Theory of dissociative tunneling ionization

We present a theoretical study of the dissociative tunneling ionization process. Analytic expressions for the nuclear kinetic energy distribution of the ionization rates are derived. A particularly simple expression for the spectrum is found by using the Born-Oppenheimer (BO) approximation in conjunction with the reflection principle. These spectra are compared to exact non-BO ab initio spectra obtained through model calculations with a quantum mechanical treatment of both the electronic and nuclear degrees of freedom. In the regime where the BO approximation is applicable, imaging of the BO nuclear wave function is demonstrated to be possible through reverse use of the reflection principle, when accounting appropriately for the electronic ionization rate. A qualitative difference between the exact and BO wave functions in the asymptotic region of large electronic distances is shown. Additionally, the behavior of the wave function across the turning line is seen to be reminiscent of light refraction. For weak fields, where the BO approximation does not apply, the weak-field asymptotic theory describes the spectrum accurately.

Figure 1

The blue/red surface is a paraboloid showing a surface of constant $\eta$. The gray paraboloid is the same for a smaller value of $\eta$. The electron is ionized in the negative $z$ direction due to its negative charge, given that the electric field points in the positive $z$ direction. The ${\mathrm{\Phi }}_{\nu }(\xi ,\phi )$ states [Eq. (16)] “live” in the constant $\eta$ paraboloids. The colors in the blue/red surface illustrate an example of the nodal structure of such a ${\mathrm{\Phi }}_{\nu }(\xi ,\phi )$ state. The curvature of the paraboloids means that the ${\mathrm{\Phi }}_{\nu }(\xi ,\phi )$ states are bound. This means that $\eta$ is the only coordinate where we have to consider the wave function at infinity, i.e., $\eta$ is the “tunneling” coordinate. For large $\eta$, the polar angle $\beta$, which specifies the orientation of the molecule, does not matter for the asymptotic form of the wave function in parabolic coordinates, though it matters for the value of the coefficients [Eq. (18)].

Figure 2

The field dressed nuclear wave function ${\left|\chi \left(R\right)\right|}^2$ (light blue shaded area in the lower $U\left(R\right)+E_e(R;F)$ BO curve) is multiplied by the electronic rate ${\mathrm{\Gamma }}_e\left(R\right)$ (dashed purple line) and reflected in the dissociative $U\left(R\right)$ BO curve to give a KER spectrum [solid blue line in upper right corner, Eq. (32)], using the relation $U\left(R\right)=\frac{k^2}{2M}$ to translate $k$ into $R$. This is compared to the exact KER spectrum $P\left(k\right)$ [red dashed line, Eq. (45)]. A field strength of $F=0.034\text{a.u.}$ was used for this calculation. The solid gray line in the lower part of the figure shows the field-free nuclear wave function ${\left|{\chi }_0\left(R\right)\right|}^2$. The surface plot in the upper part of the figure shows the continuum states $g(R,k)$ of the $U\left(R\right)$ potential, these are solutions of Eq. (8b).

Figure 3

KER Spectra normalized to their maxima. Solid (red) line: $P\left(k\right)$ [Eq. (45)]. Dashed dotted (blue) line: BO combined with reflection principle [Eq. (32)]. Short dashed (green) line: WFAT [1D equivalent of Eq. (40)]. The insets show the normalized KER spectra on a linear scale. The critical field for use of BO [Eq. (49)] is for ${\mathrm{H}}_2{}^{+}$: $F_{\text{BO}}=0.0315\text{a.u}$. (a) $P_{\text{max}}^{\text{BO}}/P_{\text{max}}^{\text{Exact}}=2.32$ and $P_{\text{max}}^{\text{WFAT}}/P_{\text{max}}^{\text{Exact}}=0.29$. (b) $P_{\text{max}}^{\text{BO}}/P_{\text{max}}^{\text{Exact}}=95.0$ and $P_{\text{max}}^{\text{WFAT}}/P_{\text{max}}^{\text{Exact}}=0.54$.

Figure 4

From the exact KER spectrum $P\left(k\right)$ (upper right corner, [Eq. (45)]) at $F=0.034\text{a.u.}$ the magnitude of the asymptotic wave function has been found by reversing the reflection principle, giving $P\left(k\right)/\left(2\pi \left|\frac{dR}{dk}\right|\right)$, using the relation $U\left(R\right)=\frac{k^2}{2M}$ to translate $k$ into $R$. From this, the field-dressed nuclear wave function has been imaged by dividing with the electronic rate ${\mathrm{\Gamma }}_e\left(R\right)$ and normalizing. In the lowest part of the plot, the short dashed (purple) line shows this imaging using the exact electronic rate ${\mathrm{\Gamma }}_e\left(R\right)=-2Im\left[E_e(R;F)\right]$, the long dashed (red) line shows it using the BO WFAT approximation ${\mathrm{\Gamma }}_e^{\text{WFAT}}\left(R\right)$ [Eq. (35)]. The solid gray line shows the field-free nuclear wave function ${\left|{\chi }_0\left(R\right)\right|}^2$. The shaded (light blue) area shows the field-dressed nuclear wave function ${\left|\chi \left(R\right)\right|}^2$. The surface plot in the upper part of the figure shows the continuum states $g(R,k)$.

Figure 5

Real part of wave function normalized with the electron density $\rho \left(z\right)=\int {\left|\mathrm{\Psi }(z,R)\right|}^{2}dR$ for two different fields strengths. Solid (blue) lines: electronic turning lines $Re\left(E_e(R;F)\right)=V(z;R)+Fqz$. Dashed (red) lines: full turning lines $Re\left(E\right(F\left)\right)=V(z;R)+U\left(R\right)+Fqz$. In the upper panels, the dashed black line shows a constant line that coincides with the maximum of the BO wave function. In the middle panels, the green line additionally shows a classical trajectory that coincides with the maximum of the exact wave function. The black dot at the end of the classical trajectory is the exit point $(z_{k_{\text{max}}},R_{k_{\text{max}}})$ determined from the maximum of the spectrum $k_{\text{max}}$ (see main text). The critical BO distance [Eq. (48)] is shown with a vertical dashed line at $-z_{\text{BO}}=-32.9\text{a.u}$.

Figure 6

Absolute value of wave function normalized with the electron density $\rho \left(z\right)=\int {\left|\mathrm{\Psi }(z,R)\right|}^{2}dR$ for $F=0.035\text{a.u}$. Solid purple lines: full turning lines $Re\left(E\right(F\left)\right)=V(z;R)+U\left(R\right)+Fqz$. The long dashed red line shows for each $z$ the $R$ at which the wave function $\left|\mathrm{\Psi }(z,R)\right|$ has its maximum. The solid pink line shows a classical trajectory [Eq. (50)]. The black dot at the end of the classical trajectory is the exit point $(z_{k_{\text{max}}},R_{k_{\text{max}}})$ determined from the maximum of the spectrum $k_{\text{max}}$ (see main text). The short dashed lines are the simple straight line estimates for the tunneling and initial classical motion described around Eq. (52).