Compliance with the Lieb–Oxford bound for the indirect Coulomb energy and for the exchange–correlation energy is investigated for a number of density-functional methods based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem to treat correlation. Furthermore, the correlation contribution to the pair density resulting from these methods is compared with highly accurate reference values for the helium atom and for the hydrogen molecule at several bond distances. For molecules, the Lieb–Oxford bound is obeyed by all considered methods. For the homogeneous electron gas, it is violated by all methods for low electron densities. The simplest considered ACFD method, the direct random phase approximation (dRPA), violates the Lieb–Oxford bound much earlier than more advanced ACFD methods that, in addition to the simple Hartree kernel, take into account the exchange kernel and an approximate correlation kernel in the calculation of the correlation energy. While the dRPA yields quite poor correlation contributions to the pair density, those from more advanced ACFD methods are physically reasonable but still leave room for improvements, particularly in the case of the stretched hydrogen molecule.

中文翻译：

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基于绝热连接波动耗散定理的密度泛函方法的Lieb-Oxford界和对相关函数。

基于绝热连接波动耗散（ACFD）定理来处理相关性，研究了许多密度泛函方法对间接库伦能量和交换相关能量与利勃-牛津界的依从性。此外，将这些方法对对密度的相关贡献与氦原子和氢分子在几个键距处的高精度参考值进行了比较。对于分子，所有考虑的方法都遵守李布-牛津界。对于均质的电子气，低电子密度的所有方法都违反了它。被认为是最简单的ACFD方法，即直接随机相位逼近（dRPA），比更高级的ACFD方法（除了简单的Hartree内核之外）更早地违反了Lieb-Oxford界，在计算相关能量时要考虑交换核和近似相关核。尽管dRPA对成对密度的相关性贡献很差，但来自更先进的ACFD方法的那些在物理上是合理的，但仍留有改进的空间，特别是在拉伸氢分子的情况下。