We investigate quantum computation with neutral atoms in optical microtraps where the qubit is implemented in the motional states of the atoms, i.e., in the two lowest vibrational states of each trap. The quantum gate operation is performed by adiabatically approaching two traps and allowing tunneling and cold collisions to take place. We demonstrate the capability of this scheme to realize a square root of swap gate, and address the problem of double occupation and excitation to other unwanted states. We expand the two-particle wave function in an orthonormal basis and analyze quantum correlations throughout the whole gate process. Fidelity of the gate operation is evaluated as a function of the degree of adiabaticity in moving the traps. Simulations are based on rubidium atoms in state-of-the-art optical microtraps with quantum gate realizations in the few tens of milliseconds duration range.