  In this work we introduce a new family of twelve-step linear multistep
methods for the integration of the Schr\"odinger equation. The new methods are
constructed by adopting a new methodology which improves the phase lag
characteristics by vanishing both the phase lag function and its first
derivatives at a specific frequency. This results in decreasing the sensitivity
of the integration method on the estimated frequency of the problem. The
efficiency of the new family of methods is proved via error analysis and
numerical applications.
