Adiabatic electronic flux density: A Born-Oppenheimer broken-symmetry ansatz

The Born-Oppenheimer approximation leads to the counterintuitive result of a vanishing electronic flux density upon vibrational dynamics in the electronic ground state. To circumvent this long known issue, we propose using pairwise antisymmetrically translated vibronic densities to generate a symmetric electronic density that can be forced to satisfy the continuity equation approximately. The so-called Born-Oppenheimer broken-symmetry ansatz yields all components of the flux density simultaneously while requiring only knowledge about the nuclear quantum dynamics on the electronic adiabatic ground-state potential energy surface. The underlying minimization procedure is transparent and computationally inexpensive, and the solution can be computed from the standard output of any quantum chemistry program. Taylor series expansion reveals that the implicit electron dynamics originates from nonadiabatic coupling to the explicit Born-Oppenheimer nuclear dynamics. Our approach is applied to the ${\mathrm{H}}_2^{+}$ molecular ion vibrating in its ${}^2{\mathrm{\Sigma }}_g^{+}$ ground state. The electronic flux density is found to have the correct nodal structure and symmetry properties at all times.

Figure 1

Schematic representation of the Born-Oppenheimer broken-symmetry ansatz. The bottom panel shows the two translated vibrational wave functions used to generate the broken-symmetry densities. Top panel: a symmetric vibronic density is created by adding two pairwise antisymmetric densities with respect to the $Q=Q^{\prime }$ axis. The coordinate $Q^{\prime }$ is introduced to emphasize that two different translations are used in Eq. (13).

Figure 2

Representative snapshots of the electron flow (top panel) and the electronic flux density (central panels) computed with the linearized BOBS ansatz [cf. Eqs. (21) and (22)] for a vibrating hydrogen molecular ion ${\mathrm{H}}_2^{+}$ oriented along the $z$ axis. The results are plotted in the $xz$ plane for three different times within the first half of the vibrational period: for $t=3\mathrm{fs}$ (left column), $t=6.45\mathrm{fs}$ (middle column), and $t=11.05\mathrm{fs}$ (right column). Note the logarithmic scale in the contour plots and the different scales in the vector plots. Bottom panels: Error of the correlated electronic continuity equation per grid point, ${\mathcal{L}}_2=\frac{1}{N}\parallel \frac{\partial }{\partial t}{\rho }_c+{\vec{\nabla }}_e·{\vec{j}}_c{\parallel }_2$, in units of $E_{\mathrm{h}}/\left(\hslash a_0^3\right)$ as a function of the correlation length squared for the three selected snapshots.

Figure 3

Detailed view of the symmetry properties of the flux density at the classical turning point, $t=11.05\mathrm{fs}$, at the topmost hydrogen atom (left panel) and close to the inversion center (right panel). The vector field is seen to be antisymmetric with respect to the molecular inversion center, and the flux density is largest pointing toward the nuclei.

Figure 4

Representative snapshot at $t=6.45\mathrm{fs}$ of the electron flow (top panels) and the electronic flux density (bottom panels) in the $xz$ plane for a vibrating hydrogen molecular ion ${\mathrm{H}}_2^{+}$ oriented along the $z$ axis. The results in the left panels are obtained from a non-Born-Oppenheimer ansatz [20], and those from the BOBS ansatz are shown in the right panels.

Figure 5

Top panel: the expectation value of the internuclear distance $〈Q〉$ (black dotted line) in units of $a_0$ as a function of time. Second panel: the squared value of the correlation length ${\delta }_q^2$ (black solid line) and the squared variance of the internuclear distance ${\sigma }^2=〈Q^2〉-{〈Q〉}^2$ (red dashed line) as a function of time. Third panel: the error of the optimization per grid point, ${\mathcal{L}}_2=\frac{1}{N}\parallel \frac{\partial }{\partial t}{\rho }_c+{\vec{\nabla }}_e·{\vec{j}}_c{\parallel }_2$ (black dashed dotted line), in units of ${10}^{-6}{E}_{\mathrm{h}}/\left(\hslash a_0^3\right)$ as a function of time. Regions where the expectation value of the internuclear distance $〈Q〉$ decreases are shaded gray. Notice the different scaling. Bottom panel: Nuclear wave packet at three times during the first half oscillation cycle: $t=3.00\mathrm{fs}$ (blue), $t=6.45\mathrm{fs}$ (green), and $t=11.05\mathrm{fs}$ (red).

Figure 6

Normalized power spectra of the correlation length squared ${\delta }_q^2$ (black) and the variance of the internuclear distance ${\sigma }^2=\langle Q^2\rangle -{\langle Q\rangle }^2$ (red) as a function of the frequency $\omega$. The dynamics was simulated up to twice the nuclear recurrence time of $T=293.16\mathrm{fs}$ with a time-step width of $\mathrm{\Delta }t=0.3\mathrm{fs}$.