Approximations for exchange scattering: I. Elastic collisions

The problem of elastic scattering of electrons by the hydrogen ground state is formulated using a variational principle. With trial functions ψ = (1/√2){ψ( r 1 )ϕ( r 2 ) ± ϕ( r 1 )ψ( r 2 )} where ψ is the ground state functions, one obtains the exchange integro-differential equations for the functions phi. Results from exact solutions of these equations are compared with results obtained using: ( a ) modified Born-Oppenheimer approximations, with trial functions ϕ( r ) = [1/ r (4π) 1/2 ] sin( kr ) + αψ( r ) and various choices for the parameter α, ( b ) the approximation of Ochkur, in which one retains only the leading term in the (1/ k ) expansion of the Oppenheimer exchange amplitudes, and ( c ) the distorted wave approximation, in which the direct potential is allowed for exactly.