1
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Gedeon J, Schmidt J, Hodgson MJP, Wetherell J, Benavides-Riveros CL, Marques MAL. Machine learning the derivative discontinuity of density-functional theory. MACHINE LEARNING: SCIENCE AND TECHNOLOGY 2022. [DOI: 10.1088/2632-2153/ac3149] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Abstract
Machine learning is a powerful tool to design accurate, highly non-local, exchange-correlation functionals for density functional theory. So far, most of those machine learned functionals are trained for systems with an integer number of particles. As such, they are unable to reproduce some crucial and fundamental aspects, such as the explicit dependency of the functionals on the particle number or the infamous derivative discontinuity at integer particle numbers. Here we propose a solution to these problems by training a neural network as the universal functional of density-functional theory that (a) depends explicitly on the number of particles with a piece-wise linearity between the integer numbers and (b) reproduces the derivative discontinuity of the exchange-correlation energy. This is achieved by using an ensemble formalism, a training set containing fractional densities, and an explicitly discontinuous formulation.
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2
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Jacob D, Stefanucci G, Kurth S. Mott Metal-Insulator Transition from Steady-State Density Functional Theory. PHYSICAL REVIEW LETTERS 2020; 125:216401. [PMID: 33274971 DOI: 10.1103/physrevlett.125.216401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2020] [Accepted: 10/15/2020] [Indexed: 06/12/2023]
Abstract
We present a computationally efficient method to obtain the spectral function of bulk systems in the framework of steady-state density functional theory (i-DFT) using an idealized scanning tunneling microscope (STM) setup. We calculate the current through the STM tip and then extract the spectral function from the finite-bias differential conductance. The fictitious noninteracting system of i-DFT features an exchange-correlation (XC) contribution to the bias which guarantees the same current as in the true interacting system. Exact properties of the XC bias are established using Fermi-liquid theory and subsequently implemented to construct approximations for the Hubbard model. We show for two different lattice structures that the Mott metal-insulator transition is captured by i-DFT.
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Affiliation(s)
- David Jacob
- Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Departamento Polímeros y Materiales Avanzados: Física, Química y Tecnología, Universidad del País Vasco UPV/EHU, Avenida Tolosa 72, E-20018 San Sebastián, Spain
- IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, E-48009 Bilbao, Spain
| | - Gianluca Stefanucci
- Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy
- INFN, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy
| | - Stefan Kurth
- Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Departamento Polímeros y Materiales Avanzados: Física, Química y Tecnología, Universidad del País Vasco UPV/EHU, Avenida Tolosa 72, E-20018 San Sebastián, Spain
- IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, E-48009 Bilbao, Spain
- Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
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3
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Grossi J, Musslimani ZH, Seidl M, Gori-Giorgi P. Kohn-Sham equations with functionals from the strictly-correlated regime: investigation with a spectral renormalization method. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:475602. [PMID: 32759484 DOI: 10.1088/1361-648x/abace2] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2020] [Accepted: 08/06/2020] [Indexed: 06/11/2023]
Abstract
We re-adapt a spectral renormalization method, introduced in nonlinear optics, to solve the Kohn-Sham (KS) equations of density functional theory, with a focus on functionals based on the strictly-correlated electrons (SCE) regime, which are particularly challenging to converge. Important aspects of the method are: (i) the eigenvalues and the density are computed simultaneously; (ii) it converges using randomized initial guesses; (iii) easy to implement. Using this method we could converge for the first time the Kohn-Sham equations with functionals that include the next leading term in the strong-interaction limit of density functional theory, the so called zero-point energy (ZPE) functional as well as with an interaction-strength-interpolation functional that includes both the exact SCE and ZPE terms. This work is the first building block for future studies on quantum systems confined in low dimensions with different statistics and long-range repulsions, such as localization properties of fermions and bosons with strong long-range repulsive interactions in the presence of a random external potential.
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Affiliation(s)
- Juri Grossi
- Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Ziad H Musslimani
- Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
- Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, United States of America
| | - Michael Seidl
- Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Paola Gori-Giorgi
- Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
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4
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Fabrizio A, Meyer B, Corminboeuf C. Machine learning models of the energy curvature vs particle number for optimal tuning of long-range corrected functionals. J Chem Phys 2020; 152:154103. [DOI: 10.1063/5.0005039] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023] Open
Affiliation(s)
- Alberto Fabrizio
- Laboratory for Computational Molecular Design, Institute of Chemical Sciences and Engineering, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
- National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
| | - Benjamin Meyer
- Laboratory for Computational Molecular Design, Institute of Chemical Sciences and Engineering, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
- National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
| | - Clemence Corminboeuf
- Laboratory for Computational Molecular Design, Institute of Chemical Sciences and Engineering, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
- National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
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5
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Gerolin A, Grossi J, Gori-Giorgi P. Kinetic Correlation Functionals from the Entropic Regularization of the Strictly Correlated Electrons Problem. J Chem Theory Comput 2020; 16:488-498. [PMID: 31855421 PMCID: PMC6964418 DOI: 10.1021/acs.jctc.9b01133] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2019] [Indexed: 11/29/2022]
Abstract
In this work, we study the entropic regularization of the strictly correlated electrons formalism, discussing the implications for density functional theory and establishing a link with earlier works on quantum kinetic energy and classical entropy. We carry out a very preliminary investigation (using simplified models) on the use of the solution of the entropic regularized problem to build approximations for the kinetic correlation functional at large coupling strengths. We also analyze lower and upper bounds to the Hohenberg-Kohn functional using the entropic regularized strictly correlated electrons problem.
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Affiliation(s)
- Augusto Gerolin
- Department of Theoretical
Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Juri Grossi
- Department of Theoretical
Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Paola Gori-Giorgi
- Department of Theoretical
Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
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6
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Biele R, D’Agosta R. Beyond the State of the Art: Novel Approaches for Thermal and Electrical Transport in Nanoscale Devices. ENTROPY 2019; 21:e21080752. [PMID: 33267466 PMCID: PMC7515281 DOI: 10.3390/e21080752] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Revised: 07/28/2019] [Accepted: 07/29/2019] [Indexed: 11/16/2022]
Abstract
Almost any interaction between two physical entities can be described through the transfer of either charge, spin, momentum, or energy. Therefore, any theory able to describe these transport phenomena can shed light on a variety of physical, chemical, and biological effects, enriching our understanding of complex, yet fundamental, natural processes, e.g., catalysis or photosynthesis. In this review, we will discuss the standard workhorses for transport in nanoscale devices, namely Boltzmann's equation and Landauer's approach. We will emphasize their strengths, but also analyze their limits, proposing theories and models useful to go beyond the state of the art in the investigation of transport in nanoscale devices.
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Affiliation(s)
- Robert Biele
- Institute for Materials Science and Max Bergmann Center of Biomaterials, TU Dresden, 01062 Dresden, Germany
- Correspondence: (R.B.); (R.D.); Tel.: +34-943-015-803 (R.D.)
| | - Roberto D’Agosta
- Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Universidad del Pais Vasco CFM CSIC-UPV/EHU-MPC and DIPC, Av. Tolosa 72, 20018 San Sebastian, Spain
- IKERBASQUE, Basque Foundation for Science Maria Diaz de Haro 3, 6 Solairua, 48013 Bilbao, Spain
- Correspondence: (R.B.); (R.D.); Tel.: +34-943-015-803 (R.D.)
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7
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Giarrusso S, Vuckovic S, Gori-Giorgi P. Response Potential in the Strong-Interaction Limit of Density Functional Theory: Analysis and Comparison with the Coupling-Constant Average. J Chem Theory Comput 2018; 14:4151-4167. [PMID: 29906106 PMCID: PMC6096453 DOI: 10.1021/acs.jctc.8b00386] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Using the formalism of the conditional amplitude, we study the response part of the exchange-correlation potential in the strong-coupling limit of density functional theory, analyzing its peculiar features and comparing it with the response potential averaged over the coupling constant for small atoms and for the hydrogen molecule. We also use a simple one-dimensional model of a stretched heteronuclear molecule to derive exact properties of the response potential in the strong-coupling limit. The simplicity of the model allows us to unveil relevant features also of the exact Kohn-Sham potential and its different components, namely the appearance of a second peak in the correlation kinetic potential on the side of the most electronegative atom.
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Affiliation(s)
- Sara Giarrusso
- Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW , Vrije Universiteit , De Boelelaan 1083 , 1081HV Amsterdam , The Netherlands
| | - Stefan Vuckovic
- Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW , Vrije Universiteit , De Boelelaan 1083 , 1081HV Amsterdam , The Netherlands
| | - Paola Gori-Giorgi
- Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW , Vrije Universiteit , De Boelelaan 1083 , 1081HV Amsterdam , The Netherlands
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8
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Vuckovic S, Gori-Giorgi P. Simple Fully Nonlocal Density Functionals for Electronic Repulsion Energy. J Phys Chem Lett 2017; 8:2799-2805. [PMID: 28581751 PMCID: PMC5502414 DOI: 10.1021/acs.jpclett.7b01113] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2017] [Accepted: 06/05/2017] [Indexed: 05/24/2023]
Abstract
From a simplified version of the mathematical structure of the strong coupling limit of the exact exchange-correlation functional, we construct an approximation for the electronic repulsion energy at physical coupling strength, which is fully nonlocal. This functional is self-interaction free and yields energy densities within the definition of the electrostatic potential of the exchange-correlation hole that are locally accurate and have the correct asymptotic behavior. The model is able to capture strong correlation effects that arise from chemical bond dissociation, without relying on error cancellation. These features, which are usually missed by standard density functional theory (DFT) functionals, are captured by the highly nonlocal structure, which goes beyond the "Jacob's ladder" framework for functional construction, by using integrals of the density as the key ingredient. Possible routes for obtaining the full exchange-correlation functional by recovering the missing kinetic component of the correlation energy are also implemented and discussed.
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9
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Fabiano E, Gori-Giorgi P, Seidl M, Della Sala F. Interaction-Strength Interpolation Method for Main-Group Chemistry: Benchmarking, Limitations, and Perspectives. J Chem Theory Comput 2016; 12:4885-4896. [DOI: 10.1021/acs.jctc.6b00713] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Eduardo Fabiano
- Euromediterranean
Center for Nanomaterial Modelling and Technology (ECMT), Istituto Nanoscienze-CNR, Via per Arnesano 16, 73100 Lecce, Italy
- Center
for Biomolecular Nanotechnologies@UNILE, Istituto Italiano di Tecnologia (IIT), Via Barsanti, 73010 Arnesano, Italy
| | - Paola Gori-Giorgi
- Department
of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling,
FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Michael Seidl
- Department
of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling,
FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Fabio Della Sala
- Euromediterranean
Center for Nanomaterial Modelling and Technology (ECMT), Istituto Nanoscienze-CNR, Via per Arnesano 16, 73100 Lecce, Italy
- Center
for Biomolecular Nanotechnologies@UNILE, Istituto Italiano di Tecnologia (IIT), Via Barsanti, 73010 Arnesano, Italy
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10
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Hopjan M, Karlsson D, Ydman S, Verdozzi C, Almbladh CO. Merging Features from Green's Functions and Time Dependent Density Functional Theory: A Route to the Description of Correlated Materials out of Equilibrium? PHYSICAL REVIEW LETTERS 2016; 116:236402. [PMID: 27341247 DOI: 10.1103/physrevlett.116.236402] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2015] [Indexed: 06/06/2023]
Abstract
We propose a description of nonequilibrium systems via a simple protocol that combines exchange-correlation potentials from density functional theory with self-energies of many-body perturbation theory. The approach, aimed to avoid double counting of interactions, is tested against exact results in Hubbard-type systems, with respect to interaction strength, perturbation speed and inhomogeneity, and system dimensionality and size. In many regimes, we find significant improvement over adiabatic time dependent density functional theory or second Born nonequilibrium Green's function approximations. We briefly discuss the reasons for the residual discrepancies, and directions for future work.
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Affiliation(s)
- M Hopjan
- Mathematical Physics Division, Department of Physics, Lund University, 22100 Lund, Sweden; and European Theoretical Spectroscopy Facility (ETSF)
| | - D Karlsson
- Mathematical Physics Division, Department of Physics, Lund University, 22100 Lund, Sweden; and European Theoretical Spectroscopy Facility (ETSF)
| | - S Ydman
- Mathematical Physics Division, Department of Physics, Lund University, 22100 Lund, Sweden; and European Theoretical Spectroscopy Facility (ETSF)
| | - C Verdozzi
- Mathematical Physics Division, Department of Physics, Lund University, 22100 Lund, Sweden; and European Theoretical Spectroscopy Facility (ETSF)
| | - C-O Almbladh
- Mathematical Physics Division, Department of Physics, Lund University, 22100 Lund, Sweden; and European Theoretical Spectroscopy Facility (ETSF)
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11
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Vuckovic S, Irons TJP, Savin A, Teale AM, Gori-Giorgi P. Exchange-Correlation Functionals via Local Interpolation along the Adiabatic Connection. J Chem Theory Comput 2016; 12:2598-610. [PMID: 27116427 PMCID: PMC4910137 DOI: 10.1021/acs.jctc.6b00177] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
![]()
The construction
of density-functional approximations is explored
by modeling the adiabatic connection locally, using
energy densities defined in terms of the electrostatic potential of
the exchange–correlation hole. These local models are more
amenable to the construction of size-consistent approximations than
their global counterparts. In this work we use accurate input local
ingredients to assess the accuracy of a range of local interpolation
models against accurate exchange–correlation energy densities.
The importance of the strictly correlated electrons (SCE) functional
describing the strong coupling limit is emphasized, enabling the corresponding
interpolated functionals to treat strong correlation effects. In addition
to exploring the performance of such models numerically for the helium
and beryllium isoelectronic series and the dissociation of the hydrogen
molecule, an approximate analytic model is presented for the initial
slope of the local adiabatic connection. Comparisons are made with
approaches based on global models, and prospects for future approximations
based on the local adiabatic connection are discussed.
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Affiliation(s)
- Stefan Vuckovic
- Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit , De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Tom J P Irons
- School of Chemistry, University of Nottingham , University Park, Nottingham NG7 2RD, United Kingdom
| | - Andreas Savin
- Laboratoire de Chimie Théorique, UPMC, Paris 06, UMR 7616, Sorbonne Universités , F-75005 Paris, France.,Laboratoire de Chimie Théorique, UMR 7616, CNRS F-75005, Paris, France
| | - Andrew M Teale
- School of Chemistry, University of Nottingham , University Park, Nottingham NG7 2RD, United Kingdom
| | - Paola Gori-Giorgi
- Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit , De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
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12
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Vuckovic S, Wagner LO, Mirtschink A, Gori-Giorgi P. Hydrogen Molecule Dissociation Curve with Functionals Based on the Strictly Correlated Regime. J Chem Theory Comput 2015; 11:3153-62. [DOI: 10.1021/acs.jctc.5b00387] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Stefan Vuckovic
- Department of Theoretical
Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Lucas O. Wagner
- Department of Theoretical
Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - André Mirtschink
- Department of Theoretical
Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
| | - Paola Gori-Giorgi
- Department of Theoretical
Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
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13
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Vieira D. Strong Correlations in Density-Functional Theory: A Model of Spin-Charge and Spin-Orbital Separations. J Chem Theory Comput 2014; 10:3641-6. [PMID: 26588509 DOI: 10.1021/ct400223d] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
It is known that the separation of electrons into spinons and chargons, the spin-charge separation, plays a decisive role when describing one-dimensional (1D) strongly correlated systems [ Phys. Rev. B 2012 , 86 , 075132 ]. In this paper, within the density-functional theory (DFT) formalism, we extend the investigation by considering a model for the third electron fractionalization: the separation into spinons, chargons and orbitons, the last associated with the electronic orbital degree of freedom. Specifically, we deal with two exact constraints of exchange-correlation (XC) density-functionals: (i) the constancy of the highest occupied (HO) Kohn-Sham (KS) eigenvalues upon fractional electron numbers and (ii) their discontinuities at integers. By means of 1D discrete Hubbard chains and 1D H2 molecules in the continuum, we find that spin-charge separation yields almost constant HO KS eigenvalues, whereas the spin-orbital counterpart can be decisive when describing derivative discontinuities of XC potentials at strong correlations.
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Affiliation(s)
- Daniel Vieira
- Departamento de Física, Universidade do Estado de Santa Catarina , Joinville, 89219-710 Santa Catarina, Brazil
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14
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Mirtschink A, Umrigar CJ, Morgan JD, Gori-Giorgi P. Energy density functionals from the strong-coupling limit applied to the anions of the He isoelectronic series. J Chem Phys 2014; 140:18A532. [DOI: 10.1063/1.4871018] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
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15
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Mori-Sánchez P, Cohen AJ. The derivative discontinuity of the exchange–correlation functional. Phys Chem Chem Phys 2014; 16:14378-87. [DOI: 10.1039/c4cp01170h] [Citation(s) in RCA: 57] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Abstract
Manifestations of the derivative discontinuity of the energy in density functional theory are demonstrated in simple systems in chemistry and physics.
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Affiliation(s)
- Paula Mori-Sánchez
- Departamento de Química and Instituto de Física de la Materia Condensada (IFIMAC)
- Universidad Autónoma de Madrid
- Madrid, Spain
| | - Aron J. Cohen
- Department of Chemistry
- University of Cambridge
- Cambridge, UK
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16
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Probing Strongly Correlated Materials in Non-equilibrium: Basic Concepts and Possible Future Trends in First Principle Approaches. Top Curr Chem (Cham) 2014; 347:347-84. [DOI: 10.1007/128_2014_528] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/11/2023]
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