A new method is given for the computation of quantum mechanical microcanonical densities of states of large molecules. The method is based on the observation that for large molecules with many vibrational degrees of freedom, the complex time partition function dephases rapidly allowing for a good short-time approximation. The short-time approximation up to third order gives an Airy function expression for the thermal density of states at a given temperature T. The microcanonical density of states is then deduced from the thermal density. The input needed for the method is the first three moments of the Hamiltonian at a series of temperatures, which adequately cover the energy range of interest. These may be computed using quantum Monte Carlo methods. The method is tested for a harmonic model of trans-stilbene, a separable anharmonic model of cyclopropane, and a separable anharmonic model of a system with 50 degrees of freedom. The short-time Airy method is found to give accurate estimates for the density of states, the integrated density of states, and RRKM microcanonical rate constants.