一个精确能量泛函的景观
Landscape of an exact energy functional
摘要：
电子结构理论面临的最大挑战之一是寻求密度泛函理论的精确泛函。它的存在被证明，但它是一个复杂的多变量功能，几乎不可能概念化。本文研究了具有二维密度矩阵宇宙的非对称二位哈伯德模型。精确的泛函变成了一个简单的两个变量的函数，其三维能量景观可以可视化和探索。在这个独特的景观上行走，倾斜到一个由单电子哈密顿量定义的角度，给出一个最小值是确切的总能量的谷。这与一些近似泛函的景观形成对比，解释了它们在强相关极限下的电子转移失败。我们展示了具体的例子，纯态密度矩阵是不$v$ -representable由于潜在的非凸性质的能源景观。对于第一次，精确的功能是计算所有数量的电子，包括分数，允许导数的不连续性被可视化和理解。所有可能系统的基本间隙仅由精确泛函的导数获得。
英文摘要：
One of the great challenges of electronic structure theory is the quest for the exact functional of density functional theory. Its existence is proven, but it is a complicated multivariable functional that is almost impossible to conceptualize. In this paper, the asymmetric two-site Hubbard model is studied, which has a two-dimensional universe of density matrices. The exact functional becomes a simple function of two variables whose three dimensional energy landscape can be visualized and explored. A walk on this unique landscape, tilted to an angle defined by the one-electron Hamiltonian, gives a valley whose minimum is the exact total energy. This is contrasted with the landscape of some approximate functionals, explaining their failure for electron transfer in the strongly correlated limit. We show concrete examples of pure-state density matrices that are not $v$-representable due to the underlying non-convex nature of the energy landscape. For the first time, the exact functional is calculated for all numbers of electrons, including fractional, allowing the derivative discontinuity to be visualized and understood. The fundamental gap for all possible systems is obtained solely from the derivatives of the exact functional.
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