This chapter discusses the detailed quantum structure of small linear molecules used to probe fundamental symmetry violation. We begin by covering angular momentum, spherical tensors, and the Wigner-Eckart theorem, before moving on to describing atomic and molecular states. We overview the various electronic, vibrational, and rotational energy scales in molecules, and we explore Hund’s cases in detail. We also detail the use of effective Hamiltonians as a powerful tool to accurately predict and model molecular structure, and introduce important concepts such as parity doubling, molecular orientation, and Renner-Teller interactions in linear polyatomic molecules. This background sets the stage for later chapters covering experimental spectroscopy and quantum control of polyatomic molecules to search for fundamental symmetry violation.