Second vs First Theorem

[Samra Mahmood]

The second theorem is crucial for the practical use of Density Functional Theory (DFT) because it introduces the variational principle, which is an independent concept. It shows how the ground-state energy can be minimized with respect to the electron density, providing a method to identify the true ground-state density. While the first theorem proves that a unique external potential and energy can be derived from the ground-state density, it does not guarantee that the density identified minimizes the energy. The second theorem fills this gap by explicitly stating that the energy functional reaches its minimum only at the true ground-state density. Therefore, the second theorem is more than just a consequence of the first; it plays a distinct role in ensuring that the correct ground-state density corresponds to the lowest energy.

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