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| documentation:standard_operators:coulomb_repulsion [2017/02/27 13:38] – Maurits W. Haverkort | documentation:standard_operators:coulomb_repulsion [2017/05/23 16:43] (current) – Maurits W. Haverkort |
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| The radial part of the operator ($\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}$) is more difficult and will be cast into parameters: | The radial part of the operator ($\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}$) is more difficult and will be cast into parameters: |
| \begin{equation} | \begin{equation} |
| R^{(k)}[\tau_1\tau_2\tau_3\tau_4]=e^2\int_0^{\infty}\int_0^{\infty}\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}R_1[r_i]R_2[r_j]R_3[r_i]R_4[r_j]\mathrm{d}r_i\mathrm{d}r_j. | R^{(k)}[\tau_1\tau_2\tau_3\tau_4]=e^2\int_0^{\infty}\int_0^{\infty}\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}R_1[r_i]R_2[r_j]R_3[r_i]R_4[r_j]r_i^2 r_j^2\mathrm{d}r_i\mathrm{d}r_j. |
| \end{equation} | \end{equation} |
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| ### | ### |
| {{:documentation:standard_operators:coulomb_diagram_ll.png?nolink&400 |}}The Coulomb repulsion between two shells which does not change the number of electrons is given by a direct term ($l_1=l_3$ and $l_2=l_4$) and an indirect or exchange term ($l_1=l_4$ and $l_2=l_3$). The direct term is given by the Slater integrals: | {{:documentation:standard_operators:coulomb_diagram_ll.png?nolink&400 |}}The Coulomb repulsion between two shells which does not change the number of electrons is given by a direct term ($n_1l_1=n_3l_3$ and $n_2l_2=n_4l_4$) and an indirect or exchange term ($n_1l_1=n_4l_4$ and $n_2l_2=n_3l_3$). We here assume that $n_1l_1\neq n_2l_2$. The direct term is given by the Slater integrals: |
| \begin{equation} | \begin{equation} |
| F^{(k)}=e^2\int_0^{\infty}\int_0^{\infty}\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}R_1[r_i]^2R_2[r_j]^2\mathrm{d}r_i\mathrm{d}r_j, | F^{(k)}=e^2\int_0^{\infty}\int_0^{\infty}\frac{\mathrm{Min}[r_i,r_j]^k}{\mathrm{Max}[r_i,r_j]^{k+1}}R_1[r_i]^2R_2[r_j]^2\mathrm{d}r_i\mathrm{d}r_j, |
