Structure factors for tunneling ionization rates of molecules

Within the weak-field asymptotic theory, the dependence of the tunneling ionization rate of a molecule in a static electric field on its orientation with respect to the field is determined by the structure factor for the highest occupied molecular orbital (HOMO). An accurate determination of this factor, and hence the ionization rate, requires accurate values of the HOMO in the asymptotic region. Techniques for calculating the structure factors for molecules in the Hartree-Fock approximation are discussed. For diatomics, grid-based numerical Hartree-Fock calculations which reproduce the correct asymptotic tail of the HOMO are possible. However, for larger molecules, to solve the Hartree-Fock equations one should resort to basis-based approaches with too rapidly decaying Gaussian basis functions. A systematic study of the possibility to reproduce the asymptotic tail of the HOMO in calculations with Gaussian basis sets is presented. We find that polarization-consistent basis sets with quadruple or pentuple-zeta quality greatly improve the tail of the HOMO, but only when used with variationally optimized exponents. This methodology is validated by considering the CO molecule for which reliable grid-based calculations can be performed. The optimized Gaussian basis sets are used to calculate the structure factors for the triatomic molecules CO${}_2$ and OCS. The results are compared with available experimental and theoretical results.

Figure 1

The value of the structure function $G_{00}(\beta ={180}^{\circ },\eta )$ [see Eq. (5)] as a function of $\eta$ for the CO HOMO of $\sigma$ symmetry. The solid blue reference curve uses the HOMO obtained by x2dhf [21, 22]. The dashed red curve uses the HOMO from gaussian [26] with the standard TZV basis set. The dotted black curve uses the HOMO from gaussian with the standard aug-cc-pVTZ basis set. The nuclei are fixed at the distance $R=2.132178$ a.u.

Figure 2

The value of the structure function $G_{00}(\beta ={180}^{\circ },\eta )$ [see Eq. (5)] for the CO HOMO of $\sigma$ symmetry. The solid blue reference curve uses the HOMO obtained by x2dhf [21, 22]. The dashed red curve uses the HOMO from gaussian [26] with an aug-pc-1 basis set. The dotted green curve uses the HOMO from gaussian with a aug-pc-2 basis set. The solid magenta curve with plus signs uses the HOMO from gaussian with a aug-pc-3 basis set. The solid black curve uses the HOMO from gaussian with a aug-pc-4 basis set. The nuclei are fixed at the distance $R=2.132178$ a.u.

Figure 3

The value of the structure function $G_{00}(\beta ={180}^{\circ },\eta )$ [see Eq. (5)] as a function of $\eta$ for the CO HOMO of $\sigma$ symmetry. The solid blue reference curve uses the HOMO obtained by x2dhf [21, 22]. The dashed red curve uses the HOMO from gaussian [26] with an optimized pc-1 basis set. The dotted green curve uses the HOMO from gaussian with an optimized pc-2 basis set. The solid magenta curve with plus signs uses the HOMO from gaussian with an optimized pc-3 basis set. The dot-dashed black curve uses the HOMO from gaussian with an optimized pc-4 basis set. The solid black curve uses the HOMO from gaussian with an optimized pc-5 basis set. The nuclei are fixed at the distance $R=2.132178$ a.u.

Figure 4

The value of the structure function $G_{00}\left(\eta \right)$ [see Eq. (5)] as a function of $\eta$ for the $1s$ electron in the He atom. The solid blue curve uses the HOMO from gaussian [26] with a basis with 5 $s$ orbitals. The dashed red curve uses the HOMO from gaussian with a basis with 9 $s$ orbitals. The dotted green curve uses the HOMO from gaussian with a basis with 13 $s$ orbitals. The solid black curve uses the HOMO from gaussian with a basis with 18 $s$ orbitals.

Figure 5

The value of the structure function $G_{00}\left(\eta \right)$ [see Eq. (5)] as a function of $\eta$ for the $2s$ electron in the Be atom. The solid blue curve uses the HOMO from gaussian [26] with a basis with 8 $s$ orbitals. The dashed red curve uses the HOMO from gaussian with a basis with 12 $s$ orbitals. The dotted green curve uses the HOMO from gaussian with a basis with 18 $s$ orbitals. The solid black curve uses the HOMO from gaussian with a basis with 24 $s$ orbitals.

Figure 6

The dependence of the structure factor $G_{00}\left(\beta \right)$ [see Eq. (4)] on the method used to obtain the HOMO of CO (x2dhf [21, 22], circles; gaussian [26] with an optimized pc-4 basis set, squares; and an optimized pc-5, diamonds) and the order $n$ of the polynomial in $1/\eta$ [see Eq. (15)] in the asymptotic expansion.

Figure 7

The dependence of the structure factor on the angle $\beta$ between the internuclear axis and the electric field for the HOMO in CO of $\sigma$ symmetry. The C (O) atom is on the negative (positive) $z$ axis in the molecular fixed frame. The solid black curve uses the HOMO wave function obtained by x2dhf [21, 22]. The dashed red curve uses the HOMO from gaussian [26] with an optimized pc-4 basis set. The dot-dashed blue curve uses the HOMO from gaussian with an optimized pc-5 basis set. The nuclei are fixed at the distance $R=2.132178$ a.u.

Figure 8

The value of the structure function $G_{00}(\beta ,\eta )$ [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 using the HOMO obtained by gaussian [26] with the optimized pc-4 basis set. The distance from C to O is $R=2.19605$ a.u.

Figure 9

The value of the structure function $G_{00}(\beta ,\eta )$ [see Eq. (5)] as a function of $\eta$ for the OCS$\left(yz\right)$ HOMO of $\pi$ symmetry and for three different orientations $\beta$ of the internuclear axis with respect to the direction of the electric field using the HOMO obtained by gaussian [26] with the optimized pc-4 basis set. The O (S) atom is on the negative (positive) $z^{\prime }$ axis in the molecular fixed frame. $z^{\prime }\left[\mathrm{O}\right]=-3.19847$ a.u., $z^{\prime }\left[\mathrm{C}\right]=-0.98957$ a.u., $z^{\prime }\left[\mathrm{S}\right]=1.97032$ a.u.

Figure 10

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the electric field for the two degenerate HOMOs in CO${}_2$ of ${\pi }_g$ symmetry. CO${}_2\left(yz\right)$ [CO${}_2\left(xz\right)$] denotes that the nodal plane of the $\pi$ orbital is in the $yz$ ($xz$) plane. Solid black curve, $m=0$; dashed red curve, $m=1$. The distance from C to O is $R=2.19605$ a.u.

Figure 11

The dependence of the structure factors on the angle $\beta$ between the internuclear axis and the electric field for the two degenerate HOMOs in OCS of ${\pi }_g$ symmetry. OCS$\left(yz\right)$ [OCS$\left(xz\right)$] denotes that the nodal plane of the $\pi$ orbital is in the $yz$ ($xz$) plane. Solid black curve, $m=0$; dashed red curve, $m=1$. The O (S) atom is on the negative (positive) $z^{\prime }$ axis in the molecular fixed frame. $z^{\prime }\left[\mathrm{O}\right]=-3.19847$ a.u., $z^{\prime }\left[\mathrm{C}\right]=-0.98957$ a.u., $z^{\prime }\left[\mathrm{S}\right]=1.97032$ a.u.