Borgoo Alex, Teale Andrew M, Tozer David J
Department of Chemistry, Centre for Theoretical and Computational Chemistry, University of Oslo, P.O. Box 1033, Blindern, Oslo N-0315, Norway.
Phys Chem Chem Phys. 2014 Jul 28;16(28):14578-83. doi: 10.1039/c4cp00170b.
Scaling relations play an important role in the understanding and development of approximate functionals in density functional theory. Recently, a number of these relationships have been redefined in terms of the Kohn-Sham orbitals [Calderín, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 86, 032510]. For density scaling the author proposed a procedure involving a multiplicative scaling of the Kohn-Sham orbitals whilst keeping their occupation numbers fixed. In the present work, the differences between this scaling with fixed occupation numbers and that of previous studies, where the particle number change implied by the scaling was accommodated through the use of the grand canonical ensemble, are examined. We introduce the terms orbital and ensemble density scaling for these approaches, respectively. The natural ambiguity of the density scaling of the non-interacting kinetic energy functional is examined and the ancillary definitions implicit in each approach are highlighted and compared. As a consequence of these differences, Calderín recovered a homogeneity of degree 1 for the non-interacting kinetic energy functional under orbital scaling, contrasting recent work by the present authors [J. Chem. Phys., 2012, 136, 034101] where the functional was found to be inhomogeneous under ensemble density scaling. Furthermore, we show that the orbital scaling result follows directly from the linearity and the single-particle nature of the kinetic energy operator. The inhomogeneity of the non-interacting kinetic energy functional under ensemble density scaling can be quantified by defining an effective homogeneity. This quantity is shown to recover the homogeneity values for important approximate forms that are exact for limiting cases such as the uniform electron gas and one-electron systems. We argue that the ensemble density scaling provides more insight into the development of new functional forms.
标度关系在密度泛函理论中近似泛函的理解和发展中起着重要作用。最近，其中一些关系已根据科恩 - 沈轨道重新定义[卡尔德林，《物理评论A：原子、分子、光学物理》，2013年，86卷，032510]。对于密度标度，作者提出了一种程序，涉及对科恩 - 沈轨道进行乘法标度，同时保持其占据数不变。在本工作中，研究了这种固定占据数的标度与先前研究的标度之间的差异，在先前研究中，标度所隐含的粒子数变化是通过使用巨正则系综来处理的。我们分别将这些方法称为轨道密度标度和系综密度标度。研究了非相互作用动能泛函密度标度的自然模糊性，并突出和比较了每种方法中隐含的辅助定义。由于这些差异，卡尔德林在轨道标度下恢复了非相互作用动能泛函的1次齐次性，这与作者近期的工作[《化学物理杂志》，2012年，136卷，034101]形成对比，在该工作中发现该泛函在系综密度标度下是非齐次的。此外，我们表明轨道标度结果直接源于动能算符的线性和单粒子性质。非相互作用动能泛函在系综密度标度下的非齐次性可以通过定义一个有效齐次性来量化。该量被证明可以恢复重要近似形式在诸如均匀电子气和单电子系统等极限情况下精确的齐次性值。我们认为系综密度标度为新泛函形式的发展提供了更多见解。
