反対称交換相互作用
δ E = ∑ m [ ⟨ n | λ L ^ i ⋅ S ^ i | m ⟩ 2 J ( m n ′ n n ′ ) S ^ i ⋅ S ^ j E n − E m + 2 J ( n n ′ m n ′ ) S ^ i ⋅ S ^ j ⟨ m | λ L ^ i ⋅ S ^ i | n ⟩ E n − E m ] + ∑ m ′ [ ⟨ m ′ | λ L ^ j ⋅ S ^ j | m ⟩ 2 J ( m ′ n n ′ n ) S ^ i ⋅ S ^ j E n ′ − E m ′ + 2 J ( n ′ n m ′ n ) S ^ i ⋅ S ^ j ⟨ m ′ | λ L ^ j ⋅ S ^ j | n ′ ⟩ E n ′ − E m ′ ] {\displaystyle {\begin{aligned}\delta E=\sum _{m}&{\Biggl [}{\frac {\langle n|\lambda {\hat {\mathbf {L} }}_{i}\cdot {\hat {\mathbf {S} }}_{i}|m\rangle 2J(mn'nn'){\hat {\mathbf {S} }}_{i}\cdot {\hat {\mathbf {S} }}_{j}}{E_{n}-E_{m}}}\\&+{\frac {2J(nn'mn'){\hat {\mathbf {S} }}_{i}\cdot {\hat {\mathbf {S} }}_{j}\langle m|\lambda {\hat {\mathbf {L} }}_{i}\cdot {\hat {\mathbf {S} }}_{i}|n\rangle }{E_{n}-E_{m}}}{\Biggr ]}\\+\sum _{m'}&{\Biggl [}{\frac {\langle m'|\lambda {\hat {\mathbf {L} }}_{j}\cdot {\hat {\mathbf {S} }}_{j}|m\rangle 2J(m'nn'n){\hat {\mathbf {S} }}_{i}\cdot {\hat {\mathbf {S} }}_{j}}{E_{n'}-E_{m'}}}\\&+{\frac {2J(n'nm'n){\hat {\mathbf {S} }}_{i}\cdot {\hat {\mathbf {S} }}_{j}\langle m'|\lambda {\hat {\mathbf {L} }}_{j}\cdot {\hat {\mathbf {S} }}_{j}|n'\rangle }{E_{n'}-E_{m'}}}{\Biggr ]}\end{aligned}}}
ここで、 J {\displaystyle J} は以下の式により与えられる交換積分を表す。
J ( n n ′ m m ′ ) = ∫ ∫ φ n ∗ ( r 1 − R ) φ n ′ ∗ ( r 2 − R ′ ) e 2 r 12 φ m ( r 2 − R ) φ m ′ ( r 1 − R ′ ) d r 1 d r 2 {\displaystyle J(nn'mm')=\int \int \phi _{n}^{*}(\mathbf {r_{1}} -\mathbf {R} )\phi _{n'}^{*}(\mathbf {r_{2}} -\mathbf {R'} ){\frac {e^{2}}{r_{12}}}\phi _{m}(\mathbf {r_{2}} -\mathbf {R} )\phi _{m'}(\mathbf {r_{1}} -\mathbf {R'} )\mathrm {d} \mathbf {r_{1}} \mathrm {d} \mathbf {r_{2}} }