This section explores the stationary electronic states of the hydrogen molecular cation obtained by solving the electronic Schrödinger equation within the linear combination of atomic orbitals (LCAO) framework. It illustrates the formation of bonding and antibonding electronic states, their corresponding potential energy surfaces, and the associated electron probability densities.
This section explores the coherent evolution of the electronic wavefunction formed by a superposition of the bonding and antibonding electronic states. The time-dependent electron probability density illustrates how quantum interference between stationary electronic states gives rise to oscillatory charge redistribution within the molecule. By varying the composition of the electronic superposition and following its evolution in time, the relationship between the stationary electronic structure and the electron dynamics can be explored interactively.
This section explores the quantum motion of the nuclei, which is fully determined by the electronic structure of the molecule. Nuclear dynamics are described by the time evolution of nuclear wave packets propagating on the electronic potential energy surfaces, where the gradients and curvature of the surfaces determine the forces acting on the nuclei and the character of the resulting motion.
This section explores the coupled evolution of electronic and nuclear degrees of freedom in the hydrogen molecular cation following ionization. The full molecular wavefunction describes the simultaneous dynamics of the electronic state and the nuclear wave packets, capturing the interplay between electronic coherence and nuclear motion.
Welcome to the Quantum Dynamics Learning Tool!
Developers: Alex Klyber, Maximillian Thomas, and Nikolay Golubev
This interactive learning tool provides a visualization of ultrafast electron-nuclear dynamics in the \(H_2^{+}\) molecular ion. The interface is organized into multiple sections, each focusing on a specific aspect of the system and the underlying dynamical processes. Context-sensitive information icons accompanying the figures and interface elements provide detailed descriptions of the quantities being displayed and their physical interpretation. A comprehensive discussion of the theoretical framework, computational methodology, and the physical concepts underlying the presented results is available in the Theory section.
Adjust the system parameters and simulation time to explore the electronic and electron-nuclear dynamics under different conditions. You can return to this page at any time by selecting the About button.
For additional interactive demonstrations and related projects, visit www.ngolubev.com!
Hydrogen Cation Bonding and Antibonding PESs
This plot displays the bonding and antibonding potential energy surfaces (PESs), \(E_B(R)\) and \(E_A(R)\), as functions of the internuclear distance \(R\). The bonding curve exhibits a minimum corresponding to the equilibrium bond length of the molecule, whereas the antibonding curve is purely repulsive, illustrating the absence of a stable nuclear configuration.
Electron Probability Density of Bonding State
This plot shows the electron probability density, \(\rho_{BB}(\mathbf{r})=|\Phi_B(\mathbf{r})|^2\), of the bonding electronic state for the selected internuclear distance. The constructive interference of the atomic orbitals leads to an increased electron density between the two nuclei, giving rise to the chemical bond. The red dots indicate the positions of the nuclei (protons) in the H2+ molecular ion.
Electron Probability Density of Antibonding State
This plot displays the electron probability density, \(\rho_{AA}(\mathbf{r})=|\Phi_A(\mathbf{r})|^2\), of the antibonding electronic state. Destructive interference between the atomic orbitals produces a nodal region between the nuclei, reducing the electron density in the bonding region and resulting in a repulsive interaction.
Transition Electron Density
This plot illustrates the transition electron density, \(\rho_{BA}(\mathbf{r})=\Phi^{*}_B(\mathbf{r})\Phi_A(\mathbf{r})\), corresponding to the off-diagonal element of the electronic density matrix. Unlike the electron densities of the states, the transition density describes the redistribution of electronic charge associated with a transition between two quantum states. It reveals where the excitation originates, where electron density is transferred, and how strongly different regions of the molecule participate in processes such as optical excitation, charge migration, and light-matter interaction. This quantity therefore plays an important role in the ultrafast quantum dynamics discussed in the following sections.
Time-Dependent Electron Probability Density
This plot displays the time-dependent electron probability density, \(\rho(\mathbf{r},t)\), corresponding to the selected coherent superposition of the bonding and antibonding states. The electron density oscillates between the two sides of the molecule as the relative phase between the stationary electronic states evolves in time, providing a direct visualization of ultrafast electron motion. By adjusting the expansion coefficients, the slider controls the contribution of the bonding and antibonding states to the superposition. When the superposition approaches a purely bonding or purely antibonding state, the electron density converges to the corresponding stationary distributions shown in the static analysis. The red dots indicate the positions of the nuclei (protons) in the H2+ molecular ion.
Dynamics of Nuclear Wave Packets
This panel displays the time evolution of the nuclear wave packets, \(\chi(\mathbf{R},t)\), in the hydrogen molecular cation following sudden ionization from the neutral hydrogen molecule. The ionization process promotes the nuclear wave packet from the neutral state to the bonding and antibonding electronic states of the cation, which then evolve according to their respective potential energy landscapes. It illustrates how the different shapes of these electronic states influence the subsequent nuclear motion, including wave packet spreading and dissociation dynamics.
Internuclear Distance
This panel displays the time evolution of the expectation value of the internuclear distance for the nuclear wave packets evolving on the bonding and antibonding electronic states. The individual curves show the nuclear motion associated with each electronic state, while their average represents the overall internuclear dynamics resulting from the coherent superposition of the states.
Nuclear Momentum
This panel displays the time evolution of the expectation value of the nuclear momentum for the nuclear wave packets evolving on the bonding and antibonding electronic states. The momentum evolution provides information about the direction and rate of nuclear motion. The average momentum highlights the overall nuclear motion of the molecular system resulting from the combined contribution of the bonding and antibonding states.
Electronic Coherence
This panel displays the time evolution of the electronic coherence between the bonding and antibonding states, characterized by the overlap of the corresponding nuclear wave packets. The coherence initially enables quantum interference between the electronic states and drives coherent electron motion. As the nuclear wave packets evolve and separate due to the different forces acting on the electronic states, the coherence decreases, leading to the gradual suppression of coherent electronic oscillations.
Dynamics of Electron Probability Density
This contour plot provides a spatial representation of the time-dependent electron probability density, highlighting the coupled electron-nuclear dynamics of the hydrogen molecular cation. Following ionization, the electron initially undergoes coherent oscillations driven by the superposition of bonding and antibonding electronic states. As the nuclear wave packets evolve and the nuclei move apart, the electronic coherence gradually diminishes, suppressing the oscillatory charge redistribution. At later times, the electron density evolves together with the separating nuclei, illustrating the transition from coherent electron motion to dynamics governed by the nuclear motion.
Time-Dependent Electron Probability Density
This panel displays the time-dependent electron probability density, \(\rho(\mathbf{r},t)\), obtained from the full molecular wavefunction by including both electronic and nuclear dynamics. At early times, when the nuclear wave packets associated with the bonding and antibonding states still strongly overlap, the electron density evolves similarly to the purely electronic case, exhibiting coherent charge oscillations driven by interference between the two electronic states. As the nuclear wave packets evolve on their respective potential energy surfaces and their overlap decreases, the electronic coherence gradually diminishes, suppressing the interference-driven oscillations. At later times, the electron density is dominated by the contributions from the individual electronic states, \(\int \rho_A |\chi_A|^2 dR\) and \(\int \rho_B |\chi_B|^2 dR\), and the electronic motion becomes primarily coupled to the ongoing nuclear dynamics.
The blue dots indicate the positions of the protons associated with the bonding-state nuclear wave packet, while the orange dots indicate the positions of the protons associated with the antibonding-state nuclear wave packet. Their separation illustrates the different nuclear evolution pathways on the corresponding electronic states and highlights the coupling between nuclear motion and the evolving electron density.
Internuclear Distance
This slider controls the separation \(R\) between the two nuclei. Adjusting this parameter updates the position of the nuclei and all electron density plots in real time, allowing the changes in the molecular electronic structure with bond length to be explored interactively.
Bohr
Expansion Coefficients
These sliders control the amplitudes of the bonding and antibonding electronic states in the time-dependent wavefunction. Changing these coefficients modifies the populations of the stationary states and the magnitude of the electronic coherence, thereby determining the strength of the time-dependent charge redistribution. Pure stationary states produce a time-independent electron density, whereas superpositions lead to coherent electron dynamics.
\(|c_\text{B}|^2\):
\(|c_\text{A}|^2\):
Time
This slider controls the evolution time of the quantum system. Changing this parameter allows the temporal evolution of the corresponding wavefunction and observables to be explored, revealing how electronic and nuclear degrees of freedom evolve and interact over time.