Recently a novel approach to find approximate exchange–correlation functionals in density-functional theory was presented (Mordovina etal 2019 J. Chem. Theory Comput. 15 5209), which relies on approximations to the interacting wave function using density-matrix embedding theory (DMET). This approximate interacting wave function is constructed by using a projection determined by an iterative procedure that makes parts of the reduced density matrix of an auxiliary system the same as the approximate interacting density matrix. If only the diagonal of both systems are connected this leads to an approximation of the interacting-to-non-interacting mapping of the Kohn–Sham approach to density-functional theory. Yet other choices are possible and allow to connect DMET with other density-functional theories such as kinetic-energy density functional theory or reduced density-matrix functional theory. In this work we give a detailed review of the basics of the DMET procedure from a density-functional perspective and show how both approaches can be used to supplement each other. We do not present a specific realization of combining density-functional methods with DMET but rather provide common grounds to facilitate future developments that encompass both approaches. We do so explicitly for the case of a one-dimensional lattice system, as this is the simplest setting where we can apply DMET and the one that was originally presented. Among others we highlight how the mappings of density-functional theories can be used to identify uniquely defined auxiliary systems and projections in DMET and how to construct approximations for different density-functional theories using DMET inspired projections. Such alternative approximation strategies become especially important for density-functional theories that are based on non-linearly coupled observables such as kinetic-energy density-functional theory, where the Kohn–Sham fields are no longer obtainable by functional differentiation of an energy expression, or for reduced density-matrix functional theories, where a straightforward Kohn–Sham construction is not feasible.

最近提出了一种在密度泛函理论中寻找近似交换相关泛函的新方法（Mordovina et al 2019 J. Chem. Theory Comput. 155209），它依赖于使用密度矩阵嵌入理论（DMET）对相互作用波函数的近似。该近似相互作用波函数是通过使用由迭代过程确定的投影来构建的，该过程使辅助系统的约简密度矩阵的一部分与近似相互作用密度矩阵相同。如果仅连接两个系统的对角线，这将导致密度泛函理论的 Kohn-Sham 方法的相互作用到非相互作用映射的近似。还有其他选择是可能的，并允许将 DMET 与其他密度泛函理论联系起来，例如动能密度泛函理论或约化密度矩阵泛函理论。在这项工作中，我们从密度泛函的角度详细回顾了 DMET 程序的基础知识，并展示了这两种方法如何相互补充。我们没有提出将密度泛函方法与 DMET 相结合的具体实现，而是提供了共同的基础，以促进包含这两种方法的未来发展。对于一维晶格系统的情况，我们明确这样做，因为这是我们可以应用 DMET 的最简单设置，也是最初提出的设置。其中，我们强调如何使用密度泛函理论的映射来识别 DMET 中唯一定义的辅助系统和投影，以及如何使用 DMET 启发的投影构建不同密度泛函理论的近似值。