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

7.8.1 Excited States via EOM-EE-CCSD

One can describe electronically excited states at a level of theory similar to that associated with coupled-cluster theory for the ground state by applying either linear response theory482 or equation-of-motion methods.866 A number of groups have demonstrated that excitation energies based on a coupled-cluster singles and doubles ground state are generally very accurate for states that are primarily single electron promotions. The error observed in calculated excitation energies to such states is typically 0.1–0.2 eV, with 0.3 eV as a conservative estimate, including both valence and Rydberg excited states. This, of course, assumes that a basis set large and flexible enough to describe the valence and Rydberg states is employed. The accuracy of excited state coupled-cluster methods is much lower for excited states that involve a substantial double excitation character, where errors may be 1 eV or even more. Such errors arise because the description of electron correlation of an excited state with substantial double excitation character requires higher truncation of the excitation operator. The description of these states can be improved by including triple excitations, as in EOM(2,3).

Q-Chem includes coupled-cluster methods for excited states based on the coupled cluster singles and doubles (CCSD) method described earlier. CCMAN also includes the optimized orbital coupled-cluster doubles (OD) variant. OD excitation energies have been shown to be essentially identical in numerical performance to CCSD excited states.500

These methods, while far more computationally expensive than TDDFT, are nevertheless useful as proven high accuracy methods for the study of excited states of small molecules. Moreover, they are capable of describing both valence and Rydberg excited states, as well as states of a charge-transfer character. Also, when studying a series of related molecules it can be very useful to compare the performance of TDDFT and coupled-cluster theory for at least a small example to understand its performance. Along similar lines, the CIS(D) method described earlier as an economical correlation energy correction to CIS excitation energies is in fact an approximation to EOM-CCSD. It is useful to assess the performance of CIS(D) for a class of problems by benchmarking against the full coupled-cluster treatment. Finally, Q-Chem includes extensions of EOM methods to treat ionized or electron attachment systems, as well as di- and triradicals.

EOM-EE     Ψ(MS=0)=R(MS=0)Ψ0(MS=0)

↿⇂
↿⇂⇂↿
↿⇂
⏟ΦiaΦi⁢ja⁢b

EOM-IP     Ψ⁢(N)=R⁢(-1)⁢Ψ0⁢(N+1)

↿⇂↿⇂
↿⇂↿↿↿⇂
↿⇂↿↿⇂↿⇂↿↿↿⇂↿
⏟Φi⏟Φi⁢ja

EOM-EA     Ψ⁢(N)=R⁢(+1)⁢Ψ0⁢(N-1)

↿⇂
↿⇂↿↿⇂↿
↿↿⇂↿↿⇂⇂↿↿↿⇂↿⇂↿⇂
⏟Φa⏟Φia⁢b

EOM-SF     Ψ(MS=0)=R(MS=-1)Ψ0(MS=1)

↿↿
↿⇂↿⇂
↿⇂⇂↿
⏟Φia
Figure 7.1: In the EOM formalism, target states Ψ are described as excitations from a reference state Ψ0: Ψ=R⁢Ψ0, where R is a general excitation operator. Different EOM models are defined by choosing the reference and the form of the operator R. In the EOM models for electronically excited states (EOM-EE, upper panel), the reference is the closed-shell ground state Hartree-Fock determinant, and the operator R conserves the number of α and β electrons. Note that two-configurational open-shell singlets can be correctly described by EOM-EE since both leading determinants appear as single electron excitations. The second and third panels present the EOM-IP/EA models. The reference states for EOM-IP/EA are determinants for N+1/N-1 electron states, and the excitation operator R is ionizing or electron-attaching, respectively. Note that both the EOM-IP and EOM-EA sets of determinants are spin-complete and balanced with respect to the target multi-configurational ground and excited states of doublet radicals. Finally, the EOM-SF method (the lowest panel) employs the high-spin triplet state as a reference, and the operator R includes spin-flip, i.e., does not conserve the number of α and β electrons. All the determinants present in the target low-spin states appear as single excitations, which ensures their balanced treatment both in the limit of large and small HOMO/LUMO gaps. Other EOM methods available in Q-Chem are EOM-2SF, EOM-DIP, and EOM-DEA.