James E Russell
Cross sections for protonium formation are being estimated using the modern Born-Oppenheimer approximation.
Protonium is a bound state of a proton and an antiproton. It can be formed by bombarding hydrogen atoms with slowly moving antiprotons.
The modern Born-Oppenheimer approximation takes into account coupling between traditional Born-Oppenheimer states. A traditional Born-Oppenheimer wave function is the product of two functions. In the case of protonium formation, one of these functions is an electronic function that depends on the momentary vector separation between the proton and antiproton, and the other is a function describing the proton-antiproton relative motion in the presence of adiabatically varying electronic motion. The coupling between traditional Born-Oppenheimer states is given by a vector potential, which is a function of the proton-antiproton separation. The vector potential is a matrix, the elements of which are specified in terms of the integral, over the electronic coordinates, of the product of the final electron function and the gradient of the initial electron function with respect to the proton-antiproton separation.