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The physics term ground state means the lowest possible energy level of something. The term is commonly used in chemistry and spectroscopy for the lowest state of excitation of an electron (i.e., no excitation, or the lowest electron shell) within an atom. The term is also commonly used for the physics of nuclei, in which nucleons (protons and neutrons) can undergo excitation and relaxation analogous to that of electrons in atoms.
The term zero-point energy (ZPE) means the same thing and is perhaps the more usual term for quantum-mechanical systems other than atoms and molecules. The term vacuum energy is used for the zero-point energy of a vacuum (i.e., of a region containing no particles), an energy level which is actually not precisely zero, a consequence of the Heisenberg uncertainty principle.
It is a common usage to declare an electron within an atom to be in the ground state despite the existence of states with levels very-slightly lower: much smaller transitions termed the fine structure and hyperfine structure may actually result in multiple states termed ground states that are very close in energy level (though only one of them is actually lowest). This usage presumably came about because with some equipment (lower spectral resolution) and under some circumstances (line broadening), individual spectral lines associated with these smaller transitions are not observable.