= Solution
<Direct exchange> is a quantum-mechanical spin-dependent <energy> arising from the <Coulomb interaction> and the required antisymmetry of a two-electron state. It requires spatial overlap of the relevant orbitals. It is not the classical magnetic dipole <force> between two small magnets, and it can be much stronger than that <force>.
For an explicit fixed-orbital illustration, take orthonormal spatial orbitals $a,b$. <Orthonormality> means $\langle a|b\rangle=0$; it does not require their spatial densities to occupy disjoint regions. The normalized spatial states are
$$
\Psi_\pm(1,2)=\frac{a(1)b(2)\pm b(1)a(2)}{\sqrt2}.
$$
The symmetric spatial state combines with the antisymmetric <spin-one-half singlet state>, and the antisymmetric spatial state combines with a symmetric <spin-one-half triplet state>, so that the complete fermionic state is antisymmetric.
For <Coulomb potential> $V_{12}=e^2/(4\pi\varepsilon|\mathbf r_1-\mathbf r_2|)$, distinguish the direct electrostatic integral and exchange integral:
$$
D=\int d^3r_1d^3r_2\,|a(1)|^2V_{12}|b(2)|^2,
$$
$$
K=\int d^3r_1d^3r_2\,a^*(1)b(1)V_{12}b^*(2)a(2).
$$
The <Coulomb exchange splitting of two orbitals> gives
$$
E_S=\epsilon_a+\epsilon_b+D+K,\qquad E_T=\epsilon_a+\epsilon_b+D-K,\qquad \boxed{E_S-E_T=2K.}
$$
The repulsive <Coulomb potential> makes $K\geq0$ in this model. Thus the triplet is favored when the orbitals have nonzero exchange overlap. Its antisymmetric spatial <wavefunction> suppresses close <Electron> encounters and reduces their repulsion. Since the <Coulomb interaction> contains no explicit <spin> operator, the <spin> dependence enters through the allowed spatial <symmetry>.
For dimensionless <spin> operators $\mathbf s_i=\boldsymbol\sigma_i/2$, whose pair <eigenvalues> are $1/4$ in the triplet and $-3/4$ in the singlet, the same splitting is represented by
$$
\boxed{H_{\rm ex}=C-2K\,\mathbf s_1\cdot\mathbf s_2.}
$$
This explicitly states the sign convention: positive $K$ here favors parallel spins. A convention $H=J\mathbf s_1\cdot\mathbf s_2$ would instead have $J=-2K$.
In a <crystal>, <direct exchange> therefore depends sensitively on orbital extent, overlap and interatomic spacing. It often decreases rapidly as localized orbitals separate. The fixed-orbital example does not prove that every material must be ferromagnetic: orbital rearrangement and competing kinetic or virtual-hopping processes alter the singlet-triplet balance. In particular, <antiferromagnetic> <superexchange> through an intermediate ligand is a different mechanism and should not be confused with direct overlap exchange.
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