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The mystery of the local density approximation: Why is it crucial in quantum computing?
In the world of quantum computing, local density approximation (LDA) undoubtedly plays a special and important role. This approximation is derived from the value of the electron density at each point in space and is used in density functional theory (DFT) to describe the exchange-correlation energy functional. First introduced by Hugh Kohn and Lu Jeu Sham in 1965, LDA still plays a key role in many advanced computing methods today.
The main advantage of the local density approximation is that its calculation depends only on the local electron density, which makes it particularly effective in predicting many complex systems.
Application of LDA in electronic structure calculation
LDA is not only used for basic electronic structure calculations, but is also widely used in the study of semiconductor materials in solid-state physics. In particular, LDA provides an important computational basis for understanding the interaction between electrons and magnetism. Central to these studies is the complexity of the systems, which are extremely sensitive to synthesis parameters and therefore require first principles analysis.
Computational methods based on LDA, such as CASTEP and DMol3, are helpful in predicting the Fermi level and band structure of doped semiconductor oxides.
Challenges of Local Density Approximation
However, LDA is not without challenges. Underestimation of the energy gap is often a common problem when using LDA or the generalized gradient approximation (GGA), which can lead to incorrect predictions of defect-mediated conductivity and carrier-mediated magnetism. Since 1998, the application of Rayleigh's theorem for eigenvalues has provided more accurate results for calculating the energy gap of materials.
The accuracy of many calculated properties depends on a correct understanding of the second theorem of density functional theory.
Basics of the uniform electron gas model
The local density approximation is based on the uniform electron gas model, which maintains the neutrality of the system by placing the interacting electrons in a positive background charge. The success of this model lies in its ability to simplify the calculation of the total energy into contributions from kinetic energy, electrostatic interaction energy, and exchange-correlation energy.
Exchange and association functions
For the exchange energy density of HEG, we already know its analytical expression. In LDA, the exchange and correlation energies are expressed using different approximations. For the exchange energy, the HEG result is applied in a localized way. In the correlation energy, there are analytical expressions for the high and low density limits.
Spin polarization extensionLDA can also be extended to spin-polarized systems, which is crucial for studying fields such as spin electronics. In this case, the exchange-correlation energy will be adjusted according to the spin density, which makes LDA well able to handle spin-polarized conditions.
This treatment of spin makes the application of DFA in spin electronics research more feasible.
Conclusion
In general, the local density approximation occupies an irreplaceable position in computational chemistry and solid-state physics. It not only provides us with a powerful tool to understand electronic structure, but also promotes the development of other methods. However, with the advancement of technology and the discovery of new materials, the limitations of LDA are becoming increasingly apparent. Whether our understanding can be deepened will be an important topic for future research.
