# Real-Time Extension of TAO-DFT

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Ground-State Theory: TAO-DFT

#### 2.1. Overview of TAO-DFT

#### 2.2. Density Representation in TAO-DFT

#### 2.3. Approximate Energy Functionals and Fictitious Temperatures in TAO-DFT

#### 2.4. Comparison of KS-DFT, TAO-DFT, and FT-DFT

#### 2.5. TAO-DFT-Related Methods

#### 2.5.1. TAO-DFT with ${E}_{\mathrm{xc}\theta}\left[\rho \right]\approx {E}_{\mathrm{xc}}\left[\rho \right]$

#### 2.5.2. KS-DFA with the rTAO Energy Correction

## 3. Real-Time Theory: RT-TAO-DFT

#### 3.1. RT-TAO Equation

#### 3.2. Matrix Representation

- Construct the initial one-electron density matrix $\mathbf{P}\left(0\right)$ (see Equation (56)) and the initial RT-TAO matrix $\mathbf{F}\left(0\right)$ (see Equation (57)) for the GS of the unperturbed physical system at time $t=0$ using TAO-DFT (i.e., the respective GS theory).
- Apply the TD field to the physical system for $t>0$, and propagate the one-electron density matrix $\mathbf{P}\left(t\right)$ and the RT-TAO matrix $\mathbf{F}\left(t\right)$ in the time domain, according to the RT-TAO equation (given by the matrix representation, e.g., see Equation (55)).
- Post-process the resulting TD observables (electron density, dipole moment, etc.).

## 4. HHG Spectra from RT-TAO-DFT

- ${\mathrm{H}}_{2}$ with an equilibrium bond length of 1.45 bohr (≈0.767 Å).
- ${\mathrm{H}}_{2}$ with a stretched bond length of 3.78 bohr (≈2.00 Å).

^{2}), the laser frequency (also called the fundamental frequency) ${\omega}_{0}$ = 1.5498 eV (corresponding to the wavelength ${\lambda}_{0}\approx $ 800 nm), and ${\sigma}_{p}$ = 500 a.u. (≈12.1 fs) are adopted.

^{2}, and the maximum potential value V

_{max}= 10 hartree.

_{2}with a stretched bond length of 3.78 bohr, obtained with spin-restricted and spin-unrestricted TD-ALDA (i.e., RT-TAO-ALDA with θ = 0), are distinctly different, yielding unphysical spin-symmetry breaking effects in all the TD properties examined. Such an unphysical spin-symmetry breaking feature of spin-unrestricted TD-ALDA is apparently undesirable for RT simulations. By contrast, the spin-symmetry breaking effects in the TD properties obtained with RT-TAO-ALDA are shown to be reducible with the increase in θ at essentially no additional computational cost. In particular, the TD properties obtained with spin-restricted and spin-unrestricted RT-TAO-ALDA (with θ = 40 mhartree) are essentially the same, yielding essentially no unphysical spin-symmetry breaking effects in all the TD properties examined. This desirable feature can be attributed to the satisfaction of spin-symmetry constraint on the singlet GS density of the stretched H

_{2}by spin-unrestricted TAO-LDA (with θ = 40 mhartree) [26].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Number of bound electrons for ${\mathrm{H}}_{2}$ with an equilibrium bond length of 1.45 bohr, obtained with spin-restricted and spin-unrestricted RT-TAO-ALDA (with various $\theta $). Here, the $\theta =0$ case corresponds to TD-ALDA.

**Figure 3.**Induced dipole moment for ${\mathrm{H}}_{2}$ with an equilibrium bond length of 1.45 bohr, obtained with spin-restricted and spin-unrestricted RT-TAO-ALDA (with various $\theta $). Here, the $\theta =0$ case corresponds to TD-ALDA.

**Figure 4.**HHG spectrum for ${\mathrm{H}}_{2}$ with an equilibrium bond length of 1.45 bohr, obtained with spin-restricted and spin-unrestricted RT-TAO-ALDA (with various $\theta $). Here, the $\theta =0$ case corresponds to TD-ALDA.

**Figure 5.**Number of bound electrons for ${\mathrm{H}}_{2}$ with a stretched bond length of 3.78 bohr, obtained with spin-restricted and spin-unrestricted RT-TAO-ALDA (with various $\theta $). Here, the $\theta =0$ case corresponds to TD-ALDA.

**Figure 6.**Induced dipole moment for ${\mathrm{H}}_{2}$ with a stretched bond length of 3.78 bohr, obtained with spin-restricted and spin-unrestricted RT-TAO-ALDA (with various $\theta $). Here, the $\theta =0$ case corresponds to TD-ALDA.

**Figure 7.**HHG spectrum for ${\mathrm{H}}_{2}$ with a stretched bond length of 3.78 bohr, obtained with spin-restricted and spin-unrestricted RT-TAO-ALDA (with various $\theta $). Here, the $\theta =0$ case corresponds to TD-ALDA.

KS-DFT | TAO-DFT | FT-DFT | |
---|---|---|---|

Electronic Temperature ${\theta}_{el}$ | 0 | 0 | ≥0 |

Fictitious Temperature $\theta $ | 0 | ≥0 | ≥0 |

Is $\theta ={\theta}_{el}$ assumed? | Yes | No | Yes |

Electronic Property | GS | GS | Thermal Equilibrium |

Electron Density | GS | GS | Thermal Equilibrium |

Density Representation | NI-PS-VR | NI-TE-VR | NI-TE-VR |

Universal Functional | Hohenberg–Kohn | Hohenberg–Kohn | Mermin |

Approximate Functional | ${E}_{\mathrm{xc}}\left[\rho \right]$ | ${E}_{\mathrm{xc}\theta}\left[\rho \right]$ | ${F}_{\mathrm{xc}}^{{\theta}_{el}}\left[{\rho}^{{\theta}_{el}}\right]$ |

TAO-DFT (with ${\mathit{E}}_{\mathbf{xc}\mathit{\theta}}\left[\mathit{\rho}\right]\approx {\mathit{E}}_{\mathbf{xc}}\left[\mathit{\rho}\right]$) | FT-DFT (with ${\mathit{F}}_{\mathbf{xc}}^{{\mathit{\theta}}_{\mathbf{el}}}\left[{\mathit{\rho}}^{{\mathit{\theta}}_{\mathbf{el}}}\right]\approx {\mathit{E}}_{\mathbf{xc}}\left[{\mathit{\rho}}^{{\mathit{\theta}}_{\mathbf{el}}}\right]$) | |
---|---|---|

Electronic Temperature ${\theta}_{el}$ | 0 | ≥0 |

Fictitious Temperature $\theta $ | ≥0 | ≥0 |

Is $\theta ={\theta}_{el}$ assumed? | No | Yes |

Electronic Property | GS | Thermal Equilibrium |

Electron Density | GS | Thermal Equilibrium |

Density Representation | NI-TE-VR | NI-TE-VR |

Approximate Functional | ${E}_{\mathrm{xc}}\left[\rho \right]$ | ${E}_{\mathrm{xc}}\left[{\rho}^{{\theta}_{el}}\right]$ |

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Tsai, H.-Y.; Chai, J.-D.
Real-Time Extension of TAO-DFT. *Molecules* **2023**, *28*, 7247.
https://doi.org/10.3390/molecules28217247

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Tsai H-Y, Chai J-D.
Real-Time Extension of TAO-DFT. *Molecules*. 2023; 28(21):7247.
https://doi.org/10.3390/molecules28217247

**Chicago/Turabian Style**

Tsai, Hung-Yi, and Jeng-Da Chai.
2023. "Real-Time Extension of TAO-DFT" *Molecules* 28, no. 21: 7247.
https://doi.org/10.3390/molecules28217247