Structure factors for tunneling ionization rates of molecules: General Hartree-Fock-based integral representation

In the leading-order approximation of the weak-field asymptotic theory (WFAT), the dependence of the tunneling ionization rate of a molecule in an electric field on its orientation with respect to the field is determined by the structure factor of the ionizing molecular orbital. The WFAT yields an expression for the structure factor in terms of a local property of the orbital in the asymptotic region. However, in general quantum chemistry approaches molecular orbitals are expanded in a Gaussian basis which does not reproduce their asymptotic behavior correctly. This hinders the application of the WFAT to polyatomic molecules, which are attracting increasing interest in strong-field physics. Recently, an integral-equation approach to the WFAT for tunneling ionization of one electron from an arbitrary potential has been developed. The structure factor is expressed in an integral form as a matrix element involving the ionizing orbital. The integral is not sensitive to the asymptotic behavior of the orbital, which resolves the difficulty mentioned above. Here, we extend the integral representation for the structure factor to many-electron systems treated within the Hartree-Fock method and show how it can be implemented on the basis of standard quantum chemistry software packages. We validate the methodology by considering noble-gas atoms and the CO molecule, for which accurate structure factors exist in the literature. We also present benchmark results for ${\mathrm{CO}}_2$ and for ${\mathrm{NH}}_3$ in the pyramidal and planar geometries.

Figure 1

Radial functions $R_1^{00}\left(r\right)$ in the partial-wave expansion (22) for noble-gas atoms Ne, Ar, Kr, and Xe (from top to bottom) with the HOMO of $p$ symmetry. The exact functions (solid lines) are obtained from Eq. (24). The fits (dashed lines) are generated as described in Sec. 3c with $r_{\text{fit}}=8$ (this value is indicated by the vertical dotted line).

Figure 2

Dependence of the structure factor for the HOMO in CO of $\sigma$ symmetry on the angle $\beta$ between the internuclear axis and the electric field. The distance between the nuclei is $R=2.132$ and the HOMO energy is $E_0=-0.554923$ [30]. In the molecular frame where ${\mathbf{D}}^{\prime }=\mathbf{0}$ the nuclear coordinates are $Z_{\text{C}}^{\prime }=0.603639$ and $Z_{\text{O}}^{\prime }=2.735639$ and the HOMO dipole is ${\mu }_z^{\prime }=-0.1043$. Broken lines show the present IR results obtained with $r_{\text{fit}}=7$ and $L_{\text{max}}=6$. The figure illustrates their convergence with respect to the basis set used in the calculations. The notation upc-$n$ ($n=1,\cdots ,4)$ refers to standard basis sets from Ref. [50]. The solid line shows the TR results from Ref. [30] extracted from the HF orbital generated using a grid method x2dhf [28].

Figure 3

Same as in Fig. 2, but illustrates the convergence of the IR results (broken lines) obtained with $r_{\text{fit}}=7$ using the upc-4 basis set [50] with respect to the number of partial waves $L_{\text{max}}$ in Eq. (30).

Figure 4

Dependence of the structure factor for the degenerate HOMO in ${\mathrm{CO}}_2$ of $\pi$ symmetry on the angle $\beta$ between the internuclear axis and the electric field. The HOMO energy is $E_0=-0.544905$. In the molecular frame where ${\mathbf{D}}^{\prime }=\mathbf{0}$ the nuclear positions are $Z_{\text{O}}^{\prime }=\pm 2.19605$ and $Z_{\text{C}}^{\prime }=0$ and the HOMO dipole is ${\mu }_z^{\prime }=0$. Broken lines show the present IR results obtained with $r_{\text{fit}}=7$ and $L_{\text{max}}=6$. The figure illustrates their convergence with respect to the basis set used in the calculations. The notation upc-$n$ ($n=1,\cdots ,3)$ refers to standard basis sets from Ref. [50]. The solid line shows the TR results from Ref. [39] extracted from the HF orbital generated using an expansion in terms of the optimized opc-4 GTO basis set.

Figure 5

Same as in Fig. 4, but illustrates the convergence of the IR results (broken lines) obtained with $r_{\text{fit}}=7$ using the upc-3 basis set [50] with respect to the number of partial waves $L_{\text{max}}$ in Eq. (30).

Figure 6

Dependence of the structure factors $G_{00}$ for the HOMO of ${\mathrm{NH}}_3$ in planar (top panel) and pyramidal (bottom panel) geometries on the Euler angles $\beta$ and $\gamma$. In the planar geometry, the nuclear positions are ${\mathbf{R}}_{\text{N}}^{\prime }=(0,0,0)$, ${\mathbf{R}}_{\text{H}}^{\prime }=(0,1.882129,0)$, ${\mathbf{R}}_{\text{H}}^{\prime }=(1.629972,-0.941065,0)$, and ${\mathbf{R}}_{\text{H}}^{\prime }=(-1.629972,-0.941065,0)$; the HOMO energy is $E_0=-0.391564$; and its dipole is ${\mathit{\mu }}^{\prime }=\mathbf{0}$. In the pyramidal geometry, in the frame where ${\mathbf{D}}^{\prime }=\mathbf{0}$, the nuclear positions are ${\mathbf{R}}_{\text{N}}^{\prime }=(0,0,-0.862442)$, ${\mathbf{R}}_{\text{H}}^{\prime }=(0,1.772461,-1.580782)$, ${\mathbf{R}}_{\text{H}}^{\prime }=(1.534996,-0.886231,-1.580782)$, and ${\mathbf{R}}_{\text{H}}^{\prime }=(-1.534996,-0.886231,-1.580782)$; the HOMO energy is $E_0=-0.429608$; and its dipole is ${\mathit{\mu }}^{\prime }=(0,0,0.635392)$. In both cases, the calculations were done using the upc-2 basis set [50] with $r_{\text{fit}}=7$ and $L_{\text{max}}=6$. The $\beta$ is the angle between the field direction, and the principal inertial axis of the molecule going through $N$ and the $\gamma$ is the angle of rotation around this axis.