Validation part two

Author: Ladme

README.md

@@ -56,26 +56,42 @@ $ gorder YOUR_INPUT_YAML_FILE
 ## Validation
 
 ### Atomistic order parameters
-🏗️ Work in progress... 🏗️
+A CHARMM36m simulation of a membrane consisting of 256 lipids was used to validate the calculation of atomistic order parameters by the `gorder` program. In total, the system contained ~64,500 atoms. The trajectory was 200 ns long and consisted of 10,000 frames. The following programs were used for validation:
+- [`OrderParameter.py` by NMR Lipids](https://github.com/NMRLipids/Databank/blob/main/Scripts/AnalyzeDatabank/calcOrderParameters.py)
+- [VMD's `calc_op.tcl` script](https://www.ks.uiuc.edu/Research/vmd/mailing_list/vmd-l/att-14731/calc_op.tcl)
+
+We also present a comparison of the calculated order parameters with the results from `gmx order`. Note that `gmx order` actually calculates united atom order parameters, so it is not suitable for atomistic systems. (However, many users still utilize it.)
+
+![Three completely overlapping curves, appearing as a single curve, and an additional dashed curve corresponding to `gmx order`.](validation/aaorder_validation.png)
+
+*All programs, except for `gmx order`, produce the same results. Minor variations due to different calculation approaches are too small to be visible in the chart. `gmx order` returns slightly different values (and no value for carbon #16) because it calculates united atom, not atomistic, order parameters.*
 
 ### Coarse-grained order parameters
-A Martini 3 simulation of a membrane consisting of 512 POPC lipids was used to validate the `gorder` program. In total, the system contained ~16,800 beads. The trajectory had a length of 1 μs and consisted of 10,000 frames. The following programs/libraries were used for validation:
+A Martini 3 simulation of a membrane consisting of 512 POPC lipids was used to validate the calculation of CG order parameters by the `gorder` program. In total, the system contained ~16,800 beads. The trajectory had a length of 1 μs and consisted of 10,000 frames. The following programs/libraries were used for validation:
 - [Martini's `do-order` script](https://cgmartini.nl/docs/downloads/tools/other-tools.html#do-order)
 - [`lipyphilic` library](https://lipyphilic.readthedocs.io/en/stable/index.html)
 - [Ladme's `order` program](https://doi.org/10.5281/zenodo.8369479)
 
 ![Four completely overlapping curves, appearing as a single curve.](validation/cgorder_validation.png)
 
-*Yes, all programs returned the same results. Minor variations due to the employed calculation approaches are too small to be visible in the chart.*
+*All programs produce the same results. Minor variations due to the employed calculation approaches are too small to be visible in the chart.*
 
 ## Benchmarking
-> **tl;dr `gorder` is fast. Like *really* fast.**
+### Atomistic order parameters
+Run time of the analyses performed in the Validation section by various programs:
+![Bar chart showing the run time of various programs.](validation/aaorder_benchmark.png)
+
+*Benchmarks were conducted on Debian 12 with an 8-core Intel Core i7-10700 CPU. Benchmarking of `gmx order` and `gorder` was performed using [`hyperfine`](https://github.com/sharkdp/hyperfine). The `NMR lipids` script and `calc_op.tcl` were dramatically slower, so only approximate values obtained using GNU's `time` are reported.*
+
+<sup>a</sup> Note that, unlike `calc_op.tcl` and `gmx order`, the NMR lipids script provides information about the order of individual C-H bonds. `gorder` also provides this information.
+
+<sup>b</sup> Note that `gmx order` calculates united atom order parameters, not atomistic order parameters. For saturated tails, this calculation is reasonably accurate; however, for unsaturated tails, it is **very** inaccurate [[1]](https://doi.org/10.1021/acs.jctc.7b00643). Additionally, it is slower and more tedious to use than `gorder`, so there is little justification for using it for atomistic systems.
 
 ### Coarse-grained order parameters
 Run time of the analyses performed in the Validation section by various programs:
 ![Bar chart showing the run time of various programs.](validation/cgorder_benchmark.png)
 
-*Benchmarks were run on 8-core Intel Core i7-10700 CPU. Benchmarking of `order` and `gorder` was performed using [`hyperfine`](https://github.com/sharkdp/hyperfine). `do-order` and `lipyphilic` were really slow, so only an approximate value obtained using GNU's `time` is reported.*
+*Benchmarks were conducted on Debian 12 with with an 8-core Intel Core i7-10700 CPU. Benchmarking of `order` and `gorder` was performed using [`hyperfine`](https://github.com/sharkdp/hyperfine). `do-order` and `lipyphilic` were dramatically slower, so only an approximate value obtained using GNU's `time` is reported.*
 
 <sup>a</sup> Note that the `do-order` script is not able to calculate order parameters for individual leaflets in the same run. In contrast, both `order` and `gorder` were run with this capability enabled.
 
@@ -88,4 +104,5 @@ Run time of the analyses performed in the Validation section by various programs
 The command line tool and the crate are both released under the MIT License.
 
 ## Limitations
-Only works for simulations with orthogonal simulation boxes.
+- `gorder` only works for simulations with orthogonal simulation boxes.
+- `gorder` is developed on Linux for Linux. While it should work on other operating systems, it is not guaranteed.