Derivation of the EOM-CC equations

As the reference function in the EOM-CC method, we take the coupled-cluster wave function for the ground state  

The operators Uk change the coefficients in front of the configurations (see p. 526). The operators Uk are (unlike the wave operator exp(r)) linear with respect to the excitations, i.e. the excitation amplitudes occur there in the first powers. For the case of the single and double excitations (EOM-CCSD) we have T in the form of the sum of single and double excitations  

By neglecting higher than single and double excitations the equation represents an approximation. 

Due to the missing deexcitation part (i.e. that which lowers the excitation rank, e.g., from doubles to singles) the operators and T commute, hence the operators Uk and expCT) also commute  

From the last equation we subtract the CC equation for the ground state [exp(-r)i7exp(f)] bo = Eo i o 

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Due to the Cl-like nature of EOM-CC, there are no contributions involving the derivatives of 1Z and in equation (101). The derivatives of the cluster amplitudes can be eliminated from using the Dalgamo-Stewart interchange theorem yielding... 

Since 9///9jc is the only derivative quantity left in equation (102), the cost of an EOM-CC gradient calculation is - as discussed for CC gradients - independent of the number of perturbations. 

The algebraic equations and efficient computational sequences were derived by smith and reported by us [33] for CCSD-, CCSDT-, and CCSDTQ-R12, their excited-state analogues via the equation-of-motion (EOM) formalisms (EOM-CC-R12 up to EOM-CCSDTQ-R12), and the so-called A equations for the analytical gradients and response properties, again up to A-CCSDTQ-R12. The full CCSD-, CCSDT-, and CCSDTQ-R12 methods [34,35] were implemented by smith into efficient computer codes that took advantage of spin, spatial, and index-permutation symmetries. 

The equation of the PCM-EOM-CC analytical gradients contains additional one-electron MO derivative integrals with respect to the corresponding gradients for isolated molecules (Stanton 1993). As these solvation terms can be evaluated with a small effort, the PCM-EOM-CC analytical gradients can be performed for aU the molecular systems for which the EOM-CCSD... 

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