The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectations values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose a method to compute the energy gradient within the VQE approach. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the cost of the circuit depth and execution time of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.