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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