Linear scaling algorithms for electronic structure calculations of large systems

    Traditional quantum chemistry methods are only applicable to small and medium-sized systems, due to the fact that their computational cost increases rapidly with the system size. We have developed two efficient linear scaling methods, which can extend ab initio quantum chemistry calculations to very large systems. 
     (1) The cluster-in-molecule (CIM) local correlation approach, which can extend electron correlation calculations (like MP2 and CCSD) to very large molecules [1,2]. The main idea of this approach is to decouple the CC (or MP) equations of the whole system into a series of CC (or MP) equations corresponding to some small clusters (a subset of LMOs), by using orthonormal LMOs for both occupied and virtual spaces. The total correlation energy is then obtained as a sum of the individual contributions from electron correlation calculations of various cluster. The most important advantage of this approach is that electron correlation calculations on different clusters can be carried out independently on different computer nodes. This feature allows the CIM method to be applicable to very large systems as long as electron correlation calculation of each cluster is computationally accessible. The CIM method has been recognized to be one of the most promising local correlation approaches for large systems, and has been extended and generalized by many groups in other countries. 
     (2) The energy-based fragmentation approach, in which the ground-state total energy of a large molecule can be directly evaluated as linear combination of ground-state energies of a series of small subsystems [3] or “electrostatically embedded” subsystems [4]. The latter approach, named as the generalized energy-based fragmentation (GEBF) approach, can be easily implemented at various theoretical levels (HF, DFT, and so on) with existing ab initio programs. With this approach, one can now perform full ab initio quantum chemistry calculations for systems with thousands of atoms on ordinary workstations, providing accurate descriptions on ground-state energies and properties, optimized geometries, and vibrational spectra [5]. This approach has been established to be a practical theoretical tool for full quantum mechanical calculations of very large systems, which are much beyond the reach of traditional quantum chemistry methods. The recent developments and applications of this approach can be found in a recent review [6].
      Very recently, the GEBF approach was extended to molecular crystals [7] under periodic boundary condition (PBC). Illustrative applications demonstrate that the PBC-GEBF method with explicitly correlated quantum chemistry methods is capable of providing accurate descriptions on the lattice energies and structures for various types of molecular crystals.

Representative Publications
[1] Li, S.*, Ma J. and Jiang Y. "Linear Scaling Local Correlation Approach for Solving the Coupled Cluster Equations of Large Systems" J. Comput. Chem. 2002, 23, 237. (Times cited: 85)
[2] Li, S.*, Shen, J., Li, W., Jiang Y. “An efficient implementation of the ‘cluster-in-molecule’ approach for local electron correlation calculations” J. Chem. Phys. 2006, 125, 074109. (Times cited: 62)
[3] Li, S.*, Li, W. and Fang, T. “An efficient fragment-based approach for predicting the ground-state energies and structures of large molecules”J. Am. Chem. Soc. 2005, 127, 7215. (Times cited: 126)
[4] Li, W., Li, S.*, Jiang Y. “Generalized energy-based fragmentation approach for computing the ground-state energies and properties of large molecules” J. Phys. Chem. A 2007, 111, 2193. (Times cited: 106)
[5] Hua, W., Fang, T., Li, W., Yu, J.G., Li, S.* “Geometry optimizations and vibrational spectra of large molecules from a generalized energy-based fragmentation approach” J. Phys. Chem. A 2008, 112, 10864. (Times cited: 56)
[6] Li S.*; Li W.; Ma J. "Generalized Energy-Based Fragmentation Approach and Its Applications to Macromolecules and Molecular Aggregates" Acc. Chem. Res. 2014, 47, 2712.
[7] Fang, T.; Li, W.; Gu, F.; Li, S.* "Accurate Prediction of Lattice Energies and Structures of Molecular Crystals with Molecular Quantum Chemistry Methods" J. Chem. Theory Comput. 2014, DOI: 10.1021/ct500833k.


Accurate electron correlation methods

    We have proposed two types of multi-reference electron correlation methods and two types of single-reference coupled cluster (CC) methods for systems with multi-reference character, which cannot be accurately described by traditional electron correlation methods.
     (1) Multi-reference electron correlation methods. In order to overcome the difficulty that the traditional single-reference CC methods are only applicable to molecules at their equilibrium structures, we have proposed the block-correlated coupled cluster (BCCC) method [1]. In this method, all orbitals in a system are divided into blocks and the tensor products of block states are used as N-electron basis sets. BCCC provides a novel electron correlation framework, which is directly applicable to molecular electronic states with quasi-degenerate orbitals. Depending on how blocks are defined, BCCC can evolve into CCSD, CAS-BCCC, etc. With the CASSCF reference wave function and the cluster operator truncated up to the four-block correlation level, the method is defined as CAS-BCCC4, which is free of the intruder-state problem, and nearly size-extensive [2]. The CAS-BCCC4 method can provide accurate descriptions for bond-breaking PESs, singlet-triplet gaps in diradicals, reaction barriers, and so on [3].
Another multi-reference method developed by our group is called as the block correlated second-order perturbation theory based on the generalized valence bond wave function (GVB-BCPT2) [4]. The method is computationally similar to the MP2 method, and is strictly size consistent. It has noticeably better performance than MP2 for systems with significant multi-reference character, and is applicable for some systems requiring large active spaces, which are beyond the capability of all CASSCF-based methods.
     (2) Single-reference CC approaches. The CCSD(T) method is considered to be the most accurate method for ab initio calculations. However, it is not applicable for describing bond-breaking processes and diradical systems. To go beyond the CCSD(T) method, we have developed a hybrid CC method, CCSD(T)-h [5]. In this approach, single and double excitations are treated at the CCSD level, but triple excitations are treated in a hybrid manner. With the concept of active orbitals, triple excitations can be divided into two subsets (“active” and “inactive”). The amplitudes of these two classes of triple excitations are obtained via solving the CCSDt and CCSD(T) equations, respectively. Another different approach is a new hierarchy of CC methods based on orbital pair excitations, in which orbitals are first grouped into pairs and then the cluster operator is truncated up to excitations involving several orbital pairs [6]. For many systems with significant multi-reference character, the overall performance of CCSD(T)-h is very close to that of the computationally expensive CCSDT, and much better than that of CCSD(T). The CC methods based on orbital pair excitations are much more accurate than the traditional CC methods with comparable computational costs.

Representative Publications
[1] Li, S. “Block-correlated coupled cluster theory: The general formulation and its application to the antiferromagnetic Heisenberg model” J. Chem. Phys. 2004, 120, 5017. (Times cited: 36)
[2] Fang, T., Li, S.* “Block correlated coupled cluster theory with a CASSCF reference function: The formulation and test applications for single bond breaking” J. Chem. Phys. 2007, 127, 204108. (Times cited: 32)
[3] Fang, T., Shen, J., Li, S.* “Block correlated coupled cluster method with a complete active-space self-consistent-field reference function: the formula for general active spaces and its applications for multi-bond breaking systems”. J. Chem. Phys. 2008, 128, 224107. (Times cited: 27)
[4] Xu, E.; Li, S* "Block correlated second order perturbation theory with a generalized valence bond reference function." J. Chem. Phys. 2013, 139, 174111. (Times cited: 2)
[5] Shen, J., Xu, E., Kou, Z., Li, S.* “A coupled cluster approach with a hybrid treatment of connected triple excitations for bond-breaking potential energy surfaces” J. Chem. Phys. 2010, 132, 114115. (Times cited: 14)
[6] Xu, E., Shen, J., Kou, Z., Li, S.* “Coupled cluster with singles, doubles, and partial higher-order excitations based on the corresponding orbitals: The formulation and test applications for bond breaking processes” J. Chem. Phys. 2010, 132, 134110. (Times cited: 11)

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