DeePTB: machine learning for large-scale electronic structure

DeePTB 是我和顾强强一起做的工作。我们当时想解决一个很实际的问题：第一性原理电子结构计算很贵，体系一大，很多事情就没法直接算了。尤其是有限温度的材料，原子一直在振动，模拟时需要取很多不同的构型。如果每个构型都重新做一次 DFT（密度泛函理论）计算，开销很快就受不了。

我们的办法是让机器学习去拟合紧束缚模型里的参数。紧束缚把电子在不同原子轨道之间的运动写成一个哈密顿矩阵，计算起来比直接做 DFT 便宜很多。传统紧束缚模型往往使用固定参数，原子环境变了，就容易不准。DeePTB 根据附近原子的排列预测这些参数，再把算出的电子能级和第一性原理结果对起来训练。

论文里我们做了一个比较大的例子：把磷化镓（GaP）的分子动力学构型扩展到一百万个原子，用学到的哈密顿量配合紧束缚传播方法，计算态密度、光学电导率等随温度变化的性质。这里的一百万原子指紧束缚模拟的规模，并不是做了一次一百万原子的全自洽 DFT。

这种方法也不是训练一次就能到处用。如果材料的成键方式和局部环境跟训练数据差得太远，就需要重新验证，必要时还得补充 DFT 数据。我后来继续做大规模物性计算，也与这件事有关：哈密顿量能很快预测出来以后，电学、光学和输运这些真正关心的物理量，依然不容易算。

DeePTB is a project I worked on with Qiangqiang Gu. We were dealing with a practical problem: first-principles electronic-structure calculations are expensive, and large systems quickly become difficult to handle. At finite temperature, atoms keep moving, so a simulation needs many different atomic configurations. Running a fresh DFT (density-functional theory) calculation for each one soon becomes too costly.

We tried learning the parameters of a tight-binding model. Tight binding represents the motion of electrons between atomic orbitals using a Hamiltonian matrix, which is much cheaper to work with than a full DFT calculation. Conventional tight-binding models often rely on fixed parameters, and those parameters can become inaccurate when the atomic environment changes. DeePTB predicts them from the arrangement of nearby atoms and trains against electronic energy levels calculated from first principles.

In the paper, we tried this on a rather large example. We expanded molecular-dynamics configurations of gallium phosphide (GaP) to a million atoms, then used the learned Hamiltonian with the tight-binding propagation method to calculate temperature-dependent properties, including the density of states and optical conductivity. The million atoms refer to the size of the tight-binding calculation. We did not perform a self-consistent DFT calculation on a million atoms.

Of course, a trained model does not work reliably everywhere. If the bonding or local atomic environments differ substantially from the training data, the predictions need checking, and more DFT data may be needed. This also led into some of my later work on large-scale property calculations. Even when we can predict a Hamiltonian quickly, calculating the electrical, optical and transport properties we care about is still difficult.