^{1}, Wenkel Liang

^{1}, Craig T. Chapman

^{1}, Christine M. Isborn

^{1}and Xiaosong Li

^{1,a)}

### Abstract

Nonperturbative electronic dynamics using the time-dependent Hartree-Fock (TDHF) and time-dependent Kohn-Sham (TDKS) theories with the adiabatic approximation is a powerful tool in obtaining insights into the interaction between a many-electron system and an external electromagnetic field. In practical applications of TDHF/TDKS using a truncated basis set, the electronic dynamics and molecular properties become gauge-dependent. Numerical simulations are carried out in the length gauge and velocity gauge to verify the extent of gauge-dependence using incomplete basis sets. Electronic dynamics of two many-electron systems, a helium atom and a carbon monoxide molecule in high-intensity linearly polarized radiation fields are performed using the TDHF and TDKS with two selected adiabatic exchange-correlation (xc) functionals. The time evolution of the expectation values of the dipole moment and harmonic spectra are calculated in the two gauges, and the basis set dependence on the gauge-invariance of these properties is investigated.

This work was supported by the U.S. National Science Foundation (CHE-CAREER 0844999), and the U.S. Department of Energy (DE-SC0006863). Computing support from the University of Washington Student Technology Fund is gratefully acknowledged. We also thank Professor David Masiello for helpful discussions.

I. INTRODUCTION

II. GAUGE INVARIANCE OF THE TDHF AND TDKS EQUATIONS

A. The many-body TDSE in a classical electromagnetic field

B. Gauge invariance of the TDHF formalism

C. The Ghosh-Dhara theorem and gauge invariance of the TDKS equations with adiabatic approximation

III. GAUGE PROPERTIES OF TDHF AND TDKS ELECTRONIC DYNAMICS IN INCOMPLETE BASIS SETS

A. Numerical methods

B. Results and discussion

1. Helium atom

2. CO molecule

IV. CONCLUSION

### Key Topics

- Electromagnetic interactions
- 16.0
- Adiabatic theorem
- 13.0
- Kinematics
- 12.0
- Wave functions
- 10.0
- Basis sets
- 7.0

## Figures

Time-dependent expectation values (in atomic units) of the dipole (first row), dipole velocity (second row), and dipole acceleration (third row) of the He atom using the d-aug-cc-pVQZ basis, obtained from TDHF (a)–(c), TDBLYP (d)–(f), and TDB3LYP (g)–(i). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

Time-dependent expectation values (in atomic units) of the dipole (first row), dipole velocity (second row), and dipole acceleration (third row) of the He atom using the d-aug-cc-pVQZ basis, obtained from TDHF (a)–(c), TDBLYP (d)–(f), and TDB3LYP (g)–(i). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

Time-dependent expectation values (in atomic units) of the dipole (first row), dipole velocity (second row), and dipole acceleration (third row) of the He atom using the aug-cc-pVTZ basis, obtained from TDHF (a)–(c), TDBLYP (d)–(f), and TDB3LYP (g)–(i). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

Time-dependent expectation values (in atomic units) of the dipole (first row), dipole velocity (second row), and dipole acceleration (third row) of the He atom using the aug-cc-pVTZ basis, obtained from TDHF (a)–(c), TDBLYP (d)–(f), and TDB3LYP (g)–(i). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

Time-dependent expectation values (in atomic units) of the dipole (first row), dipole velocity (second row), and dipole acceleration (third row) of the He atom using the 6-311++G** basis, obtained from TDHF (a)–(c), TDBLYP (d)–(f), and TDB3LYP (g)–(i). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

Time-dependent expectation values (in atomic units) of the dipole (first row), dipole velocity (second row), and dipole acceleration (third row) of the He atom using the 6-311++G** basis, obtained from TDHF (a)–(c), TDBLYP (d)–(f), and TDB3LYP (g)–(i). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

TDHF simulations of expectation values (in atomic units) of the dipole moment of the CO molecule in the basis of (a) 6-311++G**, (b) aug-cc-pVTZ, and (c) d-aug-cc-pVQZ, and the corresponding harmonic spectra (d)–(f). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

TDHF simulations of expectation values (in atomic units) of the dipole moment of the CO molecule in the basis of (a) 6-311++G**, (b) aug-cc-pVTZ, and (c) d-aug-cc-pVQZ, and the corresponding harmonic spectra (d)–(f). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

TDBLYP simulations of expectation values (in atomic units) of the dipole moment of the CO molecule in the basis of (a) 6-311++G**, (b) aug-cc-pVTZ, and (c) d-aug-cc-pVQZ, and the corresponding harmonic spectra (d)–(f). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

TDBLYP simulations of expectation values (in atomic units) of the dipole moment of the CO molecule in the basis of (a) 6-311++G**, (b) aug-cc-pVTZ, and (c) d-aug-cc-pVQZ, and the corresponding harmonic spectra (d)–(f). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

TDB3LYP simulations of expectation values (in atomic units) of the dipole moment of the CO molecule in the basis of (a) 6-311++G**, (b) aug-cc-pVTZ, and (c) d-aug-cc-pVQZ, and the corresponding harmonic spectra (d)–(f). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

TDB3LYP simulations of expectation values (in atomic units) of the dipole moment of the CO molecule in the basis of (a) 6-311++G**, (b) aug-cc-pVTZ, and (c) d-aug-cc-pVQZ, and the corresponding harmonic spectra (d)–(f). The red solid and blue dashed lines refer to the length gauge (LG) and the velocity gauge (VG), respectively.

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