CCSD amplitude equations

A similar matrix element is needed for the evaluation of the CCSD amplitudes equations see... 

The CC2 wave function and energy are correct to the same orders in the fluctuation potential as the MP2 wave function and energy. However, we would expect the CC2 wave function to be slightly more accurate and more robust than the MPl wave function since the CC2 wave function has the additional merit of fulfilling the CCSD amplitudes equations to first order. The improvements are expected to be small, however. A direct comparison is given after the discussion of the CC3 approximation. 

The CCSD energy is given by the general CC equation (4.53), and amplitude equations are derived by multiplying (4.50) with a singly excited determinant and integrating (analogously to eq. (4.54)). 

Owing to its complexity, the CC-R12 method was initially realized in various approximate forms. The first implementation of the CCSD-R12 method including noniterative connected triples [CCSD(T)-R12] was reported by Noga et al. [31,32,57-60] within the SA. The use of the same basis set for the orbital expansion and the RI in the SA rendered many diagrammatic terms to vanish and, thereby, drastically simplified the CCSD-R12 amplitude equations, easing its implementation effort. However, the simplified equations also meant that large basis sets (such as uncontracted quintuple- basis set) were needed to obtain reliable results and, therefore, the SA CCSD-R12 method was useful only in limited circumstances. 

Subsequently, Klopper and coworkers developed the CCSD(R12) and CCSD(T)(R12) methods [61-63] in which the use of the SA was avoided, while maintaining the simplicity of the equations. The "(R12)" approximation retains the terms that are at most linear in ff and thus simplifies the amplitude equations considerably. Equations (20)—(22) are, therefore, replaced by [61]... 

Figure 1 A computational sequence of the CCSD-R12 geminal amplitude equation. For the definitions of symbols, see ref. 33.

Every term in the coupled cluster amplitude equations that is nonlinear in T may be factored into linear components. As a result, each step of the iterative solution of the CCSD equations scales at worst as ca. 0(X ) (where X is the number of molecular orbitals). The full CCSDT method in which all Tycon-taining terms are included requires an iterative 0(X ) algorithm, whereas the CCSD(T) method, which is designed to approximate CCSDT, requires a noniterative O(X ) algorithm. The inclusion of all T4 clusters in the CCSDTQ method scales as... 

Scuseria, Janssen, and Schaefer, for example, developed a set of intermediates based on their reformulation of the CCSD amplitude and energy equations in a unitary group formalism designed to offer special efficiency when the refer-... 

As already pointed out, we have not implemented the CR-CC(2,4) approach yet and there remain open issues, such as the role of terms that couple triples and quadruples in a system of equations for the relevant (CCSD) and CCSD) amplitudes, as defined by Eq. (76), where niA = 2 and mg = 4 (the... 

In view of these results it is of interest to explore the performance of the AL version of the RMR CCSD method. Since the latter requires the solution of CCSD-like equations, one can expect a similar performance and savings as in the standard CCSD case. Moreover, in generating the and Tf amplitudes, one obtains the and amplitudes as a byproduct. These can then be exploited using the split-amplitude strategy in obtaining AL-RMR CCSD. 

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