粗粒化对密度泛函理论的驯服
Taming Density Functional Theory by Coarse-Graining
摘要：
量子密度泛函理论的标准（"细粒度“）解释，其中密度是用无限精细的空间分辨率来指定的，在数学上是难以驾驭的。在这里，粗粒度版本的DFT，具有有限的空间分辨率，其关系的细粒度理论的$L^1\cap L^3$制定的Lieb，进行了研究，显示它不仅在数学上表现良好，但符合DFT的精神，实际上（计算）足够接近标准的解释，以准确地反映其非病理性质的目的。粗粒度的解释被证明是一个很好的模型，正式DFT的意义上说：所有的密度（合奏）-V-representable的内禀能量泛函$F$是一个连续的函数的密度和代表的外部潜在的（方向）功能的衍生物的内在能量。而且，代表电位$v[ρ]$是准连续的，因为$v[ρ]ρ$作为$ρ$的函数是连续的。粗粒化尺度趋于零的限制进行了研究，看看是否收敛到非病理方面的细粒度的理论是足够的，以证明作为一个很好的近似粗粒化。找到了合适的内禀能量、密度和表示势的极限行为：内禀能量单调收敛，粗粒密度一致强收敛于其低内禀能量的细粒密度，密度的$L^{3/2}+L^\infty$表示性等价于存在一个粗粒势/基态密度对的收敛序列.
英文摘要：
The standard (``fine-grained'') interpretation of quantum density functional theory, in which densities are specified with infinitely-fine spatial resolution, is mathematically unruly. Here, a coarse-grained version of DFT, featuring limited spatial resolution, and its relation to the fine-grained theory in the $L^1\cap L^3$ formulation of Lieb, is studied, with the object of showing it to be not only mathematically well-behaved, but consonant with the spirit of DFT, practically (computationally) adequate and sufficiently close to the standard interpretation as to accurately reflect its non-pathological properties. The coarse-grained interpretation is shown to be a good model of formal DFT in the sense that: all densities are (ensemble)-V-representable; the intrinsic energy functional $F$ is a continuous function of the density and the representing external potential is the (directional) functional derivative of the intrinsic energy. Also, the representing potential $v[ρ]$ is quasi-continuous, in that $v[ρ]ρ$ is continuous as a function of $ρ$. The limit of coarse-graining scale going to zero is studied to see if convergence to the non-pathological aspects of the fine-grained theory is adequate to justify regarding coarse-graining as a good approximation. Suitable limiting behaviors or intrinsic energy, densities and representing potentials are found. Intrinsic energy converges monotonically, coarse-grained densities converge uniformly strongly to their low-intrinsic-energy fine-grainings, and $L^{3/2}+L^\infty$ representability of a density is equivalent to the existence of a convergent sequence of coarse-grained potential/ground-state density pairs.
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