Probing electron correlation in molecules via quantum fluxes

We present quantum simulations of a vibrating hydrogen molecule H${}_2$ and address the issue of electron correlation. After appropriately setting the frame and the observer plane, we were able to determine precisely the number of electrons and nuclei which actually flow by evaluating electronic and nuclear fluxes. This calculation is repeated for three levels of quantum chemistry, for which we account for no correlation, Hartree-Fock, static correlation, and dynamic correlation. Exciting each of these systems with the same amount of energy, we show that the electron correlation can be revealed with the knowledge of quantum fluxes. This is evidenced by a clear sensitivity of these fluxes to electron correlation. In particular, we find that this correlation remarkably enhances more electronic yield than the nuclear one. It turns out that less electrons accompany the nuclei in Hartree-Fock than in the correlation cases.

Figure 1

Potential-energy surfaces of H${}_2$ in the electronic ground state using HF (black), CAS(2,2) (dashed), and CAS(2,8) [red (gray)] calculations, on which the initial wave packets (1), (2), and (3) are launched, respectively. The excitation energy $E_{\mathrm{ex}}=0.1231E_h$ is the same for each case.

Figure 2

The mean molecular bond length $\langle R\rangle$ (a), the nuclear flux (b), and the corresponding nuclear yield (c) as functions of time, for the three levels of computations: HF (black), CAS(2,2) (dashed), and CAS(2,8) [red (gray)]. For all computations, we set the observer at $R_{\mathrm{ob}}=2.5a_0$ (dotted). Here $t_{\mathrm{del}}$, indicated by the arrow, is the time delay for which the nuclei start passing the observer plane.

Figure 3

The electronic flux (a) and the corresponding electronic yield (b) as functions of time, for the three levels of computation: HF (black), CAS(2,2) (dashed), and CAS(2,8) [red (gray)]. For all computations, we set the observer at $R_{\mathrm{ob}}=2.5a_0$ (dotted), exactly as in Fig. 2.

Figure 4

Snapshots of the electronic densities at time $t=0$ (a), at the times the electronic flux is maximum (b), and at the times the electronic yields are maximum (c). In each panel, there are three levels of computation: HF (black), CAS(2,2) (dashed), and CAS(2,8) [red (gray)]. The dashed lines indicate the location of the observer $R_{\text{ob}}$ in $z$ subspace

Figure 5

The electronic flux (red) and the corresponding nuclear flux (black) as functions of time, for the three levels of computation: HF, CAS(2,2), and CAS(2,8). For all computations, we set the observer at $R_{\mathrm{ob}}=2.5a_0$, exactly as in Fig. 2. In the three right panels, the zoom-in is for the time intervals $\Delta t$, for which electron and nuclear fluxes are counterintuitively opposed.