# Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Results and Discussion

## 4. Materials and Methods

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hohenberg, P.; Kohn, W. Inhomogeous Electron Gas. Phys. Rev. B
**1964**, 136, 864–871. [Google Scholar] [CrossRef][Green Version] - Yonei, K.; Tomishima, Y. On the Weizsäcker Correction to the Thomas-Fermi Theory of the Atom. J. Phys. Soc. Jpn.
**1965**, 20, 1051–1057. [Google Scholar] [CrossRef] - Yang, W. Gradient correction in Thomas-Fermi theory. Phys. Rev. A
**1986**, 34, 4575–4585. [Google Scholar] [CrossRef] - Karasiev, V.; Trickey, S.B. Frank Discussion of the Status of Ground-state Orbital-free DFT. Adv. Quantum Chem.
**2015**, 71, 221–245. [Google Scholar] - Dreizler, R.M.; Gross, E.K.U. Density Functional Theory; Springer: Berlin/Heidelberg, Germany, 1990. [Google Scholar]
- Kohn, W.; Sham, L.J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. A
**1965**, 140, 1133–1138. [Google Scholar] [CrossRef][Green Version] - Burke, K. Perspective on density functional theory. J. Chem. Phys.
**2012**, 136, 150901. [Google Scholar] [CrossRef] [PubMed] - Becke, A.D. Perspective: Fifty years of density-functional theory in chemical physics. J. Chem. Phys.
**2014**, 140, 18A301. [Google Scholar] [CrossRef][Green Version] - Nagy, A. A thermal orbital-free density functional approach. J. Phys. Chem.
**2019**, 151, 014103. [Google Scholar] [CrossRef] [PubMed] - Levämäki, H.; Nagy, A.; Kokko, K.; Vitos, L. Cusp relation for the Pauli potential. Phys. Rev. A
**2014**, 90, 062515. [Google Scholar] [CrossRef][Green Version] - March, N.H. The local potential determining the square root of the ground-state electron density of atoms and molecules from the Schrödinger equation. Phys. Lett. A
**1986**, 113, 476–478. [Google Scholar] [CrossRef] - von Weizsäcker, C.F. Zur Theorie der Kernmassen. Z. Phys.
**1935**, 96, 431–458. [Google Scholar] [CrossRef] - Levy, M.; Ou-Yang, H. Exact properties of the Pauli potential for the square root of the electron density and the kinetic energy functional. Phys. Rev. A
**1988**, 38, 625–629. [Google Scholar] [CrossRef] - Nagy, A. Analysis of the Pauli potential of atoms and ions. Acta Phys. Hung.
**1991**, 70, 321–331. [Google Scholar] - Nagy, A.; March, N.H. The exact form of the Pauli potential for the ground state of two- and three-level atoms and Ions. Int. J. Quantum Chem.
**1991**, 39, 615–623. [Google Scholar] [CrossRef] - Nagy, A.; March, N.H. Relation between the Pauli potential and the Pauli energy density in an inhomogeneous electron liquid. Phys. Chem. Liq.
**1992**, 25, 37–42. [Google Scholar] [CrossRef] - Holas, A.; March, N.H. Exact theorems concerning non-interacting kinetic energy density functional in D dimensions and their implications for gradient expansions. Int. J. Quantum Chem.
**1995**, 56, 371–383. [Google Scholar] [CrossRef] - Amovilli, C.; March, N.H. Kinetic energy density in terms of electron density for closed-shell atoms in a bare Coulomb field. Int. J. Quantum Chem.
**1998**, 66, 281–283. [Google Scholar] [CrossRef] - Nagy, A. Alternative descriptors of Coulomb systems and their relationship to the kinetic energy. Chem. Phys. Lett.
**2008**, 460, 343–346. [Google Scholar] [CrossRef][Green Version] - Nagy, A. The Pauli potential from the differential virial theorem. Int. J. Quantum Chem.
**2010**, 110, 2117–2120. [Google Scholar] [CrossRef] - Nagy, A. Functional derivative of the kinetic energy functional for spherically symmetric systems. J. Chem. Phys.
**2011**, 135, 044106. [Google Scholar] [CrossRef][Green Version] - Tsirelson, V.G.; Stash, A.I.; Karasiev, V.V.; Liu, S. Pauli potential and Pauli charge from experimental electron density. Comput. Theor. Chem.
**2013**, 106, 92–99. [Google Scholar] [CrossRef] - Trickey, S.; Karasiev, V.V.; Vela, A. Positivity constraints and information-theoretical kinetic energy functionals. Phys. Rev. B
**2011**, 84, 075146. [Google Scholar] [CrossRef][Green Version] - Karasiev, V.; Chakraborty, D.; Trickey, S.B. Progress on new approaches to old ideas: Orbital-free Density Functionals. In Many-Electron Approaches in Physics, Chemistry and Mathematics; Delle Site, L., Bach, V., Eds.; Springer: Heidelberg, Germany, 2014; pp. 113–134. [Google Scholar]
- Thomas, L.H. The calculation of atomic fields. Proc. Camb. Philos. Soc.
**1927**, 23, 542–548. [Google Scholar] [CrossRef][Green Version] - Fermi, E. Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente. Z. Phys.
**1928**, 48, 73–79. [Google Scholar] [CrossRef] - Kirzhnits, D.A. Quantum Corrections to the Thomas-Fermi Equation. Sov. Phys. JETP
**1957**, 5, 64–71. [Google Scholar] - Hodges, C.H. Quantum Corrections to the Thomas-Fermi Approximation—The Kirzhnits Method. Can. J. Phys.
**1973**, 51, 1428–1437. [Google Scholar] [CrossRef] - Murphy, D.R. Sixth-order term of the gradient expansion of the kinetic-energy density functional. Phys. Rev. A
**1981**, 24, 1682–1688. [Google Scholar] [CrossRef] - Yang, W.; Parr, R.G.; Lee, C. Various functionals for the kinetic energy density of an atom or molecule. Phys. Rev. A
**1986**, 34, 4586–4590. [Google Scholar] [CrossRef] - Lee, C.L.; Ghosh, S.K. Density gradient expansion of the kinetic-energy functional for molecules. Phys. Rev. A
**1986**, 33, 3506–3507. [Google Scholar] [CrossRef] [PubMed] - Kozlowski, P.M.; Nalewajski, R.F. A Graph Approach to the Gradient Expansion of Density Functionals. Int. J. Quantum Chem.
**1986**, 30, 219–226. [Google Scholar] [CrossRef] - Lee, H.; Lee, C.; Parr, R.G. Conjoint gradient correction to the Hartree-Fock kinetic- and exchange-energy density functionals. Phys. Rev. A
**1991**, 44, 768–771. [Google Scholar] [CrossRef] [PubMed] - Thakkar, A.J. Comparison of kinetic-energy density functionals. Phys. Rev. A
**1992**, 46, 6920–6924. [Google Scholar] [CrossRef] [PubMed] - Liu, S.; Parr, R.G. Expansion of density functionals in terms of homogeneous functionals: Justification and nonlocal representation of the kinetic energy, exchange energy and classical Coulomb repulsion energy for atoms. Phys. Rev. A
**1997**, 55, 1792–1798. [Google Scholar] [CrossRef] - Tran, F.; Wesolowski, T.A. Link between the Kinetic- and Exchange-Energy Functionals in the Generalized Gradient Approximation. Int. J. Quantum Chem.
**2002**, 89, 441–446. [Google Scholar] [CrossRef] - Ayers, P.W.; Lucks, J.B.; Parr, R.G. Constructing exact density functionals from the moments of the electron density. Acta Chim. Phys. Debrecina
**2002**, 34, 223–248. [Google Scholar] - Chai, J.D.; Weeks, J.D. Modified Statistical Treatment of Kinetic Energy in the Thomas-Fermi Model. J. Phys. Chem. B
**2004**, 108, 6870–6876. [Google Scholar] [CrossRef][Green Version] - Ghiringhelli, L.M.; Delle Site, L. Design of kinetic functionals for many body electron systems: Combining analytical theory with Monte Carlo sampling of electronic configurations. Phys. Rev. B
**2008**, 77, 073104. [Google Scholar] [CrossRef][Green Version] - Lee, D.; Constantin, L.A.; Perdew, J.P.; Burke, K. Condition on the Kohn-Sham kinetic energy and modern parametrization of the Thomas-Fermi density. J. Chem. Phys.
**2009**, 130, 034107. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ghiringhelli, L.M.; Hamilton, I.P.; Delle Site, L. Interacting electrons, spin statistics, and information theory. J. Chem. Phys.
**2010**, 132, 014106. [Google Scholar] [CrossRef][Green Version] - Salazar, E.X.; Guarderas, P.F.; Ludeña, E.V.; Cornejo, M.H.; Karasiev, V.V. Study of some simple approximations to the non-interacting kinetic energy functional. Int. J. Quantum Chem.
**2016**, 116, 1313–1321. [Google Scholar] [CrossRef][Green Version] - Ludeña, E.V.; Salazar, E.X.; Cornejo, M.H.; Arroyo, D.E.; Karasiev, V.V. The Liu-Parr power series expansion of the Pauli kinetic energy functional with the incorporation of shell-inducing traits: Atoms. Int. J. Quantum Chem.
**2018**, 118, e25601. [Google Scholar] [CrossRef] - Wang, Y.A.; Carter, E.A. Orbital-free kinetic-energy density functional theory. In Theoretical Methods in Condensed Phase Chemistry; Schwarz, S.D., Ed.; Kluwer: New York, NY, USA, 2000; pp. 117–184. [Google Scholar]
- Ho, G.S.; Lignères, V.L.; Carter, E.A. Introducing PROFESS: A new program for orbital-free density functional calculations. Comput. Phys. Comm.
**2008**, 179, 839–854. [Google Scholar] [CrossRef] - Shin, I.; Carter, E.A. Enhanced von Weizsäcker Wang-Govind-Carter kinetic energy density functional for semiconductors. J. Chem. Phys.
**2014**, 140, 18A531. [Google Scholar] [CrossRef] - Witt, W.C.; del Rio, B.G.; Dieterich, J.M.; Carter, E.A. Orbital-free density functional theory for materials research. J. Mat. Res.
**2018**, 33, 777–795. [Google Scholar] [CrossRef] - Lehtomäki, J.; Makkonen, I.; Caro, M.A.; Harju, A.; Lopez-Acevedo, O. Orbital-free density functioal theory implementation with the projector augmented-wave method. J. Chem. Phys.
**2014**, 141, 234102. [Google Scholar] [CrossRef] [PubMed] - Ghosh, S.; Suryanarayana, P. Higher-order finite-difference formulation of Periodic Orbital-free Density Functional Theory. J. Comput. Phys.
**2016**, 307, 634–652. [Google Scholar] [CrossRef][Green Version] - Kraisler, E.; Schild, A. Discontinous behavior of the Pauli potential in density functional theory as a function of the electron number. Phys. Rev. Res.
**2020**, 2, 013159. [Google Scholar] [CrossRef][Green Version] - Kocák, J.; Kraisler, E.; Schild, A. On the relationship between the Kohn-Sham potential, the Pauli potential, and the Exact Eelectron Factorization. arXiv
**2020**, arXiv:2010.14885. [Google Scholar] - Finzel, K. Chemical bonding without orbitals. Comput. Theor. Chem.
**2018**, 1144, 50–55. [Google Scholar] [CrossRef] - Finzel, K. The first order atomic fragment approach—An orbital-free implementation of density functional theory. J. Chem. Phys.
**2019**, 151, 024109. [Google Scholar] [CrossRef] - Finzel, K. Equilibrium bond lengths from orbital-free density functional theory. Molecules
**2020**, 25, 1771. [Google Scholar] [CrossRef][Green Version] - Finzel, K.; Baranov, A.I. A simple model for the Slater exchange potential and its performance for solids. Int. J. Quantum Chem.
**2016**, 117, 40–47. [Google Scholar] [CrossRef] - Finzel, K. A fragment-based approximation of the Pauli kinetic energy. Theor. Chem. Acc.
**2018**, 137, 182. [Google Scholar] [CrossRef] - Finzel, K.; Kohout, M. A study of the basis set dependence of the bifunctional expression of the non-interacting kinetic energy for atomic systems. Comput. Theor. Chem.
**2019**, 1155, 56–60. [Google Scholar] [CrossRef] - Levy, M.; Perdew, J.P. Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms. Phys. Rev. A
**1985**, 32, 2010–2021. [Google Scholar] [CrossRef] - Lewis, G.N. The atom and the molecule. J. Am. Chem. Soc.
**1916**, 38, 762–785. [Google Scholar] [CrossRef][Green Version] - Kossel, W. Über Molekülbildung als Frage des Atombaus. Ann. Phys.
**1916**, 49, 229–362. [Google Scholar] [CrossRef][Green Version] - Pauling, L. The nature of the chemical bond. IV. The energy of single bonds and the relative electronegativity of atoms. J. Am. Chem. Soc.
**1932**, 54, 3570–3582. [Google Scholar] [CrossRef] - Ayers, P.W. Proof-of-principle functionals for the shape function. Phys. Rev. A
**2005**, 71, 062506-1–062506-8. [Google Scholar] [CrossRef] - Kutzelnigg, W. Einführung in die Theoretische Chemie; Wiley-VCH Verlag GmbH: Weinheim, Germany, 2002. [Google Scholar]
- Finzel, K. Analytical shell models for light atoms. Int. J. Quantum Chem.
**2021**, 121, e26212. [Google Scholar] [CrossRef][Green Version] - ADF 2017.01. SCM, Theoretical Chemistry; Vrije Universiteit: Amsterdam, The Netherlands, 2017; Available online: http://www.scm.com (accessed on 11 March 2021).
- Szabo, A.; Ostlund, N.S. Modern Quantum Chemistry—Introduction to Advanced Electronic Structure Theory; Dover Publications, Inc.: Mineola, NK, USA, 1996. [Google Scholar]
- Huber, K.P.; Herzberg, G. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules; Van Nostrand: Princenton, NJ, USA, 1979. [Google Scholar]
- Røeggen, I.; Veseth, L. Interatomic potential for the ${\Xi}^{1}{\Sigma}_{g}^{+}$ state of Be
_{2}, revisited. Int. J. Quantum Chem.**2005**, 101, 201–210. [Google Scholar] [CrossRef] - Kohout, M.; Savin, A. Atomic Shell Structure and Electron Numbers. Int. J. Quantum Chem.
**1996**, 60, 875–882. [Google Scholar] [CrossRef][Green Version] - Zener, C. Analytic atomic wave functions. Phys. Rev.
**1930**, 36, 51–56. [Google Scholar] [CrossRef] - Slater, J.C. Atomic shielding constants. Phys. Rev.
**1930**, 36, 57–64. [Google Scholar] [CrossRef]

**Figure 1.**Pauli potential and components for second-row dimers. The first column depicts the Pauli potential (PP) using the bare atomic fragment approach. In the second column, the molecular PP evaluated from Kohn–Sham orbitals is compiled, together with its components ${t}_{\mathrm{P}}\left(\overrightarrow{r}\right)/\rho \left(\overrightarrow{r}\right)$ and ${\sum}_{i}\left(\right)open="("\; close=")">{\u03f5}_{M}-{\u03f5}_{i}$, cf. Equation (8), shown in columns three and four, respectively. First row: Ne${}_{2}$ color scale from 0.0 (blue) to 41.0 (white). Second row: F${}_{2}$ color scale from 0.0 (blue) to 36.0 (white). Third row: O${}_{2}$ color scale from 0.0 (blue) to 26.0 (white). Fourth row: N${}_{2}$ color scale from 0.0 (blue) to 18.5 (white). Fifth row: C${}_{2}$ color scale from 0.0 (blue) to 14.3 (white). Sixth row: B${}_{2}$ color scale from 0.0 (blue) to 8.9 (white). Seventh row: Be${}_{2}$ color scale from 0.0 (blue) to 5.2 (white). Eighth row: Li${}_{2}$ color scale from 0.0 (blue) to 2.4 (white). Orthoslices are shown within the range of 5 × 8 bohr for all dimers.

**Figure 2.**Equilibrium bond length for second-row homonuclear dimers. Black squares: Experimental values [67,68], green circles: Kohn–Sham calculations from ADF/LDA/QZ4P level, blue diamonds: OF-DFT using the bare atomic fragment approach (some of the data previously published here [54]), red triangles: OF-DFT using the deformation potential and electron counts from MO theory.

**Table 1.**Equilibrium bond distances for N${}_{2}$ from OF-DFT approaches, including various amounts of constructive versus destructive interactions between the atoms. According to molecular orbital (MO), theory N${}_{2}$ exhibits eight constructive and two destructive interaction terms. The corresponding entry is indicated in bold. The equilibrium bond distance for the bare atomic fragment approach was already published elsewhere [54].

N_{2} | Constructive | ||||||
---|---|---|---|---|---|---|---|

0 | 2 | 4 | 6 | 8 | 10 | ||

destructive | 0 | 2.12 | 2.01 | 1.93 | 1.86 | 1.80 | 1.76 |

2 | nb | nb | 2.89 | 2.56 | 2.38 | 2.25 | |

4 | nb | nb | nb | nb | 3.28 | 2.93 | |

6 | nb | nb | nb | nb | nb | / | |

8 | nb | nb | nb | nb | / | / | |

10 | nb | nb | nb | / | / | / |

**Table 2.**Equilibrium bond distances for the second-row homonuclear dimers from OF-DFT using the deformation potentials in accordance with molecular orbital (MO) theory.

X_{2} | Constructive | |||||
---|---|---|---|---|---|---|

0 | 2 | 4 | 6 | 8 | ||

destructive | 0 | Ω | Li_{2}nc | |||

2 | nb | Be_{2}nb | B_{2}4.12 | C_{2}3.02 | N_{2}2.38 | |

4 | nb | nb | nb | nb | O_{2}2.77 | |

6 | nb | nb | nb | nb | F_{2}nb | |

8 | nb | nb | nb | nb | Ne_{2}nb |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Finzel, K.
Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory. *Molecules* **2021**, *26*, 1539.
https://doi.org/10.3390/molecules26061539

**AMA Style**

Finzel K.
Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory. *Molecules*. 2021; 26(6):1539.
https://doi.org/10.3390/molecules26061539

**Chicago/Turabian Style**

Finzel, Kati.
2021. "Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory" *Molecules* 26, no. 6: 1539.
https://doi.org/10.3390/molecules26061539