Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals

Previous Article
An interface problem: The two-layer shallow water equations
DCDS Home
This Issue
Next Article
Rational approximations of semigroups without scaling and squaring

November 2013, 33(11&12): 5319-5325. doi: 10.3934/dcds.2013.33.5319

Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals

John P. Perdew ^1, and Adrienn Ruzsinszky ^1,

1.	Department of Physics and Quantum Theory Group, Tulane University, New Orleans, LA 70123, United States, United States

Received December 2011 Published May 2013

Abstract
Related Papers

The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, ``nature's glue". (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.

Keywords: kinetic energy, Thomas-Fermi, density functional., exchange energy, averaging.

Mathematics Subject Classification: 41, 45, 46, 49, 8.

Citation: John P. Perdew, Adrienn Ruzsinszky. Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5319-5325. doi: 10.3934/dcds.2013.33.5319

References:

[1]	A. Messiah, "Quantum Mechanics," Dover, 1999.
[2]	L. H. Thomas, The calculation of atomic fields, Proc. Cambridge Philos. Soc., 23 (1926), 542-548. doi: 10.1017/S0305004100011683.
[3]	E. Fermi, Un metodo statistico per la determinazione di alcune proprieta dell atomo, Rend. Accad. Naz. Licei, 6 (1927), 602-607.
[4]	J. A. Goldstein and G. R. Rieder, Some extensions of Thomas-Fermi theory, Lecture Notes in Mathematics, 1223 (1986), 110-121. doi: 10.1007/BFb0099187.
[5]	J. A. Goldstein and G. R. Rieder, Recent rigorous results in Thomas-Fermi theory, Lecture Notes in Mathematics, 1394 (1989), 68-82. doi: 10.1007/BFb0086753.
[6]	P. Benilan, J. A. Goldstein and G. R. Rieder, Nonlinear elliptic system arising in electron-density theory, Communications in Partial Differential Equations, 17 (1992), 2079-2092. doi: 10.1080/03605309208820914.
[7]	G. R. Rieder, J. A. Goldstein and N. Naima, A convexified energy functional for the Fermi-Amaldi correction, Discrete and Continuous Systems, 28 (2010), 41-65. doi: 10.3934/dcds.2010.28.41.
[8]	P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev., 136 (1964), B864-B871. doi: 10.1103/PhysRev.136.B864.
[9]	W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation, Phys. Rev., 140 (1965), A11333-A1138. doi: 10.1103/PhysRev.140.A1133.
[10]	S. Kurth and J. P. Perdew, Role of the exchange-correlation energy: Nature's glue, Int. J. Quantum Chem., 77 (2000), 819-830. doi: 10.1002/(SICI)1097-461X(2000)77:5<814::AID-QUA3>3.0.CO;2-F.
[11]	J. P. Perdew, L. A. Constantin, E. Sagvolden and K. Burke, Relevance of the slowly-varying electron gas to atoms, molecules, and solids, Phys. Rev. Lett., 97 (2006), 223002, 4 pages. doi: 10.1103/PhysRevLett.97.223002.
[12]	J. Schwinger, Thomas-Fermi model: The leading correction, Phys. Rev. A, 22 (1980), 1827-1832; Thomas-Fermi model: The second correction, ibid., 24 (1981), 2353-2361. doi: 10.1103/PhysRevA.22.1827.
[13]	B. G. Englert and J. Schwinger, Statistical atom: Some quantum improvements, Phys. Rev. A, 29 (1984), 2339-2352; Semiclassical atom, ibid., 32 (1985), 26-35. doi: 10.1103/PhysRevA.29.2339.
[14]	E. H. Lieb, The stability of matter, Rev. Mod. Phys., 48 (1976), 553-569. doi: 10.1103/RevModPhys.48.553.
[15]	L. A. Constantin, J. C. Snyder, J. P. Perdew and K. Burke, Ionization potentials in the limit of large atomic number, J. Chem. Phys., 133 (2010), 241103, 4 pages. doi: 10.1063/1.3522767.
[16]	J. P. Perdew and S. Kurth, Density functionals for non-relativistic Coulomb systems in the new century, in "A Primer in Density Functional Theory" ( eds. C. Fiolhais, F. Nogueira and M. Marques), Lecture Notes in Physics, 620 (2003), 1-55. doi: 10.1007/3-540-37072-2_1.
[17]	J. P. Perdew and L. A. Constantin, Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy, Phys. Rev. B, 75 (2007), 155109, 9 pages. doi: 10.1103/PhysRevB.75.155109.
[18]	A. Cangi, D. Lee, P. Elliott, K. Burke and E. K. U. Gross, Electronic structure via potential functional approximations, Phys. Rev. Lett., 106 (2011), 236404, 4 pages. doi: 10.1103/PhysRevLett.106.236404.
[19]	D. C. Langreth and J. P. Perdew, The exchange-correlation energy of a metallic surface, Solid State Commun., 17 (1975), 1425-1429. doi: 10.1016/0038-1098(75)90618-3.
[20]	O. Gunnarsson and B. I. Lundqvist, Exchange and correlation in atoms, molecules, and solids, Phys. Rev. B, 13 (1976), 4274-4298. doi: 10.1016/0375-9601(76)90557-0.
[21]	D. C. Langreth and J. P. Perdew, Exchange-correlation energy of a metallic surface: Wavevector analysis, Phys. Rev. B, 15 (1977), 2884-2901.

show all references

References:

[1]	A. Messiah, "Quantum Mechanics," Dover, 1999.
[2]	L. H. Thomas, The calculation of atomic fields, Proc. Cambridge Philos. Soc., 23 (1926), 542-548. doi: 10.1017/S0305004100011683.
[3]	E. Fermi, Un metodo statistico per la determinazione di alcune proprieta dell atomo, Rend. Accad. Naz. Licei, 6 (1927), 602-607.
[4]	J. A. Goldstein and G. R. Rieder, Some extensions of Thomas-Fermi theory, Lecture Notes in Mathematics, 1223 (1986), 110-121. doi: 10.1007/BFb0099187.
[5]	J. A. Goldstein and G. R. Rieder, Recent rigorous results in Thomas-Fermi theory, Lecture Notes in Mathematics, 1394 (1989), 68-82. doi: 10.1007/BFb0086753.
[6]	P. Benilan, J. A. Goldstein and G. R. Rieder, Nonlinear elliptic system arising in electron-density theory, Communications in Partial Differential Equations, 17 (1992), 2079-2092. doi: 10.1080/03605309208820914.
[7]	G. R. Rieder, J. A. Goldstein and N. Naima, A convexified energy functional for the Fermi-Amaldi correction, Discrete and Continuous Systems, 28 (2010), 41-65. doi: 10.3934/dcds.2010.28.41.
[8]	P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev., 136 (1964), B864-B871. doi: 10.1103/PhysRev.136.B864.
[9]	W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation, Phys. Rev., 140 (1965), A11333-A1138. doi: 10.1103/PhysRev.140.A1133.
[10]	S. Kurth and J. P. Perdew, Role of the exchange-correlation energy: Nature's glue, Int. J. Quantum Chem., 77 (2000), 819-830. doi: 10.1002/(SICI)1097-461X(2000)77:5<814::AID-QUA3>3.0.CO;2-F.
[11]	J. P. Perdew, L. A. Constantin, E. Sagvolden and K. Burke, Relevance of the slowly-varying electron gas to atoms, molecules, and solids, Phys. Rev. Lett., 97 (2006), 223002, 4 pages. doi: 10.1103/PhysRevLett.97.223002.
[12]	J. Schwinger, Thomas-Fermi model: The leading correction, Phys. Rev. A, 22 (1980), 1827-1832; Thomas-Fermi model: The second correction, ibid., 24 (1981), 2353-2361. doi: 10.1103/PhysRevA.22.1827.
[13]	B. G. Englert and J. Schwinger, Statistical atom: Some quantum improvements, Phys. Rev. A, 29 (1984), 2339-2352; Semiclassical atom, ibid., 32 (1985), 26-35. doi: 10.1103/PhysRevA.29.2339.
[14]	E. H. Lieb, The stability of matter, Rev. Mod. Phys., 48 (1976), 553-569. doi: 10.1103/RevModPhys.48.553.
[15]	L. A. Constantin, J. C. Snyder, J. P. Perdew and K. Burke, Ionization potentials in the limit of large atomic number, J. Chem. Phys., 133 (2010), 241103, 4 pages. doi: 10.1063/1.3522767.
[16]	J. P. Perdew and S. Kurth, Density functionals for non-relativistic Coulomb systems in the new century, in "A Primer in Density Functional Theory" ( eds. C. Fiolhais, F. Nogueira and M. Marques), Lecture Notes in Physics, 620 (2003), 1-55. doi: 10.1007/3-540-37072-2_1.
[17]	J. P. Perdew and L. A. Constantin, Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy, Phys. Rev. B, 75 (2007), 155109, 9 pages. doi: 10.1103/PhysRevB.75.155109.
[18]	A. Cangi, D. Lee, P. Elliott, K. Burke and E. K. U. Gross, Electronic structure via potential functional approximations, Phys. Rev. Lett., 106 (2011), 236404, 4 pages. doi: 10.1103/PhysRevLett.106.236404.
[19]	D. C. Langreth and J. P. Perdew, The exchange-correlation energy of a metallic surface, Solid State Commun., 17 (1975), 1425-1429. doi: 10.1016/0038-1098(75)90618-3.
[20]	O. Gunnarsson and B. I. Lundqvist, Exchange and correlation in atoms, molecules, and solids, Phys. Rev. B, 13 (1976), 4274-4298. doi: 10.1016/0375-9601(76)90557-0.
[21]	D. C. Langreth and J. P. Perdew, Exchange-correlation energy of a metallic surface: Wavevector analysis, Phys. Rev. B, 15 (1977), 2884-2901.

[1]	Gisèle Ruiz Goldstein, Jerome A. Goldstein, Naima Naheed. A convexified energy functional for the Fermi-Amaldi correction. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 41-65. doi: 10.3934/dcds.2010.28.41
[2]	Wei Deng, Baisheng Yan. On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics. Evolution Equations and Control Theory, 2013, 2 (4) : 599-620. doi: 10.3934/eect.2013.2.599
[3]	Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911
[4]	Sergio Grillo, Leandro Salomone, Marcela Zuccalli. Explicit solutions of the kinetic and potential matching conditions of the energy shaping method. Journal of Geometric Mechanics, 2021, 13 (4) : 629-646. doi: 10.3934/jgm.2021022
[5]	Simone Creo, Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. Approximation of a nonlinear fractal energy functional on varying Hilbert spaces. Communications on Pure and Applied Analysis, 2018, 17 (2) : 647-669. doi: 10.3934/cpaa.2018035
[6]	Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic and Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1
[7]	Rafael O. Ruggiero. On the density of mechanical Lagrangians in $T^{2}$ without continuous invariant graphs in all supercritical energy levels. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 661-679. doi: 10.3934/dcdsb.2008.10.661
[8]	Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119
[9]	Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure and Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[10]	Julius Fergy T. Rabago, Hideyuki Azegami. A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem. Evolution Equations and Control Theory, 2019, 8 (4) : 785-824. doi: 10.3934/eect.2019038
[11]	Jorge Cortés. Energy conserving nonholonomic integrators. Conference Publications, 2003, 2003 (Special) : 189-199. doi: 10.3934/proc.2003.2003.189
[12]	Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063
[13]	Luisa Berchialla, Luigi Galgani, Antonio Giorgilli. Localization of energy in FPU chains. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 855-866. doi: 10.3934/dcds.2004.11.855
[14]	Michele Carriero, Antonio Leaci, Franco Tomarelli. Uniform density estimates for Blake & Zisserman functional. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1129-1150. doi: 10.3934/dcds.2011.31.1129
[15]	Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic and Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85
[16]	Patricia Bauman, Andrea C. Rubiano. Energy-minimizing nematic elastomers. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 259-282. doi: 10.3934/dcdss.2015.8.259
[17]	Antonin Chambolle, Francesco Doveri. Minimizing movements of the Mumford and Shah energy. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153
[18]	Gisèle Ruiz Goldstein, Jerome A. Goldstein, Fabiana Travessini De Cezaro. Equipartition of energy for nonautonomous wave equations. Discrete and Continuous Dynamical Systems - S, 2017, 10 (1) : 75-85. doi: 10.3934/dcdss.2017004
[19]	Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. On a system of semirelativistic equations in the energy space. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1343-1355. doi: 10.3934/cpaa.2015.14.1343
[20]	Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
John P. Perdew Adrienn Ruzsinszky
on Google Scholar
John P. Perdew Adrienn Ruzsinszky

[Back to Top]