# Excitons in Solids from Time-Dependent Density-Functional Theory: Assessing the Tamm-Dancoff Approximation

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Dyson Equation

#### 2.2. Casida Equation

#### 2.3. Local-Field Effect

#### 2.4. LRC Kernel: Head-Only vs. Diagonal

#### 2.5. Tamm-Dancoff Approximation

#### 2.6. Band-Gap Corrections: LDA vs. Scissors Shift

## 3. Computational Details

## 4. Results and Discussion

#### 4.1. Overview of LRC-Type Kernels

#### 4.2. Effect of the LRC Kernel on Optical Spectra

#### 4.3. TDA and Exciton Binding Energies

#### 4.4. Comparison of Dyson and Full Casida Equations

#### 4.5. Limitations of Our Findings

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Long-range-corrected (LRC) kernel strengths α (see Equation (1)) of LRC-type kernels for various materials.

**Figure 2.**Experimental [24,25] and calculated optical absorption spectra of GaAs (

**top**) and solid Ne (

**bottom**). For the LRC kernel, $\alpha =A{\alpha}_{0-\mathrm{Boot}}$ is used, where ${\alpha}_{0-\mathrm{Boot}}=0.064$ (25.3) for GaAs (solid Ne). The spectra are shifted to align the LDA gap with the experimental gap ${E}_{\mathrm{g}}^{\mathrm{exp}}$; the $GW$ gap ${E}_{\mathrm{g}}^{GW}$ is shown only for comparison. A Lorentzian broadening of 0.15 eV (0.2 eV) is used for GaAs (solid Ne). Note that $A=0.9$ and 1.1 approximately correspond to Bootstrap and RPA-Bootstrap kernels, respectively.

**Figure 3.**Calculated exciton binding energies ${E}_{\mathrm{b}}$ of solid Ne as a function of scaling factor A. For the LRC kernel, $\alpha =A{\alpha}_{0-\mathrm{Boot}}$ is used, where ${\alpha}_{0-\mathrm{Boot}}=25.3$.

Casida Equation | GaAs | α-GaN | β-GaN | AlN | MgO | LiF | Ar | Ne | |
---|---|---|---|---|---|---|---|---|---|

Exp. | 3.27 | 20.4 | 26.0 | 48.0 | 80.0 | 1600 | 1900 | 4080 | |

RPA-Boot | TDA | 0.334 | 0.927 | 0.875 | 0.00 | 1.72 | 33.3 | 37.7 | 666 |

0-Boot | TDA | 0.285 | 0.811 | 0.720 | 0.00 | 1.43 | 22.4 | 10.8 | 128 |

Boot | TDA | 0.267 | 0.651 | 0.562 | 0.00 | 1.03 | 10.7 | 7.70 | 39.7 |

JGM | TDA | 0.137 | 0.387 | 0.226 | 0.00 | 0.348 | 9.12 | 12.9 | 5.30 |

LRC | TDA | 0.636 | 1.16 | 1.14 | 0.00 | 0.747 | 1.61 | 1.46 | 1.01 |

RPA-Boot | Full | 0.344 | 1.06 | 1.01 | 0.00 | 2.12 | 94.7 | 96.0 | 2400 |

0-Boot | Full | 0.293 | 0.919 | 0.829 | 0.00 | 1.72 | 43.2 | 13.7 | 612 |

Boot | Full | 0.278 | 0.735 | 0.649 | 0.00 | 1.20 | 14.8 | 9.14 | 101 |

JGM | Full | 0.141 | 0.438 | 0.279 | 0.00 | 0.397 | 12.1 | 17.1 | 5.96 |

LRC | Full | 0.670 | 1.33 | 1.32 | 0.00 | 0.855 | 1.89 | 1.54 | 1.06 |

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Byun, Y.-M.; Ullrich, C.A.
Excitons in Solids from Time-Dependent Density-Functional Theory: Assessing the Tamm-Dancoff Approximation. *Computation* **2017**, *5*, 9.
https://doi.org/10.3390/computation5010009

**AMA Style**

Byun Y-M, Ullrich CA.
Excitons in Solids from Time-Dependent Density-Functional Theory: Assessing the Tamm-Dancoff Approximation. *Computation*. 2017; 5(1):9.
https://doi.org/10.3390/computation5010009

**Chicago/Turabian Style**

Byun, Young-Moo, and Carsten A. Ullrich.
2017. "Excitons in Solids from Time-Dependent Density-Functional Theory: Assessing the Tamm-Dancoff Approximation" *Computation* 5, no. 1: 9.
https://doi.org/10.3390/computation5010009