強結合近似
∫ d 3 r ψ ∗ ( r ) ψ ( r ) = ∑ R n b ∗ ( R n ) ∑ R l b ( R l ) ∫ d 3 r φ ∗ ( r − R n ) φ ( r − R l ) = b ∗ ( 0 ) b ( 0 ) ∑ R n e − i k ⋅ R n ∑ R l e i k ⋅ R l ∫ d 3 r φ ∗ ( r − R n ) φ ( r − R l ) = N b ∗ ( 0 ) b ( 0 ) ∑ R p e − i k ⋅ R p ∫ d 3 r φ ∗ ( r − R p ) φ ( r ) = N b ∗ ( 0 ) b ( 0 ) ∑ R p e i k ⋅ R p ∫ d 3 r φ ∗ ( r ) φ ( r − R p ) {\displaystyle {\begin{aligned}\int d^{3}r~\psi ^{*}({\boldsymbol {r}})\psi ({\boldsymbol {r}})&=\sum _{{\boldsymbol {R}}_{n}}b^{*}({\boldsymbol {R}}_{n})\sum _{{\boldsymbol {R}}_{\ell }}b({\boldsymbol {R}}_{\ell })\int d^{3}r~\varphi ^{*}({\boldsymbol {r}}-{\boldsymbol {R}}_{n})\varphi ({\boldsymbol {r}}-{\boldsymbol {R}}_{\ell })\\&=b^{*}(0)b(0)\sum _{{\boldsymbol {R}}_{n}}e^{-i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{n}}\sum _{{\boldsymbol {R}}_{\ell }}e^{i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{\ell }}\int d^{3}r~\varphi ^{*}({\boldsymbol {r}}-{\boldsymbol {R}}_{n})\varphi ({\boldsymbol {r}}-{\boldsymbol {R}}_{\ell })\\&=Nb^{*}(0)b(0)\sum _{{\boldsymbol {R}}_{p}}e^{-i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{p}}\int d^{3}r~\varphi ^{*}({\boldsymbol {r}}-{\boldsymbol {R}}_{p})\varphi ({\boldsymbol {r}})\\&=Nb^{*}(0)b(0)\sum _{{\boldsymbol {R}}_{p}}e^{i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{p}}\int d^{3}r~\varphi ^{*}({\boldsymbol {r}})\varphi ({\boldsymbol {r}}-{\boldsymbol {R}}_{p})\end{aligned}}}
H | k ⟩ = 1 N ∑ n e i n k a H | n ⟩ ⟨ k | H | k ⟩ = 1 N ∑ m , n e i ( n − m ) k a ⟨ m | H | n ⟩ = 1 N ∑ n ⟨ n | H | n ⟩ + 1 N ∑ n ⟨ n − 1 | H | n ⟩ e + i k a + 1 N ∑ n ⟨ n + 1 | H | n ⟩ e − i k a = E 0 − 2 Δ cos  k a {\displaystyle {\begin{aligned}H|k\rangle &={\frac {1}{\sqrt {N}}}\sum _{n}e^{inka}H|n\rangle \\\langle k|H|k\rangle &={\frac {1}{N}}\sum _{m,n}e^{i(n-m)ka}\langle m|H|n\rangle \\&={\frac {1}{N}}\sum _{n}\langle n|H|n\rangle +{\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}+{\frac {1}{N}}\sum _{n}\langle n+1|H|n\rangle e^{-ika}\\&=E_{0}-2\Delta \cos ka\end{aligned}}}
したがってこの状態 | k ⟩ {\displaystyle |k\rangle } のエネルギーは次のようなよく知られたエネルギー分散を持つ。
E ( k ) = E 0 − 2 Δ cos  k a 1 + 2 S cos  k a {\displaystyle E(k)={\frac {E_{0}-2\Delta \cos ka}{1+2S\cos ka}}}
ε m ( k ) = E m − N | b ( 0 ) | 2 ( β m + ∑ R n ≠ 0 ∑ l γ m , l ( R n ) e i k ⋅ R n ) = E m − β m + ∑ R n ≠ 0 ∑ l e i k ⋅ R n γ m , l ( R n ) 1 + ∑ R n ≠ 0 ∑ l e i k ⋅ R n α m , l ( R n ) {\displaystyle {\begin{aligned}\varepsilon _{m}({\boldsymbol {k}})&=E_{m}-N|b(0)|^{2}\left(\beta _{m}+\sum _{{\boldsymbol {R}}_{n}\neq 0}\sum _{l}\gamma _{m,l}({\boldsymbol {R}}_{n})e^{i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{n}}\right)\\&=E_{m}-{\frac {\beta _{m}+\sum _{{\boldsymbol {R}}_{n}\neq 0}\sum _{l}e^{i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{n}}\gamma _{m,l}({\boldsymbol {R}}_{n})}{1+\sum _{{\boldsymbol {R}}_{n}\neq 0}\sum _{l}e^{i{\boldsymbol {k}}\cdot {\boldsymbol {R}}_{n}}\alpha _{m,l}({\boldsymbol {R}}_{n})}}\end{aligned}}}
ここで、 Em は m 番目の原子準位であり、 αm,l, βm, γm,l は強結合行列要素と呼ばれる。