اثر اشتارک

${\displaystyle E_{\mathrm{i}\mathrm{n}\mathrm{t}}=\int \rho (\mathbf{r})V(\mathbf{r})d{\mathbf{r}}^3}$

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${\displaystyle E_{\mathrm{i}\mathrm{n}\mathrm{t}}=-\sum _{i=1}^3{F}_i\int \rho (\mathbf{r})r_{i}d{\mathbf{r}}^{\mathbf{3}}\equiv -\sum _{i=1}^3{F}_i{\mu }_i=-\mathbf{F}\cdot \mathit{\mu }}$

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${\displaystyle V_{\mathrm{i}\mathrm{n}\mathrm{t}}=-\mathbf{F}\cdot \mathit{\mu }}$

${\displaystyle ({\mathbf{V}}_{\mathrm{i}\mathrm{n}\mathrm{t}}{)}_{kl}=⟨{\psi }_k^0|V_{\mathrm{i}\mathrm{n}\mathrm{t}}|{\psi }_l^0⟩=-\mathbf{F}\cdot ⟨{\psi }_k^0|\mathit{\mu }|{\psi }_l^0⟩,k,l=1,\dots ,g.}$

${\displaystyle E^{(1)}=-\mathbf{F}\cdot ⟨{\psi }_1^0|\mathit{\mu }|{\psi }_1^0⟩=-\mathbf{F}\cdot ⟨\mathit{\mu }⟩}$

${\displaystyle n^2=\sum _{\ell =0}^{n-1}(2\ell +1)}$

${\displaystyle 16=1+3+5+7⟹n=4\text{contains}s\oplus p\oplus d\oplus f}$

${\displaystyle E^{(2)}=\sum _{k>0}\frac{⟨{\psi }_0^0|V_{\mathrm{i}\mathrm{n}\mathrm{t}}|{\psi }_k^0⟩⟨{\psi }_k^0|V_{\mathrm{i}\mathrm{n}\mathrm{t}}|{\psi }_0^0⟩}{E_0^{(0)}-E_k^{(0)}}=-\frac{1}{2}\sum _{i,j=1}^3{F}_i{\alpha }_{ij}{F}_j}$

${\displaystyle {\alpha }_{ij}\equiv -2\sum _{k>0}\frac{⟨{\psi }_0^0|{\mu }_i|{\psi }_k^0⟩⟨{\psi }_k^0|{\mu }_j|{\psi }_0^0⟩}{E_0^{(0)}-E_k^{(0)}}.}$

${\displaystyle {\alpha }_{ij}={\alpha }_0{\delta }_{ij}⟹E^{(2)}=-\frac{1}{2}{\alpha }_0{F}^2,}$