# On the v-Representabilty Problem in Density Functional Theory: Application to Non-Interacting Systems

^{*}

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Background

#### 2.2. Functional Differentiation

#### 2.3. Illustration of the Formalism

^{®}[22] to accomplish that task. We tested cases up to $N=6$, and obtained every time the corresponding potential up to a constant. That is not surprising because we have shown in Appendix A that the formalism to calculate those functional derivatives leads to the potential. Equation (A9) also obtains a constant that does not have a physical meaning. Applying l’Hopital’s rule twice at the right boundary for the particles in a box example we obtain the constant:

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Functional Differentiation of the Kinetic Energy (Non-Interacting)

## Appendix B. Cioslowski’s Procedure

## References

- Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev.
**1964**, 136, B864–B871. [Google Scholar] [CrossRef] - Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev.
**1965**, 140, A1133–A1138. [Google Scholar] [CrossRef] - Levy, M. Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc. Natl. Acad. Sci. USA
**1979**, 76, 6062–6065. [Google Scholar] [CrossRef] [PubMed] - Levy, M. Electron densities in search of Hamiltonians. Phys. Rev. A
**1982**, 26, 1200–1208. [Google Scholar] [CrossRef] - Shimony, A.; Feshbach, H. (Eds.) Density functionals for coulomb systems. In Physics as Natural Philosophy: Essays in Honor of Laszlo Tisza on His 75th Birthday; MIT Press: Cambridge, MA, USA, 1982; pp. 111–149.
- Dreizler, R.; Providencia, J. (Eds.) Density Functional Methods in Physics; Nato ASI Series B123; Plenum Press: Berlin/Heidelberg, Germany, 1985.
- Kohn, W. v-representability and density functional theory. Phys. Rev. Lett.
**1983**, 51, 1596–1598. [Google Scholar] [CrossRef] - Englisch, H.; Englisch, R. Hohenberg–Kohn theorem and non-v-representable densities. Phys. A Stat. Mech. Appl.
**1983**, 121, 253–268. [Google Scholar] [CrossRef] - Chen, J.; Stott, M.J. v-representability for noninteracting systems. Phys. Rev. A
**1993**, 47, 153–160. [Google Scholar] [CrossRef] [PubMed] - Parr, R.G.; Yang, C.Y. Density Functional Theory of Atoms and Molecules; Oxford University Press: Oxford, UK, 1989. [Google Scholar]
- Dreitzler, R.M.; Gross, E.K.U. Density Functional Theory; Springer Verlag: Berlin, Germany; New York, NY, USA, 1990. [Google Scholar]
- Gonis, A.; Zhang, X.G.; Däne, M.; Stocks, G.; Nicholson, D. Reformulation of density functional theory for N-representable densities and the resolution of the v-representability problem. J. Phys. Chem. Solids
**2016**, 89, 23–31. [Google Scholar] [CrossRef] - Gonis, A.; Däne, M.; Nicholson, D.; Stocks, G. Computationally simple, analytic, closed form solution of the Coulomb self-interaction problem in Kohn–Sham density functional theory. Solid State Commun.
**2012**, 152, 771–774. [Google Scholar] [CrossRef] - Däne, M.; Gonis, A.; Nicholson, D.; Stocks, G. On a solution of the self-interaction problem in Kohn–Sham density functional theory. J. Phys. Chem. Solids
**2014**, 75, 1160–1178. [Google Scholar] [CrossRef] - Däne, M.; Gonis, A.; Nicholson, D.; Stocks, G. Solving the self-interaction problem in Kohn–Sham density functional theory: Application to atoms. J. Phys. Chem. Solids
**2015**, 79, 55–65. [Google Scholar] [CrossRef] - Cioslowski, J. Density functionals for the energy of electronic systems: Explicit variational construction. Phys. Rev. Lett.
**1988**, 60, 2141–2143. [Google Scholar] [CrossRef] [PubMed] - Cioslowski, J. Density driven self-consistent field method. II. Construction of all one-particle wavefunctions that are orthonormal and sum up to a given density. Int. J. Quantum Chem.
**1989**, 36, 255–262. [Google Scholar] [CrossRef] - Lieb, E.H. Density functionals for coulomb systems. In Physics as Natural Philosophy, Essays in Honor of Laszlo Tisza; Shimony, A., Feshbach, H., Eds.; MIT Press: Cambridge, MA, USA, 1982. [Google Scholar]
- Macke, W. Zur wellenmechanischen Behandlung von Vielkörperproblemen. Ann. Phys.
**1955**, 452, 1–9. (In German) [Google Scholar] [CrossRef] - Harriman, J.E. Orthonormal orbitals for the representation of an arbitrary density. Phys. Rev. A
**1981**, 24, 680–682. [Google Scholar] [CrossRef] - Zumbach, G.; Maschke, K. New approach to the calculation of density functionals. Phys. Rev. A
**1983**, 28, 544–554. [Google Scholar] [CrossRef] - Mathematica; Version 10.4; Wolfram Research, Inc.: Champaign, IL, USA, 2016.

**Figure 1.**The six lowest in energy wave functions (

**left**) and the corresponding single particle densities (

**right**) for the particle-in-a-box model (L = 1). For illustrative purposes, all curves are shifted by the corresponding energies (see Equation (16)) in atomic units $\hslash =m=1$.

**Figure 2.**The six lowest in energy wave functions (

**left**) and the corresponding single particle densities (

**right**) for the Harmonic oscillator. All curves are shifted by the corresponding energies (see Equation (19)). The quadratic potential is shown in black.

**Figure 3.**Results for the square well example, as described in the text. The given density is shown in the top row, where the new orbitals obtained though Cioslowski’s procedure [16] using initial functions from Table 1. The potential, obtained using functional differentiation of the expression of the kinetic energy, is plotted in the third row. The three rows at the bottom show the results of the test (see Equation (21)), where the dashed line corresponds to the left-hand side of the equation, and the solid line to the right-hand side, equal to the new orbital, both normalized. Some of the dashed functions are multiplied by $-1$ for better comparison, corresponding to a negative α in Equation (21).

Column | Test Orbital 1 | Test Orbital 2 | Test Orbital 3 |
---|---|---|---|

A | $\sqrt{2}sin\left(\pi x\right)$ | $\sqrt{2}sin\left(2\pi x\right)$ | $\sqrt{2}sin\left(3\pi x\right)$ |

B | 1 | $cos\left(2\pi x\right)$ | $cos\left(4\pi x\right)$ |

C | 1 | x | ${x}^{2}$ |

D | $\frac{1}{4}+x$ | ${\left(\right)}^{\frac{1}{2}}$ | ${\left(\right)}^{\frac{3}{4}}$ |

Column | Test Orbital 1 | Test Orbital 2 | Test Orbital 3 |
---|---|---|---|

A | $\frac{1}{\sqrt[4]{\pi}}{e}^{-\frac{1}{2}{x}^{2}}$ | $\frac{\sqrt{2}}{\sqrt[4]{\pi}}{e}^{-\frac{1}{2}{x}^{2}}x$ | $\frac{1}{\sqrt{2}\sqrt[4]{\pi}}{e}^{-\frac{1}{2}{x}^{2}}\left(\right)open="("\; close=")">2{x}^{2}-1$ |

B | 1 | x | ${x}^{2}$ |

C | 1 | $\frac{x}{6}$ | $\frac{{x}^{2}}{18}-1$ |

D | 1 | $cos\left(\right)open="("\; close=")">\frac{\pi x}{6}$ | $cos\left(\right)open="("\; close=")">\frac{\pi x}{3}$ |

A | B | C | D | |
---|---|---|---|---|

orbital 1 | 0 | 0.5134 | 0.2508 | 0.0148 |

orbital 2 | 0 | 0.7795 | 0.1833 | 0.1158 |

orbital 3 | 0 | 1.9006 | 0.1694 | 0.0428 |

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Däne, M.; Gonis, A.
On the v-Representabilty Problem in Density Functional Theory: Application to Non-Interacting Systems. *Computation* **2016**, *4*, 24.
https://doi.org/10.3390/computation4030024

**AMA Style**

Däne M, Gonis A.
On the v-Representabilty Problem in Density Functional Theory: Application to Non-Interacting Systems. *Computation*. 2016; 4(3):24.
https://doi.org/10.3390/computation4030024

**Chicago/Turabian Style**

Däne, Markus, and Antonios Gonis.
2016. "On the v-Representabilty Problem in Density Functional Theory: Application to Non-Interacting Systems" *Computation* 4, no. 3: 24.
https://doi.org/10.3390/computation4030024