The Power of Hellmann–Feynman Theorem: Kohn–Sham DFT Energy Derivatives with Respect to the Parameters of the Exchange-Correlation Functional at Linear Cost

Efficient methods for computing derivatives with respect to the parameters of scientific models are crucial for applications in machine learning. These methods are important when training is done using gradient-based optimization algorithms or when the model is integrated with deep learning, as they help speed up calculations during the backpropagation pass. In the present work, we applied the Hellmann-Feynman theorem to calculate the derivatives of the Kohn-Sham DFT energies with respect to the parameters of the exchange-correlation functional. This approach was implemented in a prototype program on the basis of Python package PySCF. Using the LDA and GGA functionals as examples, we have shown that this approach scales approximately linear with the system size for a series of <i>n</i>-alkanes (<i>C</i>_<i>n</i><i>H</i>_2<i>n</i>+2, <i>n</i> = 4...64) with a double-zeta basis set. We demonstrated a significant speedup in the derivative calculations in comparison with the widely used automatic differentiation approach such as PyTorch-based DQC, which has a computational complexity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2.0</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2.5</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math>.