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Shaped laser pulses are a powerful tool to induce population transfer betweenelectronic molecular states, and time-dependent perturbation theory is suitablefor a description of such a transfer in weak external fields. The applicationof perturbation theory in numerical simulations of field matter interactionscan lead to divergences. In a recent paper [K. Renziehausen et. al., J. Phys.B: At. Mol. Opt. Phys., 42:195402, 2009] we explained that the arising error inthe norm of the wave function can be split into two parts. The first part isrelated to numerical errors caused by the discretisation of time that isrequired in the simulation and can be suppressed for a sufficiently small timestep or abolished for an adequate numerical implementation of perturbationtheory. The second part may cause divergences and is associated with theperturbative expansion order. We presented numerical evidence without anyanalytical proof. Here we are focussing on the derivation of analyticalexpressions to interpret the behavior of what we have called in the abovementioned paper 'simple algorithm'. The derivation of analytical expressionsfor the interpretation of what we have called in the above mentioned paper'improved algorithm' are given in another paper [K. Renziehausen. arXiv, inpreparation, 2012]. Moreover, we introduce here a gedankenexperiment toillustrate the influence of the different orders on the field-moleculeinteraction.

On the divergence of time-dependent perturbation theory applied to  laser-induced molecular transitions: Analytical calculations for the simple  algorithm