Validity of the Born-Oppenheimer approximation in the indirect-dissociative-recombination process

An alternative method is introduced to solve a simple two-dimensional model describing vibrational excitation and dissociation processes during the electron-molecule collisions. The model works with one electronic and one nuclear degree of freedom. The two-dimensional $R$ matrix can be constructed simultaneously on the electronic and nuclear surfaces using all three forms developed previously for electron-atom and electron-molecule collisions. These are the eigenchannel $R$-matrix form, inversion technique of Nesbet and Robicheaux, and the Wigner-Eisenbud-type form using expansion over the poles of the symmetrized Hamiltonian. The 2D $R$-matrix method is employed to solve a simple model tailored to describe the dissociative recombination and the vibrational excitation of ${{\mathrm{H}}_2}^{+}$ cation in the singlet ungerade symmetry ${}^1\mathrm{\Sigma }_u$. These results then serve as a (near-exact) benchmark for the following calculation in which the $R$-matrix states are replaced by their Born-Oppenheimer approximations. The accuracy of this approach and its correction with the first-order nonadiabatic couplings are discussed.

Figure 1

Partitioning of the configuration space into inner and outer regions.

Figure 2

DR cross sections into final $n=1$ (broken curves) and $n=2$ states (full lines). Black lines are exact results, while the blue lines show calculations from the Born-Oppenheimer $R$ matrix and $r_0=6\mathrm{bohr}$.

Figure 3

DR cross sections into final $n=1$ (broken curves) and $n=2$ states (full lines). Black lines are exact results, while the blue lines show calculations from the Born-Oppenheimer $R$ matrix and $r_0=12\mathrm{bohr}$. Red dot-dashed curves represent results with first-order nonadiabatic couplings included.

Figure 4

Same as in Fig. 3 with the radius $r_0=20\mathrm{bohr}$.

Figure 5

Fixed-nuclei $R$-matrix poles ${\overline{E}}_k\left(R\right)$ (28) as a function of the internuclear distance $R$. Energy curves are shown for three sizes $r_0$ of the $R$-matrix box.

Figure 6

First-order electronic coupling terms ${\langle {\psi }_k^{\prime }|{\psi }_{k^{\prime }}\rangle }_r$ of Eq. (30), where $k^{\prime }=1$ and $k=2$,3,4. Data for three $R$-matrix box sizes $r_0$ are displayed.

Figure 7

Vibrationally inelastic cross sections. Full black curves represent the exact results, while the red dashed show calculations from the Born-Oppenheimer $R$ matrix and $r_0=20\mathrm{bohr}$.

Figure 8

Propagation of the 2D $R$ matrix from Box A to Box B. The two boxes share the surface ${\mathcal{S}}_4$. Functions $v^{\left(\alpha \right)}$ denote a complete set of orthonormal functions defined on the respective surface ${\mathcal{S}}_{\alpha }$.

Figure 9

Matrix elements of the Block B $R$ matrix constructed on surfaces ${\mathcal{S}}_2, {\mathcal{S}}_3$, and ${\mathcal{S}}_4$.