Finite-basis projector energy model

Let $S=S^{\ast }\succ 0$ be an $n\times n$ finite-basis overlap matrix and supply its Cholesky factor $L$ with $S=L{L}^{\ast }$, $L$ invertible (Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal, A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique). If $h_0=h_0^{\ast }$ is the coefficient-basis one-body matrix, set $h=L^{-1}{h}_0{L}^{-\ast }$, where $L^{-\ast }=(L^{\ast }{)}^{-1}$; this is the Cholesky whitening whose generalized Ritz quotient is preserved by Positive-definite overlap whitening preserves a generalized Ritz pencil. For coefficient vectors use $y=L^{\ast }c$; the interaction map and chosen exchange-correlation functional below are supplied in these orthonormal coordinates. Fix $1\le r<n$ and the projector domain ${\mathcal{P}}_r$ of Rank-r orthogonal projector domain.

With the whitened Hermitian matrix $h=L^{-1}{h}_0{L}^{-\ast }$ fixed above, supply a real-linear self-adjoint map $G:{\mathbb{H}}_n\to {\mathbb{H}}_n$ for the Frobenius inner product, and a real-valued $C^2$ functional $E_{xc}$ on an open neighborhood of ${\mathcal{P}}_r$. Define $E(P):=\mathrm{tr}(hP)+\frac{1}{2}\mathrm{tr}(PG(P))+E_{xc}(P),F(P):=h+G(P)+\mathrm{\nabla }{E}_{xc}(P)\mathrm{.}$ Here $F(P)$ is the Hermitian Frobenius-gradient formula justified by The finite projector energy is real and has the stated Frobenius gradient ↗.

The finite-basis Hartree–Fock model is the special case $E_{xc}=0$ with a supplied symmetry-compatible Coulomb/exchange tensor defining $G$. A Kohn–Sham-like model supplies its density map and chosen differentiable exchange-correlation functional. These are finite-dimensional mathematical models; this definition makes no assertion that the selected functional is an exact continuum physical energy.