A mathematical aspect of Hohenberg-Kohn theorem
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摘要
The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of matter.In this article, we study the Hohenberg-Kohn theorem for a class of external potentials based on a unique continuation principle.
The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of matter.In this article, we study the Hohenberg-Kohn theorem for a class of external potentials based on a unique continuation principle.
The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of matter.In this article, we study the Hohenberg-Kohn theorem for a class of external potentials based on a unique continuation principle.
引文
1 Adams R A, Fournier J J F. Sobolev Spaces, 2nd ed. Amsterdam:Academic Press, 2003
2 Ayers P W, Golden S, Levy M. Generalizations of the Hohenberg-Kohn theorem, I:Legendre transform constructions of variational principles for density matrices and electron distrbution functions. J Chem Phys, 2006, 124:054101
3 Boosting materials modelling. Nature Materials, 2016, 15:365,http://www.nature.com/nmat/journal/v15/n4/pdf/nmat4619.pdf
4 Eschrig H. The Fundamentals of Density Functional Theory. Leipzig:Eagle, 2003
5 Fournais S, Hoffmann-Ostenhof M, Hoffmann-Ostenhof T, et al. Positivity of the spherically averaged atomic oneelectron density. Math Z, 2008, 259:123-130
6 Hadjisavvas N, Theophilou A. Rigorous formulation of the Kohn-Sham theory. Phys Rev A(3), 1984, 30:2183-2186
7 Hohenberg P, Kohn W. The inhomogeneous electron gas. Phys Rev, 1964, 136:864-871
8 Jerison D, Kenig C E. Unique continuation and absence of positive eigenvalues for Schrodinger operators. Ann of Math(2), 1985, 121:463-494
9 Kohn W, Sham L J. Self-consistent equations including exchange and correlation effects. Phys Rev A, 1965, 140:4743-4754
10 Kryachko E S. On the original proof by reductio ad absurdum of the Hohenberg-Kohn theorem for many-electron Coulomb systems. Int J Quantum Chem, 2005, 103:818-823
11 Kryachko E S. On the proof by reductio ad absurdum of the Hohenberg-Kohn theorem for ensembles of fractionally occupied states of Coulomb systems. Int J Quantum Chem, 2006, 106:1795-1798
12 Kryachko E S, Ludena E V. Density functional theory:Foundations reviewed. Phys Rep, 2014, 544:123-239
13 Kvaal S, Helgaker T. Ground-state densities from the Rayleigh-Ritz variation principle and from density functional theory. J Chem Phys, 2015, 143:184106
14 Lammert P E. Differentiability of Lieb functional in electronic density functional theory. Int J Quantum Chem, 2007,107:1943-1953
15 Levy M. University variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solutions of the v-representability problems. Proc Natl Acad Sci USA, 1979, 76:6062-6065
16 Levy M. Electron densities in search of Hamiltonians. Phys Rev A, 1982, 26:1200-1208
17 Lieb E H. Density functionals for Coulomb systems. Int J Quantum Chem, 1983, 24:243-277
18 Martin R. Electronic Structure:Basic Theory and Practical Methods. London:Cambridge University Press, 2004
19 Parr R G, Yang W T. Density-Functional Theory of Atoms and Molecules. Oxford:Oxford University Press, 1989
20 Pino R, Bokanowski O, Ludena E V, et al. A re-statement of the Hohenberg-Kohn theorem and its extension to finite subspaces. Theor Chem Account, 2007, 118:557-561
21 Reed M, Simon B. Methods of Modern Mathematical Physics IV:Analysis of Operators. San Diego:Academic Press,1978
22 Redner S. Citation statistics from 110 years of physical review. Physics Today, 2005, 58:49-54
23 Schechter M, Simon B. Unique continuation for Schrodinger operators with unbounded potentials. J Math Anal Appl,1980, 77:482-492
24 Szczepanik W, Dulak M, Wesolowski T A. Comment on"On the original proof by reductio ad absurdum of the Hohenberg-Kohn theorem for many-electron Coulomb systems". Int J Quantum Chem, 2007, 107:762-763
25 van Noorden R, Maher B, Nuzzo R. The top 100 papers. Nature, 2014, 514:550-553
26 Wolff T H. Recent work on sharp estimates in second-order elliptic unique continuation problems. J Geom Anal, 1993,3:621-650
27 Zhou A. Hohenberg-Kohn theorem for Coulomb type systems and its generalization. J Math Chem, 2012, 50:2746-2754
28 Zhou A. Some open mathematical problems in electronic structure models and calculations(in Chinese). Sci Sin Math,2015, 45:929-938
1)We do not know if the particle wavefunction does not vanish on all sets of positive measure when the external potential is in L_(loc)~3(R~3)∩L~(3/2)(R~3)+L~∞(R3).
2 Ayers P W, Golden S, Levy M. Generalizations of the Hohenberg-Kohn theorem, I:Legendre transform constructions of variational principles for density matrices and electron distrbution functions. J Chem Phys, 2006, 124:054101
3 Boosting materials modelling. Nature Materials, 2016, 15:365,http://www.nature.com/nmat/journal/v15/n4/pdf/nmat4619.pdf
4 Eschrig H. The Fundamentals of Density Functional Theory. Leipzig:Eagle, 2003
5 Fournais S, Hoffmann-Ostenhof M, Hoffmann-Ostenhof T, et al. Positivity of the spherically averaged atomic oneelectron density. Math Z, 2008, 259:123-130
6 Hadjisavvas N, Theophilou A. Rigorous formulation of the Kohn-Sham theory. Phys Rev A(3), 1984, 30:2183-2186
7 Hohenberg P, Kohn W. The inhomogeneous electron gas. Phys Rev, 1964, 136:864-871
8 Jerison D, Kenig C E. Unique continuation and absence of positive eigenvalues for Schrodinger operators. Ann of Math(2), 1985, 121:463-494
9 Kohn W, Sham L J. Self-consistent equations including exchange and correlation effects. Phys Rev A, 1965, 140:4743-4754
10 Kryachko E S. On the original proof by reductio ad absurdum of the Hohenberg-Kohn theorem for many-electron Coulomb systems. Int J Quantum Chem, 2005, 103:818-823
11 Kryachko E S. On the proof by reductio ad absurdum of the Hohenberg-Kohn theorem for ensembles of fractionally occupied states of Coulomb systems. Int J Quantum Chem, 2006, 106:1795-1798
12 Kryachko E S, Ludena E V. Density functional theory:Foundations reviewed. Phys Rep, 2014, 544:123-239
13 Kvaal S, Helgaker T. Ground-state densities from the Rayleigh-Ritz variation principle and from density functional theory. J Chem Phys, 2015, 143:184106
14 Lammert P E. Differentiability of Lieb functional in electronic density functional theory. Int J Quantum Chem, 2007,107:1943-1953
15 Levy M. University variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solutions of the v-representability problems. Proc Natl Acad Sci USA, 1979, 76:6062-6065
16 Levy M. Electron densities in search of Hamiltonians. Phys Rev A, 1982, 26:1200-1208
17 Lieb E H. Density functionals for Coulomb systems. Int J Quantum Chem, 1983, 24:243-277
18 Martin R. Electronic Structure:Basic Theory and Practical Methods. London:Cambridge University Press, 2004
19 Parr R G, Yang W T. Density-Functional Theory of Atoms and Molecules. Oxford:Oxford University Press, 1989
20 Pino R, Bokanowski O, Ludena E V, et al. A re-statement of the Hohenberg-Kohn theorem and its extension to finite subspaces. Theor Chem Account, 2007, 118:557-561
21 Reed M, Simon B. Methods of Modern Mathematical Physics IV:Analysis of Operators. San Diego:Academic Press,1978
22 Redner S. Citation statistics from 110 years of physical review. Physics Today, 2005, 58:49-54
23 Schechter M, Simon B. Unique continuation for Schrodinger operators with unbounded potentials. J Math Anal Appl,1980, 77:482-492
24 Szczepanik W, Dulak M, Wesolowski T A. Comment on"On the original proof by reductio ad absurdum of the Hohenberg-Kohn theorem for many-electron Coulomb systems". Int J Quantum Chem, 2007, 107:762-763
25 van Noorden R, Maher B, Nuzzo R. The top 100 papers. Nature, 2014, 514:550-553
26 Wolff T H. Recent work on sharp estimates in second-order elliptic unique continuation problems. J Geom Anal, 1993,3:621-650
27 Zhou A. Hohenberg-Kohn theorem for Coulomb type systems and its generalization. J Math Chem, 2012, 50:2746-2754
28 Zhou A. Some open mathematical problems in electronic structure models and calculations(in Chinese). Sci Sin Math,2015, 45:929-938
1)We do not know if the particle wavefunction does not vanish on all sets of positive measure when the external potential is in L_(loc)~3(R~3)∩L~(3/2)(R~3)+L~∞(R3).