Nature of the 1¹B_u and 2¹A_g Excited States of Butadiene and the Goldilocks Principle of Basis Set Diffuseness

Butadiene, being the simplest conjugated organic molecule, has been studied extensively by experiments and various high-level theoretical methods. Previous studies conclude that the complete active space (CAS) self-consistent field (SCF) method was unable to obtain the correct 1¹B_u and 2¹A_g state ordering and that it introduces artificial valence-Rydberg mixing into the 1¹B_u state because of the lack of external correlation. Basis sets and initial guesses specifically constructed for this problem were able to improve the vertical excitation energy of the 1¹B_u state but did not resolve the controversy of the nature of the 1¹B_u state. In the present work, we demonstrate that, using standard intermediately diffuse basis sets such as jul-cc-pVTZ and ma-TZVP, CASSCF is able to obtain the 1¹B_u and 2¹A_g states of trans-butadiene with much improved vertical excitation energy and reasonable wave function characteristics, and it provides a good reference wave function (capable of yielding quantitative excitation energies for excited states) for three methods that treat electron correlation in different ways, namely, multiconfiguration pair-density functional theory (MC-PDFT), CAS second-order perturbation theory (PT2), and multistate (MS) CASPT2. We demonstrate that a combined analysis of the orbital second moments, state second moments, and MC-PDFT energy components is a very useful approach in determining excited-state characteristics and assigning states, and we show that basis sets without diffuse functions or with very diffuse basis functions are not stable or accurate in predicting the excited states of butadiene. We show that the 2¹A_g state is valence-like and has an atypical double/single excitation character and the 1¹B_u state has a certain degree of Rydberg character that is not artificial, settling the decades of controversy of the characters of these states.