In this paper, we introduce a new symmetric explicit four-step
method with variable coefficients for the numerical solution of
second-order linear periodic and oscillatory initial value problems
of ordinary differential equations. For the first time in the literature,
we generate an explicit method with the most important singularly
P-stability property. The method is multiderivative and has
algebraic order eight and infinite order of phase-lag. The numerical
results for some chemical (e.g. orbit problems of Stiefel and Bettis)
as well as quantum chemistry problems (i.e. systems of coupled
differential equations) indicated that the new method is superior,
efficient, accurate and stable.