The main bottleneck of a stochastic or deterministic configuration interaction method is determining the relative weights or importance of each determinant or configuration, which requires large scale matrix diagonalization. Therefore, these methods can be improved significantly from a computational standpoint if the relative importance of each configuration in the ground and excited states of molecular/model systems can be learned using machine learning techniques such as artificial neural networks (ANNs). We have used neural networks to train the configuration interaction coefficients obtained from full configuration interaction and Monte Carlo configuration interaction methods and have tested different input descriptors and outputs to find the more efficient training techniques. These ANNs have been used to calculate the ground states of one- and two-dimensional Heisenberg spin chains along with Heisenberg ladder systems, which are good approximations of polyaromatic hydrocarbons. We find excellent efficiency of training and the model this trained was used to calculate the variational ground state energies of the systems.
The main bottleneck of a stochastic or deterministic configuration interaction method is determining the relative weights or importance of each determinant or configuration, which requires large scale matrix diagonalization. Therefore, these methods can be improved significantly from a computational standpoint if the relative importance of each configuration in the ground and excited states of molecular/model systems can be learned using machine learning techniques such as artificial neural networks (ANNs). We have used neural networks to train the configuration interaction coefficients obtained from full configuration interaction and Monte Carlo configuration interaction methods and have tested different input descriptors and outputs to find the more efficient training techniques. These ANNs have been used to calculate the ground states of one- and two-dimensional Heisenberg spin chains along with Heisenberg ladder systems, which are good approximations of polyaromatic hydrocarbons. We find excellent efficiency of training and the model this trained was used to calculate the variational ground state energies of the systems.
We introduce a machine learning-based approach to selective configuration interaction, dubbed Chembot, that utilizes many novel choices for its model design and training. These choices include the use of a support vector machine to select important configurations, the use of the charge density matrix and configuration energy as features, and heuristics to improve the quality of training data. We test Chembot’s ability to obtain near full configuration interaction quality energies and find that it definitively outperforms its purely Stochastic cousin Monte Carlo configuration interaction by requiring fewer iterations to converge, fewer determinants in the variational space, and fewer important configurations to achieve the same energy. In addition, Chembot at times requires fewer determinants in its variational space than the heat-bath configuration interaction method to achieve the same energy. We demystify Chembot’s innards and then showcase our claims on the set of small but challenging systems: the hydrogen ring (H 4 ), stretched methylene (H 2 C), and stretched water (H 2 O).
Machine learning (ML) has been used to optimize the matrix product state (MPS) ansatz for the wavefunction of strongly correlated systems. The ML optimization of MPS has been tested for the Heisenberg Hamiltonian on one-dimensional and ladder lattices, which correspond to conjugated molecular systems. The input descriptors and output for the supervised ML are lattice configurations and configuration interaction coefficients, respectively. Efficient learning can be achieved from data over the full Hilbert space via exact diagonalization or full configuration interaction, as well as over a much smaller sub-space via Monte Carlo Configuration Interaction. We show that this circumvents the need to calculate energy and operator expectation values and is therefore a computationally efficient alternative to variational optimization.
Selective configuration interaction methods approximate correlated molecular ground- and excited states by considering only the most relevant Slater determinants in the expansion. While a recently proposed neural-network-assisted approach efficiently identifies such determinants, the procedure typically relies on canonical Hartree-Fock orbitals, which are optimized only at the mean-field level. Here we assess approximate natural orbitals - eigenfunctions of the one-particle density matrix computed from intermediate many-body eigenstates - as an alternative. Across our benchmarks for H2O, NH3, CO, and C3H8 we see a consistent reduction in the required determinants for a given accuracy of the computed correlation energy compared to full configuration interaction calculations. Our results confirm that even approximate natural orbitals constitute a simple yet powerful strategy to enhance the efficiency of neural-network-assisted configuration interaction calculations.
The ground state of second-quantized quantum chemistry Hamiltonians provides access to an important set of chemical properties. Wave functions based on machine-learning architectures have shown promise in approximating these ground states in a variety of physical systems. In this paper, we show how to achieve state-of-the-art energies for molecular Hamiltonians using the the neural network backflow (NNBF) wave function. To accomplish this, we optimize this ansatz with a variant of the deterministic optimization scheme based on selected configuration interaction introduced by Li et al., [J. Chem. Theory Comput. 19, 8156 (2023)], which we find works better than standard Markov chain Monte Carlo sampling. For the molecules we studied, NNBF gives lower energy states than both Coupled Cluster with Single and Double excitations and other neural network quantum states. We systematically explore the role of network size as well as optimization parameters in improving the energy. We find that, while the number of hidden layers and determinants play a minor role in improving the energy, there are significant improvements in the energy from increasing the number of hidden units as well as the batch size used in optimization, with the batch size playing a more important role.
We use artificial neural networks (ANNs) based on the Boltzmann machine (BM) architectures as an encoder of ab initio molecular many-electron wave functions represented with the complete active space configuration interaction (CAS-CI) model. As first introduced by the work of Carleo and Troyer for physical systems, the coefficients of the electronic configurations in the CI expansion are parametrized with the BMs as a function of their occupancies that act as descriptors. This ANN-based wave function ansatz is referred to as the neural-network quantum state (NQS). The machine learning is used for training the BMs in terms of finding a variationally optimal form of the ground-state wave function on the basis of the energy minimization. It is relevant to reinforcement learning and does not use any reference data nor prior knowledge of the wave function, while the Hamiltonian is given based on a user-specified chemical structure in the first-principles manner. Carleo and Troyer used the restricted Boltzmann machine (RBM), which has hidden units, for the neural network architecture of NQS, while, in this study, we further introduce its replacement with the BM that has only visible units but with different orders of connectivity. For this hidden-node free BM, the second- and third-order BMs based on quadratic and cubic energy functions, respectively, were implemented. We denote these second- and third-order BMs as BM2 and BM3, respectively. The pilot implementation of the NQS solver into an exact diagonalization module of the quantum chemistry program was made to assess the capability of variants of the BM-based NQS. The test calculations were performed by determining the CAS-CI wave functions of illustrative molecular systems, indocyanine green, and dinitrogen dissociation. The simulated energies have been shown to converge to CAS-CI energy in most cases by improving RBM with an increasing number of hidden nodes. BM3 systematically yields lower energies than BM2, reproducing the CAS-CI energies of dinitrogen across potential energy curves within an error of 50 μE h .
We propose the concept of machine learning configuration interaction (MLCI) whereby an artificial neural network is trained on-the-fly to predict important new configurations in an iterative selected configuration interaction procedure. We demonstrate that the neural network can discriminate between important and unimportant configurations, that it has not been trained on, much better than by chance. MLCI is then used to find compact wave functions for carbon monoxide at both stretched and equilibrium geometries. We also consider the multireference problem of the water molecule with elongated bonds. Results are contrasted with those from other ways of selecting configurations: first-order perturbation, random selection, and Monte Carlo configuration interaction. Compared with these other serial calculations, this prototype MLCI is competitive in its accuracy, converges in significantly fewer iterations than the stochastic approaches, and requires less time for the higher-accuracy computations.