Coupled-cluster diagrams

Figure 1 Some basic components of coupled cluster diagrams (a) hole lines ...

Finally, a similar vertical type of factorization is applicable to coupled-cluster diagrams, since they are, by definition, denominator factorized. Figure 27 presents an example of the CC diagrams and its decomposition, for computational efficiency, into independently calculated parts. The first part (A) is computed with an n6 dependence. The second one (B) is of n5, and finally the fully factorized diagram b requires an n1 basis set dependence. 

Automated derivation of the general-order coupled cluster diagrams... 

Various theoretical methods and approaches have been used to model properties and reactivities of metalloporphyrins. They range from the early use of qualitative molecular orbital diagrams (24,25), linear combination of atomic orbitals to yield molecular orbitals (LCAO-MO) calculations (26-30), molecular mechanics (31,32) and semi-empirical methods (33-35), and self-consistent field method (SCF) calculations (36-43) to the methods commonly used nowadays (molecular dynamic simulations (31,44,45), density functional theory (DFT) (35,46-49), Moller-Plesset perturbation theory ( ) (50-53), configuration interaction (Cl) (35,42,54-56), coupled cluster (CC) (57,58), and CASSCF/CASPT2 (59-63)). 

The coupled cluster (CC) method is actually related to both the perturbation (Section 5.4.2) and the Cl approaches (Section 5.4.3). Like perturbation theory, CC theory is connected to the linked cluster theorem (linked diagram theorem) [101], which proves that MP calculations are size-consistent (see below). Like standard Cl it expresses the correlated wavefunction as a sum of the HF ground state determinant and determinants representing the promotion of electrons from this into virtual MOs. As with the Mpller-Plesset equations, the derivation of the CC equations is complicated. The basic idea is to express the correlated wave-function Tasa sum of determinants by allowing a series of operators 7), 73,... to act on the HF wavefunction ... 

In spite of the method s present utility and popularity, the quantum chemical community was slow to accept coupled cluster theory, perhaps because the earliest researchers in the field used elegant but unfamiliar mathematical tools such as Feynman-like diagrams and second quantization to derive working equations. Nearly 10 years after the essential contributions of Paldus and Cizek, Hurley presented a re-derivation of the coupled cluster doubles (CCD) equa-... 

Many varieties of diagrams have used throughout the chemical physics literature for many years (e.g., see Refs. 1, 2, 80, 117, and 119). The diagrammatic formalism we have chosen here has been frequently used in work by the Bartlett group among others and is particularly straightforward for conventional coupled cluster and many-body perturbation theories. 

It is possible for groups of three or more lines to be identified as equivalent, though this can happen only in many-body perturbation theory, expectation value coupled cluster theory, or unitary coupled cluster theory. For such diagrams, a prefactor of where n is the number of electron lines, must be included. 

The present work details the derivation of a full coupled-cluster model, including single, double, and triple excitation operators. Second quantization and time-independent diagrams are used to facilitate the derivation the treatment of (diagram) degeneracy and permutational symmetry is adapted from time-dependent methods. Implicit formulas are presented in terms of products of one- and two-electron integrals, over (molecular) spin-orbitals and cluster coefficients. Final formulas are obtained that restrict random access requirements to rank 2 modified integrals and sequential access requirements to the rank 3 cluster coefficients. 

Foundations to the CC methods were laid by Coester and Kuemmel,1 Cizek,2 Hubbard,3 Sinanoglu,4 and Primas,5 while Cizek2 first presented the CC equations in explicit form. Also Hubbard3 called attention to the equivalence of CC methods and infinite-order many-body perturbation theory (MBPT) methods. From this latter viewpoint, the CC method is a device to sum to infinity certain classes of MBPT diagrams or all possible MBPT diagrams when the full set of coupled-cluster equations is solved. The latter possibility would require solving a series of coupled equations involving up to IV-fold excitations for N electrons. Practical applications require the truncation of the cluster operators to low N values. 

Another way of introducing cluster operators is to define the operator 7) to sum only connected /-fold excitation diagrams in P mbpt, and by virtue of defining Cl = exp(T) the disconnected but linked mbpt diagrams are summed as the quadratic and higher terms in the exp(T) expansion. This is the essential relationship of MBPT to coupled-cluster theory. 

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