  The quantum entanglement entropy of the electrons in one-dimensional hydrogen
molecule is quantified locally using an appropriate partitioning of the
two-dimensional configuration space. Both the global and the local entanglement
entropy exhibit a monotonic increase when increasing the inter-nuclear
distance, while the local entropy remains peaked at the middle between the
nuclei with its width decreasing. Our findings show that at the inter-nuclear
distance where stable hydrogen molecule is formed, the quantum entropy shows no
peculiarity thus indicating that the entropy and the energy measures display
different sensitivity with respect to the interaction between the two identical
electrons involved. One possible explanation is that the calculation of the
quantum entropy does not account explicitly for the distance between the
nuclei, which contrasts to the total energy calculation where the energy
minimum depends decisively on that distance. The numerically exact and the
time-dependent quantum Monte Carlo calculations show close results.
