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A finite-basis projector energy attains a global minimum

Statement

For every finite-basis projector energy $E$ defined in Finite-basis projector energy model, there is $P_{\min} \in P_{r}$ such that $E (P_{\min}) \leq E (P)$ for every $P \in P_{r}$ . This is a finite-dimensional variational existence result and makes no continuum Hartree–Fock or Kohn–Sham existence claim.

Facts & Assumptions

Given: The model data of Finite-basis projector energy model and $1 \leq r < n$ .

[F1]

The model energy is real and $C^{2}$ on a neighborhood of $P_{r}$ (Finite-basis projector energy model, The finite projector energy is real and has the stated Frobenius gradient).

[F2]

The nonempty domain $P_{r}$ is compact (The finite rank-r projector domain is nonempty and compact).

[F3]

A real continuous function on a nonempty compact metric space attains its minimum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Proof

1.1F1F2given

By [F1], the restriction $E ∣_{P_{r}}$ is continuous. The domain with its Frobenius metric is nonempty and compact by [F2].

2.1F3step 1.1∎

Apply [F3] to $E ∣_{P_{r}}$ . It yields $P_{\min} \in P_{r}$ with $E (P_{\min}) \leq E (P)$ for every $P$ in the domain. The argument uses only the specified finite matrix model.

Depends on

Used by

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Sources

Eric Cancès and Claude Le Bris, On the convergence of SCF algorithms for the Hartree-Fock equations, ESAIM: Mathematical Modelling and Numerical Analysis 34 (2000), §2.1 (standard reference, not scraped)
Alan Edelman, Tomás A. Arias, and Steven T. Smith, The Geometry of Algorithms with Orthogonality Constraints, SIAM Journal on Matrix Analysis and Applications 20 (1998), §2.5 (standard reference, not scraped)