phys. 154, 094103 (2021)]. Such huge fluctuating states are sampled from the highly excited states of clusters of 12 boron atoms (B12), which have densely quasi-degenerate electronic excited-state manifold, each adiabatic state of which gets promptly mixed with other states through the frequent and enduring nonadiabatic interactions within the manifold. Yet, the wavepacket states are expected to be of very long lifetimes. This excited-state electronic wavepacket dynamics is extremely interesting but very hard to analyze since they are usually represented in large time-dependent configuration interaction wavefunctions and/or in some other complicated forms. We have found that ENO gives an invariant energy orbital picture to characterize not only the static highly correlated electronic wavefunctions but also those time-dependent electronic wavefunctions. Hence, we first demonstrate how the ENO representation works for some general cases, choosing proton transfer in water dimer and electron-deficient multicenter chemical bonding in diborane in the ground state. We then penetrate with ENO deep into the analysis of the essential nature of nonadiabatic electron wavepacket dynamics in the excited states and show the mechanism of the coexistence of huge electronic fluctuation and rather strong chemical bonds under very random electron flows within the molecule. To quantify the intra-molecular energy flow associated with the huge electronic-state fluctuation, we define and numerically demonstrate what we call the electronic energy flux.

Real-time electronic energy current and quantum energy flux in molecules.

Intra- and inter-molecular electronic energy current is formulated by defining the probability current of electronic energy, called the energy flux. Among vast possible applications to electronic energy transfer phenomena, including chemical reaction dynamics, here we present a first numerical example from highly excited nonadiabatic electron wavepacket dynamics of a boron cluster B12.

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.

Suppression of Charge Recombination by Auxiliary Atoms in Photoinduced Charge Separation Dynamics with Mn Oxides: A Theoretical Study.

Charge separation is one of the most crucial processes in photochemical dynamics of energy conversion, widely observed ranging from water splitting in photosystem II (PSII) of plants to photoinduced oxidation reduction processes. Several basic principles, with respect to charge separation, are known, each of which suffers inherent charge recombination channels that suppress the separation efficiency. We found a charge separation mechanism in the photoinduced excited-state proton transfer dynamics from Mn oxides to organic acceptors. This mechanism is referred to as coupled proton and electron wave-packet transfer (CPEWT), which is essentially a synchronous transfer of electron wave-packets and protons through mutually different spatial channels to separated destinations passing through nonadiabatic regions, such as conical intersections, and avoided crossings. CPEWT also applies to collision-induced ground-state water splitting dynamics catalyzed by Mn4CaO5 cluster. For the present photoinduced charge separation dynamics by Mn oxides, we identified a dynamical mechanism of charge recombination. It takes place by passing across nonadiabatic regions, which are different from those for charge separations and lead to the excited states of the initial state before photoabsorption. This article is an overview of our work on photoinduced charge separation and associated charge recombination with an additional study. After reviewing the basic mechanisms of charge separation and recombination, we herein studied substituent effects on the suppression of such charge recombination by doping auxiliary atoms. Our illustrative systems are X-Mn(OH)2 tied to N-methylformamidine, with X=OH, Be(OH)3, Mg(OH)3, Ca(OH)3, Sr(OH)3 along with Al(OH)4 and Zn(OH)3. We found that the competence of suppression of charge recombination depends significantly on the substituents. The present study should serve as a useful guiding principle in designing the relevant photocatalysts.

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.

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.

On the molecular electronic flux: Role of nonadiabaticity and violation of conservation.

Analysis of electron flux within and in between molecules is crucial in the study of real-time dynamics of molecular electron wavepacket evolution such as those in attosecond laser chemistry and ultrafast chemical reaction dynamics. We here address two mutually correlated issues on the conservation law of molecular electronic flux, which serves as a key consistency condition for electron dynamics. The first one is about a close relation between "weak" nonadiabaticity and the electron dynamics in low-energy chemical reactions. We show that the electronic flux in adiabatic reactions can be consistently reproduced by taking account of nonadiabaticity. Such nonadiabaticity is usually weak in the sense that it does not have a major effect on nuclear dynamics, whereas it plays an important role in electronic dynamics. Our discussion is based on a nonadiabatic extension of the electronic wavefunction similar in idea to the complete adiabatic formalism developed by Nafie [J. Chem. Phys. 79, 4950 (1983)], which has also recently been reformulated by Patchkovskii [J. Chem. Phys. 137, 084109 (2012)]. We give straightforward proof of the theoretical assertion presented by Nafie using a time-dependent mixed quantum-classical framework and a standard perturbation expansion. Explicitly taking account of the flux conservation, we show that the nonadiabatically induced flux realizes the adiabatic time evolution of the electronic density. In other words, the divergence of the nonadiabatic flux equals the time derivative of the electronic density along an adiabatic time evolution of the target molecule. The second issue is about the accurate computationability of the flux. The calculation of flux needs an accurate representation of the (relative) quantum phase, in addition to the amplitude factor, of a total wavefunction and demands special attention for practical calculations. This paper is the first one to approach this issue directly and show how the difficulties arise explicitly. In doing so, we reveal that a number of widely accepted truncation techniques for static property calculations are potential sources of numerical flux non-conservation. We also theoretically propose alternative strategies to realize better flux conservation.

Energy natural orbitals.

We propose and numerically demonstrate that highly correlated electronic wavefunctions such as those of configuration interaction, the cluster expansion, and so on, and electron wavepackets superposed thereof can be analyzed in terms of one-electron functions, which we call energy natural orbitals (ENOs). As the name suggests, ENOs are members of the broad family of natural orbitals defined by Löwdin, in that they are eigenfunctions of the energy density operator. One of the major characteristics is that the (orbital) energies of all the ENOs are summed up exactly equal to the total electronic energy of a wavefunction under study. Another outstanding feature is that the population of each ENO varies as the chemical reaction proceeds, keeping the total population constant though. The study of ENOs has been driven by the need for new methods to analyze extremely complicated nonadiabatic electron wavepackets such as those embedded in highly quasi-degenerate excited-state manifolds. Yet, ENOs can be applied to scrutinize many other chemical reactions, ranging from the ordinary concerted reactions, nonadiabatic reactions, and Woodward-Hoffman forbidden reactions, to excited-state reactions. We here present the properties of ENOs and a couple of case studies of numerical realization, one of which is about the mechanism of nonadiabatic electron transfer.

Charge separation and successive reconfigurations of electronic and protonic states in a water-splitting catalytic cycle with the Mn4CaO5 cluster. On the mechanism of water splitting in PSII.

Much insight into the basic mechanisms of photoexcited and collision-induced ground-state water splitting has been accumulated in our nonadiabatic electron wavepacket dynamics studies based on a building-block approach reaching up to systems of binuclear Mn oxo complexes. We here extend the study to a ground-state water-splitting catalytic cycle with tetranuclear Mn oxo complex Mn4CaO5, or Mn3Ca(H2O)2(OH)4-OH-Mn(4)(H2O)2, where Mn3Ca(H2O)2(OH)4 is fixed to a skewed cubic structure by µ-hydroxo bridges and is tied to the terminal group Mn(4)(H2O)2. We show using the method of real-time nonadiabatic electron wavepacket dynamics that four charge separation steps always take place only through the terminal group Mn(4)(H2O)2 alone, thereby producing 4 electrons and 4 protons which are transported to the acceptors. Each of the three charge separation steps is followed by a reloading process from the skewed cubic structure, by which electrons and protons are refilled to the vacant terminal group so that the next charge separation dynamics can resume. After the fourth charge separation an oxygen molecule is generated. It is emphasized that the mechanisms of O2 generation should depend on the multiple channels of reloading.

Binuclear Mn oxo complex as a self-contained photocatalyst in water-splitting cycle: Role of additional Mn oxides as a buffer of electrons and protons.

We theoretically propose a photoinduced water-splitting cycle catalyzed by a binuclear Mn oxo complex. In our "bottom-up approach" to this problem, we once proposed a working minimal model of water-splitting cycle in terms of a mononuclear Mn oxo complex as a catalyst along with water clusters [K. Yamamoto and K. Takatsuka, Phys. Chem. Chem. Phys. 20, 6708 (2018)]. However, this catalyst is not self-contained in that the cycle additionally needs buffering molecules for electrons and protons in order to reload the Mn complex with electrons and protons, which are lost by photoinduced charge separation processes. We here show that a binuclear Mn oxo complex works as a self-contained photocatalyst without further assistant of additional reagents and propose another catalytic cycle in terms of this photocatalyst. Besides charge separation and proton relay transfer, the proposed cycle consists of other fundamental chemical dynamics including electron-proton reloading, radical relay-transfer, and Mn reduction. The feasibility of the present water-splitting cycle is examined by means of full dimensional nonadiabatic electron-wavepacket dynamics based on multireference electronic wavefunctions and energy profiles estimated with rather accurate quantum chemical methods for all the metastable states appearing in the cycle.

Relativistic theory of electron-nucleus-radiation coupled dynamics in molecules: Wavepacket approach.

We propose a general theoretical scheme of relativistic electron-nucleus coupled dynamics of molecules in radiation fields, which is derived from quantum electrodynamical formalism. Aiming at applications to field-induced dynamics in ultrastrong laser pulses to the magnitude of 1016 W/cm2 or even larger, we derive a nonperturbative formulation of relativistic dynamics using the Tamm-Dancoff expansion scheme, which results in, within the lowest order expansion, a time-dependent Schrödinger equation with the Coulombic and retarded transversal photon-exchange interactions. We also discuss a wavepacket type nuclear dynamics adapted for such dynamics.

On the Elementary Chemical Mechanisms of Unidirectional Proton Transfers: A Nonadiabatic Electron-Wavepacket Dynamics Study.

We propose a set of chemical reaction mechanisms of unidirectional proton transfers, which may possibly work as an elementary process in chemical and biological systems. Being theoretically derived based on our series of studies on charge separation dynamics in water splitting by Mn oxides, the present mechanisms have been constructed after careful exploration over the accumulated biological studies on cytochrome c oxidase (CcO) and bacteriorhodopsin. In particular, we have focused on the biochemical findings in the literature that unidirectional transfers of approximately two protons are driven by one electron passage through the reaction center (binuclear center) in CcO, whereas no such dissipative electron transfer is believed to be demanded in the proton transport in bacteriorhodopsin. The proposed basic mechanisms of unidirectional proton transfers are further reduced to two elementary dynamical processes, namely, what we call the coupled proton and electron-wavepacket transfer (CPEWT) and the inverse CPEWT. To show that the proposed mechanisms can indeed be materialized in a molecular level, we construct model systems with possible molecules that are rather familiar in biological chemistry, for which we perform the ab initio calculations of full-dimensional nonadiabatic electron-wavepacket dynamics coupled with all nuclear motions including proton transfers.

Chemical bonding and nonadiabatic electron wavepacket dynamics in densely quasi-degenerate excited electronic state manifold of boron clusters.

Formation of chemical bonds is theoretically discerned by the presence of static nuclear configuration on a potential energy surface given within the Born-Oppenheimer framework. We here study dynamical chemical bonding for molecules residing in the electronic excited states that are in a densely quasi-degenerate electronic state manifold and thereby keep undergoing extremely frequent nonadiabatic transitions. For this type of the states, the notion of global potential energy surfaces based on the adiabatic representation loses the usual sense. Nonetheless, chemical bonding exists and associated chemical reactions certainly proceed, for which we call chemistry without potential surfaces. As such, we investigate the highly excited states of boron clusters, which have extraordinarily long lifetimes with neither ionization nor dissociation. The dynamical chemical bonds keep rearranging themselves without converging to a static structure, the vivid electron dynamics of which is tracked by means of the nonadiabatic electron wavepacket dynamics theory. To characterize the dynamical bonding theoretically, we propose the notion of hyper-resonance.

Electronic and nuclear fluxes induced by quantum interference in the adiabatic and nonadiabatic dynamics in the Born-Huang representation.

We perform an electronic and nuclear flux analysis for nonadiabatic dynamics and its corresponding adiabatic counterpart, both of the wavefunctions of which are represented in the Born-Huang expansion. It is well known that the electronic-nuclear configurations (terms) in the expansion of the total wavefunction interfere each other through the nonadiabatic interactions and give birth to electronic and nuclear fluxes. Interestingly, even in the adiabatic dynamics without such nonadiabatic interactions, a wavefunction composed of more than one adiabatic state can undergo interference among the components and give the electronic and nuclear fluxes. That is, the individual pieces of the wavepacket components associated with the electronic wavefunctions in the adiabatic representation can propagate in time independently with no nonadiabatic interaction, and yet they can interfere among themselves to generate the specific types of electronic and nuclear fluxes. We refer to the dynamics of this class of total wavefunction as multiple-configuration adiabatic Born-Huang dynamics. A systematic way to distinguish the electronic and nuclear fluxes arising from nonadiabatic and the corresponding adiabatic dynamics is discussed, which leads to the deeper insight about the nonadiabatic dynamics and quantum interference in molecular processes. The so-called adiabatic flux will also be discussed.

Electronic and nuclear flux analysis on nonadiabatic electron transfer reaction: A view from single-configuration adiabatic born-huang representation.

A detailed flux analysis on nonadiabatically coupled electronic and nuclear dynamics in the intramolecular electron transfer of LiF is presented. Full quantum dynamics both of electrons and nuclei within two-state model has uncovered interesting features of the individual fluxes (current of probability density) and correlation between them. In particular, a spatiotemporal oscillatory pattern of electronic flux has been revealed, which reflects the coherence coming from spatiotemporal differential overlap between nuclear wavepackets running on covalent and ionic potential curves. In this regard, a theoretical analogy between the nonadiabatic transitions and the Rabi oscillation is surveyed. We also present a flux-flux correlation between the nuclear and electronic motions, which quantifies the extent of deviation of the actual electronic and nuclear coupled dynamics from the Born-Oppenheimer adiabatic limit, which is composed only of a single product of the adiabatic electronic and nuclear wavefunctions. © 2018 Wiley Periodicals, Inc.

Ab initio calculation of femtosecond-time-resolved photoelectron spectra of NO2 after excitation to the A-band.

We present calculations of time-dependent photoelectron spectra of NO2 after excitation to the A-band for comparison with extreme-ultraviolet (XUV) time-resolved photoelectron spectroscopy. We employ newly calculated potential energy surfaces of the two lowest-lying coupled 2A' states obtained from multi-reference configuration-interaction calculations to propagate the photo-excited wave packet using a split-step-operator method. The propagation includes the nonadiabatic coupling of the potential surfaces as well as the explicit interaction with the pump pulse centered at 3.1 eV (400 nm). A semiclassical approach to calculate the time-dependent photoelectron spectrum arising from the ionization to the eight energetically lowest-lying states of the cation allows us to reproduce the static experimental spectrum up to a binding energy of 16 eV and enables direct comparisons with XUV time-resolved photoelectron spectroscopy.