An alternative to the D -matrix approach would be a direct calculation of the Pancharatnam phase. This is especially useful if one is interested only in the geometric phases of a single adiabatic state. In this approach, one takes a number N + 1 of points ( n = 0 , … … , N ) along the loop R ( t n ) with t 0 = 0 and t N = 1 , then using only the j -th adiabatic states ψ ψ j [ R ( t n ) ] computes the Pancharatnam product of overlaps: I j ( Γ Γ , N ) = ∏ ∏ n = 0 N − − 1 ⟨ ⟨ ψ ψ j [ R ( t n ) ] | ψ ψ j [ R ( t n + 1 ) ] ⟩ ⟩ .