# Gauge-Invariant Formulation of Time-Dependent Configuration Interaction Singles Method

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. System Hamiltonian and Gauge Transformation

#### 2.2. Review of CI Coefficient-Based TDCIS with Fixed Orbitals

#### 2.3. Review of Channel Orbital-Based TDCIS with Fixed Orbitals

#### 2.4. Channel Orbital-Based TDCIS in the Velocity Gauge with Rotated Orbitals

#### 2.5. Evaluation of the Time Derivative of an Observable

## 3. Numerical Examples

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of the dipole moment of 1D-He exposed to a laser pulse with a wavelength of 750 nm and an intensity of (

**a**) 5 × 10${}^{14}$ W/cm${}^{2}$ and (

**b**) 1 × 10${}^{15}$ W/cm${}^{2}$. Comparison of the results with time-dependent configuration interaction singles (TDCIS) in the length gauge (LG), velocity gauge (VG), and rotated velocity-gauge (rVG) with that of the time-dependent Schrödinger equation (TDSE).

**Figure 2.**Time evolution of the dipole acceleration of 1D-He exposed to a laser pulse with a wavelength of 750 nm and an intensity of (

**a**) 5 × 10${}^{14}$ W/cm${}^{2}$ and (

**b**) 1 × 10${}^{15}$ W/cm${}^{2}$. Comparison of the results with TDCIS in the LG adopting Equations (44) and (45) with that of TDSE.

**Figure 3.**Time evolution of the dipole acceleration of 1D-He exposed to a laser pulse with a wavelength of 750 nm and an intensity of (

**a**) 5 × 10${}^{14}$ W/cm${}^{2}$ and (

**b**) 1 × 10${}^{15}$ W/cm${}^{2}$. Comparison of the results with TDCIS in the LG, VG, and rVG with that of TDSE.

**Figure 4.**High-harmonic spectrum of 1D-He exposed to a laser pulse with a wavelength of 750 nm and an intensity of (

**a**) 5 × 10${}^{14}$ W/cm${}^{2}$ and (

**b**) 1 × 10${}^{15}$ W/cm${}^{2}$. Comparison of the results with TDCIS in the LG, VG, and rVG with that of TDSE.

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**MDPI and ACS Style**

Sato, T.; Teramura, T.; Ishikawa, K.L. Gauge-Invariant Formulation of Time-Dependent Configuration Interaction Singles Method. *Appl. Sci.* **2018**, *8*, 433.
https://doi.org/10.3390/app8030433

**AMA Style**

Sato T, Teramura T, Ishikawa KL. Gauge-Invariant Formulation of Time-Dependent Configuration Interaction Singles Method. *Applied Sciences*. 2018; 8(3):433.
https://doi.org/10.3390/app8030433

**Chicago/Turabian Style**

Sato, Takeshi, Takuma Teramura, and Kenichi L. Ishikawa. 2018. "Gauge-Invariant Formulation of Time-Dependent Configuration Interaction Singles Method" *Applied Sciences* 8, no. 3: 433.
https://doi.org/10.3390/app8030433