In this article, we describe a fully computational laboratory exercise that results in an increase of students' understanding of what quantum chemical geometry optimization calculations are doing to find minimum energy structures. This laboratory exercise was conducted several times over multiple years at a small private undergraduate institution, St. Bonaventure University. Through this experiment, physical chemistry undergraduate students are exposed to chemical problems for which computations provide a necessary supplement to chemical intuition, thus cementing the importance of computational work in contemporary chemistry. Students apply their understanding of geometry optimizations to problems of complex 3-D molecular structures that stretch their intuition, including the geometries and isomers of "closo"-carboranes and of the hexamer of the cocatalyst methylaluminoxane. Students are also exposed to vibrational frequency calculations as a diagnostic tool for determining whether structures represent energetic minima or transition states, and they are exposed to the vibrational zero-point energy correction.