# Equilibrium Bond Lengths from 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

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**Figure 1.**Pauli potentials for the N2 molecule from the atomic fragment approach using closed shell fragments (first column) together with the first order level (third column) and KS data (second column). (

**a**) relative height field using closed-shell (cs) atomic fragments, (

**b**) relative height field using molecular Kohn-Sham (KS) orbitals, (

**c**) relative height field using groundstate (gs) atomic fragments, (

**d**) orthoslice using cs atomic fragments, (

**e**) orthoslice using molecular KS orbitals, (

**f**) orthoslice using gs atomic fragments, (

**g**) difference between the molecular and the cs Pauli potential (in black: isolines of the KS Pauli potential indicating regions of high contribution to the Pauli kinetic energy), (

**h**) difference between the cs and the gs Pauli potential, (

**i**) difference between the molecular and the gs Pauli potential (in black: isolines of the KS Pauli potential indicating regions of high contribution to the Pauli kinetic energy).

**Figure 2.**Energetic binding curve for N${}_{2}$ together with its kinetic and potential components. All data are evaluated as the difference between the molecular data and the groundstate atoms. Black: binding energy E, red: total kinetic energy ${T}_{s}$, orange: Pauli kinetic energy ${T}_{\mathrm{P}}$, green: von Weizsäcker kinetic energy ${T}_{\mathrm{W}}$, blue: total potential energy V. The dashed lines mark the minima of the respective curves. Inset: binding energy E around the equilibrium distance.

**Table 1.**Bond distances (in Bohr) obtained from experiment, Kohn-Sham (KS) data and orbital-free density functional theory (OF-DFT) using the atomic fragment approach of zeroth and first order level, together with the bond distances from the recent work using the atomic groundstate fragments (gs PP) and closed-shell atomic fragments (cs PP) in order to approximate the molecular Pauli potential, respectively. The respective relative errors to the experimental data $\delta $ are given in the subsequent columns. Their absolute average, the mean absolute percentage error (MAPE) is given in the last row. The zeroth order level has been performed with frozen fragments. Equilibrium bond length from first order level and the recent work are evaluated at valence optimized level [51]. Bond distances from KS data are obtained with ADF [54] using LDA(Xonly) level with the QZ4P basis sets.

OF-DFT | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Recent Work | |||||||||||

Exp. [55,56] | KS | $\mathit{\delta}$ | 0th [48] | $\mathit{\delta}$ | 1th [51] | $\mathit{\delta}$ | gs PP | $\mathit{\delta}$ | cs PP | $\mathit{\delta}$ | |

N${}_{2}$ | 2.07 | 2.09 | 0.8 | 2.9 | 39.8 | 2.30 | 10.9 | 2.26 | 8.7 | 2.12 | 2.1 |

O${}_{2}$ | 2.28 | 2.31 | 1.0 | 2.6 | 13.9 | 1.85 | −18.9 | 1.84 | −19.1 | 1.83 | −19.6 |

CO | 2.13 | 2.15 | 0.9 | 3.0 | 40.7 | 2.20 | 3.2 | 2.14 | 0.4 | 2.09 | −2.0 |

Be${}_{2}$ | 4.63 | 4.69 | 1.2 | 4.4 | −5.0 | 4.15 | −10.4 | 4.14 | −10.7 | 4.14 | −10.7 |

B${}_{2}$ | 3.00 | 3.08 | 2.6 | – | – | – | – | 3.16 | 5.3 | 3.01 | 0.0 |

C${}_{2}$ | 2.35 | 2.66 | 13.1 | – | – | – | – | 2.64 | 12.4 | 2.49 | 6.2 |

NO | 2.17 | 2.19 | 0.7 | – | – | – | – | 2.02 | −7.0 | 1.96 | −9.9 |

CN | 2.21 | 2.23 | 0.5 | – | – | – | – | 2.42 | 9.3 | 2.28 | 3.1 |

BeO | 2.52 | 2.54 | 1.2 | – | – | – | – | 2.53 | 0.6 | 2.52 | 0.3 |

MAPE | 2.4 | 24.9 | 10.8 | 8.2 | 6.0 |

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Finzel, K. Equilibrium Bond Lengths from Orbital-Free Density Functional Theory. *Molecules* **2020**, *25*, 1771.
https://doi.org/10.3390/molecules25081771

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Finzel K. Equilibrium Bond Lengths from Orbital-Free Density Functional Theory. *Molecules*. 2020; 25(8):1771.
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Finzel, Kati. 2020. "Equilibrium Bond Lengths from Orbital-Free Density Functional Theory" *Molecules* 25, no. 8: 1771.
https://doi.org/10.3390/molecules25081771