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 wavefunctions 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.

In the subsection "Comparison to standard perturbation theory" the expression for the total energy of Hooke's atom to first order should readConsequently results for the error in energy from standard first-order perturbation ͓dotted curve in Fig. 13 ͑see Fig. 1 below͔͒ are modified.Standard first-order perturbation still has the lowest accuracy and no other results are affected.In addition: ͑i͒ In the last equation of the subsection "Position-space information entropy" the constant term in the expression for S when neglecting off-diagonal terms should read ln N / ln 2 instead of 1 / ln 2;͑ii͒ The y-axis in Fig. 5 and Fig. 12 should be labeled S n / ln 2; ͑iii͒ In the sixth paragraph of the section "Overview and Conclusions" it should be տ 0.03, instead of տ 0.003. The results and conclusions of the paper remain unaffected. Energy % Error ω (Hartrees) 0 10 20 30 40 50 0.001 0.01 0.1 1 LDA Perturbed LDA KS Perturbed Exact KS Standard Perturbation FIG. 1. Relative error of the approximate energy from LDA, first-order perturbation of the LDA KS equations, first-order perturbation of the exact KS equations and standard first-order perturbation, compared to the exact energy plotted against .

Approximate natural orbitals are investigated as a way to improve a Monte Carlo configuration interaction (MCCI) calculation. We introduce a way to approximate the natural orbitals in MCCI and test these and approximate natural orbitals from Møller-Plesset perturbation theory and quadratic configuration interaction with single and double substitutions in MCCI calculations of single-point energies. The efficiency and accuracy of approximate natural orbitals in MCCI potential curve calculations for the double hydrogen dissociation of water, the dissociation of carbon monoxide, and the dissociation of the nitrogen molecule are then considered in comparison with standard MCCI when using full configuration interaction as a benchmark. We also use the method to produce a potential curve for water in an aug-cc-pVTZ basis. A new way to quantify the accuracy of a potential curve is put forward that takes into account all of the points and that the curve can be shifted by a constant. We adapt a second-order perturbation scheme to work with MCCI (MCCIPT2) and improve the efficiency of the removal of duplicate states in the method. MCCIPT2 is tested in the calculation of a potential curve for the dissociation of nitrogen using both Slater determinants and configuration state functions.

Hilbert space combines the properties of two different types of mathematical spaces: vector space and metric space. While the vector-space aspects are widely used, the metric-space aspects are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres on which maximum and minimum distances between states can be defined and geometrically interpreted. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground states and to that of particle densities, which are metric, but not Hilbert, spaces. The Hohenberg-Kohn mapping between densities and ground states, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities.

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We review a range of multireference diagnostics for quantum chemistry and discuss them in terms of choices of the molecular orbitals. We show how an approach1 of P.-O. Löwdin can also be viewed as quantifying the electron correlation via the spatial entanglement relative to a single determinant. We consider three example systems from quantum chemistry that exhibit three different combinations of multireference character and correlation: not strongly multireference and not strongly correlated, strongly multireference but not strongly correlated, and strongly multireference together with strong correlation. We find that a multireference measure (MR) does not change substantially with the cutoff used for a Monte Carlo configuration interaction calculation and investigate the effect of using natural orbitals. We see that a coupled-cluster singles and doubles diagnostic and a density-functional theory diagnostic give a correct general prediction of the multireference character for these systems. We also look at the issue of multireference character for a collection of noninteracting hydrogen molecules and the effect of basis size on the multireference character of a stretched hydrogen molecule.

The concept of machine learning configuration interaction (MLCI) [J. Chem. Theory Comput. 2018, 14, 5739], where an artificial neural network (ANN) learns on the fly to select important configurations, is further developed so that accurate ab initio potential energy curves can be efficiently calculated. This development includes employing the artificial neural network also as a hash function for the efficient deletion of duplicates on the fly so that the singles and doubles space does not need to be stored and this barrier to scalability is removed. In addition configuration state functions are introduced into the approach so that pure spin states are guaranteed, and the transferability of data between geometries is exploited. This improved approach is demonstrated on potential energy curves for the nitrogen molecule, water, and carbon monoxide. The results are compared with full configuration interaction values, when available, and different transfer protocols are investigated. It is shown that, for all of the considered systems, accurate potential energy curves can now be efficiently computed with MLCI. For the potential curves of N 2 and CO, MLCI can achieve lower errors than stochastically selecting configurations while also using substantially less processor hours.

We present two methods of calculating the spatial entanglement of an interacting electron system within the framework of density-functional theory. These methods are tested on the model system of Hooke's atom for which the spatial entanglement can be calculated exactly. We analyse how the strength of the confining potential affects the spatial entanglement and how accurately the methods that we introduced reproduce the exact trends. We also compare the results with the outcomes of standard first-order perturbation methods. The accuracies of energies and densities when using these methods are also considered.PACS numbers: 03.67.-a, 71.15.Mb