Weak-field asymptotic theory of tunneling ionization in many-electron atomic and molecular systems

The weak-field asymptotic theory of tunneling ionization in an external static uniform electric field [Tolstikhin et al., Phys. Rev. A 84, 053423 (2011)] is extended to many-electron atomic and molecular systems treated in the frozen-nuclei approximation. The leading-order term in the asymptotic expansion of the ionization rate $\Gamma$ in the value of the field $F$ for $F\to 0$ is obtained. The resulting formulas express $\Gamma$ in terms of properties of the unperturbed system. The most essential difference from the one-electron case, through which the many-electron character of the present theory reveals itself, is that the structure factor for a given ionization channel, defining the dependence of the ionization rate into this channel on the orientation of the system with respect to the field, is determined by the corresponding Dyson orbital. The theory is illustrated by calculations for several few-electron systems. The asymptotic results are compared with accurate fully correlated calculations of tunneling ionization rates available in the literature.

Figure 1

Dyson orbitals for H${}^{-}$ and three lowest states of He, see Table 1, calculated with the present accurate two-electron wave functions. Thin solid lines: $1s$ HF orbital of He($1s^2{}^{1}S$) multiplied by $\sqrt{2}$ (cyan) and $2s$ HF orbital of He($1s2s{}^{3}S$) (light green) calculated using the program of Ref. [70].

Figure 2

Ionization rate of He in the ground state divided by the field factor (61) for the dominant ionization channel $\mathrm{He}\left(1s^2{}^{1}S\right)\to {\mathrm{He}}^{+}\left(1s\right)$ with $n_{\xi }=m=0$. Solid symbols: fully correlated calculations from Refs. [36, 37, 38]. Solid (black) line with an open circle at $F=0$: present leading-order ME-WFAT results for ionization into the final state He${}^{+}\left(1s\right)$, Eq. (63), given by a constant $g^2=8.595$; see Table 1. Dashed (blue) line: SAEA results for ionization of one electron from the $1s$ state in the potential (66) calculated by the method of Ref. [31] and multiplied by 2.

Figure 3

Dyson orbital for H${}_2$, see Table 1, as a function of the distance $r$ from the center of the molecule along the internuclear axis for the internuclear distance $R=1.4009$. Dashed (black) line: results obtained with an accurate two-electron wave function from Ref. [69]. Thin solid (red) line: $1s\sigma$ HF orbital of ${\mathrm{H}}_2\left(1s{\sigma }^2{}^1{\Sigma }_g{}^{+}\right)$ multiplied by $\sqrt{2}$ calculated using the program of Ref. [76]. Vertical dotted line at $r=R/2$ indicates the position of one of the nuclei.

Figure 4

Ionization rate of H${}_2$ aligned along the field in the ground electronic state divided by the field factor (61) for the dominant ionization channel ${\mathrm{H}}_2\left(1s{\sigma }^2{}^1{\Sigma }_g{}^{+}\right)\to {\mathrm{H}}_2{}^{+}\left(1s\sigma \right)$ with $n_{\xi }=m=0$. Solid circles: fully correlated calculations from Ref. [18] for the internuclear distance $R=1.4$. Solid line with an open circle at $F=0$: present leading-order ME-WFAT results for ionization into the final state H${}_2{}^{+}\left(1s\sigma \right)$, Eq. (63), given by a constant $g^2=7.45$ calculated for $R=1.4009$; see Table 1.

Figure 5

Ionization rate of Li in the ground state divided by the field factor (61) for the dominant ionization channel $\mathrm{Li}\left(1s^{2}2s{}^{2}S\right)\to {\mathrm{Li}}^{+}\left(1s^2{}^{1}S\right)$ with $n_{\xi }=m=0$. Solid symbols: fully correlated calculations from Ref. [39]. Solid (black) line with an open circle at $F=0$: present leading-order ME-WFAT results for ionization into the final state Li${}^{+}\left(1s^2{}^{1}S\right)$, Eq. (63), obtained in the single-determinant unrelaxed HF approximation and given by a constant $g_{\mathrm{HF}}^2=0.225$; see Table 1. Dashed (blue) line: SAEA results for ionization of one electron from the $2s$ state in the potential (69) calculated by the method of Ref. [31].