Renormalized CCSD approaches

Kowalski, K. Piecuch, P The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches, ... 

For these and other reasons, much attention was given to the so-called state-selective or state-specific (SS) MR CC approaches. These are basically of two types (i) essentially SR CCSD methods that employ MR CC Ansatz to select a subset of important higher-than-pair clusters that are then incorporated either in a standard way [163,164], or implicitly [109-117], or via the so-called externally corrected (ec) approaches of either the amplimde [214-219] or energy [220,221] type, and (ii) those actually exploiting Bloch equations, but focusing on one state at a time [222]. The energy-correcting ec CC approaches [220,221] are in fact very closely related to the renormalized CCSD(T) method of Kowalski and Piecuch mentioned earlier [146,147]. 

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 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conventional CCSD(T) method and its completely renormalized CR-CC(2,3) analog [102] (see, also, W. Li and P. Piecuch, unpublished work).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. 

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. 

The CR-CCSD(T) method provides somewhat better results, when compared with the CR-CCSD[T] approach, so in the numerical examples described in the next section we focus on the results of the CR-CCSD(T) calculations, but we need the CR-CCSD[T] theory to understand the connection between the CR-CC methods and their higher-order QMMCC counterparts (see the discussion below). It is also useful to consider the renormalized CCSD[T] and CCSD(T) methods (the R-CCSD[T] and R-CCSD(T) approaches), which can be viewed as simplified variants of the CR-CCSD[T] and CR-CCSD(T) approaches in which the 0203(2) moments in the CR-CCSD[T] and CR-CCSD(T) formulas, Eqs. (41) and (39), are replaced by their lowest-order estimates, ( j j i3 (EjvT2)c )- The R-CCSD[T] and R-CCSD(T) energies are defined as follows [11-13,30,31,33,35,37] ... 

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... 

In our view, the MMCC theory represents an interesting development in the area of new CC methods for molecular PESs. The MMCC-bas renormalized CCSD(T), CCSD(TQ), and CCSDT(Q) methods and the noniterative MMCC approaches to excited states provide highly accurate results for ground and excited-state PESs, while preserving the simplicity and the black-box character of the noniterative perturbative CC schemes. In this chapter, we review the MMCC theory and new CC i pnndmations that result firom it and show the examples of the MMCC and renormalized CC calculations for ground and excited state PESs of several benchmark molecules, including HF, F2, N2, and CH" ". The review of the previously published numerical results (7,16-20) is combined with the presentation of new results for the C2, N2, and H2O molecules. 

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... 

---