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Article
Improved Cluster-in-Molecule Local Correlation Approach for Electron Correlation Calculation of Large Systems Yang Guo, Wei Li, and Shuhua Li J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp501976x • Publication Date (Web): 25 Jun 2014 Downloaded from http://pubs.acs.org on June 30, 2014
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
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The Journal of Physical Chemistry
Improved Cluster-in-molecule Local Correlation Approach for Electron Correlation Calculation of Large Systems Yang Guo, Wei Li, and Shuhua Li* School of Chemistry and Chemical Engineering, Key Laboratory of Mesoscopic Chemistry of MOE, Institute of Theoretical and Computational Chemistry, Nanjing University, Nanjing 210093, P. R. China. Email: [email protected] Tel: +86-25-83686465
Abstract: An improved cluster-in-molecule (CIM) local correlation approach is developed to allow electron correlation calculations of large systems more accurate and faster. We have proposed a refined strategy of constructing virtual LMOs of various clusters, which is suitable for basis sets of various types. To recover medium-range electron correlation, which is important for quantitative descriptions of large systems, we find that a larger distance threshold ( ξ ) is necessary for highly accurate results. Our illustrative calculations show that the present CIM-MP2 (second-order Møller-Plesser perturbation theory, MP2) or CIM-CCSD (coupled cluster singles and doubles, CCSD) scheme with a suitable ξ value is capable of recovering more than 99.8% correlation energies for a wide range of systems at different basis sets. Furthermore, the present CIM-MP2 scheme can provide reliable relative energy differences as the conventional MP2 method for secondary structures of polypeptides.
Keywords: linear scaling, MP2, CCSD, localized molecular orbitals
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1. Introduction Post-Hartree–Fock (Post-HF) methods such as Møller-Plesser perturbation (MP) theory and coupled cluster (CC) theory have been widely used in studying small or medium-sized molecules. However, it is difficult to extend their applications to large molecules, since the computational cost of these methods increases steeply with the system size. For large systems, a number of lower-scaling local correlation algorithms for solving the MP or CC equations have been proposed.1-40 The basic principle of these methods is that the pair correlation energy between localized molecular orbitals (LMOs) decays rapidly with the inter-orbital distance, and only significant pairs should be considered. The first local correlation method was proposed by Pulay and coworkers,1-3 and later generalized by Werner and coworkers.5,6,33-35 In their approach, orthogonal LMOs are used to represent the occupied space and projected atomic orbitals (PAOs) are employed to represent the virtual space. Other kinds of local correlation algorithms have also been developed. Förner and coworkers suggested a LMO-based local correlation ansatz,7,8 in which both occupied and virtual spaces are represented in terms of orthogonal LMOs. Head-Gordon and coworkers proposed an approach, in which both occupied and virtual spaces are represented in terms of nonorthogonal projected orbitals.17-19 Scuseria and Ayala developed an AO-based local correlation approach, in which the MP or CC equations can be directly reformulated in the AO basis.20,21 Since 2009, Neese and coworkers have also proposed a series of local pair natural orbital (LPNO) based CEPA, CCSD, and CCSD(T) methods for more realistic systems,36-39 in which the virtual space is
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The Journal of Physical Chemistry
represented by pair natural orbitals. Based on the divide-and-conquer method, Nakai and coworkers also developed a series of linear scaling local correlation methods at various theoretical levels.26,28,40 It should be mentioned that the linear-scaling atomic-orbital based MP2 method developed by Ochsenfeld and coworkers can give very accurate correlation energies for very large systems through rigorous screening schemes of two-electron integrals.22-25 However, the related techniques are not directly applicable for simplifying CC calculations. However, the original LMO-based local correlation approach was not successful,7,8 due to the low accuracy of this approach. Following this direction, we have proposed a significantly refined approach, cluster-in-molecule (CIM) local correlation approach, in 2002.9 The accuracy of the CIM approach is competitive with other local correlation approaches. In the CIM approach, the correlation energy of a large system is expressed as the summation of the correlation contributions from all occupied LMOs. The correlation energy of each occupied LMO can be obtained approximately from solving the MP or CC equation of a small cluster, which consists of a subset of occupied and virtual LMOs. Since the correlation calculations on different clusters can be carried out independently, the CIM approach can achieve very high parallelization efficiency, and thus the wall-time of the CIM approach can be lower than most of other types of local correlation approaches mentioned above for sufficiently large systems. During the recent years, many further developments within the CIM framework have been made. For example, with the concept of quasi-canonical MOs (QCMOs),
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the LMO-based MP equations can be directly solved without iteration, which allows CIM-MP2 and CIM-CCSD(T) calculations faster and applicable for quite large systems.12 A multi-level extension of the CIM approach, in which different orbital groups can be treated with different electron correlation methods, was also suggested.41 In order to further improve the accuracy of the CIM approach for large systems, we recently suggested a different strategy for constructing clusters.11 The main idea is to build some two-atom-centered clusters, from which the medium-range correlation effect (being critical for very large systems) can be taken into account. With this modification, we found that the CIM approach can provide accurate predictions on conformational energy differences of large systems. In addition, Jørgensen and coworkers developed the divide-expend-consolidate (DEC) local correlation algorithm.15,16,42-44 In their approach, the total correlation energy is also expanded as the summation of individual terms. Each term can be obtained from post-HF calculation on an ‘atomic fragment’ (or a dimer of two atomic fragments), which is very much like a cluster in the CIM approach. In 2011, Kallay and coworkers developed a local pair natural orbital (LNO) CIM (LNO-CIM) approach.45 Recently, they further developed a two-level domain construction algorithm to achieve linear scaling.46 The main difference between their recent LNO-CIM approach and the previous CIM approach is that the LNOs calculated from the MP2 density matrix within a given MO domain are used to construct a set of occupied and virtual LMOs for subsequent CC calculations. It should be mentioned that the incremental scheme developed by Stoll and
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coworkers47,48 is another local correlation method that represents both occupied and virtual spaces with orthogonal LMOs. In the incremental method, the correlation energy of a system can be considered as many-body expansion of various building blocks. Recently, Friedrich and coworkers13,14 developed a fully automated incremental scheme for molecules. They have demonstrated that for a variety of medium-sized systems this approach can reproduce the conventional correlation energies very well at various theory levels if the incremental series is truncated in a proper way. In this work, our purpose is to propose a more effective strategy for constructing clusters so that the applicability and accuracy of the CIM approach can be enhanced. In the latest CIM scheme, some two-atom-centered clusters (in addition to one-atom-centered clusters) are constructed to recover more medium-range electron correlation. However, with the QCMOs, we need to do a two-step index transformation from QCMO-based amplitudes to extract pair correlation energies between two LMOs. Such transformation within each cluster is time consuming for quite large clusters. To avoid this problem, we will only build one-atom-centered clusters, and increase the size of clusters with a larger distance threshold. More importantly, to extend the CIM approach to large basis sets (especially with diffuse functions),49 a more effective way of constructing virtual LMOs of various clusters will be described. This paper is organized as follows. In section 2, the basic idea of the CIM approach and a new way of constructing LMO-based clusters will be presented. In
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section 3, the applicability of the present CIM scheme is validated, and illustrative applications of this CIM approach are reported. Finally, a brief summary is given in section 4.
2. Methodology The total correlation energy of a system at the MP2 or CCSD level can be expressed as the summation of the correlation energies of all occupied LMOs: occ
Ecor = ∑ ∆Ei .
(1)
i
At the MP2 level, ∆Ei =
1 ∑ < ij || ab > tijab . 2 j ,a (tijab + tiat bj − t ajtib ) . 2 j, a
Article
Improved Cluster-in-Molecule Local Correlation Approach for Electron Correlation Calculation of Large Systems Yang Guo, Wei Li, and Shuhua Li J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp501976x • Publication Date (Web): 25 Jun 2014 Downloaded from http://pubs.acs.org on June 30, 2014
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 38
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The Journal of Physical Chemistry
Improved Cluster-in-molecule Local Correlation Approach for Electron Correlation Calculation of Large Systems Yang Guo, Wei Li, and Shuhua Li* School of Chemistry and Chemical Engineering, Key Laboratory of Mesoscopic Chemistry of MOE, Institute of Theoretical and Computational Chemistry, Nanjing University, Nanjing 210093, P. R. China. Email: [email protected] Tel: +86-25-83686465
Abstract: An improved cluster-in-molecule (CIM) local correlation approach is developed to allow electron correlation calculations of large systems more accurate and faster. We have proposed a refined strategy of constructing virtual LMOs of various clusters, which is suitable for basis sets of various types. To recover medium-range electron correlation, which is important for quantitative descriptions of large systems, we find that a larger distance threshold ( ξ ) is necessary for highly accurate results. Our illustrative calculations show that the present CIM-MP2 (second-order Møller-Plesser perturbation theory, MP2) or CIM-CCSD (coupled cluster singles and doubles, CCSD) scheme with a suitable ξ value is capable of recovering more than 99.8% correlation energies for a wide range of systems at different basis sets. Furthermore, the present CIM-MP2 scheme can provide reliable relative energy differences as the conventional MP2 method for secondary structures of polypeptides.
Keywords: linear scaling, MP2, CCSD, localized molecular orbitals
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1. Introduction Post-Hartree–Fock (Post-HF) methods such as Møller-Plesser perturbation (MP) theory and coupled cluster (CC) theory have been widely used in studying small or medium-sized molecules. However, it is difficult to extend their applications to large molecules, since the computational cost of these methods increases steeply with the system size. For large systems, a number of lower-scaling local correlation algorithms for solving the MP or CC equations have been proposed.1-40 The basic principle of these methods is that the pair correlation energy between localized molecular orbitals (LMOs) decays rapidly with the inter-orbital distance, and only significant pairs should be considered. The first local correlation method was proposed by Pulay and coworkers,1-3 and later generalized by Werner and coworkers.5,6,33-35 In their approach, orthogonal LMOs are used to represent the occupied space and projected atomic orbitals (PAOs) are employed to represent the virtual space. Other kinds of local correlation algorithms have also been developed. Förner and coworkers suggested a LMO-based local correlation ansatz,7,8 in which both occupied and virtual spaces are represented in terms of orthogonal LMOs. Head-Gordon and coworkers proposed an approach, in which both occupied and virtual spaces are represented in terms of nonorthogonal projected orbitals.17-19 Scuseria and Ayala developed an AO-based local correlation approach, in which the MP or CC equations can be directly reformulated in the AO basis.20,21 Since 2009, Neese and coworkers have also proposed a series of local pair natural orbital (LPNO) based CEPA, CCSD, and CCSD(T) methods for more realistic systems,36-39 in which the virtual space is
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The Journal of Physical Chemistry
represented by pair natural orbitals. Based on the divide-and-conquer method, Nakai and coworkers also developed a series of linear scaling local correlation methods at various theoretical levels.26,28,40 It should be mentioned that the linear-scaling atomic-orbital based MP2 method developed by Ochsenfeld and coworkers can give very accurate correlation energies for very large systems through rigorous screening schemes of two-electron integrals.22-25 However, the related techniques are not directly applicable for simplifying CC calculations. However, the original LMO-based local correlation approach was not successful,7,8 due to the low accuracy of this approach. Following this direction, we have proposed a significantly refined approach, cluster-in-molecule (CIM) local correlation approach, in 2002.9 The accuracy of the CIM approach is competitive with other local correlation approaches. In the CIM approach, the correlation energy of a large system is expressed as the summation of the correlation contributions from all occupied LMOs. The correlation energy of each occupied LMO can be obtained approximately from solving the MP or CC equation of a small cluster, which consists of a subset of occupied and virtual LMOs. Since the correlation calculations on different clusters can be carried out independently, the CIM approach can achieve very high parallelization efficiency, and thus the wall-time of the CIM approach can be lower than most of other types of local correlation approaches mentioned above for sufficiently large systems. During the recent years, many further developments within the CIM framework have been made. For example, with the concept of quasi-canonical MOs (QCMOs),
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the LMO-based MP equations can be directly solved without iteration, which allows CIM-MP2 and CIM-CCSD(T) calculations faster and applicable for quite large systems.12 A multi-level extension of the CIM approach, in which different orbital groups can be treated with different electron correlation methods, was also suggested.41 In order to further improve the accuracy of the CIM approach for large systems, we recently suggested a different strategy for constructing clusters.11 The main idea is to build some two-atom-centered clusters, from which the medium-range correlation effect (being critical for very large systems) can be taken into account. With this modification, we found that the CIM approach can provide accurate predictions on conformational energy differences of large systems. In addition, Jørgensen and coworkers developed the divide-expend-consolidate (DEC) local correlation algorithm.15,16,42-44 In their approach, the total correlation energy is also expanded as the summation of individual terms. Each term can be obtained from post-HF calculation on an ‘atomic fragment’ (or a dimer of two atomic fragments), which is very much like a cluster in the CIM approach. In 2011, Kallay and coworkers developed a local pair natural orbital (LNO) CIM (LNO-CIM) approach.45 Recently, they further developed a two-level domain construction algorithm to achieve linear scaling.46 The main difference between their recent LNO-CIM approach and the previous CIM approach is that the LNOs calculated from the MP2 density matrix within a given MO domain are used to construct a set of occupied and virtual LMOs for subsequent CC calculations. It should be mentioned that the incremental scheme developed by Stoll and
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coworkers47,48 is another local correlation method that represents both occupied and virtual spaces with orthogonal LMOs. In the incremental method, the correlation energy of a system can be considered as many-body expansion of various building blocks. Recently, Friedrich and coworkers13,14 developed a fully automated incremental scheme for molecules. They have demonstrated that for a variety of medium-sized systems this approach can reproduce the conventional correlation energies very well at various theory levels if the incremental series is truncated in a proper way. In this work, our purpose is to propose a more effective strategy for constructing clusters so that the applicability and accuracy of the CIM approach can be enhanced. In the latest CIM scheme, some two-atom-centered clusters (in addition to one-atom-centered clusters) are constructed to recover more medium-range electron correlation. However, with the QCMOs, we need to do a two-step index transformation from QCMO-based amplitudes to extract pair correlation energies between two LMOs. Such transformation within each cluster is time consuming for quite large clusters. To avoid this problem, we will only build one-atom-centered clusters, and increase the size of clusters with a larger distance threshold. More importantly, to extend the CIM approach to large basis sets (especially with diffuse functions),49 a more effective way of constructing virtual LMOs of various clusters will be described. This paper is organized as follows. In section 2, the basic idea of the CIM approach and a new way of constructing LMO-based clusters will be presented. In
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section 3, the applicability of the present CIM scheme is validated, and illustrative applications of this CIM approach are reported. Finally, a brief summary is given in section 4.
2. Methodology The total correlation energy of a system at the MP2 or CCSD level can be expressed as the summation of the correlation energies of all occupied LMOs: occ
Ecor = ∑ ∆Ei .
(1)
i
At the MP2 level, ∆Ei =
1 ∑ < ij || ab > tijab . 2 j ,a (tijab + tiat bj − t ajtib ) . 2 j, a
