In this Letter we consider the possibility to use the Curl equation as a first-order differential equation to calculate the non-adiabatic coupling terms, tau. For this purpose we treat a planar case characterized by two (polar) coordinates, theta and q and their corresponding non-adiabatic coupling terms, namely, tau(0)(q, theta) and tau(q)(q, theta), The main difficulty is that we have only one Curl equation and two unknown functions and consequently we cannot get the complete solution. To partially overcome this difficulty we employ gauge transformations which may enable to obtain approximate solutions A(theta)(q, theta) and A(q)(q, theta). In the present Letter we show how the gauge transformation leads to decoupled (second-order) differential equations for these two functions. Next is shown how these equations are solved subject to boundary conditions obtained from ab initio calculations. (C) 2002 Elsevier Science B.V. All rights reserved.