H or D) are the symmetry numbers for the reactants and transition states. The MX are the molecular masses of the corresponding species, and the IqX (q = x, y, or z) terms are the moments of inertia about the three principal axes. The uiX are directly proportional to the corresponding vibrational frequencies, νi, and the vibrational zero-point energy (see below). The integers N and N‡ are the number of atoms in the reactants and the transition states, respectively. The complicated expression given above can be represented as the product of four separate factors: . For the special case of deuterium isotope effects, we will argue that the first three terms can be treated as equal to or well approximated by unity. The first factor S (containing the σX) is the ratio of the symmetry numbers for the various species. This will be a rational number (a ratio of integers) that depends on the number of molecular and bond rotations leading to the permutation of identical atoms or groups in the reactants and the transition state. For systems of low symmetry, all σX (reactant and transition state) will be unity; thus S can often be neglected. The MMI factor (containing the MX and IqX) refers to the ratio of the molecular masses and the moments of inertia. Since hydrogen and deuterium tend to be much lighter compared to most reactants and transition states, there is little difference in the molecular masses and moments of inertia between H and D containing molecules, so the MMI factor is usually also approximated as unity. The EXC factor (containing the product of vibrational partition functions) corrects for the kinetic isotope effect caused by the reactions of vibrationally excited molecules. The fraction of molecules with enough energy to have excited state A–H/D bond vibrations is generally small for reactions at or near room temperature (bonds to hydrogen usually vibrate at 1000 cm−1 or higher, so exp(-ui) = exp(-hνi/kBT) < 0.01 at 298 K, resulting in negligible contributions from the 1–exp(-ui) factors). Hence, for hydrogen/deuterium kinetic isotope effects, the observed values are typically dominated by the last factor, ZPE (an exponential function of vibrational zero-point energy differences), consisting of contributions from the zero-point energy differences for each of the vibrational modes of the reactants and transition state, which can be represented as follows: , where we define and . The sums in the exponent of the