7.8 Coupled-Cluster Excited-State and Open-Shell Methods

7.8.18 EOM(2,3) Methods for Higher-Accuracy and Problematic Situations (CCMAN only)

In the EOM-CC(2,3) approach,379 the transformed Hamiltonian H¯ is diagonalized in the basis of the reference, singly, doubly, and triply excited determinants, i.e., the excitation operator R is truncated at triple excitations. The excitation operator T, however, is truncated at double excitation level, and its amplitudes are found from the CCSD equations, just like for EOM-CCSD [or EOM-CC(2,2)] method.

The accuracy of the EOM-CC(2,3) method closely follows that of full EOM-CCSDT [which can be also called EOM-CC(3,3)], whereas computational cost of the former model is less.

The inclusion of triple excitations is necessary for achieving chemical accuracy (1 kcal/mol) for ground state properties. It is even more so for excited states. In particular, triple excitations are crucial for doubly excited states,379 excited states of some radicals and SF calculations (diradicals, triradicals, bond-breaking) when a reference open-shell state is heavily spin-contaminated. Accuracy of EOM-CCSD and EOM-CC(2,3) is compared in Table  7.3.

System EOM-CCSD EOM-CC(2,3)
Singly-excited electronic states 0.1–0.2 eV 0.01 eV
Doubly-excited electronic states 1 eV 0.1–0.2 eV
Severe spin-contamination of the reference 0.5 eV 0.1 eV
Breaking single bond (EOM-SF) 0.1–0.2 eV 0.01 eV
Breaking double bond (EOM-2SF) 1 eV 0.1–0.2 eV
Table 7.3: Performance of the EOM-CCSD and EOM-CC(2,3) methods

The applicability of the EOM-EE/SF-CC(2,3) models to larger systems can be extended by using their active-space variants, in which triple excitations are restricted to semi-internal ones.

Since the computational scaling of EOM-CC(2,3) method is 𝒪(N8), these calculations can be performed only for relatively small systems. Moderate size molecules (10 heavy atoms) can be tackled by either using the active space implementation or tiny basis sets. To achieve high accuracy for these systems, energy additivity schemes can be used. For example, one can extrapolate EOM-CCSDT/large basis set values by combining large basis set EOM-CCSD calculations with small basis set EOM-CCSDT ones.

Running the full EOM-CC(2,3) calculations is straightforward, however, the calculations are expensive with the bottlenecks being storage of the data on a hard drive and the CPU time. Calculations with around 80 basis functions are possible for a molecule consisting of four first row atoms (NO dimer). The number of basis functions can be larger for smaller systems.

Note:  In EE calculations, one needs to always solve for at least one low-spin root in the first symmetry irrep in order to obtain the correlated EOM energy of the reference. The triples correction to the total reference energy must be used to evaluate EOM-(2,3) excitation energies.

Note:  EOM-CC(2,3) works for EOM-EE, EOM-SF, and EOM-IP/EA. In EOM-IP, “triples” correspond to 3h2p excitations, and the computational scaling of EOM-IP-CC(2,3) is less.