The similarities and differences between carbon and silicon have attracted the curiosity of chemists for centuries. Similarities and analogies can be found in their saturated compounds, but carbon exhibits a cornucopia of unsaturated compounds that silicon (and most other elements) cannot replicate. While this qualitative difference is empirically well known, quantum chemistry has previously only described quantitative differences related to orbital overlap, steric effects, or orbital energies. We study C-2 and Si-2 and their hydrides X2H2n (X= C, Si; n= 1, 2, 3) by first-principles quantum chemical calculation, and find a qualitative difference in the topologies of the core electrons: carbon has the propensity to alter its core electron topology when forming unsaturated compounds, and silicon has not. We draw a connection between the core electron topologies and ionization energies, and identify other elements we expect to have similarly flexible core topologies as carbon.