The methodology and results of a detailed four-element lens optimization analysis based on electron trajectory numerical simulations are presented for a hemispherical deflector analyzer (HDA), whose entry aperture size is determined by the injection lens itself and is therefore virtual. Trajectory calculations were performed using both the boundary-element method (BEM) and the finite-difference method (FDM) and results from these two different approaches were benchmarked against each other, to probe and confirm the accuracy of our results. Since the first and last electrode are held at fixed potentials, the two intermediate adjustable lens electrode voltages were varied over the entire available voltage space in a direct, systematic, brute-force approach, while minima in beam spot size on the 2-D position sensitive detector (PSD) at the exit of the HDA were investigated using a beam shaping approach. Lens voltages demonstrating improved energy resolution for the combined lens/HDA/PSD spectrograph system were sought with and without pre-retardation. The optimal voltages were then tested experimentally on the modeled HDA system using a hot-wire electron gun. The measured energy resolution was found to be in good overall agreement with our simulations., particularly at the highest resolution (-0.05%) working conditions. These simulations also provide a detailed insight to the distinctive trajectory optics-and positions of the first and second image planes, when the PSD has to be placed some distance away from the HDA exit plane, and is therefore not at theideal optics conjugate image position. The substantial time savings afforded over usual trial-and-error experimentation should make this type of make-do simulation approach attractive to experimentalists. (C) 2016 Elsevier B.V. All rights reserved.