J. Phys. Chem. A 2010, 114, 6721–6727
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Multilevel Extension of the Cluster-in-Molecule Local Correlation Methodology: Merging Coupled-Cluster and Møller-Plesset Perturbation Theories Wei Li and Piotr Piecuch* Department of Chemistry, Michigan State UniVersity, East Lansing, Michigan 48824 ReceiVed: April 29, 2010
A multilevel extension of the local correlation “cluster-in-molecule” (CIM) framework, which enables one to combine different quantum chemistry methods to treat different regions in a large molecular system without splitting it into ad hoc fragments and saturating dangling bonds, is proposed. The resulting schemes combine higher-level methods, such as the completely renormalized coupled-cluster (CC) approach with singles, doubles, and noniterative triples, termed CR-CC(2,3), to treat the reactive part of a large molecular system, and lowerorder methods, such as the second-order Møller-Plesset perturbation theory (MP2), to handle the chemically inactive regions. The multilevel CIM-CC/MP2 approaches preserve the key features of all CIM methods, such as the use of orthonormal localized orbitals and coarse-grain parallelism, while substantially reducing the already relatively low costs of the single-level CIM-CC calculations. Illustrative calculations include bond breaking in dodecane and the reactions of the bis(2,4,4-trimethylpentyl)dithiophosphinic acid with one and two water molecules. 1. Introduction Historically, the development of ab initio quantum chemistry methods, including high-accuracy methods based on coupledcluster (CC) theory,1 has focused on small molecules with a few atoms.2 To generalize the CC approaches, including, in particular, the CC method with singles and doubles (CCSD)3,4 and its CCSD(T)5 and CR-CC(2,3)6,7 extensions, which account for triply excited clusters via corrections to CCSD energies, and other higher-level ab initio methods to considerably larger systems, one must attack the polynomial scaling laws that define the dependence of the CPU time characterizing such approaches on the system size. This can be done intrinsically by switching from the delocalized Hartree-Fock (HF) basis to localized molecular orbitals (LMOs) or atomic orbitals (AOs) and by limiting the electronic excitations to orbital domains4,8-38 that reflect on the locality of electron correlation39,40 or by exploiting approaches in which the energy is assembled from the energies of intuitively defined fragments (see, e.g., refs 41-47). Our approach to the intrinsically local CC methods, which interest us in this paper and which has resulted in the development of the linear scaling, local correlation CCSD, CCSD(T), and CRCC(2,3) schemes,31-34 exploits the “cluster-in-molecule” (CIM) ansatz.22,23,31-34 This article describes a multilevel extension of CIM, which enables one to combine different methods of quantum chemistry to treat different regions in a large molecular system without splitting it into ad hoc fragments and then saturating dangling bonds, as is done in ONIOM and related schemes.48,49 The resulting approaches combine higher-level methods based on CC theory, such as as CR-CC(2,3), which improves the CCSD(T) results for bond breaking and biradicals,6,7 to treat the reactive part of a large molecular system, with lower-order approaches, such as the second-order Møller-Plesset perturbation theory (MP2), to handle the chemically inactive regions. They preserve the key features of the CIM formalism, such as * To whom correspondence should be addressed. E-mail: piecuch@ chemistry.msu.edu.
the use of orthonormal LMOs to define local orbital domains called CIM subsystems, the coarse-grain parallelism, and the noniterative character of the local MPn energies and local triples corrections of CR-CC(2,3) and CCSD(T) through the use of quasi-canonical MOs (QCMOs).31-34 At the same time, they substantially reduce the already relatively low costs of the singlelevel CIM-CC calculations, which are, in turn, much less expensive than the canonical CC calculations using delocalized HF MOs.31-34 In particular, if the reactive part of a large molecular system corresponding to the CIM orbital subsystem(s) treated by the CC approach has a fixed size and the system is grown by adding the CIM subsystems treated by MP2, the size dependence of the computer costs of such multilevel CIM-CC calculations, abbreviated as CIM-CC/MP2, is virtually none. The multilevel CIM-CR-CC(2,3)/MP2 and CIM-CCSD/MP2 schemes reported in this paper, and their CIM-CCSD(T)/ MP2 analog, which we do not discuss since CIM-CR-CC(2,3)/ MP2 is more robust in bond breaking situations without a substantial computer cost increase, are alternatives to the recently developed LMOMO approach,50 which is a hybrid scheme that enables one to combine different quantum chemistry methods within the local correlation methodology of refs 12-15. The multilevel CIM framework discussed in this work is simpler than LMOMO. As in the case of the single-level CIM-CC and CIM-MP2 methods,22,23,31-34 multilevel CIM uses orthonormal LMOs, so that we can take advantage of the conventional computer codes developed for orthonormal MO bases; LMOMO relies on nonorthogonal orbitals that complicate the programming effort. We do not have to classify orbital pairs as close, weak, distant, and very distant pairs, and then neglect some pairs while keeping the others, and we do not assign orbital pairs to high- and low-level methods using distance or other criteria, as is done in LMOMO. Instead, we utilize the flexibility of CIM, which allows us to determine the correlation energy contributions corresponding to individual occupied LMOs using different theory levels.
10.1021/jp1038738 2010 American Chemical Society Published on Web 05/24/2010
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J. Phys. Chem. A, Vol. 114, No. 24, 2010
Li and Piecuch
2. Theory and Algorithmic Details 2.1. Overview of the Key Concepts of the CIM-CC and CIM-MP Formalisms. To introduce the multilevel CIM-CC/ MP2 approaches, we recall that the basic idea of all CIM-CC methods and their CIM-MP analogs is the observation that the total correlation energy of a large system or its specific component, such as the triples correction of CCSD(T) or CRCC(2,3), can be obtained as a sum of contributions from the occupied orthonormal LMOs and their respective occupied and unoccupied orbital domains that define the CIM subsystems. If the occupied and unoccupied LMOs, obtained with one of the localization schemes (we use the Boys localization51), are designated by i′, j′, ... and a′, b′, ..., respectively, and if the analogous unprimed symbols label the QCMOs of each CIM subsystem {P}, which are obtained by diagonalizing the occupied-occupied and unoccupied-unoccupied blocks of the Fock matrix in the AO space of {P}, the CC and MPn correlation energies can be calculated as31-34
∆E )
∑ δEi′
δEi′ ) (1/Mi′)
i′
∑ δEi′({Pi′})
{Pi′}
(1) where {Pi′} represents any orbital subsystem {P} that contains the specific occupied LMO φi′ as the so-called central orbital and Mi′ designates the number of subsystems {P} that contain φi′ as a central orbital. The difference between various CIMCC and CIM-MP approaches lies in the definitions of the individual δEi′({Pi′}) contributions in eq 1. In the case of CIM-CCSD,
δEi′({Pi′}) )
∑
fi′ati′a + (1/2)
a∈{Pi′}
∑
ab i′j Vi′j τab≡
j,a
