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categories:
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- ABACUS
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## Abstract
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Recently, Zhao Haotian, a PhD candidate at the University of Science and Technology of China, and Professor He Lixin proposed and implemented a Hybrid Gauge Real-Time Time-Dependent Density Functional Theory (Hybrid gauge rt-TDDFT) applicable to atomic orbital basis sets in the domestic open-source density functional theory software ABACUS. Based on the traditional velocity gauge, this method introduces a time-varying phase dependent on the vector potential, effectively overcoming the systematic errors caused by the incompleteness of the local basis set and providing consistent and reliable simulation results in both periodic and non-periodic systems. At the same time, this method significantly improves the calculation efficiency in periodic systems, offering a new solution that combines accuracy and efficiency for first-principles real-time dynamics simulations under the action of an electric field.
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In rt-TDDFT, we can simulate the dynamic response of a system under external excitations such as an electric field by adding a time-dependent external field term to the Hamiltonian. The most commonly used is the length gauge, and its Hamiltonian form is relatively simple, containing only an additional scalar potential term:
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```math
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H = H_0 + E(t) \cdot r
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```
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where $`H_0`$is the Hamiltonian of the system itself, and $`E(t)\cdot r`$ describes the interaction between the electric field and electrons. The corresponding time-dependent Kohn-Sham equation is:
where H<sub>0</sub> is the Hamiltonian of the system itself, and E(t) ⋅ r describes the interaction between the electric field and electrons. The corresponding time-dependent Kohn-Sham equation is:
where V<sub>KS</sub> represents the Kohn-Sham effective potential, including electron-ion interaction, Hartree potential, exchange-correlation potential, etc. The length gauge is applicable to non-periodic systems. However, in periodic systems, this form destroys the periodic structure of the system, so it cannot be directly applied to calculations of crystal materials and other systems with periodic potential fields.To solve this problem, people usually use the method of gauge transformation to convert the Hamiltonian into the velocity gauge form. The electric field is derived from the time-dependent vector potential A(t):
where $`V_{KS}`$represents the Kohn-Sham effective potential, including electron-ion interaction, Hartree potential, exchange-correlation potential, etc. The length gauge is applicable to non-periodic systems. However, in periodic systems, this form destroys the periodic structure of the system, so it cannot be directly applied to calculations of crystal materials and other systems with periodic potential fields.To solve this problem, people usually use the method of gauge transformation to convert the Hamiltonian into the velocity gauge form. The electric field is derived from the time-dependent vector potential $`A(t)`$:
Theoretically, the length gauge and the velocity gauge are essentially equivalent, and they can be transformed into each other through the following gauge transformation:
However, in actual calculations, because we use finite basis sets (such as atomic orbitals), this phase relationship cannot be perfectly expressed, resulting in differences in calculation results under the two gauges. For example, in the atomic orbital basis set, the wave functions of both the length gauge and the velocity gauge are written in the form of a linear combination of atomic orbitals (LCAO):
Obviously, the wave function expression in the velocity gauge ignores the phase change inside the atomic orbitals caused by the vector potential, thus introducing systematic errors in the simulation, especially in the calculation of key physical quantities such as current and energy. Although the errors can be reduced by increasing the basis set, this will significantly increase the computational cost and severely limit the application efficiency of rt-TDDFT in periodic systems.
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To compensate for the phase effect introduced by the gauge transformation, we introduce a phase factor dependent on the vector potential in front of each atomic orbital:
The above transformation can be regarded as a gauge transformation. Since this gauge contains both the vector potential $`A(t)`$ and the scalar field $`E(t)`$, we call it the hybrid gauge. The Hamiltonian form in the hybrid gauge is similar to that in the length gauge, but its external field term $`\boldsymbol{E}(t)\cdot\Delta\boldsymbol{r}_{\nu i}`$remains periodic, avoiding the defect of the length gauge that violates Bloch's theorem in periodic systems while retaining its advantage of simple calculation. Compared with the velocity gauge, the hybrid gauge also has significant advantages in computational efficiency, and it is more efficient in dealing with non-local potentials. Other physical quantities, such as charge density and current density, can also be processed in a similar way. Since there is a strict gauge transformation relationship between the hybrid gauge and the length gauge, the calculation results of the two are theoretically completely equivalent.
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The above transformation can be regarded as a gauge transformation. Since this gauge contains both the vector potential A(t) and the scalar field E(t), we call it the hybrid gauge. The Hamiltonian form in the hybrid gauge is similar to that in the length gauge, but its external field term <strong>E</strong>(t) ⋅Δ<strong>r</strong><sub>νi</sub> remains periodic, avoiding the defect of the length gauge that violates Bloch's theorem in periodic systems while retaining its advantage of simple calculation. Compared with the velocity gauge, the hybrid gauge also has significant advantages in computational efficiency, and it is more efficient in dealing with non-local potentials. Other physical quantities, such as charge density and current density, can also be processed in a similar way. Since there is a strict gauge transformation relationship between the hybrid gauge and the length gauge, the calculation results of the two are theoretically completely equivalent.
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## Results
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For non-periodic systems, all three gauges can be applied. The figure above shows the response current of the $`H_{2}CO_{3}`$ molecule under Gaussian pulse excitation and its corresponding absorption spectrum. It can be seen that the results given by the length gauge and the hybrid gauge are completely consistent, verifying their theoretical equivalence. In contrast, there is a significant deviation between the response current calculated by the velocity gauge and those of the length gauge and the hybrid gauge. In particular, its corresponding spectrum shows significant divergence behavior in the low - frequency region, reflecting the systematic errors introduced by the velocity gauge in the local basis set. Although this error can be alleviated by increasing the basis set size, the resulting computational cost also increases significantly. Therefore, the authors proposed an empirical correction method that can significantly reduce the errors caused by the velocity gauge while maintaining a low computational cost.
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For non-periodic systems, all three gauges can be applied. The figure above shows the response current of the H<sub>2</sub>CO<sub>3</sub> molecule under Gaussian pulse excitation and its corresponding absorption spectrum. It can be seen that the results given by the length gauge and the hybrid gauge are completely consistent, verifying their theoretical equivalence. In contrast, there is a significant deviation between the response current calculated by the velocity gauge and those of the length gauge and the hybrid gauge. In particular, its corresponding spectrum shows significant divergence behavior in the low - frequency region, reflecting the systematic errors introduced by the velocity gauge in the local basis set. Although this error can be alleviated by increasing the basis set size, the resulting computational cost also increases significantly. Therefore, the authors proposed an empirical correction method that can significantly reduce the errors caused by the velocity gauge while maintaining a low computational cost.
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### Periodic Systems
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To test the applicability of the hybrid gauge in periodic systems and under strong fields, we also tested the linear and nonlinear response behaviors of the Si system.
The figure above shows the response current of the Si unit cell under pulse excitation and its corresponding dielectric function. It can be seen from the results that the velocity gauge has obvious systematic errors, but through the correction scheme we proposed, these errors can be effectively alleviated. In contrast, the hybrid gauge can stably provide more accurate and physically consistent results.
The figure above shows the response current of the Si unit cell under strong Gaussian pulse excitation and its corresponding high - order harmonic spectrum. As the maximum field strength increases from $`10^{10},W/cm^{2}`$ to $`10^{13},W/cm^{2}`$, the current signal gradually deviates from the fundamental frequency of the external field, exhibiting an increasingly enhanced nonlinear response characteristic. Meanwhile, the 1st, 3rd, 5th, and 7th harmonic signals appear successively in the high - order harmonic spectrum.
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The figure above shows the response current of the Si unit cell under strong Gaussian pulse excitation and its corresponding high - order harmonic spectrum. As the maximum field strength increases from 10<sup>10</sup>, W/cm<sup>2</sup> to 10<sup>13</sup>, W/cm<sup>2</sup>, the current signal gradually deviates from the fundamental frequency of the external field, exhibiting an increasingly enhanced nonlinear response characteristic. Meanwhile, the 1st, 3rd, 5th, and 7th harmonic signals appear successively in the high - order harmonic spectrum.
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The above results indicate that the hybrid gauge is applicable in different physical scenarios, including non - periodic and periodic systems, as well as strong - field and weak - field perturbations. It demonstrates good versatility and accuracy. Compared with the velocity gauge, it shows higher computational accuracy and efficiency in periodic systems. It can achieve stable and reliable real - time dynamics simulations under small basis set conditions, providing solid theoretical and technical support for the subsequent development of rt - TDDFT methods based on NAO.
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