Coupled cluster equation

It is the need to remove the unlinked clusters and the introduction of Feynman diagrams which make MBPT [and CC theory] appear unfamiliar to quantum 

In this section we construct working equations for the coupled cluster singles and doubles (CCSD) method. Beginning from the approximation 7 = 7 + T2, we use algebraic and diagrammatic techniques to obtain programmable 

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In the above the coupled cluster equations have been derived by multiplying the Schrddinger equation with ( o. and An alternative way of deriving the... 

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

The coupled-cluster equations may be derived by variation of the functional... 

The method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

The Method of Moments of Coupled-Cluster Equations An Overview of the Ground-State Formalism... 

Generalization of the method of moments of coupled-cluster equations to excited electronic states Exact formalism... 

NONITERATIVE CORRECTIONS TO COUPLED-CLUSTER AND EQUATION-OF-MOTION COUPLED-CLUSTER ENERGIES DEFINING THE EXACT METHOD OF MOMENTS OF COUPLED-CLUSTER EQUATIONS... 

Kowalski, K. Piecuch, P The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches, ... 

The projective techniques described above for solving the coupled cluster equations represent a particularly convenient way of obtaining the amplitudes that define the coupled cluster wavefunction, e o However, the asymmetric energy formula shown in Eq. [50] does not conform to any variational conditions in which the energy is determined from an expectation value equation. As a result, the computed energy will not be an upper bound to the exact energy in the event that the cluster operator, T, is truncated. But the exponential ansatz does not require that we solve the coupled cluster equations in this manner. We could, instead, construct a variational solution by requiring that the amplitudes minimize the expression ... 

The concepts of normal ordering and Wick s theorem provide the mathematical tools needed to derive programmable coupled cluster equations from the more formal expressions given in Eqs. [50] and [51]. If we truncate the cluster operator such that T = Tj + T2 insert it into the similarity-transformed normal-ordered Hamiltonian, H = e lij e, we obtain... 

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