Exchange versus direct Coulomb terms: the pair-energy split of the two-orbital spin-half model

Adopt the state and composition postulate P1 and the identical-particle and spin postulate P6 in the readings of Declared species sectors and their preparations: the framework's reading of P6, and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Declare the following two-orbital spin-half model:

(a) the species is a spin-half fermion with an arbitrary model-assigned mass parameter $M_{\mathrm{s}\mathrm{p}}>0\mathrm{k}\mathrm{g}$, unused in this computation and with no empirical value asserted; its orbital factor is $K:={\mathrm{span}}_{\mathbb{C}}\{u,v\}$ where $u,v\in W^{1,2}({\mathbb{R}}^3;\mathbb{C})\cap L^2({\mathbb{R}}^3;\mathbb{C})$ are real-valued, normalized and orthogonal in $L^2({\mathbb{R}}^3)$ (Integer-order Sobolev spaces and their norms, The space $L^p(\mu )$ as the quotient by null functions), with $uv\mathrm{\ne }0$ in $L^1$; the internal factor is $V={\mathbb{C}}^2$ with basis $(e_{+},e_{-})$; the one-particle space is $K\widehat{\otimes }V$ and $n=2$ particles are declared in the antisymmetric sector ${\mathfrak{H}}_2^{-}=\mathrm{ran}{P}_{-}^{(K\otimes V)}$ (The two spin-half fermion sector: the spin-singlet and spin-triplet parts and their orbital symmetry);

(b) write $E:=K\widehat{\otimes }K$ and let $J:E\to L^2({\mathbb{R}}^6)$ be the restriction of the Euclidean product unitary ${\mathrm{\Theta }}_6$ (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})), so $J(a\otimes b)(x,y)=a(x)b(y)$. Here $L^2({\mathbb{R}}^r)$ consists of almost-everywhere classes $[f]$ of complex Lebesgue-measurable functions with ${\int }_{{\mathbb{R}}^r}\mathrm{\mid }f{\mathrm{\mid }}^{2}dx<\mathrm{\infty }$ (The space $L^p(\mu )$ as the quotient by null functions). In SI units, let $x,y\in {\mathbb{R}}^3$ be spatial coordinates measured in metres and define $W:{\mathbb{R}}^3\to \mathbb{R}$ by $W(z)=\kappa \mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ for $z\mathrm{\ne }0$ and $W(0)=0$. The value at zero is an arbitrary representative on a measure-zero set. Here $\kappa >0$ is an adopted parameter of unit $\mathrm{J}\mathrm{m}$, with no relation to a charge assumed, so $W$ has unit $\mathrm{J}$. Define the multiplication operator $M_W:D(M_W)\subseteq L^2({\mathbb{R}}^6)\to L^2({\mathbb{R}}^6),D(M_W):=\{[g]\in L^2({\mathbb{R}}^6):[(x,y)\mapsto W(x-y)g(x,y)]\in L^2({\mathbb{R}}^6)\},M_W[g]:=[(x,y)\mapsto W(x-y)g(x,y)]$ where $g$ is any representative. This is independent of the representative because $W$ is finite and measurable everywhere. The declared pair interaction is the compression of this multiplier to $E$: for $z,z^{\mathrm{\prime }}\in E$, set $q(z,z^{\mathrm{\prime }}):={\int }_{{\mathbb{R}}^6}W(x-y)(Jz)(x,y)\overline{(J{z}^{\mathrm{\prime }})(x,y)}dxdy\mathrm{.}$ The normalized position-space orbitals have units ${\mathrm{m}}^{-3\mathrm{/}2}$, and $D$, $X$ and the pair-interaction expectations below have units $\mathrm{J}$. All $Jz$ lie in $D(M_W)$, as verified below. For the orthonormal basis $(b_1,b_2,b_3,b_4):=(u\otimes u,u\otimes v,v\otimes u,v\otimes v)$ of $E$, define $Az:={\sum }_{j=1}^{4}q(z,b_j)b_j\mathrm{.}$ Thus $⟨Az,z^{\mathrm{\prime }}⟩=q(z,z^{\mathrm{\prime }})$; $A$ is a bounded self-adjoint pair-energy operator on $E$, has units $\mathrm{J}$, and has real matrix entries in this basis. It commutes with spatial exchange. The model's sector interaction is the restriction to ${\mathfrak{H}}_2^{-}$ of ${\mathrm{\Theta }}^{-1}(A\otimes I_{V\otimes V})\mathrm{\Theta }$, with $\mathrm{\Theta }$ the finite-dimensional regrouping unitary of the symmetry-decomposition theorem. This restriction is well defined because the conjugated operator commutes with total exchange. On a factored preparation with unit $\chi$ its expectation is $q(\phi ,\phi )=⟨M_{W}J\phi ,J\phi ⟩$;

(c) the declared pair preparations are the images under the inverse regrouping map of the symmetry-decomposition theorem (Regrouping two factors: the symmetry decomposition of a two-particle sector over an orbital and an internal factor) of the factored vectors ${\mathrm{\Psi }}_T:={\mathrm{\Theta }}^{-1}({\phi }_{-}\otimes (e_{+}\otimes e_{+})),{\mathrm{\Psi }}_S:={\mathrm{\Theta }}^{-1}({\phi }_{+}\otimes {\sigma }_{-}),{\phi }_{\pm }:=\frac{u\otimes v\pm v\otimes u}{\sqrt{2}},{\sigma }_{-}:=\frac{e_{+}\otimes e_{-}-e_{-}\otimes e_{+}}{\sqrt{2}},$ which are unit vectors of ${\mathfrak{H}}_2^{-}$ by the cited thought experiment. ${\mathrm{\Psi }}_T$ is the parallel-spin (spin-triplet) preparation and ${\mathrm{\Psi }}_S$ the spin-singlet preparation of the declared two-orbital model.

The example computes the declared pair-interaction expectations $q({\phi }_{\pm },{\phi }_{\pm })$ of the two preparations and shows that the exchange integral $X$ separates them strictly.

• What "exchange" names. In this model the name refers to the difference $2X$ between the two pair expectations, a property of the declared orbitals and interaction; the calculation does not attribute a mechanical force to it and does not by itself identify the energy of any atom.
• Nonzero orbital overlap. Orthogonality of the real orbitals gives $\int uv=0$; a nonzero product consequently changes sign on sets of positive measure. The Gaussian-square argument proves $X>0$ for every nonzero product, without a sign assumption. If $uv=0$ a.e., then $X=0$ and the pair expectations coincide.
• Finite orbital model. The interaction is the compression of the Coulomb form to $K\widehat{\otimes }K$. The full multiplier need not preserve this finite spatial subspace. The separate maps $\mathrm{\Theta }$ and ${\mathrm{\Theta }}_6$ respectively regroup orbital and spin factors and identify spatial tensors with Euclidean wavefunctions.