The linear variational method is a standard computational method in quantum mechanics and quantum
chemistry. As taught in most classes, the general guidance is to include as many basis functions as
practical in the variational wave function. However, if it is desired to study the patterns of
energy change accompanying the change of system parameters such as the shape and strength of the
potential energy, the problem becomes more complicated. We use one-dimensional systems with a
particle in a rectangular or in a harmonic potential confined in an infinite rectangular box to
illustrate situations where a variational calculation can give incorrect results. These situations
result when the energy of the lowest eigenvalue is strongly dependent on the parameters that
describe the shape and strength of the potential. The numerical examples described in this work are
provided as cautionary notes for practitioners of numerical variational calculations.