Consider the collision between a projectile with a diatomic molecule with the geometry shown in

Fig. 1. The origin of the coordinate system is taken at the center C of the molecule AB. We will

**Fig. 1.**
Coordinate system in the laboratory frame for ion-H_{2} collisions.

**Fig. 1.**
Coordinate system in the laboratory frame for ion-H_{2} collisions.

assume that there is only one active electron which has coordinates

**r**_{A} and

**r**_{B} with respect to A and B, respectively, and

**r**_{P} with respect to the projectile. In the perturbation approximation, the amplitude a

_{fi}(b) for transition from the initial state |i> to final state |f> is obtained from

where V

_{P} is the electron-projectile interaction. In (1), the time-dependent initial state wavefunction is given by

and the final state time-dependent wavefunction is given by

where y

_{f} is the electronic wavefunction centered at the projectile. Since the projectile is moving at a velocity v with respect to the origin of the target molecule, a plane-wave electron translational factor is included in the time-dependent wavefunction. Atomic units are used throughout this paper unless otherwise noted. If one writes the initial state electronic wavefunction for the diatomic molecule as

then the scattering amplitudes can be expressed explicitly as

where w= e

_{f} - e

_{i}. The quantities above are defined with respect to the origin at C. We note that the first integral is identical to the electron capture amplitude between the projectile with atom A except for the constant phase factor, and the second integral is similar to the amplitude for capture from atom B, except for a phase factor. To find the phase factor, assume that the time integration with respect to C is from –T to +T, where t=0 is taken to be when the projectile is at the distance of closest approach. By shifting the integration from d

**r** to d

**r**_{A}, the first integral can be written as

where

$\overrightarrow{\rho}$ is the internuclear distance vector. We next transform the time integration so that t’=0 when the projectile is at the distance of closest approach with respect to center A. The time t for the wavefront of the projectile to travel from A to C is t= (r cos q )/(2v) where q is the angle of the molecular axis with respect to the beam direction. Such a shift results in an additional phase factor exp [ iv(r/2) cos q -i (w+v

^{2}/2)t ]. A similar procedure can be applied to the second integral in (5). Retaining only the relative phase, the electron capture amplitude can be expressed as

where

**b**_{A} and

**b**_{B} are the impact parameters with respect to atom A and atom B, respectively. Thus equation (7) expresses the electron capture amplitudes in ion-molecule collisions in terms of electron capture amplitudes in ion-atom collisions.

To obtain alignment-dependent electron capture cross section, we need to integrate over the impact parameter plane for each molecular alignment angle. Choose z-axis to be along the beam direction, the impact parameter plane is defined to be the xy-plane with

(see

Fig. 2). Let q and f be the azimuthal angles of the molecule, then the position of atom

**Fig. 2.**
The impact parameter plane in an ion-molecule collision. The projectile is moving out of the plane.

**Fig. 2.**
The impact parameter plane in an ion-molecule collision. The projectile is moving out of the plane.

A is given by

and

${\overrightarrow{R}}_{B}=-{\overrightarrow{R}}_{A}$. This allows us to calculate b

_{A} and b

_{B}, i.e., the impact parameters needed for atoms A and B, respectively,

For each aligned molecule, the probability at a given b is obtained from

where P(b,a) is obtained from (8a) or its unitarized version (8b). The cross section for electron capture from an aligned molecule is then obtained from

and electron capture cross section averaged over all the alignments of the molecule is given by