Content deleted Content added
→Total energy: Restored source. Last edit deleted duplicated equation, but also this source which was not duplicated. |
Change sign, otherwise it doesn't conform to the energy functional derivative above and the HF equation below. →Mathematical formulation |
||
Line 95:
\delta E[\phi_k^*(x_k)] &= \sum_{i=1}^N \int\text{d}\mathbf{x}_i \, h^1(\mathbf{x}_i) \phi_i(\mathbf{x}_i) \delta(\mathbf{x}_i -\mathbf{x}_k) \delta_{ik}\\ &+ \sum_{i=1}^N\sum_{j=1}^N \int \mathrm{d}\mathbf{x}_i \int \text{d}\mathbf{x}_j\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{r}_i-\mathbf{r}_j|}\phi_i(\mathbf{x}_i)\phi_j(\mathbf{x}_j) \delta(\mathbf{x}_i-\mathbf{x}_k)\delta_{ik}\\
&- \sum_{i=1}^N\sum_{j=1}^N \int \text{d}\mathbf{x}_i \int \text{d}\mathbf{x}_j\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{r}_i-\mathbf{r}_j|}\phi_i(\mathbf{x}_j)\phi_j(\mathbf{x}_i) \delta(\mathbf{x}_i-\mathbf{x}_k)\delta_{ik}\\
&
&= h^1(\mathbf{x}_k) \phi_k(\mathbf{x}_k)\\
&+ \sum_{j=1}^N \int \text{d}\mathbf{x}_j\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{r}_k-\mathbf{r}_j|}\phi_k(\mathbf{x}_k)\phi_j(\mathbf{x}_j)\\
&- \sum_{j=1}^N \int \text{d}\mathbf{x}_j\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{r}_k-\mathbf{r}_j|}\phi_k(\mathbf{x}_j)\phi_j(\mathbf{x}_k)\\
&
\end{aligned}</math>
| |||