Manual: Add sub-directories to methods section

docs/methods/embedding/index.md

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+# Embedding
+
+```{toctree}
+---
+titlesonly:
+maxdepth: 1
+---
+kim-gordon
+qm_qm
+```

docs/methods/embedding/kim-gordon.md

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+# Kim-Gordon
+
+## Introduction
+
+This method is based on density embedding. Let's introduce first the subtraction scheme definition
+of the density embedding method:
+
+$ E_{tot} = E_{HK}[\rho_{tot}] - \sum_{A}E_{HK}[\rho_{A}] + \sum_{A}E_{KS}[\rho_{A}] $.
+
+The total electronic density $\rho_{tot} = \sum_{A}\rho_{A}$ is the sum over all the subsystems $A$
+of the subsystem densities $\rho_{A}$. The energy functionals $E_{HK}$ and $E_{KS}$ are the
+Hohenberg–Kohn and the Kohn–Sham functionals, respectively.
+
+$$
+E_{HK}[\rho] = T_{HK}[\rho] + E_{ext}^{HK}[\rho] + \frac{1}{2} \int\int \frac{\rho(r)\rho(r')}{r-r'}drdr' + E_{XC}[\rho] \\
+E_{KS}[P] = T_{S}[P] + E_{ext}[P] + \frac{1}{2} \int\int \frac{\rho(r)\rho(r')}{r-r'}drdr' + E_{XC}[\rho]
+$$
+
+where $P$ is the reduced one-particle density matrix of the system. First of all, it's important to
+introduce the restriction that the external energy functional in the Hohenberg–Kohn energy is linear
+in the density.
+
+$$ E_{ext}^{HK}[\rho_{tot}] = \sum_{A}E_{ext}^{HK}[\rho_{A}] $$
+
+Now, calling the classical Coulomb term $E_{hxc}[\rho]$ and defining the non-additive kinetic energy
+as $T_{nadd}[\rho,{\rho_{A}}] = T_{HK}[\rho]-\sum_{A}T_{HK}[\rho_{A}]$, the obtained equation is:
+
+$$ E_{tot}[{P_{A}}] =\sum_{A}(T_{S}[P_{A}] + E_{ext}[P_{A}]) + E_{hxc}[\rho] + T_{nadd}[{P_{A}}] $$
+
+To avoid the integration of the kinetic energy functional for each subsystem, an atomic potential
+approximation can be applied. For a local potential:
+
+$$
+T_{nadd} = T_{S}[\rho]-\sum_{A}T_{S}[\rho_{A}] = \\
+\int\rho\mu[\rho]dr - \sum_{a}\int\rho_{A}\mu[\rho_{A}]dr = \\
+\sum_{a}\int\rho_{A}(\mu[\rho]-\mu[\rho_{A}])dr
+$$
+
+Doing a linearization approximation for the functional $\mu[\rho]$
+
+$$
+\mu[\rho]-\mu[\rho_{A}] \sim \sum_{B\neq A} \frac{\partial \mu[\rho_{A}]}{\partial \rho} \rho_{B} = \mu'[\rho_{A}] \\
+T_{nadd} = \sum_{A}T_{S}\sum_{B\neq A}\int\mu'[\rho_{A}]\rho_{A}\rho_{B}dr
+$$
+
+A further approximation of the derivative functional in atomic contributions is:
+
+$$ \mu'[\rho_{A}]\rho_{A} = V^{K}[\rho_{A}] \sim \sum_{a \in A}V_{a}^{K}(R_{a}) $$
+
+The realization that a typical kinetic energy functional is proportional to $\rho^{5/3}$ leads to a
+model for the final atomic local potential of the form:
+
+$$ V_{a}^{K}(R_{a}) = N_{a}\rho_{a}^{2/3} $$
+
+where $\rho_{a}$ is a model atomic density. Such local potential can help to speed up the underlying
+embedding calculation.
+
+## Tutorial
+
+The division of the total system into subsystems is a critical point, in order to do that properly
+it is important to specify which is the 'minimum unit', that can be defined in the TOPOLOGY section:
+
+```none
+  &SUBSYS
+    &CELL
+      ABC 9.8528 9.8528 9.8528
+    &END CELL
+    &COORD
+ O   2.28039789       9.14653873       5.08869600       1
+ H   1.76201904       9.82042885       5.52845383       1
+ H   3.09598708       9.10708809       5.58818579       1
+ O   1.25170302       2.40626097       7.76990795       2
+ H  0.554129004       2.98263407       8.08202362       2
+ H   1.77125704       2.95477891       7.18218088       2
+ O   1.59630203       6.92012787      0.656695008       3
+ H   2.11214805       6.12632084      0.798135996       3
+ H   1.77638900       7.46326399       1.42402995       3
+ ...
+    &END COORD
+    &TOPOLOGY
+      CONN_FILE_FORMAT USER
+    &END
+```
+
+This strategy is based on the fourth column in the COORD section. At this point the code is able to
+find the best combination of 'minimum units' through the
+[COLORING_METHOD](#CP2K_INPUT.FORCE_EVAL.DFT.KG_METHOD.COLORING_METHOD) in order to simplify the
+calculation. Another suggestion is to run KG calculations using
+[linear scaling DFT](../dft/linear_scaling), replacing the [SCF](#CP2K_INPUT.FORCE_EVAL.DFT.SCF)
+section with the [LS_SCF](#CP2K_INPUT.FORCE_EVAL.DFT.LS_SCF) section:
+
+```none
+&LS_SCF
+  MAX_SCF     40
+  EPS_FILTER  1.0E-6
+  EPS_SCF     1.0E-7
+  MU         -0.1
+  PURIFICATION_METHOD TRS4
+&END
+```
+
+This speeds up the calculation, especially increasing the dimension of the system.
+
+```{note}
+Keep in mind: all the keywords have to be activated in the QS section as well:
+
+    &QS
+      LS_SCF
+      KG_METHOD
+      ...
+    &END QS
+
+```
+
+Once all these passages are done, one has to choose the
+[TNADD_METHOD](#CP2K_INPUT.FORCE_EVAL.DFT.KG_METHOD.TNADD_METHOD). For the first type of
+calculation, discussed in the previous section, the keyword to select is `EMBEDDING` (default).
+Inside the [](#CP2K_INPUT.FORCE_EVAL.DFT.KG_METHOD) section the XC functional can be selected:
+
+```none
+&XC
+  &XC_FUNCTIONAL
+    &KE_GGA
+      FUNCTIONAL T92   #example
+    &END
+  &END
+&END
+```
+
+And in the same section others corrections can be added (example:
+[VDW_POTENTIAL](#CP2K_INPUT.FORCE_EVAL.DFT.KG_METHOD.XC.VDW_POTENTIAL)). For the second type of
+calculation the keyword to select is ATOMIC. This method implies a supplemental atomic potential
+(create a file which contains all the required potentials). Potential templates can be found inside
+the "tests > QS > regtest-kg" folder of CP2K and they can be generated directly from the code (look
+at "tests > ATOM > regtest-pseudo > O_KG.inp"). It's important to point out that this method is
+still in the experimental stage and further investigations are needed.
+
+```{note}
+Keep in mind: there is also the possibility to completely avoid the $T_{nadd}$ selecting `NONE` as
+[TNADD_METHOD](#CP2K_INPUT.FORCE_EVAL.DFT.KG_METHOD.TNADD_METHOD), but in this way the result of
+the calculation is going to be wrong, since one term is missing.
+```

docs/methods/embedding/qm_qm.md

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+# Quantum Embedding Theories
+
+Unfortunately no one has gotten around to writing this page yet :-(
+
+In the meantime, the following links might be helpful:
+
+- <https://www.cp2k.org/_media/events:2017_dev_meeting:rybkin_cp2kdev-meeting_zurich2017-7.pdf>