Direct Comparison of Many-Body Methods for Realistic Electronic Hamiltonians

A large collaboration carefully benchmarks 20 first-principles many-body electronic structure methods on a test set of seven transition metal atoms and their ions and monoxides. Good agreement is attained between three systematically converged methods, resulting in experiment-free reference values. These reference values are used to assess the accuracy of modern emerging and scalable approaches to the many-electron problem. The most accurate methods obtain energies indistinguishable from experimental results, with the agreement mainly limited by the experimental uncertainties. A comparison between methods enables a unique perspective on calculations of many-body systems of electrons.

Popular Summary

Quantum mechanics is an amazingly successful theory that enables accurate predictions of the behavior of atoms, molecules, and many materials. However, it becomes exponentially difficult to solve the equations as the number of particles increases, so researchers have developed sophisticated methods to approximate the many-body electron problem. If these approaches are reliable, they could be used to solve many problems, from drug design to high-temperature superconductivity. Here, we run controlled tests of over 20 many-body quantum techniques for 22 electronic systems, which allows for a deeper understanding of the approximations therein and establishes the way forward for improvement of the approximations.

While describing general many-body systems of this size is hopeless, realistic molecules and materials are simpler than they could be. Methods that take advantage of this simplicity tend to perform better. Using statistical comparisons to examine trends in the approximations, we discover that some of the techniques achieve extremely efficient representations of realistic many-body physics that can obtain very accurate results with much less computation than one would expect. We also find that it is possible to study the performance of complex quantum-mechanics calculations without needing to refer to experiment, which opens the door to more studies of this kind that lack reference data.

The next step is to develop methods that can take advantage of the regularities in realistic systems, which will result in faster and more accurate calculations of quantum systems. Further down the line, systematic understanding of the errors of these calculations will provide controlled predictions for the design of new materials.

Figure 1

Cluster analysis of electronic structure methods in this work, using the Seaborn [64] function clustermap. The matrix values are the logarithm of the rms deviation of the total energy in hartrees [Eq. (1)] between the two methods. The methods are ordered according to the “complete” distance metric $d(m,n)=\max ({\sigma }_{m,n})$ as defined in Eq. (1). The lines on the left are dendrograms which quantify the relationship between the two methods. Closely matching methods are linked with short links, and grouped by their similarity.

Figure 2

Kernel density estimation [65, 66, 67] of the percent of the SHCI-computed correlation energy within each basis obtained by each of the methods in the benchmark set. All basis sets available are plotted; individual data points are indicated by small lines.

Figure 3

Kernel density estimation plot of the dissociation energy and ionization potential of molecules and atoms to SHCI reference calculations. Methods are ordered according to the clustering in Fig. 1.

Figure 4

Cancellation of error for many methods in this study, computed by comparing the rms error in the total energy to the rms error in the (a) ionization energy of the atoms and (b) dissociation energy for molecules. HF and CISD are excluded from the comparison for more detail in the more accurate methods; they are off the scale here. The red dashed line corresponds to no cancellation of error.

Figure 5

Comparison of $5z$ dissociation energies of the transition metal molecules obtained from the more accurate methods used in this work to experiment. The $x$ axis is the year the experimental result was published, and the width of the bars indicates statistical or systematic uncertainties.