Reduced-Cost Four-Component Relativistic Double Ionization Potential Equation-of-Motion Coupled-Cluster Approaches with 4-Hole–2-Particle Excitations and Three-Body Clusters

The double ionization potential (DIP) equation-of-motion (EOM) coupled-cluster (CC) method with 4-hole-2-particle (4<i>h</i>-2<i>p</i>) excitations on top of the CC with singles, doubles, and triples calculation, abbreviated as DIP-EOMCCSDT(4<i>h</i>-2<i>p</i>), along with its perturbative DIP-EOMCCSD(T)(a)(4<i>h</i>-2<i>p</i>) approximation, are extended to a relativistic four-component (4c) framework. In addition, we introduce and test a new computationally practical DIP-EOMCC approach, which we call DIP-EOMCCSD(T)(<i>ã</i>)(4<i>h</i>-2<i>p</i>), that approximates the treatment of 4<i>h</i>-2<i>p</i> correlations within the DIP-EOMCCSD(T)(a)(4<i>h</i>-2<i>p</i>) method and reduces the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>8</mml:mn></mml:mrow></mml:msup></mml:math> scaling characterizing DIP-EOMCCSDT(4<i>h</i>-2<i>p</i>) and DIP-EOMCCSD(T)(a)(4<i>h</i>-2<i>p</i>) to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>7</mml:mn></mml:mrow></mml:msup></mml:math> with the system size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>. Further improvements in computational efficiency are obtained using the frozen natural spinor (FNS) approximation to reduce the numbers of unoccupied spinors entering the correlated steps of the DIP-EOMCC calculations according to a well-defined occupation-number-based threshold. The resulting 4c-FNS-DIP-EOMCC approaches are used to compute DIPs for the series of inert gas atoms from argon to radon as well as the vertical DIPs in Cl₂, Br₂, HBr, and HI, which have been experimentally examined in the past. We demonstrate that, when using complete basis set extrapolations and FNS truncation threshold of 10^-4.5, the 4c-FNS-DIP-EOMCCSD(T)(<i>ã</i>)(4<i>h</i>-2<i>p</i>) calculations are capable of predicting DIPs in agreement with experimental data, improving upon their nonrelativistic and spin-free scalar-relativistic counterparts, particularly when examining DIPs characterized by stronger spin-orbit coupling effects.