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A new two-step method, for the numerical solution of problems in quantum chemistry, is introduced in the present paper. The new algorithm is conditionally P-Stable and an economical scheme. It has vanished phase–lag and it’s first to fourth derivatives. We use for the new scheme the symbol LOWPF4DECN2ST. We apply the newly introduced method to problems in Quantum Chemistry. The new scheme is defined as economical since it uses 4 function evaluations per step in order to achieve an algebraic order (AOR) of 10. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About 1.8 digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.