Dynamical adiabatic theory of atomic collisions: The structure of hidden, avoided, and $L_3$ crossings

Inelastic transitions in slow ion-atom collisions are described using an adiabatic representation of the collision system, where the transition probabilities are determined by the branch points of the potential energy curves in the complex plane of internuclear separation $R$. As an example, the ${\mathrm{HeH}}^{2+}$ system is treated, in which the evolution of the branch points related to hidden, avoided, and $L_3$ crossings in the dynamical adiabatic basis of three-body Coulomb problem as functions of parameter $\omega =\rho v$ ($\rho$ is the impact parameter and $v$ the impact velocity) is studied. Rearrangement of hidden couplings due to the passage of $L_3$ branch points with the increase of $\omega$ is discussed.

Figure 1

The 90 lowest eigenvalues (DAPEC) for symmetric (${\pi }_3=1$) states of the (HeH)${}^{2+}$ collision system ($Z_A=1,Z_B=2$) as functions of the internuclear separation $R$ at real $R$ axis [Im($R)=0$] for $\omega =1$ a.u. (solid curves) and the lowest 20 DAPEC for $\omega =0$ [dashed (blue) curves].

Figure 2

Segment of low-lying scaled DAPEC represented in terms of “effective united-atom principal quantum number” [Eq. (15)] (solid curves). Also shown [dashed (blue) curves] are the standard PEC of the (HeH)${}^{2+}$ molecular ion labeled by united-atom quantum numbers.

Figure 3

Analytic continuations of $E_1(R,\omega =0)$ and $E_2(R,\omega =0)$ from the real $R$ axis into the complex $R$ plane along the paths: $\mathrm{Re}R=$ const. The (exaggerated) spherical symbols are points on the surfaces corresponding to branch point $R_b(1,2;\omega =0)$. All quantities are in atomic units.

Figure 4

Trajectory of the branch point $Q_2$ which starts as $R_b(1,2;\omega )$ for $\omega \in (0,0.75)$ but transforms into $R_b(1,5;\omega )$ in the interval $\omega \in (0.75,0.8)$. In all figures, branch points $R_b(j,j^{\prime };\omega )$ are labeled as $\omega (j-j^{\prime })$.

Figure 5

Full circles represent the trajectory of the $L_3$ branch point which starts as $R_b(2,3;\omega )$ for $\omega \in (0,0.3)$, transforms into $R_b(2,4;\omega )$ in the interval $\omega \in (0.3,0.5)$, and into $R_b(2,5;\omega )$ in the interval $\omega \in (0.75,0.8)$. Full triangles are part of the trajectory of another $L_3$ branch point $R_b(3,4;\omega )$. Full squares labeled (1-2,5) are the replica of the $Q_2$ branch points from Fig. 4.

Figure 6

Analytic continuations of $E_1(R,\omega =0.8),E_2(R,\omega =0.8)$, and $E_5(R,\omega =0.8)$ from the real $R$ axis into the complex $R$ plane along the paths: $\mathrm{Re}R=$ const. The spherical symbols are points on the surfaces corresponding to branch points: in the first column $R_b(1,5;\omega =0.8)$, in the second column $R_b(2,5;\omega =0.8)$, and in the third column $R_b(1,5;\omega =0.8)$ and $R_b(2,5;\omega =0.8)$. All quantities are in atomic units.

Figure 7

Full triangles labeled (3-4) correspond to $L_3$ branch points originating from $2s\sigma -2p\pi$ rotational coupling in the united-atom limit at $\omega =0$. Full triangles labeled (3-4)${}^{\prime }$ correspond to $L_3$ branch points originating from $3d\sigma -2p\pi$ exact crossing at $\omega =0$. Full squares labeled $Q_2$(1-2,5) are the replica of Fig. 4 and full circles labeled (2-3,4,5) are the replica of Fig. 5. Also shown are the positions of $S_{p\sigma }$ series and $Q_1$ branch points.

Figure 8

Trajectory of the branch point which starts as the first member of $S_{p\sigma }$ series $R_b(2,5;0)$ but after a series of transformations ends up as $R_b(3,14;0.0275)$.

Figure 9

Positions of the various pairs of branch points responsible for the transformations along the trajectory shown in Fig. 8. This trajectory is here contained within the uppermost square symbol.

Figure 10

Trajectory of the $Q_1$ branch points $R_b(2,3;\omega )$ for $\omega \in (0,2)$ which starts at $\omega =0$ connecting the $2p\sigma$ and $3d\sigma$ PEC.