Coupled-cluster equations constructing

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 ... 

In this section we examine the fundamental relationship between many-body perturbation theory (MBPT) and coupled cluster theory. As originally pointed out by Bartlett, this connection allows one to construct finite-order perturbation theory energies and wavefunctions via iterations of the coupled cluster equations. The essential aspects of MBPT have been discussed in Volume 5 of Reviews in Computational Chemistry,as well as in numerous other texts. We therefore only summarize the main points of MBPT and focus on its intimate link to coupled cluster theory, as well as how MBPT can be used to construct energy corrections for higher order cluster operators such as the popular (T) correction for connected triple excitations. 

The above graphical constructs represent individual normal-ordered operators. Using these graphical representations one can derive all unique combinations (operator contractions) which contribute to the considered matrix element of an operator product. The rules for the diagram manipulations are standard and can be found elsewhere [13,51]. General-order coupled cluster equations can be derived from the general-order coupled cluster functional ... 

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

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... 

The construction of the coupled cluster amplitude equations is somewhat more complicated than the energy equation in that the latter requires only reference expectation values of the second-quantized operators. For the amplitude equations, we now require matrix elements between the reference, o, on the right and specific excited determinants on the left. We must therefore convert these into reference expectation value expressions by writing the excited determinants as excitation operator strings acting on Oq. For example, a doubly excited bra determinant may be written as... 

The first-order T2 amplitudes, which are required for Eq. [207], may be determined by left-projecting the first-order variant of Eq. [202] involving by a doubly excited determinant, as we did earlier in the construction of the coupled cluster amplitude equations,... 

For the coupled cluster methods, which are non-variational, the initial values of the A s are nonzero, and 0) does not correspond to the unperturbed reference state but, in most applications, to the Hartree-Fock state. Tire initial values of the parameters are found in an iterative optimization of the coupled cluster state, and the time-dependent values of the parameters were determined from the coupled-cluster time-dependent Schrodinger equation by Koch and Jprgensen [35], The coupled cluster state is not norm conserving, but the inno roduct of the coupled cluster state vector CC(f)) and a constructed dual vector (CC(f) remains a constant of time... 

A continuous transition between BWCC and the Jeziorski-Monkhorst approach [57] can also be constructed [127] as well as a transition between BWCC and Mukherjee s state-specific RS coupled-cluster method [130]. However, because of the complexity of the coupling terms arising in the equations this is not yet implemented. 

It has been shown that, using projected dipolar oscillator orbitals to represent the virtual space in a localized orbital context, the equations involved in long-range ring coupled cluster doubles type RPA calculations can be formulated without explicit knowledge of the virtual orbital set. The P(X) virtuals have been constructed directly from the localized occupied orbitals. The matrix elements and the electron... 

Sous J, Goel P, Nooijen M. Similarity transformed equation of motion coupled cluster theory revisited a benchmark study of valence excited states. Mol Phys. 2013 112 616-638. Trofimov AB, Krivdina IL, Weller J, Schimer J. Algebraic-diagrammatic construction propagator approach to molecular response properties. Chem Phys. 2006 329 1-10. 

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