Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1704.02044

A Stochastic Formulation of the Resolution of Identity: Application to Second Order Møller-Plesset Perturbation Theory

Tyler Y. Takeshita, Wibe Albert de Jong, Daniel Neuhauser, Roi Baer, Eran Rabani

Published Apr 6, 2017• 1 citation
arXivHTMLPDF

Abstract

A stochastic orbital approach to the resolution of identity (RI) approximation for 4-index 2-electron electron repulsion integrals (ERIs) is presented. The stochastic RI-ERIs are then applied to Møller-Plesset perturbation theory (MP2) utilizing a \textit{multiple stochastic orbital approach}. The introduction of multiple stochastic orbitals results in an $N^3$ scaling for both the stochastic RI-ERIs and stochastic RI-MP2. We demonstrate that this method exhibits a small prefactor and an observed scaling of $N^{2.4}$ for a range of water clusters, already outperforming MP2 for clusters with as few as 21 water molecules.

DOI
10.48550/arxiv.1704.02044

Related papers

Efficient parallel algorithm of second‐order Møller–Plesset perturbation theory with resolution‐of‐identity approximation (RI‐MP2)

Michio Katouda, Shigeru Nagase

Abstract An efficient parallel algorithm is developed for second‐order Møller–Plesset perturbation theory with the resolution‐of‐identity approximation of two‐electron repulsion integrals (RI‐MP2) to perform MP2 energy calculations of large molecules on distributed memory processors. Benchmark calculations are carried out for taxol (C47H51NO14), valinomycin (C54H90N6O18), and two‐layer nanographene sheets (C96H24)2, which show the high parallel efficiency of the developed algorithm. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

International Journal of Quantum Chemistry•Jan 1, 2009•71 citations

Range-separated stochastic resolution of identity: Formulation and application to second-order Green’s function theory

Wenjie Dou, Ming Chen, Tyler Y. Takeshita +3 more

We develop a range-separated stochastic resolution of identity (RS-SRI) approach for the four-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic RI and the remaining terms are treated using a SRI. The approach is implemented within a second-order Green’s function formalism with an improved O(N3) scaling with the size of the basis set, N. Moreover, the RS approach greatly reduces the statistical error compared to the full stochastic version [T. Y. Takeshita et al., J. Chem. Phys. 151, 044114 (2019)], resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains and water clusters.

arXiv:2005.12338•The Journal of Chemical Physics•Aug 21, 2020•14 citations

Sparsity of the electron repulsion integral tensor using different localized virtual orbital representations in local second-order Møller–Plesset theory

Zhenling Wang, Abdulrahman Aldossary, Martin Head‐Gordon

Utilizing localized orbitals, local correlation theory can reduce the unphysically high system-size scaling of post-Hartree-Fock (post-HF) methods to linear scaling in insulating molecules. The sparsity of the four-index electron repulsion integral (ERI) tensor is central to achieving this reduction. For second-order Møller-Plesset theory (MP2), one of the simplest post-HF methods, only the (ia|jb) ERIs are needed, coupling occupied orbitals i, j and virtuals a, b. In this paper, we compare the numerical sparsity (called the "ragged list") and two other approaches revealing the low-rank sparsity of the ERI. The ragged list requires only one set of (localized) virtual orbitals, and we find that the orthogonal valence virtual-hard virtual set of virtuals originally proposed by Subotnik et al. gives the sparsest ERI tensor. To further compress the ERI tensor, the pair natural orbital (PNO) type representation uses different sets of virtual orbitals for different occupied orbital pairs, while the occupied-specific virtual (OSV) approach uses different virtuals for each occupied orbital. Our results indicate that while the low-rank PNO representation achieves significant rank reduction, it also requires more memory than the ragged list. The OSV approach requires similar memory to that of the ragged list, but it involves greater algorithmic complexity. An approximation (called the "fixed sparsity pattern") for solving the local MP2 equations using the numerically sparse ERI tensor is proposed and tested to be sufficiently accurate and to have highly controllable error. A low-scaling local MP2 algorithm based on the ragged list and the fixed sparsity pattern is therefore promising.

The Journal of Chemical Physics•Jan 19, 2023•13 citations

Stochastic resolution of identity second-order Matsubara Green’s function theory

Tyler Y. Takeshita, Wenjie Dou, Daniel G. A. Smith +4 more

We develop a stochastic resolution of identity representation to the second-order Matsubara Green’s function (sRI-GF2) theory. Using a stochastic resolution of the Coulomb integrals, the second order Born self-energy in GF2 is decoupled and reduced to matrix products/contractions, which reduces the computational cost from O(N5) to O(N3) (with N being the number of atomic orbitals). The current approach can be viewed as an extension to our previous work on stochastic resolution of identity second order Møller-Plesset perturbation theory [T. Y. Takeshita et al., J. Chem. Theory Comput. 13, 4605 (2017)] and offers an alternative to previous stochastic GF2 formulations [D. Neuhauser et al., J. Chem. Theory Comput. 13, 5396 (2017)]. We show that sRI-GF2 recovers the deterministic GF2 results for small systems, is computationally faster than deterministic GF2 for N > 80, and is a practical approach to describe weak correlations in systems with 103 electrons and more.

The Journal of Chemical Physics•Jul 28, 2019•18 citations

Enabling large-scale correlated electronic structure calculations

Giuseppe M. J. Barca, Jorge Luis Galvez Vallejo, David Poole +4 more

Second-order Møller-Plesset perturbation theory using the Resolution-of-the-Identity approximation (RI-MP2) is a state-of-the-art approach to accurately estimate many-body electronic correlation effects. This is critical for predicting the physicochemical properties of complex molecular systems; however, the scale of these calculations is limited by their extremely high computational cost. In this paper, a novel many-GPU algorithm and implementation of a molecular-fragmentation-based RI-MP2 method are presented that enable correlated calculations on over 180,000 electrons and 45,000 atoms using up to the entire Summit supercomputer in 12 minutes. The implementation demonstrates remarkable speedups with respect to other current GPU and CPU codes, excellent strong scalability on Summit achieving 89.1% parallel efficiency on 4600 nodes, and shows nearly-ideal weak scaling up to 612 nodes. This work makes feasible ab initio correlated quantum chemistry calculations on significantly larger molecular scales than before on both large supercomputing systems and on commodity clusters, with a potential for major impact on progress in chemical, physical, biological and engineering sciences.

International Conference for High Performance Computing, Networking, Storage and Analysis (SC)•Oct 21, 2021•18 citations

Lower-scaling advanced correlation methods with local Resolution of Identity strategies

Arvid Conrad Ihrig

First-principles methods are a valuable tool to study the electronic structure of systems and explore the quantum-mechanical mechanisms driving them on the microscopic scale. Semi-local Density-Functional Approximations (DFAs) are still very popular due to the good balance between accuracy and computational efficiency. However, the instantaneous electronic interactions are described with an averaged potential in these methods, which is often insufficient to describe charge transfers or subtle many-body correlations. A promising way to improve the description of such interactions is the usage of advanced correlation methods which explicitly treat the electronic interactions in a many-body picture. Well known methods form this class are e.g. the Random- Phase Approximation (RPA), 2nd Order Møller-Plesset Perturbation Theory (MP2) and Coupled-Cluster Theory (CC). Unfortunately, the time and memory requirements for all these methods suffer from a steep scaling with system size in their canonical formulations. Several lower-scaling reformulations exist for these methods, but the at least cubic memory requirements and the increased algorithmic complexity prevented any efficient parallelization of these methods so far. In my thesis I developed a new lower-scaling technique for advanced correlation methods to address this challenge. The key components of this technique are the Laplace-Transformation (LT) and a new local Resolution of Identity (RI) strategy, named RI-LVL. Together, they significantly reduce the memory requirement without sacrificing accuracy and allow for an efficient parallelization by means of à priori known sparsity patterns. In combination with established integral screening techniques, the computational scaling can also be reduced considerably. The potential of this new approach is demonstrated at the example of an in-memory LT-MP2 with a MPI/OpenMP hybrid parallelization. It features an at worst cubic computational scaling, a quadratic memory scaling and a high parallel efficiency. Based on the Numeric Atom-Centered Orbital (NAO) framework used in the Fritz Haber Institute ab initio molecular simulations (FHI-aims) code, this implementation can handle both molecular and periodic systems with the same numerical footing. I will also discuss in detail how the combination of the LT and RI- LVL can be applied to RPA calculations. The accuracy and performance of my LT- MP2 is demonstrated in detail for both molecular and periodic systems. In addition, the convergence of the MP2 level adsorption energy of water on a rutile titanium dioxide (TiO2) surface in the low coverage limit is explored. This new LT-MP2 recovers more than 99.9% of the canonical MP2 correlation energy and can handle TiO2 surfaces with up to 270 atoms in the unit cell.

•Jan 1, 2017•0 citations
A stochastic orbital approach to the resolution of identity (RI) approximation for 4-index 2-electron electron repulsion integrals (ERIs) is presented. The stochastic RI-ERIs are then applied to Møller-Plesset perturbation theory (MP2) utilizing a \textit{multiple stochastic orbital approach}. The introduction of multiple stochastic orbitals results in an N3N^3N3 scaling for both the stochastic RI-ERIs and stochastic RI-MP2. We demonstrate that this method exhibits a small prefactor and an observed scaling of N2.4N^{2.4}N2.4 for a range of water clusters, already outperforming MP2 for clusters with as few as 21 water molecules.
Abstract An efficient parallel algorithm is developed for second‐order Møller–Plesset perturbation theory with the resolution‐of‐identity approximation of two‐electron repulsion integrals (RI‐MP2) to perform MP2 energy calculations of large molecules on distributed memory processors. Benchmark calculations are carried out for taxol (C47H51NO14), valinomycin (C54H90N6O18), and two‐layer nanographene sheets (C96H24)2, which show the high parallel efficiency of the developed algorithm. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
We develop a range-separated stochastic resolution of identity (RS-SRI) approach for the four-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic RI and the remaining terms are treated using a SRI. The approach is implemented within a second-order Green’s function formalism with an improved O(N3) scaling with the size of the basis set, N. Moreover, the RS approach greatly reduces the statistical error compared to the full stochastic version [T. Y. Takeshita et al., J. Chem. Phys. 151, 044114 (2019)], resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains and water clusters.
Utilizing localized orbitals, local correlation theory can reduce the unphysically high system-size scaling of post-Hartree-Fock (post-HF) methods to linear scaling in insulating molecules. The sparsity of the four-index electron repulsion integral (ERI) tensor is central to achieving this reduction. For second-order Møller-Plesset theory (MP2), one of the simplest post-HF methods, only the (ia|jb) ERIs are needed, coupling occupied orbitals i, j and virtuals a, b. In this paper, we compare the numerical sparsity (called the "ragged list") and two other approaches revealing the low-rank sparsity of the ERI. The ragged list requires only one set of (localized) virtual orbitals, and we find that the orthogonal valence virtual-hard virtual set of virtuals originally proposed by Subotnik et al. gives the sparsest ERI tensor. To further compress the ERI tensor, the pair natural orbital (PNO) type representation uses different sets of virtual orbitals for different occupied orbital pairs, while the occupied-specific virtual (OSV) approach uses different virtuals for each occupied orbital. Our results indicate that while the low-rank PNO representation achieves significant rank reduction, it also requires more memory than the ragged list. The OSV approach requires similar memory to that of the ragged list, but it involves greater algorithmic complexity. An approximation (called the "fixed sparsity pattern") for solving the local MP2 equations using the numerically sparse ERI tensor is proposed and tested to be sufficiently accurate and to have highly controllable error. A low-scaling local MP2 algorithm based on the ragged list and the fixed sparsity pattern is therefore promising.
We develop a stochastic resolution of identity representation to the second-order Matsubara Green’s function (sRI-GF2) theory. Using a stochastic resolution of the Coulomb integrals, the second order Born self-energy in GF2 is decoupled and reduced to matrix products/contractions, which reduces the computational cost from O(N5) to O(N3) (with N being the number of atomic orbitals). The current approach can be viewed as an extension to our previous work on stochastic resolution of identity second order Møller-Plesset perturbation theory [T. Y. Takeshita et al., J. Chem. Theory Comput. 13, 4605 (2017)] and offers an alternative to previous stochastic GF2 formulations [D. Neuhauser et al., J. Chem. Theory Comput. 13, 5396 (2017)]. We show that sRI-GF2 recovers the deterministic GF2 results for small systems, is computationally faster than deterministic GF2 for N > 80, and is a practical approach to describe weak correlations in systems with 103 electrons and more.
Second-order Møller-Plesset perturbation theory using the Resolution-of-the-Identity approximation (RI-MP2) is a state-of-the-art approach to accurately estimate many-body electronic correlation effects. This is critical for predicting the physicochemical properties of complex molecular systems; however, the scale of these calculations is limited by their extremely high computational cost. In this paper, a novel many-GPU algorithm and implementation of a molecular-fragmentation-based RI-MP2 method are presented that enable correlated calculations on over 180,000 electrons and 45,000 atoms using up to the entire Summit supercomputer in 12 minutes. The implementation demonstrates remarkable speedups with respect to other current GPU and CPU codes, excellent strong scalability on Summit achieving 89.1% parallel efficiency on 4600 nodes, and shows nearly-ideal weak scaling up to 612 nodes. This work makes feasible ab initio correlated quantum chemistry calculations on significantly larger molecular scales than before on both large supercomputing systems and on commodity clusters, with a potential for major impact on progress in chemical, physical, biological and engineering sciences.
First-principles methods are a valuable tool to study the electronic structure of systems and explore the quantum-mechanical mechanisms driving them on the microscopic scale. Semi-local Density-Functional Approximations (DFAs) are still very popular due to the good balance between accuracy and computational efficiency. However, the instantaneous electronic interactions are described with an averaged potential in these methods, which is often insufficient to describe charge transfers or subtle many-body correlations. A promising way to improve the description of such interactions is the usage of advanced correlation methods which explicitly treat the electronic interactions in a many-body picture. Well known methods form this class are e.g. the Random- Phase Approximation (RPA), 2nd Order Møller-Plesset Perturbation Theory (MP2) and Coupled-Cluster Theory (CC). Unfortunately, the time and memory requirements for all these methods suffer from a steep scaling with system size in their canonical formulations. Several lower-scaling reformulations exist for these methods, but the at least cubic memory requirements and the increased algorithmic complexity prevented any efficient parallelization of these methods so far. In my thesis I developed a new lower-scaling technique for advanced correlation methods to address this challenge. The key components of this technique are the Laplace-Transformation (LT) and a new local Resolution of Identity (RI) strategy, named RI-LVL. Together, they significantly reduce the memory requirement without sacrificing accuracy and allow for an efficient parallelization by means of à priori known sparsity patterns. In combination with established integral screening techniques, the computational scaling can also be reduced considerably. The potential of this new approach is demonstrated at the example of an in-memory LT-MP2 with a MPI/OpenMP hybrid parallelization. It features an at worst cubic computational scaling, a quadratic memory scaling and a high parallel efficiency. Based on the Numeric Atom-Centered Orbital (NAO) framework used in the Fritz Haber Institute ab initio molecular simulations (FHI-aims) code, this implementation can handle both molecular and periodic systems with the same numerical footing. I will also discuss in detail how the combination of the LT and RI- LVL can be applied to RPA calculations. The accuracy and performance of my LT- MP2 is demonstrated in detail for both molecular and periodic systems. In addition, the convergence of the MP2 level adsorption energy of water on a rutile titanium dioxide (TiO2) surface in the low coverage limit is explored. This new LT-MP2 recovers more than 99.9% of the canonical MP2 correlation energy and can handle TiO2 surfaces with up to 270 atoms in the unit cell.