Update ABACUS_15_04_2025.md (#242)

Author: cloudcloud666

source/_posts/ABACUS_15_04_2025.md

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 ## Abstract
 
 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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 ## Introduction
 
 Real-time time-dependent density functional theory (rt-TDDFT) is an important theoretical method for studying the electronic dynamics of materials in excited states and under the drive of external fields. It is widely used in frontier fields such as nonlinear optical responses, ultrafast spectroscopy, and charge and energy transport. Compared with linear response methods, rt-TDDFT can directly track the quantum dynamics trajectory of electrons during time evolution, making it particularly suitable for strong-field excitation and non-linear processes far from equilibrium. Therefore, it has become a key means for studying phenomena such as light-matter interaction and carrier relaxation. However, conducting rt-TDDFT calculations in periodic systems still faces challenges. The direct application of the traditional velocity gauge in numerical atomic orbital (NAO) basis sets ignores the internal phase changes of orbitals caused by the vector potential, resulting in systematic errors that seriously affect the calculation accuracy of key physical quantities such as current responses and nonlinear optics. To solve this problem, Zhao Haotian and He Lixin proposed the hybrid gauge algorithm. By explicitly introducing the local phase correction induced by the vector potential, it effectively overcomes the limitations of the velocity gauge in local basis sets and constructs a rigorous and efficient theoretical framework, providing a new solution for the study of ultrafast electron dynamics in periodic systems.
 
 ## Hybrid Gauge Theory
 
 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
 H = H_0 + E(t) \cdot r
 ```
 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:
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 ```math
 i\frac{\partial}{\partial t}\psi_{i}(r,t) = \left[-\frac{1}{2}\nabla^{2} + V_{KS}[\rho](r,t) + E(t) \cdot r\right]\psi_{i}(r,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 $`A(t)`$:
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 ```math
 E(t)=-\frac{dA(t)}{dt}
 ```
@@ -108,4 +114,4 @@ The above results indicate that the hybrid gauge is applicable in different phys
 ## Summary
 
 In this paper, a Hybrid Gauge Real-Time Time-Dependent Density Functional Theory (Hybrid gauge rt-TDDFT) based on atomic orbital basis sets is proposed and implemented. It effectively overcomes the systematic error problem of the velocity gauge in local basis sets, and demonstrates excellent accuracy and computational efficiency under various physical conditions, such as non-periodic and periodic systems, strong fields and weak fields. This method not only provides a reliable means for real-time electron dynamics simulations with small basis sets, but also lays a solid foundation for the further development and expansion of the domestic first-principles software ABACUS.
-The relevant results were recently published in Journal of Chemical Theory and Computation, and the link to the paper is as follows: https://doi.org/10.1021/acs.jctc.5c00111
+The relevant results were recently published in Journal of Chemical Theory and Computation, and the link to the paper is as follows: https://doi.org/10.1021/acs.jctc.5c00111