Use of fixed-atom constraints invariably results in a large number of imaginary vibrational frequencies upon structure relaxation. These have been noted in numerous publications, yet a satisfactory explanation has never been provided. The present work demonstrates unequivocally that imaginary frequencies arise from a discrepancy between the curvature of the potential energy surface and the Hessian that is diagonalized to obtain harmonic frequencies. Their magnitude is typically small, being variously reported (for the modes exhibiting negative curvature) as |ν| ∼ 10 cm -1 , 37,38 |ν| 20 cm -1 , 39-42 |ν| 30 cm -1 , |ν| 40 cm -1 , 47-52 |ν| ≤ 50 cm -1 , |ν| ≤ 60 cm -1 , 53 or |ν| < 100 cm -1 . When the imaginary frequencies are small, a case can be made to obtain zero-point energy (ZPE) corrections to reaction enthalpies and barrier heights simply by ignoring them, including only the real-valued frequencies in the ZPE calculation. This may be a workable strategy to obtain T = 0 K thermochemistry and kinetics, but the harmonic partition function affording the finitetemperature vibrational entropy diverges in the presence of imaginary frequencies. Its numerical value is also sensitive to low-frequency modes; for example, a single 30 cm -1 frequency contributes 1.7 kcal/mol to ∆G • 298 . 61 Alternatives to the harmonic approximation have been suggested as better approximations for the entropy associated with low-frequency motion, yet these alternatives still require harmonic frequencies as inputs. In the context of quantum-chemical cluster modeling of enzymatic reactions, it has been argued that entropic effects are generally small (meaning ∆H • ≈ ∆G • ), yet this assumption cannot be systematically tested so long as cluster models are beset by imaginary frequencies.