Low-Data Deep Quantum Chemical Learning for Accurate MP2 and Coupled-Cluster Correlations

Abstract

Accurate ab initio prediction of electronic energies is very expensive for macromolecules by explicitly solving post-Hartree–Fock equations. We here exploit the physically justified local correlation feature in a compact basis of small molecules and construct an expressive low-data deep neural network (dNN) model to obtain machine-learned electron correlation energies on par with MP2 and CCSD levels of theory for more complex molecules and different datasets that are not represented in the training set. We show that our dNN-powered model is data efficient and makes highly transferable predictions across alkanes of various lengths, organic molecules with non-covalent and biomolecular interactions, as well as water clusters of different sizes and morphologies. In particular, by training 800 (H2O)8 clusters with the local correlation descriptors, accurate MP2/cc-pVTZ correlation energies up to (H2O)128 can be predicted with a small random error within chemical accuracy from exact values, while a majority of prediction deviations are attributed to an intrinsically systematic error. Our results reveal that an extremely compact local correlation feature set, which is poor for any direct post-Hartree–Fock calculations, has however a prominent advantage in reserving important electron correlation patterns for making accurate transferable predictions across distinct molecular compositions, bond types, and geometries.

Making quantum chemistry compressive and expressive: Toward practical ab‐initio simulation

“…Bearing this in mind, in our electronic T-dNN model, we express the correlated descriptors 143 taking only a few OSVs, and we have found that the electron correlation characters are well reserved for making transferable prediction. Based on numerical experimentation, we define the feature amplitudes that respect the unique physical nature of the electron correlations according to the 2p-2h excitation patterns: vertical (vt, e…”

“…The basic idea is that, when the orbital‐based ab‐initio descriptors are expressed in sufficiently reduced low‐rank basis via single‐ (Section 2.2) or many‐particle (Section 2.3) rotations, the expense of computing these descriptors can be significantly lowered and the model transferability can be enhanced by learning simple molecules. Hence, we have developed a transferable deep neural network (T‐dNN) model 143 using OSV‐based descriptors for predicting chemically accurate MP2 and CCSD correlation energies from a small training set containing small molecules. The low‐rank OSV algorithm of these descriptors virtually compresses a global correlating environment for all electrons in the molecule into many local correlating environments, each for one electron, which simultaneously encodes the long‐range correlation and retains the transferable feature.…”

“…Most importantly, the intrinsically low‐dimensional structure of the compressive input space may increase its inhomogeneity, favor better feature classification and selection, and prevent the ever‐growing scale of the electronic T‐dNN model 266 . We have provided a comprehensive study 143 and shown that the T‐dNN prediction demonstrates an excellent transferability and data efficiency for a broad range of chemical systems, including alkanes, organic molecule and biomolecular interactions, and water clusters of various sizes and morphologies.…”

“…Bearing this in mind, in our electronic T‐dNN model, we express the correlated descriptors 143 taking only a few OSVs, and we have found that the electron correlation characters are well reserved for making transferable prediction. Based on numerical experimentation, we define the feature amplitudes that respect the unique physical nature of the electron correlations according to the 2p–2h excitation patterns: vertical (vt, $$$ {\tilde{\mathbf{T}}}_{ij,\overline{\mu}\overline{\nu}}^{\left(\mathrm{vt}\right)}=\frac{\left[i{\overline{\mu}}_i|j{\overline{\nu}}_j\right]}{\varepsilon_{ij}} $$$), exchange (ex, $$$ {\tilde{\mathbf{T}}}_{ij,\overline{\mu}\overline{\nu}}^{\left(\mathrm{ex}\right)}=\frac{\left[i{\overline{\mu}}_j|j{\overline{\nu}}_i\right]}{\varepsilon_{ij}} $$$) and charge transfer (ct1, $$$ {\tilde{\mathbf{T}}}_{ij,\overline{\mu}\overline{\nu}}^{\left(\mathrm{ct}1\right)}=\frac{\left[i{\overline{\mu}}_i|j{\overline{\nu}}_i\right]}{\varepsilon_{ij}} $$$ for type 1 and ct2, …”

“…129,130 However, as the many-electron wavefunction does not change smoothly as a function of atomic positions, especially when state degeneracies and crossings occur, there may be a considerable challenge in directly predicting wavefunctions with ML models. A plethora of alternative machine learning approaches in the past decade [131][132][133][134][135][136][137][138][139][140][141][142][143][144] utilize the atomistic and electronic localities to further enhance the feature expressibility based on physically relevant knowledge that can be extracted directly from local environments within atomic subsystems or configuration subspaces.…”