Linear scaling local correlation

Linear Scaling Local Correlation Extensions of the Standard and Renormalized Coupled-Cluster Methods... 

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conventional CCSD(T) method and its completely renormalized CR-CC(2,3) analog [102] (see, also, W. Li and P. Piecuch, unpublished work).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

Schutz, M. Low-order scaling local electron correlation methods. III. Linear scaling local perturbative triples correction (T), J. Chem. Phys. 2000,113,9986-10001. 

A difficulty with this local approach to dynamical correlation is that, in Moller-Plesset theory, for example, the zero-order Fock operator is no longer diagonal in the space of the Slater determinants, making the application of such theories slightly more complicated than theories based on canonical orbitals. Currently, the development of local correlation methods is an active area of research [57-63]. The diatomics-inmolecules (DIM) method and the triatomics-in-molecules (TRIM) method, for instance, recover typically 95% and 99.7%, respectively, of the full MP2 correlation energy [63]. By means of a linear scaling local variant of the CCSDT method,... 

Low-order scaling local electron correlation methods. I. Linear scaling local MP2 ... 

The work of Schutz, Hetzer and Wemer " begins a series of papers exploring the development of local electron correlation methods with low-order scaling by presenting a linear scaling local MP2 . They describe a novel multipole approximation based on a sphtting of the Coulomb operator into two terms... 

Low-order scaling local correlation methods II Splitting the Coulomb operator in linear scaling local second-order Moller-Plesset perturbation theory ... 

Linear-scaling electron-correlation methods were developed by 1) combining the Pulay-Saebo local correlation variant [136] with integral-direct techniques [131] and consequently exploiting the spatial locahty of the electron-correlation effect 2) Laplace-transform techniques suggested in [137] and apphed in [129]. 

Electron Correlation Methods. IV. Linear Scaling Local Coupled-Cluster (LCCSD). 

Order Scaling Local Correlation Methods If Sphtting the Coulomb Operator in Linear Scaling Local Second-Order MoUer-Plesset Perturbation Theory. 

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... 

Of all local CC methods to date, the approaches that have received the most attention are those of Hampel and Werner [55] and Schiitz and Werner [56-60]. These authors reported the fully operational, low-order [55] or linear scaling implementations of the local CCSD [56, 60], CCSD(T) [57, 58], and CCSDT-1 [59] methods. Their implementations of local CC methods exploit the local correlation formalism of Pulay [82] and Pulay and Saeb0 [83-87], in which one solves the CC equations in a basis of orthonormal occupied LMOs obtained with one of the conventional MO localization schemes [88-90] and non-orthogonal unoccupied orbitals constructed from the projected AOs (PAOs), while dividing the large system of interest into orbital domains to which excitations defining the CC ansatz are restricted. 

As expected, the performance of all competing order-N methods depends on the system under investigation, the accuracy needed, the amount of experimental information available, the questions that need to be answered, and also computer-related parameters (processors, parallel architectures, etc.). All approaches have their pluses and minuses. It is also clear that the increased mathematical effort will "pay off" if at all) only beyond a critical number of atoms (around 100-1000 or so) below that, the normal route with cubic scaling is faster. Nonetheless, the same locality arguments may be used to derive linear-scaling methods for the extremely efficient calculation of electronic correlation (see Section 2.13). 

Two relevant topics have been ignored completely in this short chapter the treatment of electron correlation with more sophisticated methods than DFT (that remains unsatisfactory from many points of view) and the related subject of excited states. Wave function-based methods for the calculation of electron correlation, like the perturbative Moller-Plesset (MP) expansion or the coupled cluster approximation, have registered an impressive advancement in the molecular context. The computational cost increases with the molecular size (as the fifth power in the most favorable cases), especially for molecules with low symmetry. That increase was the main disadvantage of these electron correlation methods, and it limited their application to tiny molecules. This scaling problem has been improved dramatically by modern reformulation of the theory by localized molecular orbitals, and now a much more favorable scaling is possible with the appropriate approximations. Linear scaling with such low prefactors has been achieved with MP schemes that the... 

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