Completely renormalized CCSD approaches

THE RENORMALIZED AND COMPLETELY RENORMALIZED CCSD(T) AND CCSD(TQ) APPROACHES AND THE QUADRATIC MMCC THEORY... 

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. 

Although calculations of entire molecular PESs involving single bond breaking require using CR-CCSD[T] and CR-CCSD(T) methods rather than their simplified renormalized versions [11-13,30,31,33,35,37], these R-CCSD[T] and R-CCSD(T) approaches allow us to understand the relationship between the standard and completely renormalized CC approaches. 

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... 

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 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... 

In this work, in addition to the CCSD approximation, we examine two different ways of correcting the CCSD energy for the effects of the connected triply excited clusters, namely, the CCSD(T) method and its completely renormalized CR-CC(2,3) extension. Since the CCSD(T) approach can be obtained as a natural approximation to CR-CC(2,3) [24, 25], we begin our brief description of both methods with the key equations of CR-CC(2,3). 

In this chapter, we have reviewed our recent effort toward the extension of the linear scaling local correlation approach of Li and coworkers [38 0], abbreviated as CIM, to the standard CCSD approach and two CC methods with a non-iterative treatment of connected triply excited clusters, including the conventional CCSD(T) method and its completely renormalized CR-CC(2,3) analog [102] (see, also, W. Li and P. Piecuch, unpublished work). The local correlation formulation of the latter method based on the CIM formalism is particularly useful, since it enables one to obtain an accurate description of single bond breaking and biradicals, where CCSD(T) fails, with an ease of a black-box calculation of the CCSD(T) type [24-26, 109-117]. At the same time, CR-CC(2,3) is as accurate as CCSD(T) in applications involving closed-shell molecules near their equilibrium geometries. 

As mentioned in the Introduction, the renormalized (R) and completely renormalized (CR) CC approaches, which represent new classes of noniterative single-reference CC methods that are capable of removing the failing of the standard CCSD(T) and similar methods at larger internuclear separations, are based on the formalism of the method of moments of CC equations (MMCC) [11-13, 30-32,36, 75, 76]. Thus, we begin our description of the R-CC and CR-CC methods and their quadratic MMCC (QMMCC) extension for multiple bond breaking with a synopsis of the general MMCC theory. 

Further improvements in the CR-CCSD(TQ),b results for N2 can be obtained by applying the completely renormalized versions of the CCSDT(Qf) approach (see Refs. [11,30] for the description of these methods). For example, variant b of the CR-CCSDT(Q) method [11,30], in which the completely renormalized corrections due to T4 clusters are added to the full CCSDT energies, provides a curve for N2 which is characterized by errors relative to full Cl that do not exceed 7.6 millihartree in the entire R = 0.75 — 2.25i e region and that are as small as 0.719 millihartree at R = Re and 1.161 millihartree a.t R = 2Re [12,34] (cf. Table 2). The question is if we can obtain similar or better improvements by adding noniterative corrections to the CCSD (rather than CCSDT) energies. 

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