The variational simulation of electronic systems requires the parametrization of the wave function amplitudes on a given single-particle basis. Working in first (second) quantization, the parametrized wave function amplitudes must be anti-symmetric (symmetric) functions of the particle configurations, while being able to capture correlations beyond single-particle Slater determinants. To date, multiple candidates have been proposed in the space of neural-network (NN) parametrizations. While much progress has been made in the design of NN-based parametrizations, there are still open questions regarding their accuracy to describe the ground-state properties of ab-initio Hamiltonians. In this talk I will discuss our recent work on the understanding of the choice of trial state defined by anti-symmetrix (symmetric) NN-wave function ansatzs for the study, in first (second) quantization, of molecular Hamiltonians projected onto a discrete basis. I will also discuss the challenges in optimazing certain families of NN-trial states and different approaches to improve the optimization behavior.