Author: Yu Fan
background
The Schrödinger equation is a basic equation of quantum mechanics. By solving this equation, most physical and chemical problems can be solved. But the problem is that the number of basis functions in the Schrödinger equation increases exponentially as the dimension of the molecular system increases. For example, a methane molecule has 5 atoms, its dimension is 9=3*5 – 6, and the number of basis functions is 10 to the 9th power. .
Since the Schrödinger equation cannot be solved exactly, the chemical properties of molecules can be predicted with high-precision approximate solutions. Configuration Interaction and Coupled-Cluster methods have higher accuracy, but the computational cost increases exponentially; although the Density Functional Theory (DFT) method has relatively low computational cost, its accuracy is limited. Due to its powerful nonlinear fitting ability of deep learning, DeepMind proposed FermiNet to achieve approximate solution of the wave function.
Electrons in molecules not only interact with the atomic nucleus and other electrons, but also need to follow the Pauli exclusion principle: two fermions cannot be in the same quantum state, and the wave function after fermion exchange has antisymmetry, that is When two fermions exchange states, the wave function must have the opposite sign. Regarding the antisymmetry of the wave function, the determinant is naturally consistent. Therefore, using the Slater determinant to represent the wave function is widely used in quantum chemistry.
**1. ** Network architecture
Figure 1 Overall architecture of FermiNet network
Figure 2 Information flow transfer between network layers
The overall network architecture of FermiNet is shown in Figure 1, and Figure 2 is a partially enlarged view of the network layer. Each electron in the network not only has a separate information flow, but when information is transmitted at the network layer, each electron will integrate the information of other electrons and the interaction between electrons, and replace the original single electron orbit with one that satisfies substitution equivariance. The multi-electron wave function (Equation 1) constitutes the final wave function (Equation 2), which has stronger expression ability than the traditional Slater determinant.
Before the network starts training, pre-training is performed to improve the stability of the training process and reduce training time. The pre-trained loss uses the solution of the Hartree-Fock equation of the STO-3G basis set as a reference, and the loss function is shown in Equation 3.
For network training, it is based on variational Monte Carlo, with the energy expectation value as the loss function, as shown in Formula 4. Specifically, the energy can be expressed by Equation 5, and the energy gradient is calculated as shown in Equation 6. In addition, in order to efficiently optimize network parameters, the KFAC second-order optimizer that approximates the natural gradient method is used.
**2, ** Experimental results
The accuracy of FermiNet surpasses the traditional VMC method (as shown in Table 1), and the accuracy is better than the CCSD(T) method under limited basis sets, because FermiNet does not use basis sets and there is no problem of basis set extrapolation.
Table 1 Ground state energy values (the bolded part is the item closest to the exact value in FermiNet, VMC and DMC)
Although CCSD(T) is very accurate for equilibrium geometries, it has limitations for molecules that are in low excited states, stretched, twisted, or otherwise out of equilibrium geometries, and is not as good as FermiNet. The results are shown in Figure 3.
Figure 3 Energy curve of molecule H4
For the nitrogen-nitrogen triple bond dissociation of nitrogen molecules, FermiNet is better than the unrestricted CCSD(T) method, as shown in Figure 4.
Figure 4 Energy curve of nitrogen-nitrogen triple bond dissociation
**3, ** Summary
FermiNet fuses information between single electrons and multiple electrons in the network layer, and replaces single electron orbits with multi-electron wave functions that satisfy substitution equivariance. It can achieve higher-precision wave function solutions, not only surpassing traditional ones in accuracy The VMC method is better than the CCSD(T) method for challenging system structures such as non-equilibrium geometric structures and nitrogen-nitrogen triple bond dissociation. The achievements of FermiNet will encourage researchers to propose new or better network architectures in the field of quantum chemistry, thereby achieving more efficient and accurate wave function solutions.
references
[1] Pfau D, Spencer J S, Matthews A G D G, et al. Ab initio solution of the many-electron Schrödinger equation with deep neural networks[J]. Physical Review Research, 2020, 2(3): 033429.
DOI: https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.033429
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