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Subramanian V, Das S, Gavini V. Tucker Tensor Approach for Accelerating Fock Exchange Computations in a Real-Space Finite-Element Discretization of Generalized Kohn-Sham Density Functional Theory. J Chem Theory Comput 2024; 20:3566-3579. [PMID: 38661129 DOI: 10.1021/acs.jctc.4c00019] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/26/2024]
Abstract
The evaluation of Fock exchange is often the computationally most expensive part of hybrid functional density functional theory calculations in a systematically improvable, complete basis. In this work, we employ a Tucker tensor based approach that substantially accelerates the evaluation of the action of Fock exchange by transforming three-dimensional convolutional integrals into a tensor product of one-dimensional convolution integrals. Our numerical implementation uses a parallelization strategy that balances the memory and communication bottlenecks, alongside overlapping compute and communication operations to enhance computational efficiency and parallel scalability. The accuracy and computational efficiency are demonstrated on various systems, including Pt clusters of various sizes and a TiO2 cluster with 3684 electrons.
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Affiliation(s)
- Vishal Subramanian
- Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Sambit Das
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Vikram Gavini
- Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States
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2
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Corzo HH, Hillers-Bendtsen AE, Barnes A, Zamani AY, Pawłowski F, Olsen J, Jørgensen P, Mikkelsen KV, Bykov D. Corrigendum: Coupled cluster theory on modern heterogeneous supercomputers. Front Chem 2023; 11:1256510. [PMID: 37654900 PMCID: PMC10466216 DOI: 10.3389/fchem.2023.1256510] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2023] [Accepted: 07/11/2023] [Indexed: 09/02/2023] Open
Abstract
[This corrects the article DOI: 10.3389/fchem.2023.1154526.].
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Affiliation(s)
| | | | | | - Abdulrahman Y. Zamani
- Department of Chemistry and Biochemistry and Center for Chemical Computation and Theory, University of California, Merced, CA, United States
| | - Filip Pawłowski
- Department of Chemistry and Biochemistry, Auburn University, Auburn, AL, United States
| | - Jeppe Olsen
- Department of Chemistry, Aarhus University, Aarhus, Denmark
| | - Poul Jørgensen
- Department of Chemistry, Aarhus University, Aarhus, Denmark
| | - Kurt V. Mikkelsen
- Department of Chemistry, University of Copenhagen, Copenhagen, Denmark
| | - Dmytro Bykov
- Oak Ridge National Laboratory, Oak Ridge, TN, United States
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3
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Corzo HH, Hillers-Bendtsen AE, Barnes A, Zamani AY, Pawłowski F, Olsen J, Jørgensen P, Mikkelsen KV, Bykov D. Coupled cluster theory on modern heterogeneous supercomputers. Front Chem 2023; 11:1154526. [PMID: 37388945 PMCID: PMC10303140 DOI: 10.3389/fchem.2023.1154526] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2023] [Accepted: 05/11/2023] [Indexed: 07/01/2023] Open
Abstract
This study examines the computational challenges in elucidating intricate chemical systems, particularly through ab-initio methodologies. This work highlights the Divide-Expand-Consolidate (DEC) approach for coupled cluster (CC) theory-a linear-scaling, massively parallel framework-as a viable solution. Detailed scrutiny of the DEC framework reveals its extensive applicability for large chemical systems, yet it also acknowledges inherent limitations. To mitigate these constraints, the cluster perturbation theory is presented as an effective remedy. Attention is then directed towards the CPS (D-3) model, explicitly derived from a CC singles parent and a doubles auxiliary excitation space, for computing excitation energies. The reviewed new algorithms for the CPS (D-3) method efficiently capitalize on multiple nodes and graphical processing units, expediting heavy tensor contractions. As a result, CPS (D-3) emerges as a scalable, rapid, and precise solution for computing molecular properties in large molecular systems, marking it an efficient contender to conventional CC models.
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Affiliation(s)
| | | | | | - Abdulrahman Y. Zamani
- Department of Chemistry and Biochemistry and Center for Chemical Computation and Theory, University of California, Merced, CA, United States
| | - Filip Pawłowski
- Department of Chemistry and Biochemistry, Auburn University, Auburn, AL, United States
| | - Jeppe Olsen
- Department of Chemistry, Aarhus University, Aarhus, Denmark
| | - Poul Jørgensen
- Department of Chemistry, Aarhus University, Aarhus, Denmark
| | - Kurt V. Mikkelsen
- Department of Chemistry, University of Copenhagen, Copenhagen, Denmark
| | - Dmytro Bykov
- Oak Ridge National Laboratory, Oak Ridge, TN, United States
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4
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Woo J, Kim WY, Choi S. System-Specific Separable Basis Based on Tucker Decomposition: Application to Density Functional Calculations. J Chem Theory Comput 2022; 18:2875-2884. [PMID: 35437014 PMCID: PMC9098162 DOI: 10.1021/acs.jctc.1c01263] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
![]()
For fast density
functional calculations, a suitable basis that
can accurately represent the orbitals within a reasonable number of
dimensions is essential. Here, we propose a new type of basis constructed
from Tucker decomposition of a finite-difference (FD) Hamiltonian
matrix, which is intended to reflect the system information implied
in the Hamiltonian matrix and satisfies orthonormality and separability
conditions. By introducing the system-specific separable basis, the
computation time for FD density functional calculations for seven
two- and three-dimensional periodic systems was reduced by a factor
of 2–71 times, while the errors in both the atomization energy
per atom and the band gap were limited to less than 0.1 eV. The accuracy
and speed of the density functional calculations with the proposed
basis can be systematically controlled by adjusting the rank size
of Tucker decomposition.
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Affiliation(s)
- Jeheon Woo
- Department of Chemistry, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
| | - Woo Youn Kim
- Department of Chemistry, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
| | - Sunghwan Choi
- National Institute of Supercomputing and Networking, Korea Institute of Science and Technology Information, Daejeon 34141, Republic of Korea
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5
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Poirier B, Jerke J. Full-dimensional Schrödinger wavefunction calculations using tensors and quantum computers: the Cartesian component-separated approach. Phys Chem Chem Phys 2022; 24:4437-4454. [PMID: 35113096 DOI: 10.1039/d1cp02036f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Traditional methods in quantum chemistry rely on Hartree-Fock-based Slater-determinant (SD) representations, whose underlying zeroth-order picture assumes separability by particle. Here, we explore a radically different approach, based on separability by Cartesian component, rather than by particle [J. Jerke and B. Poirier, J. Chem. Phys., 2018, 148, 104101]. The approach appears to be very well suited for 3D grid-based methods in quantum chemistry, and thereby also for so-called "first-quantized" quantum computing. We first present an overview of the approach as implemented on classical computers, including numerical results that justify performance claims. In particular, we perform numerical calculations with four explicit electrons that are equivalent to full-CI matrix diagonalization with nearly 1015 SDs. We then present an implementation for quantum computers for which the number of quantum gates (and to a lesser extent, the number of qubits) can be dramatically reduced, in comparison with other quantum circuitry that has been envisioned for implementing first-quantized "quantum computational chemistry" (QCC).
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Affiliation(s)
- Bill Poirier
- Department of Chemistry and Biochemistry and Department of Physics, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA.
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6
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Albareda G, Lively K, Sato SA, Kelly A, Rubio A. Conditional Wave Function Theory: A Unified Treatment of Molecular Structure and Nonadiabatic Dynamics. J Chem Theory Comput 2021; 17:7321-7340. [PMID: 34752108 PMCID: PMC8675140 DOI: 10.1021/acs.jctc.1c00772] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2021] [Indexed: 11/28/2022]
Abstract
We demonstrate that a conditional wave function theory enables a unified and efficient treatment of the equilibrium structure and nonadiabatic dynamics of correlated electron-ion systems. The conditional decomposition of the many-body wave function formally recasts the full interacting wave function of a closed system as a set of lower-dimensional (conditional) coupled "slices". We formulate a variational wave function ansatz based on a set of conditional wave function slices and demonstrate its accuracy by determining the structural and time-dependent response properties of the hydrogen molecule. We then extend this approach to include time-dependent conditional wave functions and address paradigmatic nonequilibrium processes including strong-field molecular ionization, laser-driven proton transfer, and nuclear quantum effects induced by a conical intersection. This work paves the road for the application of conditional wave function theory in equilibrium and out-of-equilibrium ab initio molecular simulations of finite and extended systems.
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Affiliation(s)
- Guillermo Albareda
- Nano-Bio
Spectroscopy Group and European Theoretical Spectroscopy Facility
(ETSF), Universidad del País Vasco
(UPV/EHU), Av. Tolosa
72, 20018 San Sebastian, Spain
- Institute
of Theoretical and Computational Chemistry, University of Barcelona, Martí i Franquès 1-11, 08028 Barcelona, Spain
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Kevin Lively
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Centre for Ultrafast Imaging, University
of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Shunsuke A. Sato
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- Center
for Computational Sciences, University of
Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan
| | - Aaron Kelly
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Centre for Ultrafast Imaging, University
of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
- Department
of Chemistry, Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada
| | - Angel Rubio
- Nano-Bio
Spectroscopy Group and European Theoretical Spectroscopy Facility
(ETSF), Universidad del País Vasco
(UPV/EHU), Av. Tolosa
72, 20018 San Sebastian, Spain
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Centre for Ultrafast Imaging, University
of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
- Center
for Computational Quantum Physics (CCQ), Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, United
States
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7
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Madsen NK, Jensen RB, Christiansen O. Calculating vibrational excitation energies using tensor-decomposed vibrational coupled-cluster response theory. J Chem Phys 2021; 154:054113. [DOI: 10.1063/5.0037240] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Affiliation(s)
- Niels Kristian Madsen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
| | - Rasmus Berg Jensen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
| | - Ove Christiansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
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8
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Ahmadi-Asl S, Cichocki A, Huy Phan A, Asante-Mensah MG, Musavian Ghazani M, Tanaka T, Oseledets I. Randomized algorithms for fast computation of low rank tensor ring model. MACHINE LEARNING: SCIENCE AND TECHNOLOGY 2020. [DOI: 10.1088/2632-2153/abad87] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
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9
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Jerke J, Karwowski J, Poirier B. Exact matrix elements for general two-body central-force interactions, expressed as sums of products. Mol Phys 2019. [DOI: 10.1080/00268976.2018.1538541] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
Affiliation(s)
- Jonathan Jerke
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Lubbock, TX, USA
| | - Jacek Karwowski
- Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University, Toruń, Poland
| | - Bill Poirier
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Lubbock, TX, USA
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10
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Schmitz G, Christiansen O. Assessment of the overlap metric in the context of RI-MP2 and atomic batched tensor decomposed MP2. Chem Phys Lett 2018. [DOI: 10.1016/j.cplett.2018.04.014] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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11
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Jerke J, Poirier B. Two-body Schrödinger wave functions in a plane-wave basis via separation of dimensions. J Chem Phys 2018; 148:104101. [DOI: 10.1063/1.5017621] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jonathan Jerke
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
- Department of Physics, Texas Southern University, Houston, Texas 77004, USA
| | - Bill Poirier
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
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12
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Posada E, Moncada F, Reyes A. The any particle molecular orbital grid-based Hartree-Fock (APMO-GBHF) approach. J Chem Phys 2018; 148:084113. [PMID: 29495795 DOI: 10.1063/1.5012521] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Edwin Posada
- Departament of Chemistry, Universidad Nacional de Colombia, Ave. Cra. 30, #45-03 Bogotá, Colombia
| | - Félix Moncada
- Departament of Chemistry, Universidad Nacional de Colombia, Ave. Cra. 30, #45-03 Bogotá, Colombia
| | - Andrés Reyes
- Departament of Chemistry, Universidad Nacional de Colombia, Ave. Cra. 30, #45-03 Bogotá, Colombia
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13
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Madsen NK, Godtliebsen IH, Losilla SA, Christiansen O. Tensor-decomposed vibrational coupled-cluster theory: Enabling large-scale, highly accurate vibrational-structure calculations. J Chem Phys 2018; 148:024103. [DOI: 10.1063/1.5001569] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
| | | | | | - Ove Christiansen
- Department of Chemistry, Aarhus University, 8000 Aarhus C, Denmark
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14
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Wendin G. Quantum information processing with superconducting circuits: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:106001. [PMID: 28682303 DOI: 10.1088/1361-6633/aa7e1a] [Citation(s) in RCA: 107] [Impact Index Per Article: 15.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for near-future useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of quantum supremacy with fifty qubits is anticipated in just a few years. Quantum supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classical-quantum computing systems are already emerging that can be used for software development and experimentation, even via web interfaces. Therefore, the time is ripe for describing some of the recent development of superconducting devices, systems and applications. As such, the discussion of superconducting qubits and circuits is limited to devices that are proven useful for current or near future applications. Consequently, the centre of interest is the practical applications of QIP, such as computation and simulation in Physics and Chemistry.
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Affiliation(s)
- G Wendin
- Department of Microtechnology and Nanoscience-MC2, Chalmers University of Technology, SE-41296 Gothenburg, Sweden
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15
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Manthe U. Wavepacket dynamics and the multi-configurational time-dependent Hartree approach. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:253001. [PMID: 28430111 DOI: 10.1088/1361-648x/aa6e96] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
Multi-configurational time-dependent Hartree (MCTDH) based approaches are efficient, accurate, and versatile methods for high-dimensional quantum dynamics simulations. Applications range from detailed investigations of polyatomic reaction processes in the gas phase to high-dimensional simulations studying the dynamics of condensed phase systems described by typical solid state physics model Hamiltonians. The present article presents an overview of the different areas of application and provides a comprehensive review of the underlying theory. The concepts and guiding ideas underlying the MCTDH approach and its multi-mode and multi-layer extensions are discussed in detail. The general structure of the equations of motion is highlighted. The representation of the Hamiltonian and the correlated discrete variable representation (CDVR), which provides an efficient multi-dimensional quadrature in MCTDH calculations, are discussed. Methods which facilitate the calculation of eigenstates, the evaluation of correlation functions, and the efficient representation of thermal ensembles in MCTDH calculations are described. Different schemes for the treatment of indistinguishable particles in MCTDH calculations and recent developments towards a unified multi-layer MCTDH theory for systems including bosons and fermions are discussed.
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Affiliation(s)
- Uwe Manthe
- Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstr. 25, D-33615 Bielefeld, Germany
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16
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Schmitz G, Madsen NK, Christiansen O. Atomic-batched tensor decomposed two-electron repulsion integrals. J Chem Phys 2017; 146:134112. [DOI: 10.1063/1.4979571] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Gunnar Schmitz
- Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark
| | | | - Ove Christiansen
- Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark
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17
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Rai P, Sargsyan K, Najm H, Hermes MR, Hirata S. Low-rank canonical-tensor decomposition of potential energy surfaces: application to grid-based diagrammatic vibrational Green's function theory. Mol Phys 2017. [DOI: 10.1080/00268976.2017.1288937] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
- Prashant Rai
- Sandia National Laboratories, Livermore, CA, USA
| | | | - Habib Najm
- Sandia National Laboratories, Livermore, CA, USA
| | - Matthew R. Hermes
- Department of Chemistry, University of Illinois at Urbana-Champaign, Urbana, IL, USA
| | - So Hirata
- Department of Chemistry, University of Illinois at Urbana-Champaign, Urbana, IL, USA
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18
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Parkkinen P, Losilla SA, Solala E, Toivanen EA, Xu WH, Sundholm D. A Generalized Grid-Based Fast Multipole Method for Integrating Helmholtz Kernels. J Chem Theory Comput 2017; 13:654-665. [PMID: 28094984 DOI: 10.1021/acs.jctc.6b01207] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
A grid-based fast multipole method (GB-FMM) for optimizing three-dimensional (3D) numerical molecular orbitals in the bubbles and cube double basis has been developed and implemented. The present GB-FMM method is a generalization of our recently published GB-FMM approach for numerically calculating electrostatic potentials and two-electron interaction energies. The orbital optimization is performed by integrating the Helmholtz kernel in the double basis. The steep part of the functions in the vicinity of the nuclei is represented by one-center bubbles functions, whereas the remaining cube part is expanded on an equidistant 3D grid. The integration of the bubbles part is treated by using one-center expansions of the Helmholtz kernel in spherical harmonics multiplied with modified spherical Bessel functions of the first and second kind, analogously to the numerical inward and outward integration approach for calculating two-electron interaction potentials in atomic structure calculations. The expressions and algorithms for massively parallel calculations on general purpose graphics processing units (GPGPU) are described. The accuracy and the correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations (HF-SCF) on H2, H2O, and CO. Our calculations show that an accuracy of 10-4 to 10-7 Eh can be reached in HF-SCF calculations on general molecules.
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Affiliation(s)
- Pauli Parkkinen
- Department of Chemistry, University of Helsinki , P.O. Box 55, A. I. Virtanens plats 1, Helsinki FIN-00014, Finland
| | - Sergio A Losilla
- Department of Chemistry, University of Helsinki , P.O. Box 55, A. I. Virtanens plats 1, Helsinki FIN-00014, Finland
| | - Eelis Solala
- Department of Chemistry, University of Helsinki , P.O. Box 55, A. I. Virtanens plats 1, Helsinki FIN-00014, Finland
| | - Elias A Toivanen
- Department of Chemistry, University of Helsinki , P.O. Box 55, A. I. Virtanens plats 1, Helsinki FIN-00014, Finland
| | - Wen-Hua Xu
- Department of Chemistry, University of Helsinki , P.O. Box 55, A. I. Virtanens plats 1, Helsinki FIN-00014, Finland
| | - Dage Sundholm
- Department of Chemistry, University of Helsinki , P.O. Box 55, A. I. Virtanens plats 1, Helsinki FIN-00014, Finland
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19
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Benner P, Khoromskaia V, Khoromskij BN. A reduced basis approach for calculation of the Bethe–Salpeter excitation energies by using low-rank tensor factorisations. Mol Phys 2016. [DOI: 10.1080/00268976.2016.1149241] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Affiliation(s)
- Peter Benner
- Max Planck Institute for Dynamics of Complex Systems, Magdeburg, Germany
| | - Venera Khoromskaia
- Max Planck Institute for Dynamics of Complex Systems, Magdeburg, Germany
- Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
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20
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Kananenka AA, Phillips JJ, Zgid D. Efficient Temperature-Dependent Green’s Functions Methods for Realistic Systems: Compact Grids for Orthogonal Polynomial Transforms. J Chem Theory Comput 2016; 12:564-71. [DOI: 10.1021/acs.jctc.5b00884] [Citation(s) in RCA: 37] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Alexei A. Kananenka
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Jordan J. Phillips
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Dominika Zgid
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
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