Vibrating H2+(2Σg+, JM = 00) Ion as a Pulsating Quantum Bubble in the Laboratory Frame

Quantum flux densities for electronic-nuclear motion: exact versus Born-Oppenheimer dynamics

We study the coupled electronic-nuclear dynamics in a model system to compare numerically exact calculations of electronic and nuclear flux densities with those obtained from the Born-Oppenheimer (BO) approximation. Within the adiabatic expansion of the total wave function, we identify the terms which contribute to the flux densities. It is found that only off-diagonal elements that involve the interaction between different electronic states contribute to the electronic flux whereas in the nuclear case the major contribution belongs to the BO electronic state. New flux densities are introduced where in both, the electronic and the nuclear case, the main contribution is contained in the component corresponding to the BO state. As a consequence, they can be determined within the BO approximation, and an excellent agreement with the exact results is found. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

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

Charge separation, reloading of electrons and protons, and O₂ generation in a catalytic cycle for water splitting with Mn₄CaO₅ in PSII.

On the Probability Density of the Nuclei in a Vibrationally Excited Molecule

For localized and oriented vibrationally excited molecules, the qualitative features of the one-body probability density of the nuclei (one-nucleus density) are investigated. Like the familiar and widely used one-electron density that represents the probability of finding an electron at a given location in space, the one-nucleus density represents the probability of finding a nucleus at a given position in space independent of the location of the other nuclei and independent of their type. In contrast to the electrons, however, the nuclei are comparably localized. Due to this localization of the individual nuclei, the one-nucleus density provides a quantum-mechanical representation of the "chemical picture" of the molecule as an object that can largely be understood in a three-dimensional space, even though its full nuclear probability density is defined on the high-dimensional configuration space of all the nuclei. We study how the nodal structure of the wavefunctions of vibrationally excited states translates to the one-nucleus density. It is found that nodes do not necessarily lead to visible changes in the one-nucleus density: Already for relatively small molecules, only certain vibrational excitations change the one-nucleus density qualitatively compared to the ground state. It turns out that there are simple rules for predicting the shape of the one-nucleus density from the normal mode coordinates. A Python module for the computation of the one-nucleus density is provided at https://gitlab.com/axelschild/mQNMc.

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.

Na2 Vibrating in the Double-Well Potential of State 2 1Σu+ (JM = 00): A Pulsating "Quantum Bubble" with Antagonistic Electronic Flux

The theory of concerted electronic and nuclear flux densities associated with the vibration and dissociation of a multielectron nonrotating homonuclear diatomic molecule (or ion) in an electronic state ^2S+1Σ_g,u⁺ (JM = 00) is presented. The electronic population density, nuclear probability density, and nuclear flux density are isotropic. A theorem of Barth , presented in this issue, shows that the electronic flux density (EFD) is also isotropic. Hence, the evolving system appears as a pulsating, or exploding, "quantum bubble". Application of the theory to Na₂ vibrating in the double-minimum potential of the 2  ¹Σ_u⁺ (JM = 00) excited state reveals that the EFD consists of two antagonistic components. One arises from electrons that flow essentially coherently with the nuclei. The other, which is oppositely directed (i.e., antagonistic) and more intense, is due to the transition in electronic structure from "Rydberg" to "ionic" type as the nuclei traverse the potential barrier between inner and outer potential wells. This "transition" component of the EFD rises and falls sharply as the nuclei cross the barrier.

Probability and Flux Densities in the Center-of-Mass Frame

For an arbitrary nonstationary wave function of a nonrelativistic closed many-body system consisting of arbitrary interacting particles, the general expressions for the time-dependent one-particle probability and flux densities in the center-of-mass frame without applying Born-Oppenheimer approximation are obtained. Even the wave function for the translation is additionally introduced; it disappears in the center-of-mass frame automatically. It is shown that for the rotational ground state the time-dependent probability and flux densities of an arbitrary particle in the center-of-mass frame are isotropic. It means that the angular dependence is absent but these densities depend on radius and time. More importantly, it is shown that the angular components of the time-dependent flux density vanish. With these statements, one can calculate the radial component of the radius- and time-dependent electronic flux density within the Born-Oppenheimer approximation via the continuity equation. Application of this theory to the pulsating or exploding "quantum bubble" of the vibrating or dissociating Na₂ molecule in the rotational ground state, respectively, is found elsewhere in this issue.

Concerted Electronic and Nuclear Fluxes During Coherent Tunnelling in Asymmetric Double-Well Potentials

The quantum theory of concerted electronic and nuclear fluxes (CENFs) during coherent periodic tunnelling from reactants (R) to products (P) and back to R in molecules with asymmetric double-well potentials is developed. The results are deduced from the solution of the time-dependent Schrödinger equation as a coherent superposition of two eigenstates; here, these are the two states of the lowest tunnelling doublet. This allows the periodic time evolutions of the resulting electronic and nuclear probability densities (EPDs and NPDs) as well as the CENFs to be expressed in terms of simple sinusodial functions. These analytical results reveal various phenomena during coherent tunnelling in asymmetric double-well potentials, e.g., all EPDs and NPDs as well as all CENFs are synchronous. Distortion of the symmetric reference to a system with an asymmetric double-well potential breaks the spatial symmetry of the EPDs and NPDs, but, surprisingly, the symmetry of the CENFs is conserved. Exemplary application to the Cope rearrangement of semibullvalene shows that tunnelling of the ideal symmetric system can be suppressed by asymmetries induced by rather small external electric fields. The amplitude for the half tunnelling, half nontunnelling border is as low as 0.218 × 10(-8) V/cm. At the same time, the delocalized eigenstates of the symmetric reference, which can be regarded as Schrödinger's cat-type states representing R and P with equal probabilities, get localized at one or the other minima of the asymmetric double-well potential, representing either R or P.

Staircase patterns of nuclear fluxes during coherent tunneling in excited doublets of symmetric double well potentials

Step-by-step flux for one-by-one transfers of the lobes of the density, from the reactant (left) to the product (right) in the excited tunneling doublet.

Quantum theory of concerted electronic and nuclear fluxes associated with adiabatic intramolecular processes

Example of concerted electronic (right) and nuclear (left) fluxes: isomerization of B₄.