Computational cost limits the applicability of post-Hartree–Fock methods such as coupled-cluster on larger molecular systems. The data-driven coupled-cluster (DDCC) method applies machine learning to predict the coupled-cluster two-electron amplitudes ( t 2 ) using data from second-order perturbation theory (MP2). One major limitation of the DDCC models is the size of training sets that increases exponentially with the system size. Effective sampling of the amplitude space can resolve this issue. Five different amplitude selection techniques that reduce the amount of data used for training were evaluated, an approach that also prevents model overfitting and increases the portability of data-driven coupled-cluster singles and doubles to more complex molecules or larger basis sets. In combination with a localized orbital formalism to predict the CCSD t 2 amplitudes, we have achieved a 10-fold error reduction for energy calculations.
Computational cost limits the applicability of post-Hartree–Fock methods such as coupled-cluster on larger molecular systems. The data-driven coupled-cluster (DDCC) method applies machine learning to predict the coupled-cluster two-electron amplitudes ( t 2 ) using data from second-order perturbation theory (MP2). One major limitation of the DDCC models is the size of training sets that increases exponentially with the system size. Effective sampling of the amplitude space can resolve this issue. Five different amplitude selection techniques that reduce the amount of data used for training were evaluated, an approach that also prevents model overfitting and increases the portability of data-driven coupled-cluster singles and doubles to more complex molecules or larger basis sets. In combination with a localized orbital formalism to predict the CCSD t 2 amplitudes, we have achieved a 10-fold error reduction for energy calculations.
ABSTRACT We introduce a hybrid data‐driven coupled‐cluster (DDCC) framework that enables efficient coupled‐cluster singles and doubles (CCSD)‐level geometry optimizations through direct prediction of wavefunction parameters. In conventional CCSD, each optimization step requires solving the nonlinear ‐amplitude equations and the linear ‐equations, both scaling as , making geometry optimizations computationally demanding. The data‐driven CC (DDCC) scheme replaces these iterative solvers with neural network predictions of the and amplitudes, thereby removing the dominant computational bottleneck. The model utilizes a physically grounded representation of electron correlation, combining excitation‐based descriptors with a molecular orbital (MO) framework defined through orbital centroids. This design enables accurate recovery of the CC wave function parameters, which in turn yields reliable nuclear gradients throughout the optimization process. Across a diverse set of thermally perturbed geometries and out‐of‐sample molecules, DDCC reproduces CCSD geometries with near‐quantitative agreement, while reducing computational cost by over an order of magnitude. These results establish DDCC as an effective and transferable strategy for accelerating high‐level electronic structure calculations, providing a practical route toward routine CCSD‐quality geometry optimization in larger and more complex molecular systems.
Solving the coupled-cluster (CC) equations is a cost-prohibitive process that exhibits poor scaling with system size. These equations are solved by determining the set of amplitudes ( t ) that minimize the system energy with respect to the coupled-cluster equations at the selected level of truncation. Here, a novel approach to predict the converged coupled-cluster singles and doubles (CCSD) amplitudes, thus the coupled-cluster wave function, is explored by using machine learning and electronic structure properties inherent to the MP2 level. Features are collected from quantum chemical data, such as orbital energies, one-electron Hamiltonian, Coulomb, and exchange terms. The data-driven CCSD (DDCCSD) is not an alchemical method because the actual iterative coupled-cluster equations are solved. However, accurate energetics can also be obtained by bypassing solving the CC equations entirely. Our preliminary data show that it is possible to achieve remarkable speedups in solving the CCSD equations, especially when the correct physics are encoded and used for training of machine learning models.
Machine learning has recently been introduced into the arsenal of tools that are available to computational chemists. In the past few years, we have seen an increase in the applicability of these tools on a plethora of applications, including the automated exploration of a large fraction of the chemical space, the reduction of repetitive computational tasks, the detection of outliers on large databases, and the acceleration of molecular simulations. An attractive application of machine learning in molecular electronic structure theory is the “recycling” of molecular wave functions for faster and more accurate completion of complex quantum chemical calculations. Along these lines, we have developed hybrid quantum chemical/machine learning workflows that utilize information from low-level wave functions for the accurate prediction of higher-level wave functions. The data-driven coupled-cluster (DDCC) family of methods is discussed in this article together with the importance of the inclusion of physical properties in such hybrid workflows. After a short introduction to the philosophy and the capabilities of DDCC, we present our recent progress in extending its applicability to larger and more complex molecular structures and data sets. A significant advantage offered by DDCC is its transferability, with respect to different molecular systems and different excitation levels. As we show here, predicted wave functions at the coupled-cluster singles and doubles level of theory can be used for the accurate prediction of the perturbative triples of the CCSD(T) scheme. We conclude with some personal considerations with respect to future directions related to the development of the next generation of such hybrid quantum chemical/machine learning models.
We consider the rank-reduced coupled-cluster theory with single and double (RR-CCSD) excitations introduced recently [Parrish et al., J. Chem. Phys. 150, 164118 (2019)]. The main feature of this method is the decomposed form of doubly excited amplitudes, which are expanded in the basis of largest magnitude eigenvectors of MP2 or MP3 amplitudes. This approach enables a substantial compression of amplitudes with only minor loss of accuracy. However, the formal scaling of the computational costs with the system size (N) is unaffected in comparison with the conventional CCSD theory (∝N6) due to the presence of some terms quadratic in amplitudes, which do not naturally factorize to a simpler form even within the rank-reduced framework. We show how to solve this problem, exploiting the fact that their effective rank increases only linearly with the system size. We provide a systematic way to approximate the problematic terms using the singular value decomposition and reduce the scaling of the RR-CCSD iterations down to the level of N5. This is combined with an iterative method of finding dominant eigenpairs of MP2 or MP3 amplitudes, which eliminates the necessity to perform the complete diagonalization, making the cost of this step proportional to the fifth power of the system size, as well. Next, we consider the evaluation of perturbative corrections to CCSD energies resulting from triply excited configurations. The triply excited amplitudes present in the CCSD(T) method are decomposed to the Tucker-3 format using the higher-order orthogonal iteration procedure. This enables us to compute the energy correction due to triple excitations non-iteratively with N6 cost. The accuracy of the resulting rank-reduced CCSD(T) method is studied for both total and relative correlation energies of a diverse set of molecules. Accuracy levels better than 99.9% can be achieved with a substantial reduction of the computational costs. Concerning the computational timings, the break-even point between the rank-reduced and conventional CCSD implementations occurs for systems with about 30–40 active electrons.
Density functional theory (DFT) is the most widely used method for calculating molecular properties; however, its accuracy is often insufficient for quantitative predictions. Coupled-cluster (CC) theory is the most successful method for achieving accuracy beyond DFT and for predicting properties that closely align with experiment. It is known as the ''gold standard'' of quantum chemistry. Unfortunately, the high computational cost of CC limits its widespread applicability. In this work, we present the Molecular Orbital Learning (MōLe) architecture, an equivariant machine learning model that directly predicts CC's core mathematical objects, the excitation amplitudes, from the mean-field Hartree-Fock molecular orbitals as inputs. We test various aspects of our model and demonstrate its remarkable data efficiency and out-of-distribution generalization to larger molecules and off-equilibrium geometries, despite being trained only on small equilibrium geometries. Finally, we also examine its ability to reduce the number of cycles required to converge CC calculations. MōLe can set the foundations for high-accuracy wavefunction-based ML architectures to accelerate molecular design and complement force-field approaches.
Calculating the electronic structure of molecules and solids has become an important pillar of modern research in diverse fields of research from biology and materials science to chemistry and physics. Unfortunately, increasingly accurate and thus reliable approximate solution schemes to the underlying Schrödinger equation scale steeply in computational cost, rendering most accurate approaches like "gold standard" coupled cluster theory, CC, quickly intractable for larger systems of interest. Here we show that this scaling can be significantly reduced by applying machine-learning to the CC correlation energy. We introduce a vector-based representation of CC wave functions and use potential energy surfaces of a small molecule test set to learn the correlation energy from this representation. Our results show that the CC correlation energy can be efficiently learned, even when the representation is constructed from approximate amplitudes provided by computationally less demanding Møller-Plesset (MP2) perturbation theory. Exploiting existing linear scaling MP2 implementations, this potentially opens the door to CC-quality molecular dynamics simulations.