Readme with more details on usage and attach the slides for theory ex…

…planation

README.md

@@ -19,29 +19,81 @@
 
 ## Usage
 
-The [examples](https://github.com/unkcpz/PseudopotentialGenerator.jl/tree/main/examples) folder provide scripts include solving atomic Schrödinger equation and pseudolize to get pseudopotentials.
+Generating a pseudopotential requires first having reference states (from real all-electrons potential) and then construct the pseudopotential that can reproduce same scattering behavior.
+
+The [examples](https://github.com/unkcpz/PseudopotentialGenerator.jl/tree/main/examples) folder includes scripts of solving atomic Schrödinger equation and pseudizing to get pseudopotentials.
+
+To solve the atomic Schrödinger equation under DFT framework, simply define the ionic configuration and call `self_consistent_field` function. 
+
+```julia
+Z = 6
+mesh = Mesh(1e-7, 50.0, 2.7e+6, 10000);
+orbs = [Orbital(Z, 1, 0, 2), Orbital(Z, 2, 0, 2), Orbital(Z, 2, 1, 2)];
+
+res = self_consistent_field(
+    Z,
+    mesh,
+    orbs,
+    mixing_beta = 0.2,
+    abstol = 1e-7,
+    maxiters_scf = 200,
+    perturb = true,
+)
+
+ae_info = (ε_lst = res.ε_lst, ϕs = res.ϕs, vae = res.v_tot, occs = res.occs, xc = res.xc);
+```
+
+The `ae_info` contains eigenstates and self consistent potential.
+
+The pseudopotential is then generated to have pseudo-states that meet the restriction with respect to the all-electrons reference.
+The [Troullier-Martins (TM)](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.43.8861) is a one of the popular scheme to construct pseudopotential.
+To generate a TM type pseudopotential, call `pseudize` function,
+
+```julia
+v_pspot, ϕ_ps = pseudize(ae_info, mesh, rc; method = :TM, kbform = false)
+```
+
+The function returns the psudopotential and all pseudo-wavefunctions.
+
+It is essential to check whether the generated pseudopotential is good enough for solid-state DFT calculations.
+One of the prevalent metric is logarithmic derivative (usually `arctan` of the derivative) of wavefunctions which reveal the discripancies between the all-electrons wavefunctions and pseudo-wavefunctions in both eigenstates and energies of scattering states.
+The arctan logarithmic derivative is computed for every angular momentum by calling `compute_atanld` in the given potential,
+
+```julia
+# For all-electrons potential
+compute_atanld(l, Z, mesh, ae_info.vae, rcx, window = [-10.0, 10.0], δ = 0.05)
+
+# For pseudo-potential which is semi-local (dependent on angular momentum l)
+compute_atanld(l, Z, mesh, v_pspot.v[nl], rcx, window = [-10.0, 10.0], δ = 0.05)
+```
+
+## Basic theory and package construction
+
+You will find presentian that covers the thoretical background and the construction detials of this package.
+
+[DFT in atom, pseudopotential generation and its Julia implementation]()
 
 ## For developers
 
-To run tests, some functions from [DFTATOM](https://github.com/certik/dftatom) is wrappered as reference which need to compiled first.
+To run tests, functions from [DFTATOM](https://github.com/certik/dftatom) are wrappered as reference which need to compiled first.
 
 ```bash
 make -C deps all
 ```
 
-Then can run tests by 
+Then run tests
 
 ```bash
 julia --project=@. -e 'using Pkg; Pkg.test()'
 ```
 
-## TODO
+## TODOs
 
 - [x] Solving atomic DFT in radial coordination
 - [x] Migrate from self-crafted ODE solver to purely using `NonlinearSolve.jl` and `OrdinaryDiffEq.jl` as solver.
-- [x] pseudolize using TM.
-- [ ] pseudolize using RRKJ.
-- [ ] pseudolize using BHS.
+- [x] pseudize using TM.
+- [ ] pseudize using RRKJ.
+- [ ] pseudize using BHS.
 - [x] compute logarithmic derivative plots.
 - [ ] Kleinman-Bylander form and its logarithmic derivative by solving integ-ODE.
 - [ ] full relativistic PP