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He X, Lu T, Rong C, Liu W, Ayers PW, Liu S. Energetic Information from Information-Theoretic Approach in Density Functional Theory as Quantitative Measures of Physicochemical Properties. J Chem Theory Comput 2024; 20:6049-6061. [PMID: 38995176 DOI: 10.1021/acs.jctc.4c00697] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 07/13/2024]
Abstract
The Hohenberg-Kohn theorem of density functional theory (DFT) stipulates that energy is a universal functional of electron density in the ground state, so energy can be thought of having encoded essential information for the density. Based on this, we recently proposed to quantify energetic information within the framework of information-theoretic approach (ITA) of DFT (J. Chem. Phys. 2022, 157, 101103). In this study, we systematically apply energetic information to a variety of chemical phenomena to validate the use of energetic information as quantitative measures of physicochemical properties. To that end, we employed six ITA quantities such as Shannon entropy and Fisher information for five energetic densities, yielding twenty-six viable energetic information quantities. Then, they are applied to correlate with physicochemical properties of molecular systems, including chemical bonding, conformational stability, intermolecular interactions, acidity, aromaticity, cooperativity, electrophilicity, nucleophilicity, and reactivity. Our results show that different quantities of energetic information often behave differently for different properties but a few of them, such as Shannon entropy of the total kinetic energy density and information gain of the Pauli energy density, stand out and strongly correlate with several properties across different categories of molecular systems. These results suggest that they can be employed as quantitative measures of physicochemical properties. This work not only enriches the body of our knowledge about the relationship between energy and information, but also provides scores of newly introduced explicit density functionals to quantify physicochemical properties, which can serve as robust features for building machine learning models in future studies.
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Affiliation(s)
- Xin He
- Institute of Frontier Chemistry, School of Chemistry and Chemical Engineering, Shandong University, Qingdao, Shandong 266237, China
| | - Tian Lu
- Beijing Kein Research Center for Natural Sciences, Beijing 100022, China
| | - Chunying Rong
- College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan 410081, China
| | - Wenjian Liu
- Institute of Frontier Chemistry, School of Chemistry and Chemical Engineering, Shandong University, Qingdao, Shandong 266237, China
| | - Paul W Ayers
- Department of Chemistry and Chemical Biology, McMaster University, Hamilton, Ontario ONL8S, Canada
| | - Shubin Liu
- Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, United States
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, United States
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He X, Lu T, Rong C, Liu S, Ayers PW, Liu W. Topological analysis of information-theoretic quantities in density functional theory. J Chem Phys 2023; 159:054112. [PMID: 37548307 DOI: 10.1063/5.0159941] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2023] [Accepted: 07/06/2023] [Indexed: 08/08/2023] Open
Abstract
We have witnessed considerable research interest in the recent literature about the development and applications of quantities from the information-theoretic approach (ITA) in density functional theory. These ITA quantities are explicit density functionals, whose local distributions in real space are continuous and well-behaved. In this work, we further develop ITA by systematically analyzing the topological behavior of its four representative quantities, Shannon entropy, two forms of Fisher information, and relative Shannon entropy (also called information gain or Kullback-Leibler divergence). Our results from their topological analyses for 103 molecular systems provide new insights into bonding interactions and physiochemical properties, such as electrophilicity, nucleophilicity, acidity, and aromaticity. We also compare our results with those from the electron density, electron localization function, localized orbital locator, and Laplacian functions. Our results offer a new methodological approach and practical tool for applications that are especially promising for elucidating chemical bonding and reactivity propensity.
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Affiliation(s)
- Xin He
- Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, China
| | - Tian Lu
- Beijing Kein Research Center for Natural Sciences, Beijing 100022, China
| | - Chunying Rong
- Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), Hunan Normal University, Changsha, Hunan 410081, China
| | - Shubin Liu
- Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, USA
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, USA
| | - Paul W Ayers
- Department of Chemistry and Chemical Biology, McMaster University, Hamilton, Ontario L8S 4M1, Canada
| | - Wenjian Liu
- Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, China
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Vázquez-Hernández H, Esquivel RO. Phenomenological description of the acidity of the citric acid and its deprotonated species: informational-theoretical study. J Mol Model 2023; 29:253. [PMID: 37464113 DOI: 10.1007/s00894-023-05589-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 05/11/2023] [Indexed: 07/20/2023]
Abstract
CONTEXT In spite of the fact that molecular acidity is a fundamental physicochemical property of molecular systems, the vast majority of theoretical studies have focused attention on monoprotic acids and on the prediction of pKa's. Polyprotic acids, represent a challenge for electronic structure calculations since the multiple acidic sites result in a vast group of species with different conformations and reactivities. In this work, Information-theoretic (IT) concepts of localizability, order and uniformity are applied to the Citric Acid and its deprotonated species through the one-electron density functionals: Shannon entropy (S), Fisher information (I) and Disequilibrium (D), respectively. We pursue the goal of characterizing the acidity of the aforementioned species with the aim to associate the IT concepts to chemical features such as the polarizability of the protonated/deprotonated species, the liability of the acidic sites, atomic electrostatic potentials, covalent bonding. IT analyses looks very promising for future studies on the acidity of specific deprotonation-sites of polyprotic acids. METHODS Density functional theory (DFT) calculations were performed with Gaussian 09 program. A sensitivity analysis of the IT-measures was performed for the citric acid and the citrate using B3LYP, B3PW91, BPW91, M05-2X, M06-2X and PBEPBE functionals with the 6-311++g(3df,2p), 6-311++g(d,p), 6.311+g(d,p) and aug-cc-pVDZ basis sets. The rest of the analysis was performed with the M05-2X/6-311+G(d,p) level of theory. Additionally, aqueous media was considered by use of the SMD solvent model. The IT-measures were calculated using a suite of programs developed in our laboratory jointly with the DGRID software package.
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Affiliation(s)
- Hazel Vázquez-Hernández
- Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. Ferrocarril San Rafael Atlixco 186, Colonia Leyes de Reforma, 09310, Mexico City, México
| | - Rodolfo O Esquivel
- Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. Ferrocarril San Rafael Atlixco 186, Colonia Leyes de Reforma, 09310, Mexico City, México.
- Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071, Granada, Spain.
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4
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Calculations of R eẤ nyi entropy, Tsallis entropy and Onicescu information energy for helium, lithium and beryllium atoms using an analytic correlated wave function. Chem Phys Lett 2023. [DOI: 10.1016/j.cplett.2023.140343] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
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5
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Chakraborty D, Chattaraj PK. Conceptual density functional theory based electronic structure principles. Chem Sci 2021; 12:6264-6279. [PMID: 34084424 PMCID: PMC8115084 DOI: 10.1039/d0sc07017c] [Citation(s) in RCA: 61] [Impact Index Per Article: 20.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/24/2020] [Accepted: 03/10/2021] [Indexed: 12/20/2022] Open
Abstract
In this review article, we intend to highlight the basic electronic structure principles and various reactivity descriptors as defined within the premise of conceptual density functional theory (CDFT). Over the past several decades, CDFT has proven its worth in providing valuable insights into various static as well as time-dependent physicochemical problems. Herein, having briefly outlined the basics of CDFT, we describe various situations where CDFT based reactivity theory could be employed in order to gain insights into the underlying mechanism of several chemical processes.
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Affiliation(s)
- Debdutta Chakraborty
- Department of Chemistry, KU Leuven Celestijnenlaan 200F-2404 3001 Leuven Belgium
| | - Pratim Kumar Chattaraj
- Department of Chemistry, Indian Institute of Technology Kharagpur 721302 West Bengal India +91 3222 255303 +91 3222 283304
- Department of Chemistry, Indian Institute of Technology Bombay Powai Mumbai 400076 India
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Alipour M, Khorrami M. Pauli energy and information-theoretic approach for evaluating dynamic and nondynamic electron correlation. Theor Chem Acc 2020. [DOI: 10.1007/s00214-020-02689-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Guégan F, Tognetti V, Martínez-Araya JI, Chermette H, Merzoud L, Toro-Labbé A, Morell C. A statistical thermodynamics view of electron density polarisation: application to chemical selectivity. Phys Chem Chem Phys 2020; 22:23553-23562. [PMID: 33073279 DOI: 10.1039/d0cp03228j] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
A fundamental link between conceptual density functional theory and statistical thermodynamics is herein drawn, showing that intermolecular electrostatic interactions can be understood in terms of effective work and heat exchange. From a more detailed analysis of the heat exchange in a perturbation theory framework, an associated entropy can be subsequently derived, which appears to be a suitable descriptor for the local polarisability of the electron density. A general rule of thumb is evidenced: the more the perturbation can be spread, both through space and among the excited states, the larger the heat exchange and entropy.
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Affiliation(s)
- Frédéric Guégan
- IC2MP UMR 7285, Université de Poitiers - CNRS, 4, rue Michel Brunet TSA, 51106-86073 Cedex 9 Poitiers, France.
| | - Vincent Tognetti
- Normandy Univ., COBRA UMR 6014 - FR 3038, Université de Rouen, INSA Rouen, CNRS, 1 rue Tesniére, 76821 Mont St Aignan, Cedex, France
| | - Jorge I Martínez-Araya
- Departamento de Ciencias Químicas, Facultad de Ciencias Exactas, Universidad Andres Bello (UNAB), Av. República 498, Santiago, Chile
| | - Henry Chermette
- Université de Lyon, Institut des Sciences Analytiques, UMR 5280, CNRS, Université Lyon 1 - 5, rue de la Doua, F-69100 Villeurbanne, France.
| | - Lynda Merzoud
- Université de Lyon, Institut des Sciences Analytiques, UMR 5280, CNRS, Université Lyon 1 - 5, rue de la Doua, F-69100 Villeurbanne, France.
| | - Alejandro Toro-Labbé
- Laboratorio de Química Teórica Computacional (QTC), Facultad de Química, Pontificia Universidad Católica de Chile, Santiago 7820436, Chile
| | - Christophe Morell
- Université de Lyon, Institut des Sciences Analytiques, UMR 5280, CNRS, Université Lyon 1 - 5, rue de la Doua, F-69100 Villeurbanne, France.
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Geerlings P, Chamorro E, Chattaraj PK, De Proft F, Gázquez JL, Liu S, Morell C, Toro-Labbé A, Vela A, Ayers P. Conceptual density functional theory: status, prospects, issues. Theor Chem Acc 2020. [DOI: 10.1007/s00214-020-2546-7] [Citation(s) in RCA: 77] [Impact Index Per Article: 19.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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9
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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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Matrodi A, Noorizadeh S. N-Derivatives of Shannon entropy density as response functions. Phys Chem Chem Phys 2020; 22:21535-21542. [PMID: 32959037 DOI: 10.1039/d0cp03808c] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022]
Abstract
The exact first and second order partial derivatives of Shannon entropy density with respect to the number of electrons at constant external potential are introduced as new descriptors for prediction of the active sites of a molecule. The derivatives, which are a measure of the inhomogeneity of electron density, are calculated both exactly (from analytical forms) and approximately (using the finite difference method) for some molecular systems. According to the maximum entropy principle, the extreme value of the first order derivative on the surface of a given molecule should determine the active sites of the molecule in electrophilic and nucleophilic attack. The second order derivative indicates where the Shannon entropy is more concentrated or depleted during the electron exchange. Although these derivatives on the surfaces of helium and neon atoms are uniform, the corresponding values for argon, krypton and xenon atoms are not. This could explain the greater tendency of heavy noble gas atoms to form stable compounds. A dual descriptor is also defined as the difference between the left and right first order derivatives of Shannon entropy density, which allows one to simultaneously predict the preferable sites for electrophilic and nucleophilic attack over the system at point r. Therefore, the reactivity of an atom in a molecule requires the non-uniformity of the first and second order derivatives of Shannon entropy density on the surface of that atom.
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Affiliation(s)
- Abdolkarim Matrodi
- Chemistry Department, Faculty of Sciences, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
| | - Siamak Noorizadeh
- Chemistry Department, Faculty of Sciences, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
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Shannon, Rényi, Tsallis Entropies and Onicescu Information Energy for Low-Lying Singly Excited States of Helium. ATOMS 2019. [DOI: 10.3390/atoms7030070] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
Knowledge of the electronic structures of atomic and molecular systems deepens our understanding of the desired system. In particular, several information-theoretic quantities, such as Shannon entropy, have been applied to quantify the extent of electron delocalization for the ground state of various systems. To explore excited states, we calculated Shannon entropy and two of its one-parameter generalizations, Rényi entropy of order α and Tsallis entropy of order α , and Onicescu Information Energy of order α for four low-lying singly excited states (1s2s 1 S e , 1s2s 3 S e , 1s3s 1 S e , and 1s3s 3 S e states) of helium. This paper compares the behavior of these three quantities of order 0.5 to 9 for the ground and four excited states. We found that, generally, a higher excited state had a larger Rényi entropy, larger Tsallis entropy, and smaller Onicescu information energy. However, this trend was not definite and the singlet–triplet reversal occurred for Rényi entropy, Tsallis entropy and Onicescu information energy at a certain range of order α .
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13
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Alipour M, Badooei Z. Toward Electron Correlation and Electronic Properties from the Perspective of Information Functional Theory. J Phys Chem A 2018; 122:6424-6437. [DOI: 10.1021/acs.jpca.8b05703] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Mojtaba Alipour
- Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71946-84795, Iran
| | - Zeinab Badooei
- Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71946-84795, Iran
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14
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Flores-Gallegos N. Tsallis’ entropy as a possible measure of the electron correlation in atomic systems. Chem Phys Lett 2018. [DOI: 10.1016/j.cplett.2017.12.014] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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15
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Ou JH, Ho YK. Shannon information entropy in position space for doubly excited states of helium with finite confinements. Chem Phys Lett 2017. [DOI: 10.1016/j.cplett.2017.10.007] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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16
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Heidar-Zadeh F, Vinogradov I, Ayers PW. Hirshfeld partitioning from non-extensive entropies. Theor Chem Acc 2017. [DOI: 10.1007/s00214-017-2077-z] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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López-Rosa S, Molina-Espíritu M, Esquivel RO, Soriano-Correa C, Dehesa JS. Study of the Chemical Space of Selected Bacteriostatic Sulfonamides from an Information Theory Point of View. Chemphyschem 2016; 17:4003-4010. [DOI: 10.1002/cphc.201600790] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2016] [Indexed: 11/08/2022]
Affiliation(s)
- Sheila López-Rosa
- Departamento de Fsica Aplicada II; Universidad de Sevilla; 41012 Sevilla Spain
- Instituto Carlos I de Fsica Teórica y Computacional; Universidad de Granada; 18071 Granada Spain
| | | | - Rodolfo O. Esquivel
- Instituto Carlos I de Fsica Teórica y Computacional; Universidad de Granada; 18071 Granada Spain
- Departamento de Química; Universidad Autónoma Metropolitana; 09340- Mexico City México
| | - Catalina Soriano-Correa
- Química Computacional, FES-Zaragoza; Universidad Nacional Autónoma de México; 09230- Iztapalapa, Mexico City México
| | - Jésus S. Dehesa
- Instituto Carlos I de Fsica Teórica y Computacional; Universidad de Granada; 18071 Granada Spain
- Departamento de Fsica Atómica, Molecular y Nuclear; Universidad de Granada; 18071 Granada Spain
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Esquivel RO, López-Rosa S, Molina-Espíritu M, Angulo JC, Dehesa JS. Information-theoretic space from simple atomic and molecular systems to biological and pharmacological molecules. Theor Chem Acc 2016. [DOI: 10.1007/s00214-016-2002-x] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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20
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Zhou XY, Rong C, Lu T, Zhou P, Liu S. Information Functional Theory: Electronic Properties as Functionals of Information for Atoms and Molecules. J Phys Chem A 2016; 120:3634-42. [PMID: 27115776 DOI: 10.1021/acs.jpca.6b01197] [Citation(s) in RCA: 49] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
How to accurately predict electronic properties of a Columbic system with the electron density obtained from experiments such as X-ray crystallography is still an unresolved problem. The information-theoretic approach recently developed in the framework of density functional reactivity theory is one of the efforts to address the issue. In this work, using 27 atoms and 41 molecules as illustrative examples, we present a study to demonstrate that one is able to satisfactorily describe such electronic properties as the total energy and its components with information-theoretic quantities like Shannon entropy, Fisher information, Ghosh-Berkowitz-Parr entropy, and Onicescu information energy. Closely related to the earlier attempt of expanding density functionals using simple homogeneous functionals, this work not only confirms Nagy's proof that Shannon entropy alone should contain all the information needed to adequately describe an electronic system but also provides a feasible pathway to map the relationship between the experimentally available electron density and various electronic properties for Columbic systems such as atoms and molecules. Extensions to other electronic properties are straightforward.
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Affiliation(s)
- Xia-Yu Zhou
- Department of Pharmacy, Changde Vocational Technical College , Changde, Hunan 415000, China
| | - Chunying Rong
- Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), College of Chemistry and Chemical Engineering, Hunan Normal University , Changsha, Hunan 410081, China
| | - Tian Lu
- Beijing Kein Research Center for Natural Sciences , Beijing 100022, P. R. China
| | - Panpan Zhou
- State Key Laboratory of Applied Organic Chemistry, Key Laboratory of Nonferrous Metal Chemistry and Resources Utilization of Gansu Province, College of Chemistry and Chemical Engineering, Lanzhou University , 222 South Tianshui Road, 730000 Lanzhou, P. R. China
| | - Shubin Liu
- Research Computing Center, University of North Carolina , Chapel Hill, North Carolina 27599-3420, United States
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Patrikeev L, Joubert L, Tognetti V. Atomic decomposition of Kohn–Sham molecular energies: the kinetic energy component. Mol Phys 2015. [DOI: 10.1080/00268976.2015.1113314] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Affiliation(s)
- Leonid Patrikeev
- Normandie Université COBRA, UMR 6014 et FR 3038, Université de Rouen, INSA Rouen, CNRS, Mont St Aignan, Cedex, France
| | - Laurent Joubert
- Normandie Université COBRA, UMR 6014 et FR 3038, Université de Rouen, INSA Rouen, CNRS, Mont St Aignan, Cedex, France
| | - Vincent Tognetti
- Normandie Université COBRA, UMR 6014 et FR 3038, Université de Rouen, INSA Rouen, CNRS, Mont St Aignan, Cedex, France
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22
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Lin CH, Ho YK. Shannon information entropy in position space for two-electron atomic systems. Chem Phys Lett 2015. [DOI: 10.1016/j.cplett.2015.05.029] [Citation(s) in RCA: 46] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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