مدل تنگ‌بست - ویکی‌پدیا، دانشنامهٔ آزاد

${\displaystyle H(\mathbf {r} )=H_{\mathrm {at} }(\mathbf {r} )+\sum _{\mathbf {R} _{n}\neq \mathbf {0} }V(\mathbf {r} -\mathbf {R} _{n})=H_{\mathrm {at} }(\mathbf {r} )+\Delta U(\mathbf {r} )\!}$

${\displaystyle \psi _{m}(\mathbf {r} )=\sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})}$

${\displaystyle \psi (\mathbf {r+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }\psi (\mathbf {r} )\,}$

${\displaystyle \sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n}+\mathbf {R} _{\ell })=e^{i\mathbf {k} \cdot \mathbf {R} _{\ell }}\sum _{\mathbf {R} _{n}}b_{m}(\mathbf {R} _{n})\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})\ }$

${\displaystyle b_{m}(\mathbf {R} _{p}+\mathbf {R} _{\ell })=e^{i\mathbf {k\cdot R_{\ell }} }b_{m}(\mathbf {R} _{p})\,}$

${\displaystyle b_{m}(\mathbf {R} _{\ell })=e^{i\mathbf {k} \cdot \mathbf {R} _{\ell }}b_{m}(\mathbf {0} )\ .}$

${\displaystyle \int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )\psi _{m}(\mathbf {r} )=1}$${\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\sum _{\mathbf {R_{\ell }} }b_{m}(\mathbf {R_{\ell }} )\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\varphi _{m}(\mathbf {r} -\mathbf {R} _{\ell })}$${\displaystyle =b_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k\cdot R_{n}} }\sum _{\mathbf {R_{\ell }} }e^{i\mathbf {k\cdot R_{\ell }} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\varphi _{m}(\mathbf {r} -\mathbf {R} _{\ell })}$${\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{p}}e^{-i\mathbf {k} \cdot \mathbf {R} _{p}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{p})\varphi _{m}(\mathbf {r} )\ }$${\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R} _{p}}e^{i\mathbf {k} \cdot \mathbf {R} _{p}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )\varphi _{m}(\mathbf {r} -\mathbf {R} _{p})\ ,}$

${\displaystyle b_{m}^{*}(0)b_{m}(0)={\frac {1}{N}}\ \cdot \ {\frac {1}{1+\sum _{\mathbf {R} _{p}\neq 0}e^{i\mathbf {k} \cdot \mathbf {R} _{p}}\alpha _{m}(\mathbf {R} _{p})}}\,}$

${\displaystyle b_{m}(0)\approx {\frac {1}{\sqrt {N}}}\,}$

${\displaystyle \psi _{m}(\mathbf {r} )\approx {\frac {1}{\sqrt {N}}}\sum _{\mathbf {R} _{n}}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\ \varphi _{m}(\mathbf {r} -\mathbf {R} _{n})\ .}$

${\displaystyle \varepsilon _{m}=\int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )H(\mathbf {r} )\psi _{m}(\mathbf {r} )}$

${\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H(\mathbf {r} )\psi _{m}(\mathbf {r} )}$

${\displaystyle =\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H_{\mathrm {at} }(\mathbf {r} )\psi _{m}(\mathbf {r} )+\sum _{\mathbf {R} _{n}}b_{m}^{*}(\mathbf {R} _{n})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}$

${\displaystyle =\sum _{\mathbf {R} _{n},\mathbf {R} _{l}}b_{m}^{*}(\mathbf {R} _{n})b_{m}(\mathbf {R} _{l})\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})H_{\mathrm {at} }(\mathbf {r} )\varphi _{m}(\mathbf {r} -\mathbf {R} _{l})+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}$

${\displaystyle =b_{m}^{*}(\mathbf {0} )b_{m}(\mathbf {0} )\ N\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )H_{\mathrm {at} }(\mathbf {r} )\varphi _{m}(\mathbf {r} )+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )}$

${\displaystyle \approx E_{m}+b_{m}^{*}(0)\sum _{\mathbf {R} _{n}}e^{-i\mathbf {k} \cdot \mathbf {R} _{n}}\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} -\mathbf {R} _{n})\Delta U(\mathbf {r} )\psi _{m}(\mathbf {r} )\ .}$

${\displaystyle \varepsilon _{m}(\mathbf {k} )=E_{m}-N\ |b_{m}(0)|^{2}\left(\beta _{m}+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}\gamma _{m,l}(\mathbf {R} _{n})e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\right)\ ,}$

${\displaystyle =E_{m}-\ {\frac {\beta _{m}+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\gamma _{m,l}(\mathbf {R} _{n})}{\ \ 1+\sum _{\mathbf {R} _{n}\neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R} _{n}}\alpha _{m,l}(\mathbf {R} _{n})}}\,}$