- Repository now using regular Git (instead of git-lfs). Please check out again.
- 1.3.0 Released (Code name “Triskaidekaphobia”)
- Update for Mathematica 13.1
- Update macOS instructions for ARM (M1, M2, etc.)
- 1.2.0 Released (Code name “u know u want it”)
- Atomic Surface Delineation
- Spherical Polar Integration
- Parallel Integration Techiques
- Cerjan-Miller-Baker approach to location of critical points
- 1.1.0 Released (Code name “Mulligan”)
- This project’s announcement was a staff pick on Wolfram’s community forum!
- Primitive Cutoffs in Mathematica-compiled versions (competitive with Fortran, so Fortran routines removed)
- Parallel routines for integration
- Bond Path Styles, dashed/dotted based on Laplacian
- Beta Sphere Tuning
- Drastic simplification of the API, moving routines into QTAIM.m
- Plots use rectilinear aspect ratio
- Better design of bounding box, allowing for easier adjustment and propagation by making a variable
- Improved coloring of critical points
Wolfram Language (Mathematica) Package for computations involving the Quantum Theory of Atoms In Molecules (QTAIM)
Q. Are atoms discernible in molecules?
A. Yes, an atom can be defined as the union of steepest ascent gradient paths through the electron density, and its attractor. These atoms are individually proper quantum systems exhibiting the same boundary conditions that are applied to the molecule as a whole. (i.e. the zero-flux condition that applies to isolated molecules and crystals)
All examples here are for water molecule (H2O).
]]
I have always wanted a high-level Mathematica interface to commonly-performed QTAIM tasks. This is for both research and teaching purposes, especially creating highly customized graphics.
Two anticipated use cases are: 1) Teaching QTAIM background and methods and 2) generating annotated graphics. QTAIM.wl is the bridge between QTAIM-computed quantities and the Mathematica ecosystem.
A key characteristic of Mathematica is that can create very high quality graphics and in such a way that they are adaptively sampled, in the space of a graphics routine, i.e. Cartesian, rather than e.g. spherical polar grids. This reduces a certain kind of artifact that I often see. (Though this can be done when appropriate.)
Also, Wolfram Language/Mathematica has a nice graphics grammar, that make composing tailored graphics easy and the results are nice.
Wolfram Engine is basically free for certain use, so there’s much less to complain about concerning Wolfram Language as an implementation. A determined person can figure it out. :-)
This is not a project to do “production” QTAIM, but rather to explain what QTAIM is and make nice graphics, and a way to investigate easily some of the numerical issues that underpin it. I always err on the side of slow and reliable. If QTAIM.wl is useful for production, that is a collateral benefit.
An example of something that I am interested in is comparing ODE methods for steepest-ascent paths. Mathematica has a broad suite of ODE algorithms that have the same interface.
Note: This step is somewhat optional, but still required for WFN file import! TODO: write wfn2wfx.f90. Also, there are a number of ways around this, including molden2aim.
An example, using MacPorts gfortran and MathLink’s mcc in my PATH variable (I may have hacked mcc script in a couple places to first find gcc-mp-12, so it might not need -lgfortran):
export PATH=/Applications/Mathematica.app/Contents/SystemFiles/Links/MathLink/DeveloperKit/MacOSX-ARM64/CompilerAdditions:${PATH}
gfortran-mp-12 -O3 -fPIC -c -o qtaim.o qtaim.f90
mcc -n qtaim.o qtaim.tm main.c -o qtaim -lgfortranon GNU/Linux, you may have to link gfortran:
# sudo apt install build-essential gcc g++ gfortran
gfortran -O3 -fPIC -c -o qtaim.o qtaim.f90
mcc -b64 qtaim.o qtaim.tm main.c -o qtaim -lgfortranI would appreciate it if someone would provide instructions for windows, but I suspect they are similar.
All this is because the electron density and gradient evaluations are so plentiful and expensive, that they must be executed in a compiled language.
Another reason is that the WFN format is of at least historical interest and it’s Fortran-formatted. So here bite the bullet and have a Fortran/MathLink dependency.
An advantage is that it is a mechanism for interfacing with other codes. One simply needs to add C/Fortran code and write the interface like the example for rho1, vide infra.
Akk this is because the electron density and gradient evaluations are so plentiful and expensive, that they must be executed in a compiled language.
Another reason is that the WFN format is abundant and it’s Fortran-formatted. So bite the bullet and have a Fortran/MathLink dependency.
An advantage is that it is a mechanism for interfacing with other codes. One simply needs to add C/Fortran code and write the interface like the example for rho1.
- WFX, WFN import
- PySCF (examples give full-valence CASSCF of water)
pip install pyscf
pip install pyscf[geomopt]
...Mathematica 12 has beautiful slice plots:
The first task is locate the nuclear critical points:
The second task is locate the bond critical points. This uses a graphical technique inside of Mathematica. There are no heuristics.
Symmetrical molecules contain a lot of information along planes.
It is easy to compose images with different quantities. The bond paths shown in the 3D model above can be projected onto the plane:
In the QTAIM, atoms are defined by regions of space that are the union of all steepest ascent paths through the electron density and the attractor.
and composite:
and in 3D:
“Lone Pairs” of electrons can be located by finding maxima in the negative of the Laplacian of the electron density. A feature of QTAIM.wl is that is uses the same techniques to find critical points in the Laplacian as it does in the electron density.
and the plane that shows the pile-up in pink, above and below the plane which protrudes toward the viewer in the 3D version:
- Critic2 / AIMAll / MultiWFN etc. are great and I recommend these results be checked against them
- Infinite-order analytical partial derivatives based on two applications of Leibniz Product Rule of Cartesian GTOs. The best way to find roots is with Hessian information. Since the Laplacian of the electron density already needs second derivatives, its Hessian needs fourth order derivatives! This is too high a derivative to hand code, and everything here is very stiff for numerical differentiation. Written in Modern Fortran.
- Molecular graphs are computed using “J.M.“‘s StackExchange for 3D critical point location. This technique generalizes to fields for which there may not be heuristics for searching, e.g. bond paths between two nuclear critical points. This includes the Laplacian field, as well all other imaginable, such as Kinetic Energy Density topology.
- Data generated adaptively depending on task. Eschews precomputed grids.
- Adams-Bashforth-Moulton (ABM) integrator with variable order/step due to Shampine and Watts (DEABM). We already know these are going start or are going to get stiff, so even better than Doermand-Prince. Need to verify this since tuning beta spheres.
- Mathematica’s infinite possibilities for styling graphics, scripting, orchestrating, extending, and exploring
- Parallelization opportunities abound because its easy to subdivide a number of these methods into subregions and then combine. (Split-Apply-Combine over different computer architectures!)
- Written in a language that is rich with “scientific methods” e.g. has e.g. Hermite Polynomials built-in.
- I can’t tell if this package is slow, or just that accurate images without jagged edges just take a long time to compute. Note that parallelization may be found at the frame/configuration/atom level as well, it does not have to be serial. Again just getting this stood up, many order-saving optimizations to come.
- Other packages for QTAIM exist and they can be interfaced e.g. through files and/or linking. These functions should work with interpolated functions as well. I am currently investigating how Critic2’s results can be utilized. Its routines cover additional domains such as solid state and this package could be “just graphics” in certain cases.
Please send suggestions/comments to:
Eric Brown [email protected]
or open an issue. I am especially interested in new chemistry that you would like to share either as early ideas or results you want to showcase in notebook form.