The finite-basis projector energy model supplies a C1 Hermitian response map

Given: A finite-basis model with Hermitian $h$, real-linear $G:{\mathbb{H}}_n\to {\mathbb{H}}_n$, and real $C^2$ functional $E_{xc}$ on an open neighborhood; a point $P_0$ at which $F(P_0)$ has a strict $r$-th spectral gap.

The model defines $F(P)=h+G(P)+\mathrm{\nabla }{E}_{xc}(P)$, with $G$ real-linear and $E_{xc}$ of class $C^2$ (Finite-basis projector energy model).

The real Frechet derivative treats complex matrix spaces as real finite-dimensional spaces (The real Frechet derivative on real and complex matrix spaces with the Frobenius norm).

A gapped block projector is $C^1$ on a neighborhood with a common separating contour (A separated block spectral projector is Frechet differentiable with a resolvent bound).

The derivative of a composition is the composition of the derivatives (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$).

A computable operator-norm ball around a Hermitian matrix with a strict occupied gap retains that gap (A spectral gap persists on a computable matrix ball with a uniform resolvent bound).

The operator norm is at most the Frobenius norm, and both are continuous finite-dimensional norms (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, The Frobenius norm $\parallel A{\parallel }_F=({\sum }_{i,j}\mathrm{\mid }{a}_{ij}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ on real or complex matrices, Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison).

The gapped response map is the unique rank-$r$ occupied projector (A strict occupied gap defines a unique rank-r spectral projector).

The $C^2$ convention in finite-dimensional real coordinates gives continuous partial derivatives through order two ($C^k$ maps and multi-index derivative notation in Euclidean space).

A map whose coordinate partial derivatives exist and are continuous is total differentiable with those derivatives (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

A real-linear map between finite-dimensional Euclidean spaces is bounded and continuous (Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$).