  Entanglement plays a central role in numerous fields of quantum science.
However, as one departs from the typical "Alice versus Bob" setting into the
world of indistinguishable fermions, it is not immediately clear how the
concept of entanglement is defined among these identical particles. Our
endeavor to recover the notion of subsystems, or mathematically speaking, the
tensor product structure of the Hilbert space, lead to two natural pictures of
defining fermionic entanglement: the particle picture and the mode picture. In
the particle picture, entanglement characterizes the deviation of a fermionic
quantum state from the non-interacting ones, e.g., single Slater determinants.
In the mode picture, we recover the notion of subsystems, by referring to the
partitioning of the orbital/mode that the fermions occupy, which allows us to
naturally adopt the formalism of entanglement between distinguishable
constituents. Both pictures reveal essential and interconnected aspects of
fermionic entanglement, and thus offer precise tools for studying electron
entanglement in highly relevant systems such as atoms and molecules. We
showcase here two applications: i) resolving the correlation paradox in the
molecular dissociation limit, ii) quantitative electronic structure analysis
with orbital entanglement.
