Figure 1

Image of the nuclear wave functions of $\mathrm{H}_2{}^{+}$. (a) Measured vibrational energy $E_{\mathrm{vib}}$ versus internuclear distance $R$ calculated from the kinetic energy release of the dissociation as described below. The potential energy curve of $\mathrm{H}_2{}^{+}(1s{\sigma }_g)$ extracted from [12] is shown as the green line. (b) Theoretical spatial density of the nine lowest vibrational states numerically calculated using the Born-Oppenheimer approximation.Reuse & Permissions

Figure 2

Single molecule imaging using the reflection approximation for $\mathrm{H}_2{}^{+}$. The molecule is promoted by a vertical electronic transition to the repulsive potential energy curve $b^3\Sigma _u{}^{+}$ (green line) which is described by the function $\epsilon ={\phi }^f(R)$. Within a few 10 fs it dissociates along this curve. After approximately 1500 ns we measure the kinetic energy release (KER) of the H ($1s$) fragments. In the standard frozen nuclei version of the reflection approximation the relationship between KER and the internuclear distance $R$ is given by ${\mathrm{KER}}_{\mathrm{frozen}}={\phi }^f(R)$. The resulting KER distribution is shown in the inset.Reuse & Permissions

Figure 3

Spatial density of the lower vibrational states of $\mathrm{H}_2{}^{+}$ with vibrational quantum number $\nu =0$ to 4. (a),(b) The internuclear distance $R$ was calculated from the measured data either by the standard frozen nuclei reflection approximation (green dots) or the moving nuclei reflection approximation (red open circles). For both approximations the positions of the nodes do not coincide with the theoretical prediction (black solid line). (c)–(g) The mean value of both approximations (blue dots) is compared to the theoretical distributions of the rotational ground state (solid line) and rotationally excited states characterized by the quantum number $J$ which is not determined in the experiment.Reuse & Permissions

Figure 4

Quantum mechanical calculation of the KER distribution. (a) Initial state wave function of $\mathrm{H}_2{}^{+}$ $\nu =2$. (b) Product of initial state and the continuum states of ${\mathrm{H}}_2(b^3\Sigma _u{}^{+})$ parametrized by KER. The frozen nuclei and moving nuclei reflection approximation constitute a strong correlation between $R$ and KER as depicted by the green solid and green dashed lines. Integrating this product over $R$ yields the KER dependent overlap shown in panel (c) The square of the overlap is compared to the experimental results in (d) and (e) shows the same for the vibrational state $\nu =3$.Reuse & Permissions