CCSD correlation energy, defined

CCSD energy, given by Eq. (13) and defining the CR-CC(2,3), CCSD(T), and other non-iterative triples CC approximations, can be rewritten as sums of contributions associated with the individual occupied spin-orbitals. The CCSD correlation energy A (ccsD) is a sum of the contributions defined by Eq. (11), whereas the... 

In order to complete our description of the theoretical and computational details behind the CIM-CCSD, CIM-CR-CC(2,3), and CIM-CCSD(T) approaches, we must provide information about the actual design of the occupied and unoccupied LMOs defining the CIM subsystems (P). This is done in the next subsection. Obviously, if we do not introduce any approximations and there is only one subsystem or orbital domain that corresponds to all orbitals of the system, Eqs. (23)-(38) become equivalent to the exact expressions, Eqs. (9) and (11) for the CCSD correlation energy and Eqs. (13) and (14) for the triples correction to CCSD. In this case, the CIM and canonical CC calculations yield identical results, i.e., and = 3 (2.3) -phe key idea of the CIM-CC... 

The basic idea of the externally corrected CCSD methods relies on the fact that the electronic Hamiltonian, defining standard ah initio models, involves at most two body terms, so that the correlation energy is fully determined by one (Ti) and two (T2) body cluster amplitudes, while the subset of CC equations determining these amplitudes involves at most three (T3) and four (T4) body connected clusters. In order to decouple this subset of singly and doubly projected CC equations from the rest of the CC chain, one simply neglects all higher than pair cluster amplitudes by setting... 

The coefficients entering Eqs. (13) and (14) are the hole-particle de-excitation amplitudes that in the exact expansion of the correlation energy in terms of the generalized moments of the CCSD equations, which produces the full Cl energy, define the three-body component of the left ground eigenstate (l of the similarity-transformed Hamiltonian obtained by diagonalizing ff(ccso) jjj gj,tire... 

The generalization of the interchange theorem [103] to the correlation problem is what makes CC analytical gradient theory viable, and, indeed, routine today. Also, the introduction of the response and the relaxed density matrices provides the non-variational CC generalizations of density matrix theory that makes it almost as easy to evaluate a property as with a normal expectation value. They are actually more general, since they apply to any energy expression whether or not it derives from a wavefunction This is essential, e.g. for CCSD(T). The difference is that we require a solution for both T and A if we want to use untruncated expressions for properties, as is absolutely necessary to define proper critical points. It is certainly true that... 

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