Abstract-This is a survey on graph visualization and navigation techniques, as used in information visualization. Graphs appear in numerous applications such as web browsing, state-transition diagrams, and data structures. The ability to visualize and to navigate in these potentially large, abstract graphs is often a crucial part of an application. Information visualization has specific requirements, which means that this survey approaches the results of traditional graph drawing from a different perspective.

The S22 test set of interaction energies for small model complexes [Phys. Chem. Chem. Phys. 8, 1985 (2006)] has been very valuable for benchmarking new and existing methods for noncovalent interactions. However, the basis sets utilized to compute the CCSD(T) interaction energies for some of the dimers are insufficient to obtain converged results. Here we consistently extrapolate all CCSD(T)/complete basis set (CBS) interaction energies using larger basis sets for the CCSD(T) component of the computation. The revised values, which we designate S22A, represent the most accurate results to date for this set of dimers. The new values appear to be within a few hundredths of 1 kcal mol(-1) of the true CCSD(T)/CBS limit at the given geometries, but the former S22 values are off by as much as 0.6 kcal mol(-1) compared to the revised values. Because some of the most promising methods for noncovalent interactions are already achieving this level of agreement (or better) compared to the S22 data, more accurate benchmark values would clearly be helpful. The MP2, SCS-MP2, SCS-CCSD, SCS(MI)-MP2, and B2PLYP-D methods have been tested against the more accurate benchmark set. The B2PLYP-D method outperforms all other methods tested here, with a mean average deviation of only 0.12 kcal mol(-1). However, the consistent, slight underestimation of the interaction energies computed by the SCS-CCSD method (an overall mean absolute deviation and mean deviation of 0.24 and -0.23 kcal mol(-1), respectively) suggests that the SCS-CCSD method has the potential to become even more accurate with a reoptimization of its parameters for noncovalent interactions.

In benchmark-quality studies of non-covalent interactions, it is common to estimate interaction energies at the complete basis set (CBS) coupled-cluster through perturbative triples [CCSD(T)] level of theory by adding to CBS second-order perturbation theory (MP2) a "coupled-cluster correction," δ(MP2)(CCSD(T)), evaluated in a modest basis set. This work illustrates that commonly used basis sets such as 6-31G*(0.25) can yield large, even wrongly signed, errors for δ(MP2)(CCSD(T)) that vary significantly by binding motif. Double-ζ basis sets show more reliable results when used with explicitly correlated methods to form a δ(MP2-F12)(CCSD(T(*))-F12) correction, yielding a mean absolute deviation of 0.11 kcal mol(-1) for the S22 test set. Examining the coupled-cluster correction for basis sets up to sextuple-ζ in quality reveals that δ(MP2)(CCSD(T)) converges monotonically only beyond a turning point at triple-ζ or quadruple-ζ quality. In consequence, CBS extrapolation of δ(MP2)(CCSD(T)) corrections before the turning point, generally CBS (aug-cc-pVDZ,aug-cc-pVTZ), are found to be unreliable and often inferior to aug-cc-pVTZ alone, especially for hydrogen-bonding systems. Using the findings of this paper, we revise some recent benchmarks for non-covalent interactions, namely the S22, NBC10, HBC6, and HSG test sets. The maximum differences in the revised benchmarks are 0.080, 0.060, 0.257, and 0.102 kcal mol(-1), respectively.

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High-quality benchmark computations are critical for the development and assessment of approximate methods to describe noncovalent interactions. Recent advances in the treatment of dispersion by density functional theory and also the development of more efficient wave function techniques to reliably address noncovalent interactions motivate new benchmark computations of increasing accuracy. This work considers focal point approximations to estimate the complete basis set limit of coupled-cluster theory through perturbative triples [CCSD(T)/CBS] and evaluates how this approach is affected by the use or absence of counterpoise (CP) correction or, as has recently gained traction, the average of those values. Current benchmark protocols for interaction energies are computed with all CP procedures and assessed against the A24 and S22B databases and also to highly converged results for formic acid, cyanogen, and benzene dimers. Whether CP correction, no correction, or the average is favored depends upon the theoretical method, basis set, and binding motif. In recent high-quality benchmark studies, interaction energies often use second-order perturbation theory with extrapolated aug-cc-pVTZ (aTZ) and aug-cc-pVQZ (aQZ) basis sets [MP2/aTQZ] combined with a "coupled-cluster correction," δ MP2 CCSD(T), evaluated in an aug-cc-pVDZ basis. For such an approach, averaging CP-corrected and uncorrected values for the MP2 component and using CP-corrected δ MP2 CCSD(T) values offers errors more balanced among binding motifs and generally more favorable overall. Other combinations of counterpoise correction are not quite as accurate. When employing MP2/aQ5Z extrapolations and an aTZ basis for δ MP2 CCSD(T) , using CP-corrected or averaged MP2 estimates are about equally effective (and slightly superior to uncorrected MP2 values), but the counterpoise treatment of δ MP2 CCSD(T) makes little difference. Focal point estimates at this level achieve benchmark quality results otherwise accessible only with CCSD(T)/aQZ or better.