Methods
Why does this work?
-
The X-ray absorption spectrum encodes the pair and many-body interactions up to about 6 angstroms from the absorbing atom:
The spectrum, \( \chi(k) \), is related to the radial distribution functions \( g(r)_n \), where \(n\) is the order, through the following integral*: \[ \begin{aligned} \langle \chi(k) \rangle = & \int_0^\infty dr \, 4\pi r^2 \rho g_2(r) \gamma^{(2)}(r,k) \\ & + \int dr_1 dr_2 d\phi \, 8\pi^2 r_1^2 r_2^2 \sin(\phi) \rho^2 g_3(r_1, r_2, \phi) \gamma^{(3)}(r_1, r_2, \phi, k) \\ & + \int dr_1 dr_2 dr_3 d\phi d\Omega \, 8\pi^3 r_1^2 r_2^2 r_3^2 \sin(\phi) \rho^3 g_4(r_1, r_2, \phi, r_3, \Omega) \gamma^{(4)}(r_1, r_2, \phi, r_3, \Omega, k) + \cdots \end{aligned} \] *A. Filipponi, A. Di Cicco, C. R. Natoli, Phys. Rev. B 52 (1995) 15122-15134.
Where:-
Pair distribution function: \( g_2(r) \) encodes the pair interactions.
Three-body distribution function: \( g_3(r_1, r_2, \phi) \) encodes the three-body interactions.
Four-body distribution function: \( g_4(r_1, r_2, \phi, r_3, \Omega) \) encodes the four-body interactions. -
The scattering between all interactions can be readily calculated on the molecular dynamics trajectories and \( \chi(k)_{\text{theory}} \) can be compared to \( \chi(k)_{\text{experiment}} \).
Figure 2. Example of scattering paths taken from the lecture by Alexei Kuzmin. You can find more details in the presentation here. -
In some cases, the agreement is very good, i.e., \( \chi(k)_{\text{theory}} \) ~= \( \chi(k)_{\text{experiment}} \), but it depends on the theory used to train the MLFF.
Figure 3. The validation of MLFFs trained on PBE, LDA, and PBEsol DFT functionals for the simulation of Pt systems. The result is quantified by the euclidean distance between the simulated and experimental spectra. PBEsol does the best, followed by LDA, and then PBE. The differences in the dynamics are quantified by the MSRD (mean squared relative displacement of the interatomic distance), where the slope is correlated to the force constant.