  Cyclobutadiene is a well-known playground for theoretical chemists and is
particularly suitable to test ground- and excited-state methods. Indeed, due to
its high spatial symmetry, especially at the $D_{4h}$ square geometry but also
in the $D_{2h}$ rectangular arrangement, the ground and excited states of
cyclobutadiene exhibit multi-configurational characters and single-reference
methods, such as adiabatic time-dependent density-functional theory (TD-DFT) or
equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle
in such situations. In this work, using a large panel of methods and basis
sets, we provide an extensive computational study of the automerization barrier
(defined as the difference between the square and rectangular ground-state
energies) and the vertical excitation energies at $D_{2h}$ and $D_{4h}$
equilibrium structures. In particular, selected configuration interaction
(SCI), multi-reference perturbation theory (CASSCF, CASPT2, and NEVPT2), and
coupled-cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed.
The spin-flip formalism, which is known to provide a qualitatively correct
description of these diradical states, is also tested within TD-DFT (combined
with numerous exchange-correlation functionals) and the algebraic diagrammatic
construction [ADC(2)-s, ADC(2)-x, and ADC(3)] schemes. A theoretical best
estimate is defined for the automerization barrier and for each vertical
transition energy.
