Excited States via Coupled Cluster Theory without Equation-of-Motion Methods: Seeking Higher Roots with Application to Doubly Excited States and Double Core Hole States

In this work, we revisited the idea of using the coupled-cluster ground state formalism to target excited states. Our main focus was targeting doubly excited states and double core hole states. Typical equation-of-motion (EOM) approaches for obtaining these states struggle without higher-order excitations than doubles. We showed that by using a non-aufbau determinant optimized via the maximum overlap method the CC ground state solver can target higher energy states. Furthermore, just with singles and doubles (i.e., CCSD), we demonstrated that the accuracy of $\Delta$CCSD and $\Delta$CCSD(T) far surpasses that of EOM-CCSD for doubly excited states. The accuracy of $\Delta$CCSD(T) is nearly exact for doubly excited states considered in this work. For double core hole states, we used an improved ansatz for greater numerical stability by freezing core hole orbitals. The improved methods, core valence separation (CVS)-$\Delta$CCSD and CVS-$\Delta$CCSD(T), were applied to the calculation of the double ionization potential of small molecules. Even without relativistic corrections, we observed qualitatively accurate results with CVS-$\Delta$CCSD and CVS-$\Delta$CCSD(T). Remaining challenges in $\Delta$CC include the description of open-shell singlet excited states with the single-reference CC ground state formalism as well as excited states with genuine multi-reference character. The tools and intuition developed in this work may serve as a stepping stone towards directly targeting arbitrary excited states using ground state CC methods.