Gauge transformation to study electronic non-adiabatic coupling terms in molecular systems

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, τ. For this purpose we treat a planar case characterized by two (polar) coordinates, θ and q and their corresponding non-adiabatic coupling terms, namely, τθ(q,θ) and τq(q,θ). 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 Aθ(q,θ) and Aq(q,θ). 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.