EOM-CC methods

The plan of this chapter is as follows. The next section briefly reviews the CC formalism for the ground state. This is necessary since the LR-CC and EOM-CC approaches start from the CC ground state description. It also introduces some notation that will be used in later sections. Next, the basics of the exact EOM-CC approach are derived, showing how an eigensystem is arrived at. After some aspects of characterizing an electronic transition, EOM-/LR-CC methods that have been developed and implemented are surveyed. The next section presents a numerical assessment of some of the main methods. Finally, a few illustrative applications are summarized. Some aspects of EOM-CC methods are discussed in Chapter 2. The symmetry-adapted cluster configuration interaction (SAC-CI) method can be related to EOM-CC methods. The SAC-CI method and several impressive applications thereof are described in Chapter 4. 

It is not necessary to base the calculation on a singlet closed-shell reference state, but the vast majority of chemical applications choose this option. Open-shell reference EOM-CC and EOM-CC methods are, however, relevant to the discussion of real singularities, since it is the (ROHF)UHF-EOM-CC excitation energies relative to the open-shell state of interest that give the energy differences in the denominators of Eq. (18). 

Krylov and her co-workers have taken the de-excitation EOM to its logical conclusion in what they call their spin-ip (SF-EOM-CC) method. EOM-CC readily lends itself... 

As discussed above, the wide applicability and predictive nature of CC methods have established a plateau in the field and led to automated program generation for the standard sequence of approximations of CC and EOM-CC methods. This alleviates any need for further development of the basic CC functional structure, at least for molecules treated in a conventional way. However, one profitable route for further development is to focus on... 

FC, PSO and SD terms were calculated with the EOM-CC method in HF, N2 and CO, where it is known that noncontact contributions are important, and results were compared with those reported in refs 164,186,187. In the last two cases, couplings were also calculated by Vahtras et Good agreement with other correlated calculations was obtained. Couplings in H2O, NH3 and HCl were also discussed and results were compared with Fukui s MBPT(2), showing that both methods reproduce the experimental 7(0,H) and /(N,H) values within 2%, but for V(H,H) significant differences with experiment were obtained. It is worth noting that / (H,Cl) = 30.03 Hz, which compares favourably with 7 om-<=<=(H,C1) = 35.03 Hz and with / (H,C1) = 37.7 Hz. ... 

The EOM-CC method) 15,16] makes use of the similarity-transformed perturbed Hamiltonian,... 

Properties computed using Eq. (11) are not size-extensive, however, because of the appearance of unlinked diagrams arising from disconnected terms implicit in Eq. (3). Such terms naturally cancel if the T and operators are not truncated, implying that the EOM-CC method is formally exact in the full-CC limit. 

A computational cost of QM/EFP calculations is typically determined by the cost of the QM calculation. Additionally, "method 2" (perturbative account of state-specific polarization) requires calculation of one-particle density matrix for each electronic state. QM/EFP schemes were implemented for a variety of electronic structure methods, such as HF, DFT, CIS, CIS(D), TD-DFT, various EOM-CC methods, and provide means to anal3 e electronic structure in the environment at the desired level of accuracy. 

The CC method is used to calculate the ground-state energy and wave function. What about the excited states This is a task for the equation-of-motion coupled cluster (EOM-CC) method, the primary goal being not the excited states themselves, but the excitation energies with respect to the ground state. 

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

The equation-of-motion coupled-cluster (EOM-CC) method is based on the CC wave function obtained for the ground state and is designed to provide the electronic excitation energies and the corresponding excited-state wave functions. 

EOM-CC method (p. 638) exchange hole (p. 597) explicit correlation (p. 584) exponentially correlated function (p. 594) Fermi hole (p. 597) frozen orbitals (p. 624)... 

EOM-CC method (p. 548) deexcitations (p. 550) many body perturbation theory (MBPT) (p.551)... 

A reliable electronic-structure model which provides a balanced description of the electronic states of interest is required for the computation of inter-state vibronic coupling constants. In particular, artifactual symmetrybreaking effects in the electronic wave function should be avoided. For excited electronic states, a CASSCF/MRCI description with appropriately chosen active space is generally to be recommended. For ionized states, the OVGF method, the ADC(3) method or the EOM-CC method have been found to be appropriate tools. 

Applying the time-dependent perturbation is straigthforward and leads to LR-CC methods. The nonlinear systems of equations include the normal T and Ti (for CCSD) operators-amplitudes and additionally single and double excitation (time-dependent) response amplitudes (for details the reader is referred to Refs. 1, 64, 88, 89 and references cited therein). An alternative approach, that, although conceptually different yields exactly the same excitation energies, is the equation-of-motion coupled cluster (EOM-CC) method [90]. The EOM-CC equations also contain the CC wave function 4 cc) (Eq. [50]) and a second (state-dependent) excitation operator R including single, double,. .. excitations (usually R is truncated in the same manner as T). The EOM equations read as... 

One characteristic of all LR/EOM-CC methods is that a non-Hermitian eigenvalue problem finally has to be solved (which cannot be reduced to a standard eigenvalue problem as Eq. [43]) yielding left and right sets of solutions, and therefore also two sets of transition moments. 

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