The CCSD Amplitude Equations

As discussed earlier, the cluster amplitudes that parameterize the coupled cluster wavefunction may be determined from the projective Schrodinger equation given in Eq. [51]. In the CCSD approximation, the single-excitation amplitudes, t- , may be determined from 

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 final matrix element therefore requires that we obtain all fully contracted Wick s theorem terms from the product of the operator string above and the elements of H. 

The leading term of H in Eq. [122] is simply the electronic Hamiltonian itself. For its contribution to the T x amplitude equation, we must therefore evaluate matrix elements of between singly excited determinants and I o, 

The two-electron component of this equation cannot produce full contractions and is therefore zero. The one-electron term, however, simplifies to a single Fock matrix element  

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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. 

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-... 

The starting point for our discussion is equation (13.2.30) for the CCSD energy and equations (13.2.40) and (13.2.41) for the CCSD amplitudes. Omitting from these equations all terms containing commutators that are quadratic or higher in f and from (13.2.41) also the term containing [[H, f i], til we arrive at the equations that define the ( CISD model ... 

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.

In this section, we derive basic equations for the monoexcited and biexcited cluster amplitudes at the CCSD level of approximation, i.e. with the cluster operators 7 being approximated by their singly and doubly excited cluster components... 

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