Completely renormalized CCSD methods

It has been demonstrated in several benchmark calculations that the CR-CCSD(T) (completely renormalized CCSD(T)) and CR-CCSD(TQ) (completely renormalized CCSD(TQ)) methods provide an excellent description of entire PESs involving single and double bond dissociation (P, 13, 15, 17-19, 21, 111), highly-excited vibrational term values near dissociation 17, 18, 21, 111), and... 

This leads to the so-called MMCC(my, ttib) schemes. The renormalized and completely renormalized CCSD(T) and CCSD(TQ) methods discussed in... 

The completely renormalized CCSD(T) method (the CR-CCSD(T) approach) is an MMCC(2,3) scheme, in which the wave function o) is replaced by the very simple, MBPT(2)[SDT]-like, expression. 

The above analysis implies that the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods can be viewed as the MMCC-based extensions of the standard CCSD[T] and CCSD(T) approaches. Very similar extensions can be formulated for other noniterative CC approaches. In particular, we can use the MMCC formalism to renormalize the CCSD(TQf) method of Ref. [23], in which the correction due to the combined effect of triples and quadruples is added to the CCSD energy. The resulting completely renormalized CCSD(TQ) (CR-CCSD(TQ)) approaches are the examples of the MMCC(2,4) approximation, defined by Eq. (35). As in the case of the CR-CCSD[T] and CR-CCSD(T) methods, we use the MBPT(2)-like expressions to define the wave function o) in the CR-CCSD(TQ) energy formulas. Two variants of the CR-CCSD(TQ) method, labelled by the extra letters a and b , are particularly useful. The CR-CCSD(TQ),a and CR-CCSD(TQ),b energies will be defined as follows [11-13,30,31,33,35] ... 

The simple relationships between the renormalized and completely renormalized CCSD[T], CCSD(T), and CCSD(TQ) methods and their standard counterparts, discussed above, imply that computer costs of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], CR-CCSD(T), R-CCSD(TQ)-n,x, and CR-CCSD(TQ),x (n = 1,2, x = a, b) calculations are essentially identical to the costs of the standard CCSD[T], CCSD(T), and CCSD(TQf) calculations. In analogy to the standard CCSD[T] and CCSD(T) methods, the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches are procedures in the noniterative steps involving triples and procedures in the iterative CCSD steps. More specifically, the CR-CCSD[T] and CR-CCSD(T) approaches are twice as expensive as the standard CCSD[T] and CCSD(T) approaches in the steps involving noniterative triples corrections, whereas the costs of the R-CCSD[T] and R-CCSD(T) calculations are the same as the costs of the CCSD[T] and CCSD(T) calculations [77]. The memory and disk storage requirements characterizing the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods are essentially identical to those characterizing the standard CCSD[T] and CCSD(T) approaches (see Ref. [77] for further details). In complete analogy to the noniterative triples corrections, the costs of the R-CCSD(TQ)-n,x calculations are identical to the costs of the CCSD(TQf) calculations (the CCSD(TQf) method... 

As mentioned in the previous section, the primary motivation behind the QMMCC approximations is the need to improve the CR-CCSD(T) and CR-CCSD(TQ) description of the more complicated types of multiple bond breaking. Let us, therefore, examine the performance of the standard and completely renormalized CCSD(T) and CCSD(TQ) methods and their QMMCC counterparts for a complicated case of triple bond breaking in N2. In the following discussion, we use the numerical results obtained for the DZ [108] model of N2 in Refs. [12,13,31, 78]. 

Different types of the MMCC(2,3), MMCC(2,4), and MMCC(3,4) approximations are obtained by making different choices for o) in eqs (35)-(37) (7,16-18). The most intriguing results are obtained when wave functions o) are defined by the low-order MBPT. The MBPT-like forms of o) lead to the renormcdized and completely renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) schemes (7,16-18). As demonstrated below, these new methods represent powerful computational tools that remove the failing of the standard CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approximations at large internuclefu separations, while preserving the simplicity and black-box character of the noniterative perturbative CC approaches. 

The idea of renormalizing the CCSD[T] and CCSD(T) methods can be extended to the CCSD(TQ) and CCSDT(Q) cases. The completely renormalized CCSD(TQ) methods, termed CR-CCSD(TQ),a and CR-CCSD(TQ),b, are examples of the MMCC(2,4) scheme, in which we correct the CCSD results by considering the projections of the CCSD equations on triply and quadruply excited configurations. The CR-CCSD(TQ),a and CR-CCSD(TQ),b energies are calculated as follows (7,16,17) ... 

The general nature of the MMCC theory, on which all renormalized and completely renormalized CC methods described here are based, allows us to proposed many other potentially useful approximations. We can, for example, introduce the MMCC(2,6) method, in which the CCSD results are corrected by considering all nonzero moments of the CCSD equations, including those corresponding to projections on pentuply and hextuply excited configurations. We can also introduce the active-space variants of the renormalized and completely renormalized CC approaches, in which we consider small subsets of the generalized moments of CC equations defined... 

We have overviewed the new approach to the many-electron correlation problem in atoms and molecules, termed the method of moments of coupled-cluster equations (MMCC). The main idea of the MMCC theory is that of the noniterative energy corrections which, when added to the ground- and excited-state energies obtained in approximate CC calculations, recover the exeict energies. We have demonstrated that the MMCC formalism leads to a number of useful approximations, including the renormalized and completely renormalized CCSD(T), CCSD(TQ), and CCSDT(Q) methods for... 

Keywords Coupled-cluster theory Local correlation methods Cluster-inmolecule formalism Linear scaling algorithms Single-reference coupled-cluster methods CCSD approach CCSD(T) approach Completely renormalized coupled-cluster approaches CR-CC(2,3) approach Large molecular systems Bond breaking Normal alkanes Water clusters... 

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