This work studies the variational quantum eigensolver algorithm, designed to determine the ground state of a quantum mechanical system by combining classical and quantum hardware. Methods of reducing the number of required qubit manipulations, prone to induce errors, for the variational quantum eigensolver are studied. We formally justify the qubit removal process as sketched by Bravyi, Gambetta, Mezzacapo and Temme [arXiv:1701.08213 (2017)]. Furthermore, different classical optimization and entangling methods, both gate based and native, are surveyed by computing ground state energies of H$_2$ and LiH. This paper aims to provide performance-based recommendations for entangling methods and classical optimization methods. Analyzing the VQE problem is complex, where the optimization algorithm, the method of entangling, and the dimensionality of the search space all interact. In specific cases however, concrete results can be shown, and an entangling method or optimization algorithm can be recommended over others. In particular we find that for high dimensionality (many qubits and/or entanglement depth) certain classical optimization algorithms outperform others in terms of energy error.