|
| 1 | +Background |
| 2 | +=========== |
| 3 | + |
| 4 | +Workflow |
| 5 | +********* |
| 6 | + |
| 7 | +The goal of MALA is to build surrogate models for electronic structure theory. |
| 8 | +These surrogate models are based on neural networks. After training such |
| 9 | +a model, it allows the fast evaluation of the total energy and atomic forces. |
| 10 | +MALA is build around Density Functional Theory, but can in |
| 11 | +principle be used with all electronic structure methods that calculate the |
| 12 | +total energy and atomic forces given atomic positions as input. |
| 13 | +Building these surrogate models requires preprocessing |
| 14 | +:doc:`preprocessing <preprocessing>` of the data, training of a |
| 15 | +:doc:`neural network <neuralnetworks>` and |
| 16 | +:doc:`postprocessing <postprocessing>` of the results. |
| 17 | +MALA is designed for the investigation of systems at non-zero temperatures and |
| 18 | +operates in a "per-grid-point" manner, meaning that every grid point of a |
| 19 | +simulation cell is passed through the network individually. |
| 20 | + |
| 21 | +Density Functional Theory |
| 22 | +************************* |
| 23 | + |
| 24 | +Density Functional Theiry is one of the most popular electronic structure |
| 25 | +calculation methods due to its combination of reasonable accuracy and |
| 26 | +computational cost. |
| 27 | +In DFT, the central quantity is the electronic density of a given system. |
| 28 | +Within the Kohn-Sham framework, this density is given by |
| 29 | + |
| 30 | +.. math:: |
| 31 | +
|
| 32 | + n(\boldsymbol{r}) = \sum_j f^\beta(\epsilon_j)\, |
| 33 | + |\phi_j(\boldsymbol{r})|^2 \; . |
| 34 | +
|
| 35 | +Here, :math:`\phi_j(\boldsymbol{r})` denote the Kohn-Sham wave functions, |
| 36 | +which are given by the Kohn-Sham equations |
| 37 | + |
| 38 | +.. math:: |
| 39 | +
|
| 40 | + \left[-\frac{1}{2}\nabla^2 + v_\mathrm{{\scriptscriptstyle S}}(\mathbf{r}; |
| 41 | + \underline{\boldsymbol{R}})\right] \phi_j(\boldsymbol{r}; |
| 42 | + \underline{\boldsymbol{R}}) = \epsilon_j \phi_j(\boldsymbol{r}; |
| 43 | + \underline{\boldsymbol{R}}) \; , |
| 44 | +
|
| 45 | +which give a system of non-interacting particles restricted to reproduce |
| 46 | +the density of the interacting system. The total energy is evaluated using |
| 47 | + |
| 48 | +.. math:: |
| 49 | +
|
| 50 | + E_\mathrm{total}(\underline{\boldsymbol{r}}) = |
| 51 | + T_\mathrm{{\scriptscriptstyle S}} |
| 52 | + [n](\underline{\boldsymbol{r}}) - |
| 53 | + S_\mathrm{{\scriptscriptstyle S}} |
| 54 | + [n](\underline{\boldsymbol{r}})/\beta |
| 55 | + + E_\mathrm{{\scriptscriptstyle H}} |
| 56 | + [n](\underline{\boldsymbol{r}}) + |
| 57 | + E_\mathrm{{\scriptscriptstyle XC}}[n](\underline{\boldsymbol{r}}) |
| 58 | + + E^{ei}[n](\underline{\boldsymbol{r}})+ E^{ii} + \mu N_e \; . |
| 59 | +
|
| 60 | +Forces and other quantities of interest can be derived from the total energy. |
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