Determination of outer molecular orbitals by collisional ionization experiments and comparison with Hartree-Fock, Kohn-Sham, and Dyson orbitals

Although the outer shapes of molecular orbitals (MO’s) are of great importance in many phenomena, they have been difficult to be probed by experiments. Here we show that metastable helium $\left({\mathrm{He}}^{*}\right)$ atoms can sensitively probe the outer properties of molecules and that an electron spectroscopic technique using velocity-selected ${\mathrm{He}}^{*}$ atoms in combination with classical trajectory simulations leads to a consistent determination of MO functions and the molecular surface. MO functions composed of linear combinations of atomic orbital functions were fitted to the observed collision energy dependences of partial ionization cross sections (CEDPICS). The obtained CEDPICS MO functions were compared with conventionally available Hartree-Fock, Kohn-Sham, and Dyson orbitals.

Figure 1

(Color online) Observed and simulated collision energy dependence of Penning ionization cross sections (CEDPICS) for ${\mathrm{N}}_2+{\mathrm{He}}^{*}\left(2{}^{3}S\right)$. Observed partial cross section for $X$ (circle), $A$ (triangle), and $B$ (square) states. Experimental statistical errors are as large as the size of the markers except for the edge of the low-collision-energy side. Simulated CEDPICS are drawn with solid or dashed lines by using optimized MO’s in a valence double-$\zeta$-type basis (CEDPICS MO) and SCF MO’s in the minimal basis set (SCF-Min), respectively.

Figure 2

(Color online) (a) The contour map of the potential energy surface around a ${\mathrm{N}}_2$ molecule in interaction with various atomic probes. The energy spacing is $50\mathrm{meV}$ from $0\mathrm{meV}\text{to}800\mathrm{meV}$. $V_{\mathrm{He}}$ and $V_{\mathrm{Li}}$ were calculated by $\mathrm{CCSD}\left(\mathrm{T}\right)∕6\text{−}311+{\mathrm{G}}^{*}$. (b) Electron density contour maps for various MO’s corresponding to the observed three ionic states for ${\mathrm{N}}_2$. The density of the $n\text{th}$ contour line increases from the outermost value of $1.0\times {10}^{-7}{\mathrm{a.u.}}^{-3}$ with a proportionality constant of $4^{n-1}$.

Figure 3

(Color online) Electron density of various kinds of MO’s for ${\mathrm{N}}_2$ as a function of distance $r$ from the center of ${\mathrm{N}}_2$. The directions are collinear and perpendicular to the molecular axis for $\sigma$ and $\pi$ MO’s, respectively. Optimized MO (CEDPICS MO, circle), and theoretical MO’s, Hartree-Fock (HF: solid line), Kohn-Sham (KS: long dashed line), SCF with the minimal basis (dotted line), and Dyson orbitals with cc-pV5Z (chained line) basis sets are shown. Shades represent less reliable regions in the collision experiments.

Figure 4

(Color online) The logarithmic derivative of electron densities, $-2\alpha$, for various kinds of MO’s as a function of distance $r$.