A new set of time-dependent deterministic sampling (TDDS) measures, based on local Shannon entropy, are presented to adaptively gauge the importance of various regions on a potential energy surface and to be employed in "on-the-fly" quantum dynamics. Shannon sampling and Shannon entropy are known constructs that have been used to analyze the information content in functions: for example, time-series data and discrete data sets such as amino acid sequences in a protein structure. Here the Shannon entropy, when combined with dynamical parameters such as the instantaneous potential, gradient and wavepacket density provides a reliable probe on active regions of a quantum mechanical potential surface. Numerical benchmarks indicate that the methods proposed are highly effective in locating regions of the potential that are both classically allowed as well as those that are classically forbidden, such as regions beyond the classical turning points which may be sampled during a quantum mechanical tunneling process. The approaches described here are utilized to improve computational efficiency in two different settings: (a) It is shown that the number of potential energy calculations required to be performed during on-the-fly quantum dynamics is fewer when the Shannon entropy based sampling functions are used. (b) Shannon entropy based TDDS functions are utilized to define a new family of grid-based electronic structure basis functions that reduce the computational complexity while maintaining accuracy. The role of both results for on-the-fly quantum/classical dynamics of electrons and nuclei is discussed.