with transformer (or multilayer perceptron)
Molecular modeling is a promising with to understand and design everything made out of molecules.
What does a molecular model looks like? atomic geometry & electronic structure
How to model a molecular? Energy calculation -> one of the method: DFT
DFT is already successful but have limitation: violation of mathematical properties in all popular approximations -> limited performance in special systems.
Eliminate some limitation in accuracy of DFT calculation:
- Solve pathological errors from violation of exact conditions for systems with fractional electrons in existing functionals to enhance the accuarcy.
- from 0 to 1 in some application system.
DFT 101
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Time-indenpendent Schrödinger Equation (TISE)
${\hat {H}}\Psi = E\Psi$ this gives energy -
TISE for molecule
${\hat {H}}\Psi =\left[{\hat {T}}+{\hat {V}}+{\hat {U}}\right]\Psi = \left[\sum_{i=1}^{N}\left(-{\frac {\hbar^{2}}{2m_{i}}}\nabla_{i}^{2}\right) + \sum_{i=1}^{N} V(\mathbf {r_i})+ \sum_{i \lt j}^{N}U\left(\mathbf {r_i},\mathbf {r_j}\right) \right]\Psi =E\Psi$ limition in solving the many-body problem limits its solution.
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Hohenberg–Kohn theorems
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electronic density can give wavefunction
$\Psi_{0}=\Psi [n_{0}]$ $O[n_{0}]={\big \langle }\Psi [n_{0}]{\big |}{\hat {O}}{\big |}\Psi [n_{0}]{\big \rangle }$ -
defines an energy functional for the system and proves that the ground-state electron density minimizes this energy functional
$E[\rho ]=T_{s}[\rho ]+\int d\mathbf {r} ,v_{\text{ext}}(\mathbf {r} )\rho (\mathbf {r} )+E_{\text{H}}[\rho ]+E_{\text{xc}}[\rho ]$
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Self-consistent field
$\left[-\frac{\hbar^2}{2m}\nabla^2+V_s(\vec r)\right] \phi_i(\vec r) = \epsilon_i \phi(\vec r)$ $n(\vec r )\equiv n_s(\vec r)=\sum_i^N \left|\phi_i(\vec r)\right|^2$ $V_s = V +\int \frac{e^2n_s(\vec r,')}{|\vec r-\vec r,'|} {\rm d}^3r'+ V_{\rm XC}[n_s(\vec r)]$ - inital guess of
$n(\vec r)$ - calculate
$\V_s$ from DFT functional - calculate
$\phi_i(\vec r)$ or that$n(\vec r)$ from K-S equation - do this until converge
- inital guess of
- Train a new functional that obeys two classes of mathematical constraints with fractional electrons.
- Only the exchange-correlation term
$E_{ex}$ was learned and interfaced to a standard Kohn-Sham DFT code (PySCF).
fixed densities of reactant and product (by B3LYP) -> reaction energy (by experiment or CCSD(T)/CBS)
Input:
Output:
Hyperparameters:
Parameters:
For
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1 for
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5 for
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9 Run same for
10
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Provide a new paradigm for DFT design.
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In general, outpreforms popular hand-made functionals in all datasets.
Especially,
- in bond breaking benchmark (BBB), accuratly described systems with fractional charge (FC) and fractional spin (FS).
- in mindless benchmark subset (MB16-43), accuratly described systems with out-of-distribution exotic geometries. (randomly generated)
In cases where traditional DFTs are expected to be bad,
- correctly described bond breaking for charged and closed-shell neutral molecules
- correctly described charge delocalization in the DNA base pair
- magnetic properties of a compressed hydrogen chain
- reaction barrier heights for a ring-opening intermediate with diradical character
The super slow speed hinders it from widely usage. (near dlpno-CCSD(T))
Improve the speed of feature calculation and convergence. Data for non-main group element.
doubt
- leaking of the training set in test set of BBB
- generalization part all have problems
response
- do leaking but have other part not leaking proving the fact
- no examples that it is really bad at generalization.
Inconsistant performance shown by a recent study [6].
- The trend of DM21 changes several times with the increase of the atomic number
Why dont they directly learn from structure - energy data?
Answer
People have had many tries on it and the ability to generilize the model is the core problem.
Why do they run the same netword twice and average the result?
Answer
To build spin symmetry.
[1] DM21 repo https://github.com/deepmind/deepmind-research/tree/master/density_functional_approximation_dm21
[2] Comment on the paper from John P. Predew https://www.science.org/doi/10.1126/science.abm2445
[3] The doubt from Gerasimov et al. https://www.science.org/doi/10.1126/science.abq3385
[4] The author's reponse of the doubt https://www.science.org/doi/10.1126/science.abq4282
[5] The deepmind blog about this paper https://www.deepmind.com/blog/simulating-matter-on-the-quantum-scale-with-ai
[6] A study shows the inconsistancy of DM21 on one-electron systems https://arxiv.org/pdf/2208.06482.pdf
[7] Intro to DFT theory https://www.youtube.com/watch?v=Ez_Fm4iTUeo