1
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Yao YF, Su NQ. Enhancing Reduced Density Matrix Functional Theory Calculations by Coupling Orbital and Occupation Optimizations. J Phys Chem A 2024; 128:7669-7679. [PMID: 39207020 DOI: 10.1021/acs.jpca.4c02182] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 09/04/2024]
Abstract
Reduced density matrix functional theory (RDMFT) calculations are usually implemented in a decoupled manner, where the orbital and occupation optimizations are repeated alternately. Typically, orbital updates are performed using the unitary optimization method, while occupations are optimized through the explicit-by-implicit (EBI) method. The EBI method addresses explicit constraints by incorporating implicit functions, effectively transforming constrained optimization scenarios into unconstrained minimizations. Although the unitary and EBI methods individually achieve robust performance in optimizing orbitals and occupations, respectively, the decoupled optimization methods often suffer from slow convergence and require dozens of alternations between the orbital and occupation optimizations. To address this issue, this work proposes a coupled optimization method that combines unitary and EBI optimizations to update orbitals and occupations simultaneously at each step. To achieve favorable convergence in coupled optimization using a simple first-order algorithm, an effective and efficient preconditioner and line search are further introduced. The superiority of the new method is demonstrated through numerous tests on different molecules, random initial guesses, different basis sets, and different functionals. It outperforms all decoupled optimization methods in terms of convergence speed, convergence results, and convergence stability. Even a large system like C60 can converge to 10-8 au in 154 iterations, which shows that the coupled optimization method can make RDMFT more practical and facilitate its wider application and further development.
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Affiliation(s)
- Yi-Fan Yao
- Center for Theoretical and Computational Chemistry, Frontiers Science Center for New Organic Matter, State Key Laboratory of Advanced Chemical Power Sources, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education), Department of Chemistry, Nankai University, Tianjin 300071, China
| | - Neil Qiang Su
- Center for Theoretical and Computational Chemistry, Frontiers Science Center for New Organic Matter, State Key Laboratory of Advanced Chemical Power Sources, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education), Department of Chemistry, Nankai University, Tianjin 300071, China
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2
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Vladaj M, Marécat Q, Senjean B, Saubanère M. Variational minimization scheme for the one-particle reduced density matrix functional theory in the ensemble N-representability domain. J Chem Phys 2024; 161:074105. [PMID: 39145549 DOI: 10.1063/5.0219898] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2024] [Accepted: 07/29/2024] [Indexed: 08/16/2024] Open
Abstract
The one-particle reduced density-matrix (1-RDM) functional theory is a promising alternative to density-functional theory (DFT) that uses the 1-RDM rather than the electronic density as a basic variable. However, long-standing challenges such as the lack of the Kohn-Sham scheme and the complexity of the pure N-representability conditions are still impeding its wild utilization. Fortunately, ensemble N-representability conditions derived in the natural orbital basis are known and trivial such that almost every functional of the 1-RDM is actually natural orbital functional, which does not perform well for all the correlation regimes. In this work, we propose a variational minimization scheme in the ensemble N-representable domain that is not restricted to the natural orbital representation of the 1-RDM. We show that splitting the minimization into the diagonal and off-diagonal parts of the 1-RDM can open the way toward the development of functionals of the orbital occupations, which remains a challenge for the generalization of site-occupation functional theory in chemistry. Our approach is tested on the uniform Hubbard model using the Müller and the Töws-Pastor functionals, as well as on the dihydrogen molecule using the Müller functional.
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Affiliation(s)
- Matthieu Vladaj
- ICGM, Université de Montpellier, CNRS, ENSCM, Montpellier, France
| | - Quentin Marécat
- ICGM, Université de Montpellier, CNRS, ENSCM, Montpellier, France
| | - Bruno Senjean
- ICGM, Université de Montpellier, CNRS, ENSCM, Montpellier, France
| | - Matthieu Saubanère
- ICGM, Université de Montpellier, CNRS, ENSCM, Montpellier, France
- Université de Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talence, France
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3
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Liu D, Yan B, Irimia M, Wang J. Potential energy curves for F2, Cl2, and Br2 with the i-DMFT method. J Chem Phys 2024; 161:044118. [PMID: 39072668 DOI: 10.1063/5.0220836] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2024] [Accepted: 07/10/2024] [Indexed: 07/30/2024] Open
Abstract
The potential energy curves for dihalogens (F2, Cl2, and Br2) are calculated with the i-DMFT method proposed recently [Wang and Baerends, Phys. Rev. Lett. 128, 013001]. All electrons are correlated in a set of self-consistent-field eigenvalue equations, with the orbital occupation numbers obeying the Fermi-Dirac distribution. The only input is the dissociation energies of the molecules, which are usually available from an experimental database. The quality of the computed potential energy curve is examined by extracting spectroscopic parameters and rotation-vibration energy levels, which are compared with experiment data and other theoretical calculations.
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Affiliation(s)
- Di Liu
- School of Science, Huzhou University, Huzhou, Zhejiang 313000, China
- Institute of Atomic and Molecular Physics, Jilin University, Changchun, Jilin 130000, China
| | - Bing Yan
- Institute of Atomic and Molecular Physics, Jilin University, Changchun, Jilin 130000, China
| | - Marinela Irimia
- International School, Huzhou University, Huzhou, Zhejiang 313000, China
| | - Jian Wang
- School of Science, Huzhou University, Huzhou, Zhejiang 313000, China
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4
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Franco L, Bonfil-Rivera IA, Huan Lew-Yee JF, Piris M, M Del Campo J, Vargas-Hernández RA. Softmax parameterization of the occupation numbers for natural orbital functionals based on electron pairing approaches. J Chem Phys 2024; 160:244107. [PMID: 38920134 DOI: 10.1063/5.0213719] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2024] [Accepted: 06/07/2024] [Indexed: 06/27/2024] Open
Abstract
Within the framework of natural orbital functional theory, having a convenient representation of the occupation numbers and orbitals becomes critical for the computational performance of the calculations. Recognizing this, we propose an innovative parametrization of the occupation numbers that takes advantage of the electron-pairing approach used in Piris natural orbital functionals through the adoption of the softmax function, a pivotal component in modern deep-learning models. Our approach not only ensures adherence to the N-representability of the first-order reduced density matrix (1RDM) but also significantly enhances the computational efficiency of 1RDM functional theory calculations. The effectiveness of this alternative parameterization approach was assessed using the W4-17-MR molecular set, which demonstrated faster and more robust convergence compared to previous implementations.
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Affiliation(s)
- Lizeth Franco
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México City C.P. 04510, Mexico
| | - Iván A Bonfil-Rivera
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México City C.P. 04510, Mexico
| | | | - Mario Piris
- Donostia International Physics Center (DIPC), 20018 Donostia, Spain
- Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU), 20018 Donostia, Spain
- IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain
| | - Jorge M Del Campo
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México City C.P. 04510, Mexico
| | - Rodrigo A Vargas-Hernández
- Department of Chemistry and Chemical Biology, McMaster University, Hamilton, Ontario L8S 4L8, Canada
- Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario L8S 4M1, Canada
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5
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Gibney D, Boyn JN, Mazziotti DA. Universal Generalization of Density Functional Theory for Static Correlation. PHYSICAL REVIEW LETTERS 2023; 131:243003. [PMID: 38181140 DOI: 10.1103/physrevlett.131.243003] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Revised: 09/27/2023] [Accepted: 11/01/2023] [Indexed: 01/07/2024]
Abstract
A major challenge for density functional theory (DFT) is its failure to treat static correlation, yielding errors in predicted charges, band gaps, van der Waals forces, and reaction barriers. Here we combine one- and two-electron reduced density matrix (1- and 2-RDM) theories with DFT to obtain a universal O(N^{3}) generalization of DFT for static correlation. Using the lowest unitary invariant of the cumulant 2-RDM, we generate a 1-RDM functional theory that corrects the convexity of any DFT functional to capture static correlation in its fractional orbital occupations. Importantly, the unitary invariant yields a predictive theory by revealing the dependence of the correction's strength upon the trace of the two-electron repulsion matrix. We apply the theory to the barrier to rotation in ethylene, the relative energies of the benzynes, as well as an 11-molecule, dissociation benchmark. By inheriting the computational efficiency of DFT without sacrificing the treatment of static correlation, the theory opens new possibilities for the prediction and interpretation of significant quantum molecular effects and phenomena.
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Affiliation(s)
- Daniel Gibney
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 USA
| | - Jan-Niklas Boyn
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 USA
| | - David A Mazziotti
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 USA
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6
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Ai W, Su NQ, Fang WH. Short-range screened density matrix functional for proper descriptions of thermochemistry, thermochemical kinetics, nonbonded interactions, and singlet diradicals. J Chem Phys 2023; 159:174110. [PMID: 37933778 DOI: 10.1063/5.0169234] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2023] [Accepted: 10/09/2023] [Indexed: 11/08/2023] Open
Abstract
Common one-electron reduced density matrix (1-RDM) functionals that depend on Coulomb and exchange-only integrals tend to underestimate dynamic correlation, preventing reduced density matrix functional theory (RDMFT) from achieving comparable accuracy to density functional theory in main-group thermochemistry and thermochemical kinetics. The recently developed ωP22 functional introduces a semi-local density functional to screen the erroneous short-range portion of 1-RDM functionals without double-counting correlation, potentially providing a better treatment of dynamic correlation around equilibrium geometries. Herein, we systematically evaluate the performance of this functional model, which consists of two parameters, on main-group thermochemistry, thermochemical kinetics, nonbonded interactions, and more. Tests on atomization energies, vibrational frequencies, and reaction barriers reveal that the ωP22 functional model can reliably predict properties at equilibrium and slightly away from equilibrium geometries. In particular, it outperforms commonly used density functionals in the prediction of reaction barriers, nonbonded interactions, and singlet diradicals, thus enhancing the predictive power of RDMFT for routine calculations of thermochemistry and thermochemical kinetics around equilibrium geometries. Further development is needed in the future to refine short- and long-range approximations in the functional model in order to achieve an excellent description of properties both near and far from equilibrium geometries.
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Affiliation(s)
- Wenna Ai
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education), Frontiers Science Center for New Organic Matter, Nankai University, Tianjin 300071, China
| | - Neil Qiang Su
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education), Frontiers Science Center for New Organic Matter, Nankai University, Tianjin 300071, China
| | - Wei-Hai Fang
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education), Frontiers Science Center for New Organic Matter, Nankai University, Tianjin 300071, China
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7
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Mazziotti DA. Quantum Many-Body Theory from a Solution of the N-Representability Problem. PHYSICAL REVIEW LETTERS 2023; 130:153001. [PMID: 37115895 DOI: 10.1103/physrevlett.130.153001] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/15/2022] [Revised: 02/13/2023] [Accepted: 03/16/2023] [Indexed: 06/19/2023]
Abstract
Here we present a many-body theory based on a solution of the N-representability problem in which the ground-state two-particle reduced density matrix (2-RDM) is determined directly without the many-particle wave function. We derive an equation that re-expresses physical constraints on higher-order RDMs to generate direct constraints on the 2-RDM, which are required for its derivation from an N-particle density matrix, known as N-representability conditions. The approach produces a complete hierarchy of 2-RDM constraints that do not depend explicitly upon the higher RDMs or the wave function. By using the two-particle part of a unitary decomposition of higher-order constraint matrices, we can solve the energy minimization by semidefinite programming in a form where the low-rank structure of these matrices can be potentially exploited. We illustrate by computing the ground-state electronic energy and properties of the H_{8} ring.
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Affiliation(s)
- David A Mazziotti
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
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8
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Huan Lew-Yee JF, Piris M, Del Campo JM. Outstanding improvement in removing the delocalization error by global natural orbital functional. J Chem Phys 2023; 158:084110. [PMID: 36859086 DOI: 10.1063/5.0137378] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/11/2023] Open
Abstract
This work assesses the performance of the recently proposed global natural orbital functional (GNOF) against the charge delocalization error. GNOF provides a good balance between static and dynamic electronic correlations leading to accurate total energies while preserving spin, even for systems with a highly multi-configurational character. Several analyses were applied to the functional, namely, (i) how the charge is distributed in super-systems of two fragments, (ii) the stability of ionization potentials while increasing the system size, and (iii) potential energy curves of a neutral and charged diatomic system. GNOF was found to practically eliminate the charge delocalization error in many of the studied systems or greatly improve the results obtained previously with PNOF7.
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Affiliation(s)
- Juan Felipe Huan Lew-Yee
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, Mexico City C.P. 04510, Mexico
| | - Mario Piris
- Donostia International Physics Center (DIPC), 20018 Donostia, Euskadi, Spain; Euskal Herriko Unibertsitatea (UPV/EHU), PK 1072, 20080 Donostia, Euskadi, Spain; and Basque Foundation for Science (IKERBASQUE), 48009 Bilbao, Euskadi, Spain
| | - Jorge M Del Campo
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, Mexico City C.P. 04510, Mexico
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9
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Lew-Yee JFH, Del Campo JM, Piris M. Electron Correlation in the Iron(II) Porphyrin by Natural Orbital Functional Approximations. J Chem Theory Comput 2023; 19:211-220. [PMID: 36579972 PMCID: PMC9996833 DOI: 10.1021/acs.jctc.2c01093] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022]
Abstract
The relative stability of the singlet, triplet, and quintet spin states of iron(II) porphyrin (FeP) represents a challenging problem for electronic structure methods. While it is currently accepted that the ground state is a triplet, multiconfigurational wave function-based methods predict a quintet, and density functional approximations vary between triplet and quintet states, leading to a prediction that highly depends on the features of the method employed. The recently proposed Global Natural Orbital Functional (GNOF) aims to provide a balanced treatment between static and dynamic correlation, and together with the previous Piris Natural Orbital Functionals (PNOFs), allowed us to explore the importance of each type of correlation in the stability order of the states of FeP with a method that conserves the spin of the system. It is noteworthy that GNOF correlates all electrons in all available orbitals for a given basis set; in the case of the FeP with a double-ζ basis set as used in this work, this means that GNOF can properly correlate 186 electrons in 465 orbitals, significantly increasing the sizes of systems amenable to multiconfigurational treatment. Results show that PNOF5, PNOF7s, and PNOF7 predict the quintet to have a lower energy than the triplet state; however, the addition of dynamic correlation via second-order Møller-Plesset corrections (NOF-MP2) turns the triplet state to be lower than the quintet state, a prediction also reproduced by GNOF that incorporates much more dynamic correlation than its predecessors.
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Affiliation(s)
- Juan Felipe Huan Lew-Yee
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México CityC.P. 04510, México
| | - Jorge M Del Campo
- Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México CityC.P. 04510, México
| | - Mario Piris
- Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU), P.K. 1072, 20080Donostia, Euskadi, Spain.,Donostia International Physics Center (DIPC), 20018Donostia, Euskadi, Spain.,IKERBASQUE, Basque Foundation for Science, 48013Bilbao, Euskadi, Spain
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10
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Mercero JM, Grande-Aztatzi R, Ugalde JM, Piris M. Natural orbital functional theory studies of all-metal aromaticity: The Al 3−anion. ADVANCES IN QUANTUM CHEMISTRY 2023. [DOI: 10.1016/bs.aiq.2023.02.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 03/28/2023]
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11
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Gibney D, Boyn JN, Mazziotti DA. Comparison of Density-Matrix Corrections to Density Functional Theory. J Chem Theory Comput 2022; 18:6600-6607. [DOI: 10.1021/acs.jctc.2c00625] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Daniel Gibney
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 United States
| | - Jan-Niklas Boyn
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 United States
| | - David A. Mazziotti
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 United States
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12
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Yao YF, Zhang Z, Fang WH, Su NQ. Explicit-by-Implicit Treatment of Natural Orbital Occupations Using First- and Second-Order Optimization Algorithms: A Comparative Study. J Phys Chem A 2022; 126:5654-5662. [PMID: 35950981 DOI: 10.1021/acs.jpca.2c02345] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
To address the convergence issues in the natural occupation optimization of reduced density matrix functional theory (RDMFT), we recently proposed the explicit-by-implicit (EBI) idea to handle the ensemble N-representability constraint (Yao et al. J. Phys. Chem. Lett. 2021, 12, 6788). This work continues to focus on these issues that can affect the reliability of the electronic structure description in RDMFT; further explores the combination of EBI, as well as the (augmented) Lagrangian methods (both LM and ALM), with both first- and second-order numerical optimization algorithms; and carefully evaluates their performances in natural occupation optimizations of various systems, including strongly correlated systems and large molecules. By comparing both converged energies and elapsed times, it can be seen that the LM and ALM have serious convergence issues for systems of different sizes. In contrast, the optimizations of EBI can converge to better energies with fewer iterations. However, due to the local convergence nature of the Newton's Method (NM) algorithm, EBI@NM still suffers from the local minimum issue for both strongly correlated systems and large molecules. Overall, the combination of EBI with the simple first-order algorithm of gradient descent (GD), namely EBI@GD, consistently provides the lowest converged energies for different types of systems, with the lowest computational scaling. These tests demonstrate the advantages of EBI in the calculations of transition states, strongly correlated systems, and large molecules. Meanwhile, the insights gained from this work are helpful to further develop more efficient algorithms for RDMFT.
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Affiliation(s)
- Yi-Fan Yao
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
| | - Zhang Zhang
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
| | - Wei-Hai Fang
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China.,Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University, Beijing 100875, China
| | - Neil Qiang Su
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
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13
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Ai W, Fang WH, Su NQ. Functional-Based Description of Electronic Dynamic and Strong Correlation: Old Issues and New Insights. J Phys Chem Lett 2022; 13:1744-1751. [PMID: 35157451 DOI: 10.1021/acs.jpclett.2c00084] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
Approximate functionals in Kohn-Sham density functional theory (KS-DFT) and reduced density matrix functional theory (RDMFT) have advantages in dealing with dynamic correlation and strong correlation, respectively; their combination can benefit from complementarity while suffering from the problem of correlation double-counting. Herein, a short-range corrected reduced density matrix (1-RDM) functional is developed to take advantage of the functionals in KS-DFT and RDMFT without double-counting. The resulting functional, denoted as ωP22, outperforms other 1-RDM functionals for the tests of thermochemistry, nonbonded interactions, and bond dissociation energy. In particular, ωP22 shows much less systematic error for systems involving fractional spins, and it can properly predict the energies at both equilibrium and dissociated distances for different single and multiple bonds, which cannot be achieved by commonly used KS-DFT and RDMFT functionals. Therefore, ωP22 is demonstrated effective in balance handling dynamic and strong correlation, and the advances in this work would create new possibilities for the development and application of approximate functionals.
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Affiliation(s)
- Wenna Ai
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
| | - Wei-Hai Fang
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
- Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University, Beijing 100875, China
| | - Neil Qiang Su
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
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14
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Wang J, Baerends EJ. Self-Consistent-Field Method for Correlated Many-Electron Systems with an Entropic Cumulant Energy. PHYSICAL REVIEW LETTERS 2022; 128:013001. [PMID: 35061466 DOI: 10.1103/physrevlett.128.013001] [Citation(s) in RCA: 16] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Accepted: 11/24/2021] [Indexed: 06/14/2023]
Abstract
A self-consistent field method is presented within density matrix functional theory. The computational cost for a correlated many-electron calculation is reduced to that of the self-consistent-field Hartree-Fock method, while the accuracy still reaches that of sophisticated configuration interaction based methods. In this method, the two-electron cumulant energy is measured with an information entropy associated with the Fermi-Dirac distribution of the occupation numbers. An eigenvalue equation for the orbitals is obtained, with the eigenvalues (orbital energies) connected to the occupation numbers through the Fermi-Dirac distribution. The occupation numbers for the strongly occupied orbitals are very close to the natural orbital occupation numbers from wave function methods. It covers in a single scheme the nondynamical correlation in weak or breaking bonds as well as the dynamical correlation at all distances. The method is well suited to large-scale potential energy surface calculation and molecular dynamics simulation.
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Affiliation(s)
- Jian Wang
- School of Science, Huzhou University, Zhejiang 313000, China
| | - Evert Jan Baerends
- Afdeling Theoretische Chemie, FEW, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
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15
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Liebert J, Castillo F, Labbé JP, Schilling C. Foundation of One-Particle Reduced Density Matrix Functional Theory for Excited States. J Chem Theory Comput 2021; 18:124-140. [PMID: 34931830 DOI: 10.1021/acs.jctc.1c00561] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
In Phys. Rev. Lett. 2021, 127, 023001 a reduced density matrix functional theory (RDMFT) was proposed for calculating energies of selected eigenstates of interacting many-Fermion systems. Here, we develop a solid foundation for this so-called w-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights w in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone's pioneering work on ground state RDMFT as the special case w = (1,0, ···). Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional's domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.
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Affiliation(s)
- Julia Liebert
- Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333, München, Germany.,Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799, München, Germany
| | - Federico Castillo
- Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany
| | - Jean-Philippe Labbé
- Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333, München, Germany.,Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799, München, Germany.,Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, 14195, Berlin, Germany
| | - Christian Schilling
- Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333, München, Germany.,Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799, München, Germany
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16
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Yao YF, Fang WH, Su NQ. Handling Ensemble N-Representability Constraint in Explicit-by-Implicit Manner. J Phys Chem Lett 2021; 12:6788-6793. [PMID: 34270236 DOI: 10.1021/acs.jpclett.1c01835] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
The convergence issues caused by the improper treatment of the ensemble N-representability constraint have severely affected the applicability and reproducibility of reduced density matrix functional theory (RDMFT). Unlike the commonly used Lagrange methods explicitly bringing the constraint into the objective functions, we present a different idea to handle the constraint in an implicit manner, which is achieved by introducing implicit functions to exactly consider the nonlinear unclear connection embedded in the explicit constraint. This explicit-by-implicit idea, denoted as EBI, thus transforms the constrained optimization problem into an unconstrained minimization problem. The tests on different systems, initial guesses, and functionals demonstrate the superiority of EBI in the treatment of the ensemble N-representability constraint. Therefore, EBI solves the convergence issues of the Lagrange methods, which is essential for further development and application of RDMFT. Besides, the idea of EBI is helpful for the treatment of different constrained problems in modern physics and chemistry.
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Affiliation(s)
- Yi-Fan Yao
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
| | - Wei-Hai Fang
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
- Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University, Beijing 100875, China
| | - Neil Qiang Su
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
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Schilling C, Pittalis S. Ensemble Reduced Density Matrix Functional Theory for Excited States and Hierarchical Generalization of Pauli's Exclusion Principle. PHYSICAL REVIEW LETTERS 2021; 127:023001. [PMID: 34296916 DOI: 10.1103/physrevlett.127.023001] [Citation(s) in RCA: 20] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2021] [Revised: 03/26/2021] [Accepted: 06/02/2021] [Indexed: 06/13/2023]
Abstract
We propose and work out a reduced density matrix functional theory (RDMFT) for calculating energies of eigenstates of interacting many-electron systems beyond the ground state. Various obstacles which historically have doomed such an approach to be unfeasible are overcome. First, we resort to a generalization of the Ritz variational principle to ensemble states with fixed weights. This in combination with the constrained search formalism allows us to establish a universal functional of the one-particle reduced density matrix. Second, we employ tools from convex analysis to circumvent the too involved N-representability constraints. Remarkably, this identifies Valone's pioneering work on RDMFT as a special case of convex relaxation and reveals that crucial information about the excitation structure is contained in the functional's domain. Third, to determine the crucial latter object, a methodology is developed which eventually leads to a generalized exclusion principle. The corresponding linear constraints are calculated for systems of arbitrary size.
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Affiliation(s)
- Christian Schilling
- Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799 München, Germany
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Gibney D, Boyn JN, Mazziotti DA. Toward a Resolution of the Static Correlation Problem in Density Functional Theory from Semidefinite Programming. J Phys Chem Lett 2021; 12:385-391. [PMID: 33356286 DOI: 10.1021/acs.jpclett.0c03371] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Kohn-Sham density functional theory (DFT) has long struggled with the accurate description of strongly correlated and open shell systems, and improvements have been minor even in the newest hybrid functionals. In this Letter we treat the static correlation in DFT when frontier orbitals are degenerate by the means of using a semidefinite programming (SDP) approach to minimize the system energy as a function of the N-representable, non-idempotent 1-electron reduced density matrix. While showing greatly improved singlet-triplet gaps for local density approximation and generalized gradient approximation (GGA) functionals, the SDP procedure reveals flaws in modern meta and hybrid GGA functionals, which show no major improvements when provided with an accurate electron density.
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Affiliation(s)
- Daniel Gibney
- The James Franck Institute and The Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, United States
| | - Jan-Niklas Boyn
- The James Franck Institute and The Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, United States
| | - David A Mazziotti
- The James Franck Institute and The Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, United States
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Benavides-Riveros CL, Wolff J, Marques MAL, Schilling C. Reduced Density Matrix Functional Theory for Bosons. PHYSICAL REVIEW LETTERS 2020; 124:180603. [PMID: 32441966 DOI: 10.1103/physrevlett.124.180603] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2020] [Accepted: 04/21/2020] [Indexed: 06/11/2023]
Abstract
Based on a generalization of Hohenberg-Kohn's theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix γ as a variable but still recovers quantum correlations in an exact way it is particularly well suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying v-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals F[γ] for this N-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. For Bose-Einstein condensates with N_{BEC}≈N condensed bosons, the respective gradient forces are found to diverge, ∇_{γ}F∝1/sqrt[1-N_{BEC}/N], providing a comprehensive explanation for the absence of complete condensation in nature.
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Affiliation(s)
- Carlos L Benavides-Riveros
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany
- NR-ISM, Division of Ultrafast Processes in Materials (FLASHit), Area della Ricerca di Roma 1, Via Salaria Km 29.3, I-00016 Monterotondo Scalo, Italy
| | - Jakob Wolff
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany
| | - Miguel A L Marques
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany
| | - Christian Schilling
- Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany
- Wolfson College, University of Oxford, Linton Rd, Oxford OX2 6UD, United Kingdom
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20
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Mitxelena I, Piris M. An efficient method for strongly correlated electrons in two-dimensions. J Chem Phys 2020; 152:064108. [PMID: 32061239 DOI: 10.1063/1.5140985] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This work deals with the problem of strongly correlated electrons in two-dimensions. We give a reduced density matrix (RDM) based tool through which the ground-state energy is given as a functional of the natural orbitals and their occupation numbers. Specifically, the Piris Natural Orbital Functional 7 (PNOF7) is used for studying the 2D Hubbard model and hydrogen square lattices. The singlet ground-state is studied, as well as the doublet mixed quantum state obtained by extracting an electron from the system. Our method satisfies two-index necessary N-representability conditions of the two-particle RDM (2RDM) and guarantees the conservation of the total spin. We show the ability of PNOF7 to describe strong correlation effects in two-dimensional (2D) systems by comparing our results with the exact diagonalization, density matrix renormalization group (DMRG), and auxiliary-field quantum Monte Carlo calculations. PNOF7 overcomes variational 2RDM methods with two- and three-index positivity N-representability conditions, reducing computational cost to mean-field scaling. Consistent results are obtained for small and large systems up to 144 electrons, weak and strong correlation regimes, and many filling situations. Unlike other methods, there is no dependence on dimensionality in the results obtained with PNOF7 and no particular difficulties have been observed to converge PNOF7 away from half-filling. Smooth double occupancy of sites is obtained, regardless of the filling. Symmetric dissociation of 2D hydrogen lattices shows that long-range nondynamic correlation dramatically affects electron detachment energies. PNOF7 compares well with DMRG along the dissociation curve.
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Affiliation(s)
- Ion Mitxelena
- Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU) and Donostia International Physics Center (DIPC), 20018 Donostia, Euskadi, Spain
| | - Mario Piris
- Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU) and Donostia International Physics Center (DIPC), 20018 Donostia, Euskadi, Spain
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Benavides-Riveros CL, Marques MAL. On the time evolution of fermionic occupation numbers. J Chem Phys 2019; 151:044112. [DOI: 10.1063/1.5109009] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
| | - Miguel A. L. Marques
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany
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Mitxelena I, Piris M, Ugalde JM. Advances in approximate natural orbital functional theory. ADVANCES IN QUANTUM CHEMISTRY 2019. [DOI: 10.1016/bs.aiq.2019.04.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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