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Normalization and Fermi-Coulomb and Coulomb hole sum rules for approximate wave functions

机译：近似波函数的归一化以及费米库伦和库伦孔和规则

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For approximate wave functions, we prove the theorem that there is a one-to-one correspondence between the constraints of normalization and of the Fermi-Coulomb and Coulomb hole charge sum rules at each electron position. This correspondence is surprising in light of the fact that normalization depends on the probability of finding an electron at some position. In contrast, the Fermi-Coulomb hole sum rule depends on the probability of two electrons staying apart because of correlations due to the Pauli exclusion principle and Coulomb repulsion, while the Coulomb hole sum rule depends on Coulomb repulsion. We demonstrate the theorem for the ground state of the He atom by the use of two different approximate wave functions that are functionals rather than functions. The first of these wave function functionals is constructed to satisfy the constraint of normalization, and the second that of the Coulomb hole sum rule for each electron position. Each is then shown to satisfy the other corresponding sum rule. The significance of the theorem for the construction of approximate "exchange-correlation" and "correlation" energy functionals of density functional theory is also discussed. (c) 2006 Wiley Periodicals, Inc.
机译：对于近似波函数，我们证明了一个定理，在每个电子位置的归一化约束与费米库仑和库仑空穴电荷和规则之间存在一对一的对应关系。考虑到归一化取决于在某个位置找到电子的概率这一事实，这种对应是令人惊讶的。相反，费米-库仑空穴和规则取决于两个电子分开的概率，这是由于保利排斥原理和库仑排斥引起的，而库仑空穴和规则取决于库仑排斥。我们通过使用两个不同的近似波函数（它们是函数而不是函数）证明了He原子基态的定理。这些波函数函数中的第一个构造为满足归一化约束，第二个波函数构造为每个电子位置的库仑空穴求和规则。然后显示每个满足另一个相应的求和规则。还讨论了该定理对于构造密度泛函理论的近似“交换相关”和“相关”能量泛函的意义。 （c）2006年Wiley Periodicals，Inc.

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