Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Exact orbital-free kinetic energy functional for general many-electron systems


The exact form of the kinetic energy functional has remained elusive in orbital-free models of density functional theory (DFT). This has been the main stumbling block for the development of a general-purpose framework on this basis. Here, we show that on the basis of a two-density model, which represents many-electron systems by mass density and spin density components, we can derive the exact form of such a functional. The exact functional is shown to contain previously suggested functionals to some extent, with the notable exception of the Thomas-Fermi kinetic energy functional.

This is a preview of subscription content, log in to check access.


  1. 1.

    M. Levy, Universal variational functionals of electron densities, first-order density matrices, and natural spinorbitals and solution of the v-representability problem, Proc. Natl. Acad. Sci. USA 76(12), 6062 (1979)

  2. 2.

    M. Levy, J. P. Perdew, and V. Sahni, Exact differential equation for the density and ionization energy of a many-particle system, Phys. Rev. A 30(5), 2745 (1984)

  3. 3.

    M. Pearson, E. Smargiassi, and P. Madden, Ab initio molecular dynamics with an orbital-free density functional, J. Phys.: Condens. Matter 5(19), 3221 (1993)

  4. 4.

    T. A. Wesolowski and Y. A. Wang, Recent Progress in Orbital Free Density Functional Theory, Vol. 6, World Scientific, 2013

  5. 5.

    J. Lehtomäki, I. Makkonen, M. A. Caro, A. Harju, and O. Lopez Acevedo, Orbital-free density functional theory implementation with the projector augmented-wave method, J. Chem. Phys. 141(23), 234102 (2014)

  6. 6.

    V. V. Karasiev and S. B. Trickey, Frank discussion of the status of ground-state orbital-free DFT, in: Advances in Quantum Chemistry, Vol. 71, Elsevier, 2015, pp 221–245

  7. 7.

    D. García-Aldea and J. Alvarellos, Approach to kinetic energy density functionals: Nonlocal terms with the structure of the von Weizsäcker functional, Phys. Rev. A 77(2), 022502 (2008)

  8. 8.

    C. Huang and E. A. Carter, Nonlocal orbital-free kinetic energy density functional for semiconductors, Phys. Rev. B 81(4), 045206 (2010)

  9. 9.

    I. Shin and E. A. Carter, Enhanced von Weizsäcker Wang-Govind-Carter kinetic energy density functional for semiconductors, J. Chem. Phys.140, 18A531 (2014)

  10. 10.

    W. Mi, A. Genova, and M. Pavanello, Nonlocal kinetic energy functionals by functional integration, J. Chem. Phys. 148(18), 184107 (2018)

  11. 11.

    L. A. Constantin, E. Fabiano, and F. Della Sala, Semilocal Pauli-Gaussian kinetic functionals for orbital-free density functional theory calculations of solids, J. Phys. Chem. Lett. 9(15), 4385 (2018)

  12. 12.

    L. A. Constantin, E. Fabiano, and F. Della Sala, Nonlocal kinetic energy functional from the jellium-with-gap model: Applications to orbital-free density functional theory, Phys. Rev. B 97(20), 205137 (2018)

  13. 13.

    M. Seidl, J. P. Perdew, and S. Kurth, Simulation of allorder density-functional perturbation theory, using the second order and the strong-correlation limit, Phys. Rev. Lett. 84(22), 5070 (2000)

  14. 14.

    T. Pope and W. Hofer, Spin in the extended electron model, Front. Phys. 12(3), 128503 (2017)

  15. 15.

    T. Pope and W. Hofer, A two-density approach to the general many-body problem and a proof of principle for small atoms and molecules, Front. Phys. 14(2), 23604 (2019)

  16. 16.

    P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136(3B), B864 (1964)

  17. 17.

    C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge University Press, 2003

  18. 18.

    S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. I. Probert, K. Refson, and M. C. Payne, First principles methods using CASTEP, Z. Kristallogr. Cryst. Mater. 220(5/6), 567 (2005)

  19. 19.

    P. Hasnip and M. Probert, Auxiliary density functionals: a new class of methods for efficient, stable density functional theory calculations, arXiv: 1503.01420 (2015)

  20. 20.

    W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140(4A), A1133 (1965)

  21. 21.

    J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Phys. Rev. B 23(10), 5048 (1981)

  22. 22.

    C. Von Weizsacker, On the theory of nuclear masses, Z. Phys. 96, 431 (1935)

Download references


The authors acknowledge EPSRC funding for the UKCP consortium (Grant No. EP/K013610/1). This work was also supported by the North East Centre for Energy Materials (NECEM). WH acknowledges support from the University of Chinese Academy of Sciences.

Author information

Correspondence to Thomas Pope or Werner Hofer.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pope, T., Hofer, W. Exact orbital-free kinetic energy functional for general many-electron systems. Front. Phys. 15, 23603 (2020).

Download citation


  • condensed matter
  • density functional theory (DFT)
  • extended electrons