Direct and exchange contributions to a two-fermion mean energy and the repulsive-Coulomb ordering

Given: the declared spin-1/2 fermionic species, $N=2$, the external and pair data of the statement, the orthonormal orbitals $u,v\in W^{2,2}$, the spin vectors ${\chi }_{\pm }$ and the vectors ${\mathrm{\Phi }}_{\pm }$.

Full spin-product transport. The unitary ${\mathrm{\Theta }}_2$ maps $(f{e}_a)\otimes (g{e}_b)$ to $(x,y)\mapsto f(x)g(y)(e_a\otimes e_b)$. The transported exchange acts as $(\mathcal{U}\mathrm{\Psi })(x,y)=R\mathrm{\Psi }(y,x)$, where $R(\zeta \otimes \eta )=\eta \otimes \zeta$ swaps the two spin factors; its antisymmetric projection is ${\mathcal{P}}_{-}=(I-\mathcal{U})\mathrm{/}2={\mathrm{\Theta }}_2{P}_{-}^{(2)}{\mathrm{\Theta }}_2^{-1}$. Finite spin components preserve the $H^2$ domain, and a permutation-invariant scalar self-adjoint operator with that domain restricts self-adjointly to this combined sector (Finite spin components of an N-fold spatial tensor product clauses (a)–(c)). Tensor pairings and norms are those of Hilbert tensor construction: inner product, completion and product bases, and the tensor exchange is that of Permutation sector projections on the n-fold tensor power.

Scalar spatial products. The scalar product unitary ${\mathrm{\Theta }}_{\mathrm{s}\mathrm{p}}$ has ${\mathrm{\Theta }}_{\mathrm{s}\mathrm{p}}(u\otimes v)(x,y)=u(x)v(y)$ (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})). Fubini applies to integrable products and the finite component pairings (Fubini's theorem for L^1 functions on a sigma-finite product).

Direct and exchange integrals, including complex orbitals. For a real bounded even $W$, the displayed $D,X$ converge absolutely, are real and satisfy $⟨M_W{f}_{\pm },f_{\pm }⟩=D\pm X$ for arbitrary complex orthonormal $u,v\in L^2$. For $W(z)=\kappa \mathrm{/}\mathrm{\mid }z\mathrm{\mid }$, $\kappa >0$, the same assertion holds for arbitrary complex orthonormal $u,v\in H^1$, with $f_{\pm }\in D(M_W)$, $D\ge 0$ and $X\ge 0$. These are Slater direct and exchange calculation clauses (c),(d); its proof 2.1 proves Coulomb positivity with only the $L^1$ overlap density, without an unclaimed $L^2$ or real-orbital hypothesis.

The exact labelled realization with bounded external addition. Put $a={\mathrm{\hslash }}^2\mathrm{/}(2m)$ and $V_b(x,y)=V_{\mathrm{e}\mathrm{x}\mathrm{t}}(x)+V_{\mathrm{e}\mathrm{x}\mathrm{t}}(y)$. In the bounded case the scalar labelled operator $T=-a{\mathrm{\Delta }}_{(x,y)}+V_b+W(x-y)$ is self-adjoint on $H^2({\mathbb{R}}^6;\mathbb{C})$ by CC realization of the bounded spatial operator on a finite-component H² domain, with the same finite-component realization on $H^2({\mathbb{R}}^6;{\mathbb{C}}^4)$. In the Coulomb case apply Fixed-N Coulomb self-adjointness clauses (b),(c) with mathematical masses ${\tilde{m}}_1={\tilde{m}}_2=m\mathrm{/}(4{\pi }^2)$, its parameter $\mathrm{\hslash }$ unchanged, zero external singular coefficients and pair coefficient $\kappa$. Its kinetic Fourier symbol is then $4{\pi }^{2}a(\mathrm{\mid }\xi {\mathrm{\mid }}^2+\mathrm{\mid }\eta {\mathrm{\mid }}^2)$, exactly the physical $-a{\mathrm{\Delta }}_{(x,y)}$, so $T_C=-a{\mathrm{\Delta }}_{(x,y)}+\kappa \mathrm{/}\mathrm{\mid }x-y\mathrm{\mid }$ is self-adjoint on $H^2$. The multiplier $M_{V_b}$ is bounded self-adjoint, with norm at most $2\mathrm{\parallel }{V}_{\mathrm{e}\mathrm{x}\mathrm{t}}{\mathrm{\parallel }}_{\mathrm{\infty }}$ and relative bound zero with respect to $T_C$; Kato-Rellich theorem therefore makes $T=T_C+M_{V_b}$ self-adjoint on the same exact domain and bounded below. In either case, $T$ preserves the spatial exchange symmetry because its Laplacian and real even pair kernel and identical external slots are exchange invariant. The full labelled operator $T_4$ acts componentwise on spin and its combined antisymmetric restriction, transported by ${\mathrm{\Theta }}_2^{-1}$, is the declared $H_2^{-}$ with exact domain ${\mathrm{\Theta }}_2^{-1}(H^2\cap \mathrm{ran}{\mathcal{P}}_{-})$ by [F1]. No scalar-only application of a permutation theorem is used for the spin-four space. The one-particle $h=-a\mathrm{\Delta }+V_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is likewise self-adjoint on scalar $H^2$ by the CC bounded realization.

Weak derivatives of spatial products. For $u,v\in H^2({\mathbb{R}}^3;\mathbb{C})$ the product $u(x)v(y)$ has weak derivatives $D_x^{\alpha }{D}_y^{\beta }(u(x)v(y))=(D^{\alpha }u)(x)(D^{\beta }v)(y)$ for $\mathrm{\mid }\alpha \mathrm{\mid }+\mathrm{\mid }\beta \mathrm{\mid }\le 2$, obtained by iterated weak-derivative testing and Fubini. Their squared $L^2({\mathbb{R}}^6)$ norms are $\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_2^2\mathrm{\parallel }{D}^{\beta }v{\mathrm{\parallel }}_2^2<\mathrm{\infty }$, including mixed first derivatives. Thus the spatial products and $f_{\pm }$ lie in $H^2({\mathbb{R}}^6)$, without claiming an exact unproved Sobolev norm constant (Integer-order Sobolev spaces and their norms, Fubini's theorem for L^1 functions on a sigma-finite product). On these products the unbounded differential slot action is $(-a{\mathrm{\Delta }}_x-a{\mathrm{\Delta }}_y+V_b)(u(x)v(y))=(hu)(x)v(y)+u(x)(hv)(y)$; every term lies in $L^2$ by its factor norms. This is an exact-domain calculation, not an application of bounded tensor-map rules to an unbounded $h$.

Finite means of domain preparations. For a unit vector $\psi$ in the domain of a self-adjoint assigned $A$, the absolute first spectral moment is finite and its mean is $⟨A\psi ,\psi ⟩$ (Mean, second moment and variance of an assigned observable in a preparation clause (a), Assigned sharp observables and the outcome law of the framework clause (e)).