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Thomas J, Chen H, Ortner C. Body-Ordered Approximations of Atomic Properties. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 2022; 246:1-60. [PMID: 36164458 PMCID: PMC9499924 DOI: 10.1007/s00205-022-01809-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/30/2021] [Accepted: 06/27/2022] [Indexed: 06/16/2023]
Abstract
We show that the local density of states (LDOS) of a wide class of tight-binding models has a weak body-order expansion. Specifically, we prove that the resulting body-order expansion for analytic observables such as the electron density or the energy has an exponential rate of convergence both at finite Fermi-temperature as well as for insulators at zero Fermi-temperature. We discuss potential consequences of this observation for modelling the potential energy landscape, as well as for solving the electronic structure problem.
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Affiliation(s)
- Jack Thomas
- Mathematics Institute, Zeeman Building, University of Warwick, Coventry, UK
| | - Huajie Chen
- School of Mathematical Sciences, Beijing Normal University, Beijing, China
| | - Christoph Ortner
- Department of Mathematics, University of British Columbia, Vancouver, Canada
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2
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Wang Z, Chern GW, Batista CD, Barros K. Gradient-based stochastic estimation of the density matrix. J Chem Phys 2018. [DOI: 10.1063/1.5017741] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Zhentao Wang
- Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
| | - Gia-Wei Chern
- Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA
| | - Cristian D. Batista
- Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
- Quantum Condensed Matter Division and Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
| | - Kipton Barros
- Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Desgranges C, Anderson PW, Delhommelle J. Classical and quantum many-body effects on the critical properties and thermodynamic regularities of silicon. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:045401. [PMID: 27875329 DOI: 10.1088/1361-648x/29/4/045401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Using molecular simulation, we determine the critical properties of Si as well as the loci for several remarkable thermodynamic contours spanning the supercritical region of the phase diagram. We consider a classical three-body potential as well as a quantum (tight-binding) many-body model, and determine the loci for the ideality contours, including the Zeno line and the H line of ideal enthalpy. The two strategies (classical or quantum) lead to strongly asymmetric binodals and to critical properties in good agreement with each other. The Zeno and H lines are found to remain linear over a wide temperature interval, despite the changes in electronic structure undergone by the fluid along these contours. We also show that the classical and quantum model yield markedly different results for the parameters defining the H line, the exponents for the power-laws underlying the line of minima for the isothermal enthalpy and for the density required to achieve ideal behavior, most notably for the enthalpy.
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Affiliation(s)
- C Desgranges
- Department of Chemistry, University of North Dakota, 151 Cornell Street Stop 9024, Grand Forks, ND 58202, USA
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4
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Cole DJ, Hine NDM. Applications of large-scale density functional theory in biology. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:393001. [PMID: 27494095 DOI: 10.1088/0953-8984/28/39/393001] [Citation(s) in RCA: 37] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Density functional theory (DFT) has become a routine tool for the computation of electronic structure in the physics, materials and chemistry fields. Yet the application of traditional DFT to problems in the biological sciences is hindered, to a large extent, by the unfavourable scaling of the computational effort with system size. Here, we review some of the major software and functionality advances that enable insightful electronic structure calculations to be performed on systems comprising many thousands of atoms. We describe some of the early applications of large-scale DFT to the computation of the electronic properties and structure of biomolecules, as well as to paradigmatic problems in enzymology, metalloproteins, photosynthesis and computer-aided drug design. With this review, we hope to demonstrate that first principles modelling of biological structure-function relationships are approaching a reality.
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Affiliation(s)
- Daniel J Cole
- Theory of Condensed Matter group, Cavendish Laboratory, 19 JJ Thomson Ave, Cambridge CB3 0HE, UK. School of Chemistry, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
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Desgranges C, Delhommelle J. Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. IV. Performance of many-body force fields and tight-binding schemes for the fluid phases of silicon. J Chem Phys 2016; 144:124510. [PMID: 27036464 DOI: 10.1063/1.4944619] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We extend Expanded Wang-Landau (EWL) simulations beyond classical systems and develop the EWL method for systems modeled with a tight-binding Hamiltonian. We then apply the method to determine the partition function and thus all thermodynamic properties, including the Gibbs free energy and entropy, of the fluid phases of Si. We compare the results from quantum many-body (QMB) tight binding models, which explicitly calculate the overlap between the atomic orbitals of neighboring atoms, to those obtained with classical many-body (CMB) force fields, which allow to recover the tetrahedral organization in condensed phases of Si through, e.g., a repulsive 3-body term that favors the ideal tetrahedral angle. Along the vapor-liquid coexistence, between 3000 K and 6000 K, the densities for the two coexisting phases are found to vary significantly (by 5 orders of magnitude for the vapor and by up to 25% for the liquid) and to provide a stringent test of the models. Transitions from vapor to liquid are predicted to occur for chemical potentials that are 10%-15% higher for CMB models than for QMB models, and a ranking of the force fields is provided by comparing the predictions for the vapor pressure to the experimental data. QMB models also reveal the formation of a gap in the electronic density of states of the coexisting liquid at high temperatures. Subjecting Si to a nanoscopic confinement has a dramatic effect on the phase diagram with, e.g. at 6000 K, a decrease in liquid densities by about 50% for both CMB and QMB models and an increase in vapor densities between 90% (CMB) and 170% (QMB). The results presented here provide a full picture of the impact of the strategy (CMB or QMB) chosen to model many-body effects on the thermodynamic properties of the fluid phases of Si.
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Affiliation(s)
- Caroline Desgranges
- Department of Chemistry, University of North Dakota, 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
| | - Jerome Delhommelle
- Department of Chemistry, University of North Dakota, 151 Cornell Street Stop 9024, Grand Forks, North Dakota 58202, USA
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Bowler DR, Miyazaki T. O(N) methods in electronic structure calculations. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2012; 75:036503. [PMID: 22790422 DOI: 10.1088/0034-4885/75/3/036503] [Citation(s) in RCA: 170] [Impact Index Per Article: 14.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
Linear-scaling methods, or O(N) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N, in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high-performance computers. The linear-scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas are then discussed. The applications of linear-scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear-scaling methods are discussed.
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Affiliation(s)
- D R Bowler
- London Centre for Nanotechnology, UCL, 17-19 Gordon St, London WC1H 0AH, UK.
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Density Matrix Methods in Linear Scaling Electronic Structure Theory. CHALLENGES AND ADVANCES IN COMPUTATIONAL CHEMISTRY AND PHYSICS 2011. [DOI: 10.1007/978-90-481-2853-2_16] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/30/2022]
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Shang H, Xiang H, Li Z, Yang J. Linear scaling electronic structure calculations with numerical atomic basis set. INT REV PHYS CHEM 2010. [DOI: 10.1080/0144235x.2010.520454] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Burger SK, Yang W. Linear-scaling quantum calculations using non-orthogonal localized molecular orbitals. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2008; 20:294209. [PMID: 25076813 PMCID: PMC4112540 DOI: 10.1088/0953-8984/20/29/294209] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2023]
Abstract
An absolute energy minimum variational principle is used for carrying out linear scaling calculations with non-orthogonal localized orbitals. Compared with results based on orthogonal localized molecular orbitals, the method is shown to give significantly more accurate results when the localized molecular orbitals are allowed to be non-orthogonal. This is made possible by introducing a second minimization for approximating the inverse overlap matrix. We also show how an exact line search may be used efficiently with the conjugate gradient method for minimizing the energy functional.
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Affiliation(s)
- Steven K Burger
- Department of Chemistry, McMaster University, 1280 Main St. West, Hamilton, ON, Canada
| | - Weitao Yang
- Department of Chemistry, Duke University, Box 90346, Durham, NC 27708-0346, USA
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Rubensson EH, Rudberg E, Sałek P. Density matrix purification with rigorous error control. J Chem Phys 2008; 128:074106. [DOI: 10.1063/1.2826343] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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11
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Nomura S, Iitaka T. Linear scaling calculation of an n-type GaAs quantum dot. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:037701. [PMID: 17930374 DOI: 10.1103/physreve.76.037701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2007] [Revised: 06/11/2007] [Indexed: 05/25/2023]
Abstract
A linear scale method for calculating electronic properties of large and complex systems is introduced within a local density approximation. The method is based on the Chebyshev polynomial expansion and the time-dependent method, which is tested on the calculation of the electronic structure of a model n-type GaAs quantum dot.
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Affiliation(s)
- Shintaro Nomura
- Institute of Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Japan.
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12
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Conte R, Arrighini GP, Guidotti C. Direct evaluation via forced oscillation method of the electronic state density of sizable clusters. J Comput Chem 2007; 28:584-93. [PMID: 17186475 DOI: 10.1002/jcc.20501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
Metal clusters described in the framework of a simple tight binding model have been studied. Application of a dynamical approach (FOM), jointly with efficient storage of the sparse Hamiltonian matrix involved, is shown to allow direct evaluation of the electronic state density of sizable cubic-symmetry aggregates.
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Affiliation(s)
- R Conte
- Scuola Normale Superiore, Piazza Dei Cavalieri 7, 56126 Pisa, Italy.
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Weber V, Niklasson AMN, Challacombe M. Higher-order response in O(N) by perturbed projection. J Chem Phys 2005; 123:044106. [PMID: 16095345 DOI: 10.1063/1.1944724] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Perturbed projection for linear scaling solution of the coupled-perturbed self-consistent-field equations [V. Weber, A.M.N. Niklasson, and M. Challacombe, Phys. Rev. Lett. 92, 193002 (2004)] is extended to the computation of higher-order static response properties. Although generally applicable, perturbed projection is further developed here in the context of the self-consistent first and second electric hyperpolarizabilities at the Hartree-Fock level of theory. Nonorthogonal, density-matrix analogs of Wigner's 2n+1 rule valid for linear one-electron perturbations are given up to fourth order. Linear scaling and locality of the higher-order response densities under perturbation by a global electric field are demonstrated for three-dimensional water clusters.
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Affiliation(s)
- Valéry Weber
- Department of Chemistry, University of Fribourg, 1700 Fribourg, Switzerland.
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Niklasson AMN, Weber V, Challacombe M. Nonorthogonal density-matrix perturbation theory. J Chem Phys 2005; 123:044107. [PMID: 16095346 DOI: 10.1063/1.1944725] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Recursive density-matrix perturbation theory [A.M.N. Niklasson and M. Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] provides an efficient framework for the linear scaling computation of materials response properties [V. Weber, A.M.N. Niklasson, and M. Challacombe, Phys. Rev. Lett. 92, 193002 (2004)]. In this article, we generalize the density-matrix perturbation theory to include properties computed with a perturbation-dependent nonorthogonal basis. Such properties include analytic derivatives of the energy with respect to nuclear displacement, as well as magnetic response computed with a field-dependent basis. The theory is developed in the context of linear scaling purification methods, which are briefly reviewed.
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Affiliation(s)
- Anders M N Niklasson
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
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Schuyler AD, Chirikjian GS, Lu JQ, Johnson HT. Random-walk statistics in moment-based O(N) tight binding and applications in carbon nanotubes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:046701. [PMID: 15903812 DOI: 10.1103/physreve.71.046701] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2004] [Indexed: 05/02/2023]
Abstract
A computational framework for a moment-based O(N) tight-binding atomistic method is presented, analyzed, and applied to the problem of electronic properties of deformed carbon nanotubes, where N is the number of atoms in the system. The moment-based approach is based on the maximum entropy and kernel polynomial methods for constructing the electronic density of states from local statistical information about the environment around individual atoms. Random-walk statistics are formally presented as the basis for several methods to collect the moments of the density of states in a computationally efficient manner. The computational complexity and accuracy of these methods are systematically analyzed. Using these methods for the problem of deformed carbon nanotubes, it is shown that the computational cost for some cases, per atom, scales as efficiently as O(M log M), where M is the desired number of moments in the expansion of the density of states. These methods are compared to other methods such as direct diagonalization and a Green's function approach.
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Affiliation(s)
- Adam D Schuyler
- Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218, USA
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Conte R, Arrighini G, Guidotti C. Direct evaluation of the density of states of simple metal clusters via forced oscillator method (FOM): a preliminary report. Chem Phys Lett 2004. [DOI: 10.1016/j.cplett.2004.07.087] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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17
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Liang W, Saravanan C, Shao Y, Baer R, Bell AT, Head-Gordon M. Improved Fermi operator expansion methods for fast electronic structure calculations. J Chem Phys 2003. [DOI: 10.1063/1.1590632] [Citation(s) in RCA: 78] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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18
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Tokmachev AM, Tchougréeff AL. Fast NDDO Method for Molecular Structure Calculations Based on Strictly Localized Geminals. J Phys Chem A 2003. [DOI: 10.1021/jp0265034] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Iitaka T, Ebisuzaki T. Calculating response functions in time domain with nonorthonormal basis sets. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:R3314-7. [PMID: 11088188 DOI: 10.1103/physreve.61.r3314] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/1999] [Indexed: 04/15/2023]
Abstract
We extend the recently proposed order-N algorithms for calculating linear- and nonlinear-response functions in time domain to the systems described by nonorthonormal basis sets.
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Affiliation(s)
- T Iitaka
- Division of Computational Science, RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
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20
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Liu S, Pérez-Jordá JM, Yang W. Nonorthogonal localized molecular orbitals in electronic structure theory. J Chem Phys 2000. [DOI: 10.1063/1.480730] [Citation(s) in RCA: 99] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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21
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Challacombe M. A simplified density matrix minimization for linear scaling self-consistent field theory. J Chem Phys 1999. [DOI: 10.1063/1.477969] [Citation(s) in RCA: 190] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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22
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Baer R, Head-Gordon M. Electronic structure of large systems: Coping with small gaps using the energy renormalization group method. J Chem Phys 1998. [DOI: 10.1063/1.477709] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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23
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Baer R, Head-Gordon M. Chebyshev expansion methods for electronic structure calculations on large molecular systems. J Chem Phys 1997. [DOI: 10.1063/1.474158] [Citation(s) in RCA: 66] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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24
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Mandelshtam VA, Taylor HS. Harmonic inversion of time signals and its applications. J Chem Phys 1997. [DOI: 10.1063/1.475324] [Citation(s) in RCA: 398] [Impact Index Per Article: 14.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
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25
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Schwegler E, Challacombe M, Head-Gordon M. Linear scaling computation of the Fock matrix. II. Rigorous bounds on exchange integrals and incremental Fock build. J Chem Phys 1997. [DOI: 10.1063/1.473833] [Citation(s) in RCA: 190] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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