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Parkkinen P, Xu WH, Solala E, Sundholm D. Density Functional Theory under the Bubbles and Cube Numerical Framework. J Chem Theory Comput 2018; 14:4237-4245. [PMID: 29944363 PMCID: PMC6150645 DOI: 10.1021/acs.jctc.8b00456] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Density functional theory within the Kohn-Sham density functional theory (KS-DFT) ansatz has been implemented into our bubbles and cube real-space molecular electronic structure framework, where functions containing steep cusps in the vicinity of the nuclei are expanded in atom-centered one-dimensional (1D) numerical grids multiplied with spherical harmonics (bubbles). The remainder, i.e., the cube, which is the cusp-free and smooth difference between the atomic one-center contributions and the exact molecular function, is represented on a three-dimensional (3D) equidistant grid by using a tractable number of grid points. The implementation of the methods is demonstrated by performing 3D numerical KS-DFT calculations on light atoms and small molecules. The accuracy is assessed by comparing the obtained energies with the best available reference energies.
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Affiliation(s)
- Pauli Parkkinen
- Department of Chemistry , University of Helsinki , P.O. Box 55 ( A. I. Virtanens plats 1 ), FIN-00014 Helsinki , Finland
| | - Wen-Hua Xu
- Department of Chemistry , University of Helsinki , P.O. Box 55 ( A. I. Virtanens plats 1 ), FIN-00014 Helsinki , Finland.,College of Chemistry and Materials Science , Northwest University , Xi'an 710069 , China
| | - Eelis Solala
- Department of Chemistry , University of Helsinki , P.O. Box 55 ( A. I. Virtanens plats 1 ), FIN-00014 Helsinki , Finland
| | - Dage Sundholm
- Department of Chemistry , University of Helsinki , P.O. Box 55 ( A. I. Virtanens plats 1 ), FIN-00014 Helsinki , Finland.,Centre for Advanced Study at the Norwegian Academy of Science and Letters , Drammensveien 78 , N-0271 Oslo , Norway
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2
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Kottmann JS, Bischoff FA. Coupled-Cluster in Real Space. 2. CC2 Excited States Using Multiresolution Analysis. J Chem Theory Comput 2017; 13:5956-5965. [DOI: 10.1021/acs.jctc.7b00695] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Jakob S. Kottmann
- Institut für Chemie, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
| | - Florian A. Bischoff
- Institut für Chemie, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
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3
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Jensen SR, Saha S, Flores-Livas JA, Huhn W, Blum V, Goedecker S, Frediani L. The Elephant in the Room of Density Functional Theory Calculations. J Phys Chem Lett 2017; 8:1449-1457. [PMID: 28291362 DOI: 10.1021/acs.jpclett.7b00255] [Citation(s) in RCA: 58] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Using multiwavelets, we have obtained total energies and corresponding atomization energies for the GGA-PBE and hybrid-PBE0 density functionals for a test set of 211 molecules with an unprecedented and guaranteed μHartree accuracy. These quasi-exact references allow us to quantify the accuracy of standard all-electron basis sets that are believed to be highly accurate for molecules, such as Gaussian-type orbitals (GTOs), all-electron numeric atom-centered orbitals (NAOs), and full-potential augmented plane wave (APW) methods. We show that NAOs are able to achieve the so-called chemical accuracy (1 kcal/mol) for the typical basis set sizes used in applications, for both total and atomization energies. For GTOs, a triple-ζ quality basis has mean errors of ∼10 kcal/mol in total energies, while chemical accuracy is almost reached for a quintuple-ζ basis. Due to systematic error cancellations, atomization energy errors are reduced by almost an order of magnitude, placing chemical accuracy within reach also for medium to large GTO bases, albeit with significant outliers. In order to check the accuracy of the computed densities, we have also investigated the dipole moments, where in general only the largest NAO and GTO bases are able to yield errors below 0.01 D. The observed errors are similar across the different functionals considered here.
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Affiliation(s)
- Stig Rune Jensen
- Centre for Theoretical and Computational Chemistry, Department of Chemistry, UiT - The Arctic University of Norway , N-9037 Tromsø, Norway
| | - Santanu Saha
- Department of Physics, Universität Basel , Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - José A Flores-Livas
- Department of Physics, Universität Basel , Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - William Huhn
- Department of Mechanical Engineering and Materials Science, Duke University , Durham, North Carolina 27708, United States
| | - Volker Blum
- Department of Mechanical Engineering and Materials Science, Duke University , Durham, North Carolina 27708, United States
| | - Stefan Goedecker
- Department of Physics, Universität Basel , Klingelbergstrasse 82, 4056 Basel, Switzerland
| | - Luca Frediani
- Centre for Theoretical and Computational Chemistry, Department of Chemistry, UiT - The Arctic University of Norway , N-9037 Tromsø, Norway
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4
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Solala E, Losilla SA, Sundholm D, Xu W, Parkkinen P. Optimization of numerical orbitals using the Helmholtz kernel. J Chem Phys 2017; 146:084102. [PMID: 28249419 DOI: 10.1063/1.4976557] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We present an integration scheme for optimizing the orbitals in numerical electronic structure calculations on general molecules. The orbital optimization is performed by integrating the Helmholtz kernel in the double bubble and cube basis, where bubbles represent the steep part of the functions in the vicinity of the nuclei, whereas the remaining cube part is expanded on an equidistant three-dimensional grid. 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 kinds. The angular part of the bubble functions can be integrated analytically, whereas the radial part is integrated numerically. The cube part is integrated using a similar method as we previously implemented for numerically integrating two-electron potentials. The behavior of the integrand of the auxiliary dimension introduced by the integral transformation of the Helmholtz kernel has also been investigated. The correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations on H2, H2O, and CO. The obtained energies are compared with reference values in the literature showing that an accuracy of 10-4 to 10-7 Eh can be obtained with our approach.
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Affiliation(s)
- Eelis Solala
- Department of Chemistry, University of Helsinki, P.O. Box 55, A.I. Virtanens plats 1, FIN-00014 Helsinki, Finland
| | - Sergio A Losilla
- Department of Chemistry, University of Helsinki, P.O. Box 55, A.I. Virtanens plats 1, FIN-00014 Helsinki, Finland
| | - Dage Sundholm
- Department of Chemistry, University of Helsinki, P.O. Box 55, A.I. Virtanens plats 1, FIN-00014 Helsinki, Finland
| | - Wenhua Xu
- Department of Chemistry, University of Helsinki, P.O. Box 55, A.I. Virtanens plats 1, FIN-00014 Helsinki, Finland
| | - Pauli Parkkinen
- Department of Chemistry, University of Helsinki, P.O. Box 55, A.I. Virtanens plats 1, FIN-00014 Helsinki, Finland
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5
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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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6
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Boffi NM, Jain M, Natan A. Efficient Computation of the Hartree-Fock Exchange in Real-Space with Projection Operators. J Chem Theory Comput 2016; 12:3614-22. [PMID: 27387558 DOI: 10.1021/acs.jctc.6b00376] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We describe an efficient projection-based real-space implementation of the nonlocal single-determinant exchange operator. Through a matrix representation of the projected operator, we show that this scheme works equally well for both occupied and virtual states. Our scheme reaches a speedup of 2 orders of magnitude and has no significant loss of accuracy compared to an implementation of the full nonlocal single-determinant exchange operator. We find excellent agreement upon comparing Hartree-Fock eigenvalues, dipoles, and polarizabilities of selected molecules calculated using our method to values in the literature. To illustrate the efficiency of this scheme we perform calculations on systems with up to 240 carbon atoms.
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Affiliation(s)
- Nicholas M Boffi
- Department of Physical Electronics, Tel-Aviv University , Tel-Aviv 69978, Israel.,Paulson School of Engineering and Applied Sciences, Harvard University , Cambridge, Massachusetts 02138, United States
| | - Manish Jain
- Department of Physics, Indian Institute of Science , Bangalore 560 012, India
| | - Amir Natan
- Department of Physical Electronics, Tel-Aviv University , Tel-Aviv 69978, Israel.,The Sackler Center for Computational Molecular and Materials Science, Tel-Aviv University , Tel-Aviv 69978, Israel
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7
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Boffi NM, Jain M, Natan A. Asymptotic behavior and interpretation of virtual states: The effects of confinement and of basis sets. J Chem Phys 2016; 144:084104. [PMID: 26931678 DOI: 10.1063/1.4942187] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A real-space high order finite difference method is used to analyze the effect of spherical domain size on the Hartree-Fock (and density functional theory) virtual eigenstates. We show the domain size dependence of both positive and negative virtual eigenvalues of the Hartree-Fock equations for small molecules. We demonstrate that positive states behave like a particle in spherical well and show how they approach zero. For the negative eigenstates, we show that large domains are needed to get the correct eigenvalues. We compare our results to those of Gaussian basis sets and draw some conclusions for real-space, basis-sets, and plane-waves calculations.
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Affiliation(s)
- Nicholas M Boffi
- Department of Physical Electronics, Tel-Aviv University, Tel-Aviv 69978, Israel
| | - Manish Jain
- Department of Physics, Indian Institute of Science, Bangalore 560012, India
| | - Amir Natan
- Department of Physical Electronics, Tel-Aviv University, Tel-Aviv 69978, Israel
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8
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Losilla SA, Watson MA, Aspuru-Guzik A, Sundholm D. Construction of the Fock Matrix on a Grid-Based Molecular Orbital Basis Using GPGPUs. J Chem Theory Comput 2015; 11:2053-62. [DOI: 10.1021/ct501128u] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023]
Affiliation(s)
- Sergio A. Losilla
- Department
of Chemistry, University of Helsinki, P.O. Box 55, FIN-00014 Helsinki, Finland
| | - Mark A. Watson
- Schrödinger, Inc, 120 West 45th Street, New York, New York 10036, United States
| | - Alán Aspuru-Guzik
- Department
of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States
| | - Dage Sundholm
- Department
of Chemistry, University of Helsinki, P.O. Box 55, FIN-00014 Helsinki, Finland
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9
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Toivanen EA, Losilla SA, Sundholm D. The grid-based fast multipole method – a massively parallel numerical scheme for calculating two-electron interaction energies. Phys Chem Chem Phys 2015; 17:31480-90. [DOI: 10.1039/c5cp01173f] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
A grid-based fast multipole method has been developed for calculating two-electron interaction energies for non-overlapping charge densities.
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Affiliation(s)
| | | | - Dage Sundholm
- Department of Chemistry
- University of Helsinki
- Finland
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10
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Frediani L, Sundholm D. Real-space numerical grid methods in quantum chemistry. Phys Chem Chem Phys 2015; 17:31357-9. [DOI: 10.1039/c5cp90198g] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
This themed issue reports on recent progress in the fast developing field of real-space numerical grid methods in quantum chemistry.
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Affiliation(s)
- Luca Frediani
- Centre for Theoretical and Computational Chemistry
- Department of Chemistry
- UiT The Arctic University of Norway
- N-9037 Tromsø
- Norway
| | - Dage Sundholm
- Department of Chemistry
- University of Helsinki
- Finland
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11
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Zuzovski M, Boag A, Natan A. An auxilliary grid method for the calculation of electrostatic terms in density functional theory on a real-space grid. Phys Chem Chem Phys 2015; 17:31550-7. [DOI: 10.1039/c5cp01090j] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The calculation of electrostatic integrals is performed using domain decomposition and auxiliary non-uniform grids for density functional theory real-space implementation.
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Affiliation(s)
- Michael Zuzovski
- Department of Physical Electronics
- Tel-Aviv University
- Tel-Aviv
- Israel
| | - Amir Boag
- Department of Physical Electronics
- Tel-Aviv University
- Tel-Aviv
- Israel
| | - Amir Natan
- Department of Physical Electronics
- Tel-Aviv University
- Tel-Aviv
- Israel
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12
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Heinäsmäki S. Translation and integration of numerical atomic orbitals in linear molecules. J Chem Phys 2014; 140:064112. [DOI: 10.1063/1.4865228] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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13
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Mehine MM, Losilla SA, Sundholm D. An efficient algorithm to calculate three-electron integrals for Gaussian-type orbitals using numerical integration. Mol Phys 2013. [DOI: 10.1080/00268976.2013.793847] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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14
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Przybytek M, Helgaker T. The accuracy of the Gaussian-and-finite-element-Coulomb (GFC) method for the calculation of Coulomb integrals. J Chem Phys 2013; 139:054114. [DOI: 10.1063/1.4817002] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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15
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Losilla SA, Mehine MM, Sundholm D. Construction of the two-electron contribution to the Fock matrix by numerical integration. Mol Phys 2012. [DOI: 10.1080/00268976.2012.720725] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
Affiliation(s)
- Sergio A. Losilla
- a Department of Chemistry , University of Helsinki , A.I. Virtanens plats 1, FIN-00014 Helsinki , Finland
| | - Mooses M. Mehine
- a Department of Chemistry , University of Helsinki , A.I. Virtanens plats 1, FIN-00014 Helsinki , Finland
| | - Dage Sundholm
- a Department of Chemistry , University of Helsinki , A.I. Virtanens plats 1, FIN-00014 Helsinki , Finland
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