Application of the weak-field asymptotic theory to the analysis of tunneling ionization of linear molecules

The recently developed weak-field asymptotic theory [Phys. Rev. A 84, 053423 (2011)] is applied to the analysis of tunneling ionization of a molecular ion (H${}_2^{+}$), several homonuclear (H${}_2$, N${}_2$, O${}_2$) and heteronuclear (CO, HF) diatomic molecules, and a linear triatomic molecule (CO${}_2$) in a static electric field. The dependence of the ionization rate on the angle between the molecular axis and the field is determined by a structure factor for the highest occupied molecular orbital. This factor is calculated using a virtually exact discrete variable representation wave function for H${}_2^{+}$, very accurate Hartree-Fock wave functions for the diatomics, and a Hartree-Fock quantum chemistry wave function for CO${}_2$. The structure factors are expanded in terms of standard functions and the associated structure coefficients, allowing the determination of the ionization rate for any orientation of the molecule with respect to the field, are tabulated. Our results, which are exact in the weak-field limit for H${}_2^{+}$ and, in addition, under the Hartree-Fock approximation for the diatomics, are compared with results from the recent literature.

Figure 1

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the electric field [see Eqs. (4) and (5)] for the ground $1s{\sigma }_g$ state of H${}_2^{+}$ with nuclei fixed at a distance $R=2$. The different channels are indicated by $(n_{\xi },m)$. The dominant channel $(0,0)$ is shown by solid black curve.

Figure 2

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the electric field [see Eqs. (4) and (5)] for the ground state in H${}_2$ of ${\sigma }_g$ symmetry. Solid (black) curve, $m=0$; dashed (red) curve, $m=1$. The nuclei are fixed at the experimental equilibrium distance.

Figure 3

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the electric field [see Eqs. (4) and (5)] for the HOMO in N${}_2$ of ${\sigma }_g$ symmetry. Solid (black) curve, $m=0$; dashed (red) curve, $m=1$. The nuclei are fixed at the experimental equilibrium distance.

Figure 4

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the electric field [see Eqs. (4) and (5)] for the HOMO in CO of $\sigma$ symmetry. The C (O) atom is on the negative (positive) $z$ axis in the molecular fixed frame. Solid (black) curve, $m=0$; dashed (red) curve, $m=1$ channel. The nuclei are fixed at the experimental equilibrium distance.

Figure 5

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the direction of the electric field [see Eqs. (4) and (5)] for the two degenerate HOMOs in O${}_2$ of ${\pi }_g$ symmetry. In the top [bottom] panel O${}_2\left(yz\right)$ [O${}_2\left(xz\right)$] denotes that the nodal plane of the $\pi$ orbital is in the $yz$ [$xz$] plane. Solid (black) curve, $m=0$; dashed (red) curve, $m=1$; dot-dashed (blue) curve, $m=2$. The nuclei are fixed at the experimental equilibrium distance.

Figure 6

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the direction of the electric field [see Eqs. (4) and (5)] for the two degenerate HOMOs in HF of $\pi$ symmetry. In the top [bottom] panel HF $\left(yz\right)$ [HF $\left(xz\right)$] denotes that the nodal plane of the $\pi$ orbital is in the $yz$ [$xz$] plane. The H [F] atom is on the negative [positive] $z$ axis in the molecular fixed frame. Solid (black) curve, $m=0$; dashed (red) curve, $m=1$; dot-dashed (blue) curve, $m=2$. The nuclei are fixed at the experimental equilibrium distance.

Figure 7

The dependence of the ratio of the rate $\Gamma$ and the function $W_{00}\left(F\right)$ [see Eq. (17)] on the angle $\beta$ between the internuclear axis and the direction of the electric field for the O${}_2\left(yz\right)$ HOMO of ${\pi }_g$ symmetry near $\beta =0^{\circ }$ and $\beta ={90}^{\circ }$, where the contribution to the rate from the $(0,0)$ channel vanishes (see Fig. 5). The different curves show the results for the field strengths indicated. The nuclei are fixed at the experimental equilibrium distance.

Figure 8

The value of the structure factor $G_{00}\left(\beta \right)$ [see Eq. (5)] as a function of $\eta$ for the O${}_2\left(yz\right)$ HOMO of ${\pi }_g$ symmetry and for three different orientations $\beta$ of the internuclear axis with respect to the direction of the electric field. Solid curves, using the HOMO wave function obtained by x2dhf [64, 65]; dashed curves, using the HOMO wave function obtained by gamess [66] with the aug-cc-TZV basis set.

Figure 9

The value of the structure factor $G_{00}\left(\beta \right)$ [see Eq. (5)] as a function of $\eta$ for the CO${}_2\left(yz\right)$ HOMO of ${\pi }_g$ symmetry and for three different orientations $\beta$ of the internuclear axis with respect to the direction of the electric field. Dashed curves, using the HOMO wave function obtained by gamess [66] with the aug-cc-TZV basis set.

Figure 10

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the electric field [see Eqs. (4) and (5)] for the two degenerate HOMOs in CO${}_2$ of ${\pi }_g$ symmetry. In the top [bottom] panel CO${}_2\left(yz\right)$ [CO${}_2\left(xz\right)$] denotes that the nodal plane of the $\pi$ orbital is in $yz$ [$xz$] plane. Solid (black) curve, $m=0$; dashed (red) curve, $m=1$; dot-dashed (blue) curve, $m=2$. The nuclei are fixed at the experimental equilibrium distance.