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The vibrational zero-point energy is the energy difference between the lowest point on the potential energy surface (equilibrium energy) and the energy of the vibrationless energy level (v=0). It is not possible to measure the ZPE. The ZPE can be approximated as half the fundamental vibrational frequencies. This assumes the vibrational frequencies are harmonic and there are no vibration-rotation interactions. Given more vibrational and rotational details, such as anharmonic contants (ωexe, ωeye, ...), vibration-ration interaction constants (αe) better approximations to the ZPE can be made.
For diatomics we use:
See the paper by Irikura: K. K. Irikura "Experimental Vibrational Zero-Point Energies: Diatomic Molecules" J. Phys. Chem. Ref. Data 36(2), 389, 2007