second expression can be interpreted as running over all vibrational modes of the reactant ground state and the transition state. Alternatively, one may interpret them as running over those modes unique to the reactant or the transition state or whose vibrational frequencies change substantially upon advancing along the reaction coordinate. The remaining pairs of reactant and transition state vibrational modes have very similar and , and cancellations occur when the sums in the exponent are calculated. Thus, in practice, deuterium KIEs are often largely dependent on a handful of key vibrational modes because of this cancellation, making qualitative analyses of kH/kD possible. As mentioned, especially for hydrogen/deuterium substitution, most kinetic isotope effects arise from the difference in zero-point energy (ZPE) between the reactants and the transition state of the isotopologues in question, and this difference can be understood qualitatively with the following description: within the Born–Oppenheimer approximation, the potential energy surface is the same for both isotopic species. However, a quantum-mechanical treatment of the energy introduces discrete vibrational levels onto this curve, and the lowest possible energy state of a molecule corresponds to the lowest vibrational energy level, which is slightly higher in energy than the minimum of the potential energy curve. This difference, referred to as the zero-point energy, is a manifestation of the Heisenberg uncertainty principle that necessitates an uncertainty in the C-H or C-D bond length. Since the heavier (in this case the deuterated) species behaves more "classically," its vibrational energy levels are closer to the classical potential energy curve, and it has a lower zero-point energy. The zero-point energy differences between the two isotopic species, at least in most cases, diminish in the transition state, since the bond force constant decreases during bond breaking. Hence, the lower zero-point energy of the deuterated species translates into a larger activation energy for its reaction, as shown in the following figure, leading to a normal kinetic isotope effect. This effect should, in principle, be taken into account all 3N−6 vibrational modes for the starting material and 3N‡−7 vibrational modes at the transition state (one mode, the one corresponding to the reaction coordinate, is missing at the transition state, since a bond breaks and there is no restorative force against the motion). The harmonic oscillator is a good approximation for a vibrating bond, at least for low-energy vibrational states. Quantum mechanics gives the vibrational zero-point energy as . Thus, we