一维双势阱中基态和激发态费米子的精确含时密度泛函解
Ground and excited-state fermions in a 1D double-well, exact and time-dependent density-functional solutions
摘要：
两个最流行的相互作用的费米子的量子力学模型进行了比较，彼此和潜在的精确解的一对接触相互作用的费米子被困在一个1D的双势阱，一个模型的原子在一个准1D光晶格或电子的氢分子在一个强磁场。一个精确的少体哈密顿量在动量空间中数值求解，得到一个高度相关的本征谱。此外，近似基态能量得到使用密度泛函理论（DFT）功能和2-网站哈伯德模型。构造了一维绝热LDA核用于含时密度泛函理论（TDDFT），并将所得的激发态谱与精确结果和Hubbard结果进行了比较。DFT示出，以提供准确的结果，井与小的分离，但未能描述本地化的相反自旋费米子到不同的网站。一个本地认知（LC）的密度泛函的基础上的有效的本地费米子数将提供一个解决这个问题的方法，这里提出的近似处理相比，毫不逊色的确切和哈伯德的结果。TDDFT激发态谱在小参数区域是准确的，非绝热效应占任何偏差。正如预期的那样，基态哈伯德模型在大分离处优于DFT，但在中间分离处由于对联合原子极限的不适当缩放而失效。在强耦合下，Hubbard和TDDFT方法都不能捕获适当的能量。
英文摘要：
Two of the most popular quantum mechanical models of interacting fermions are compared to each other and to potentially exact solutions for a pair of contact-interacting fermions trapped in a 1D double-well potential, a model of atoms in a quasi-1D optical lattice or electrons of a Hydrogen molecule in a strong magnetic field. An exact few-body Hamiltonian is solved numerically in momentum space yielding a highly-correlated eigenspectrum. Additionally, approximate ground-state energies are obtained using both density functional theory (DFT) functional and 2-site Hubbard models. A 1D adiabatic LDA kernel is constructed for use in time-dependent density functional theory (TDDFT), and the resulting excited-state spectrum is compared to the exact and Hubbard results. DFT is shown to give accurate results for wells with small separations but fails to describe localization of opposite spin fermions to different sites. A locally cognizant (LC) density functional based on an effective local fermion number would provide a solution to this problem, and an approximate treatment presented here compares favorably with the exact and Hubbard results. The TDDFT excited-state spectrum is accurate in the small parameter regime with non-adiabatic effects accounting for any deviations. As expected, the ground-state Hubbard model outperforms DFT at large separations but breaks down at intermediate separations due to improper scaling to the united-atom limit. At strong coupling, both Hubbard and TDDFT methods fail to capture the appropriate energetics.
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