7.8.2 EOM-XX-CCSD and CI Suite of Methods

Q-Chem features the most complete set of EOM-CCSD models,⁵⁰⁴ enabling accurate, robust, and efficient calculations of electronically excited states (EOM-EE-CCSD or EOM-EE-OD); ^{830, 481, 866, 500, 559}; ground and excited states of diradicals and triradicals (EOM-SF-CCSD and EOM-SF-OD);^{501, 559} ionization potentials and electron attachment energies, as well as problematic doublet radicals and cation or anion radicals (EOM-IP/EA-CCSD).^{847, 868, 673} The EOM-DIP-CCSD, EOM-2SF-CCSD, and EOM-DEA-CCSD methods are available as well. Conceptually, EOM is very similar to configuration interaction (CI): target EOM states are found by diagonalizing the similarity transformed Hamiltonian $\overline{H}=e^{-T}H{e}^T$,

$$\overline{H}R=ER,$$

(7.45)

where $T$ and $R$ are general excitation operators with respect to the reference determinant $|{\mathrm{\Phi }}_0⟩$. In the EOM-CCSD models, $T$ and $R$ are truncated at single and double excitations, and the amplitudes $T$ satisfy the CC equations for the reference state $|{\mathrm{\Phi }}_0⟩$:

	$⟨{\mathrm{\Phi }}_i^a|\overline{H}|{\mathrm{\Phi }}_0⟩$	$=$	$0$		(7.46)
	$⟨{\mathrm{\Phi }}_{ij}^{ab}|\overline{H}|{\mathrm{\Phi }}_0⟩$	$=$	$0$		(7.47)

The computational scaling of EOM-CCSD and CISD methods is identical, i.e., $𝒪(N^6)$, however EOM-CCSD is numerically superior to CISD because correlation effects are “folded in” in the transformed Hamiltonian, and because EOM-CCSD is rigorously size-intensive.

By combining different types of excitation operators and references $|{\mathrm{\Phi }}_0⟩$, different groups of target states can be accessed as explained in Fig. 7.1. For example, electronically excited states can be described when the reference $|{\mathrm{\Phi }}_0⟩$ corresponds to the ground state wave function, and operators $R$ conserve the number of electrons and a total spin.⁸⁶⁶ In the ionized/electron attached EOM models,^{868, 673} operators $R$ are not electron conserving (i.e., include different number of creation and annihilation operators)—these models can accurately treat ground and excited states of doublet radicals and some other open-shell systems. For example, singly ionized EOM methods, i.e., EOM-IP-CCSD and EOM-EA-CCSD, have proven very useful for doublet radicals whose theoretical treatment is often plagued by symmetry breaking. Finally, the EOM-SF method^{501, 559} in which the excitation operators include spin-flip allows one to access diradicals, triradicals, and bond-breaking.⁵⁰⁵

Q-Chem features EOM-EE/SF/IP/EA/DIP/DSF-CCSD methods for both closed and open-shell references (RHF/UHF/ROHF), including frozen core/virtual options. For EE, SF, IP, and EA, a more economical flavor of EOM-CCSD is available (EOM-MP2 family of methods). All EOM models take full advantage of molecular point group symmetry. Analytic gradients are available for RHF and UHF references, for the full orbital space, and with frozen core/virtual orbitals.⁵⁶⁰ Properties calculations (permanent and transition dipole moments and angular momentum projections, $⟨S^2⟩$, $⟨R^2⟩$, etc.) are also available. The current implementation of the EOM-XX-CCSD methods enables calculations of medium-size molecules, e.g., up to 15–20 heavy atoms. Using RI approximation 6.8.5 or Cholesky decomposition 6.8.6 helps to reduce integral transformation time and disk usage enabling calculations on much larger systems. EOM-MP2 and EOM-MP2t variants are also less computationally demanding. The computational cost of EOM-IP calculations can be considerably reduced (with negligible decline in accuracy) by truncating virtual orbital space using FNO scheme (see Section 7.8.9).