Breaking bonds with the left eigenstate completely renormalized coupled-cluster method

The recently developed [P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105 (2005)] size-extensive left eigenstate completely renormalized (CR) coupled-cluster (CC) singles (S), doubles (D), and noniterative triples (T) approach, termed CR-CC(2,3) and abbreviated in this paper as CCL, is compared with the full configuration interaction (FCI) method for all possible types of single bond-breaking reactions between C, H, Si, and Cl (except H2) and the H2Si[Double Bond]SiH2 double bond-breaking reaction. The CCL method is in excellent agreement with FCI in the entire region R=1-3Re for all of the studied single bond-breaking reactions, where R and Re are the bond distance and the equilibrium bond length, respectively. The CCL method recovers the FCI results to within approximately 1 mhartree in the region R=1-3Re of the H-SiH3, H-Cl, H3Si-SiH3, Cl-CH3, H-CH3, and H3C-SiH3 bonds. The maximum errors are -2.1, 1.6, and 1.6 mhartree in the R=1-3Re region of the H3C-CH3, Cl-Cl, and H3Si-Cl bonds, respectively, while the discrepancy for the H2Si[Double Bond]SiH2 double bond-breaking reaction is 6.6 (8.5) mhartree at R=2(3)Re. CCL also predicts more accurate relative energies than the conventional CCSD and CCSD(T) approaches, and the predecessor of CR-CC(2,3) termed CR-CCSD(T).     

On the accuracy of explicitly correlated coupled-cluster interaction energies--have orbital results been beaten yet?

The basis set convergence of weak interaction energies for dimers of noble gases helium through krypton is studied for six variants of the explicitly correlated, frozen geminal coupled-cluster singles, doubles, and noniterative triples [CCSD(T)-F12] approach: the CCSD(T)-F12a, CCSD(T)-F12b, and CCSD(T)(F12*) methods with scaled and unscaled triples. These dimers were chosen because CCSD(T) complete-basis-set (CBS) limit benchmarks are available for them to a particularly high precision. The dependence of interaction energies on the auxiliary basis sets has been investigated and it was found that the default resolution-of-identity sets cc-pVXZ/JKFIT are far from adequate in this case. Overall, employing the explicitly correlated approach clearly speeds up the basis set convergence of CCSD(T) interaction energies, however, quite surprisingly, the improvement is not as large as the one achieved by a simple addition of bond functions to the orbital basis set. Bond functions substantially improve the CCSD(T)-F12 interaction energies as well. For small and moderate bases with bond functions, the accuracy delivered by the CCSD(T)-F12 approach cannot be matched by conventional CCSD(T). However, the latter method in the largest available bases still delivers the CBS limit to a better precision than CCSD(T)-F12 in the largest bases available for that approach. Our calculations suggest that the primary reason for the limited accuracy of the large-basis CCSD(T)-F12 treatment are the approximations made at the CCSD-F12 level and the non-explicitly correlated treatment of triples. In contrast, the explicitly correlated second-order Mo?ller-Plesset perturbation theory (MP2-F12) approach is able to pinpoint the complete-basis-set limit MP2 interaction energies of rare gas dimers to a better precision than conventional MP2. Finally, we report and analyze an unexpected failure of the CCSD(T)-F12 method to deliver the core-core and core-valence correlation corrections to interaction energies consistently and accurately.     

Testing the parametric two-electron reduced-density-matrix method with improved functionals: application to the conversion of hydrogen peroxide to oxywater

Parametrization of the two-electron reduced density matrix (2-RDM) has recently enabled the direct calculation of electronic energies and 2-RDMs at the computational cost of configuration interaction with single and double excitations. While the original Kollmar energy functional yields energies slightly better than those from coupled cluster with single-double excitations, a general family of energy functionals has recently been developed whose energies approach those from coupled cluster with triple excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. In this paper we test the parametric 2-RDM method with one of these improved functionals through its application to the conversion of hydrogen peroxide to oxywater. Previous work has predicted the barrier from oxywater to hydrogen peroxide with zero-point energy correction to be 3.3-to-3.9 kcal/mol from coupled cluster with perturbative triple excitations [CCSD(T)] and -2.3 kcal/mol from complete active-space second-order perturbation theory (CASPT2) in augmented polarized triple-zeta basis sets. Using a larger basis set than previously employed for this reaction-an augmented polarized quadruple-zeta basis set (aug-cc-pVQZ)-with extrapolation to the complete basis-set limit, we examined the barrier with two parametric 2-RDM methods and three coupled cluster methods. In the basis-set limit the M parametric 2-RDM method predicts an activation energy of 2.1 kcal/mol while the CCSD(T) barrier becomes 4.2 kcal/mol. The dissociation energy of hydrogen peroxide to hydroxyl radicals is also compared to the activation energy for oxywater formation. We report energies, optimal geometries, dipole moments, and natural occupation numbers. Computed 2-RDMs nearly satisfy necessary N-representability conditions.     

New constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix

Three strict constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix (the 1-matrix) are obtained by combining its invariance and stationary properties with the extended Koopmans' theorem. The constraints relate the quantities derived from the functional pertaining to an N-electron system with those of its (N-1)-electron congener. Together with the N-representability requirement for the 1-matrix of the congener, identities involving the electron-electron repulsion energies of the two systems and their derivatives with respect to the 1-matrices seriously narrow down the choices for potential approximate density-matrix functionals. This fact is well illustrated in the case of two-electron systems, where the validity of the new constraints is confirmed and found to originate from a nontrivial cancellation among different terms. Thus, the constraints provide a new tool for the construction and testing of new functionals that complements the previously known conditions such as the reproduction of the homogeneous gas energies and momentum distributions, convexity, and the N-representability of the associated 2-matrices.     

Variational formulation of perturbative explicitly-correlated coupled-cluster methods

We present a variational formulation of the recently-proposed CCSD(2)(R12) method [Valeev, Phys. Chem. Chem. Phys., 2008, 10, 106]. The centerpiece of this approach is the CCSD(2)(R12) Lagrangian obtained via L?wdin partitioning of the coupled-cluster singles and doubles (CCSD) Hamiltonian. Extremization of the Lagrangian yields the second-order basis set incompleteness correction for the CCSD energy. We also developed a simpler Hylleraas-type functional that only depends on one set of geminal amplitudes by applying screening approximations. This functional is used to develop a diagonal orbital-invariant version of the method in which the geminal amplitudes are fixed at the values determined by the first-order cusp conditions. Extension of the variational method to include perturbatively the effect of connected triples produces the method that approximates the complete basis-set limit of the standard CCSD plus perturbative triples [CCSD(T)] method. For a set of 20 small closed-shell molecules, the method recovered at least 94.5/97.3% of the CBS CCSD(T) correlation energy with the aug-cc-pVDZ/aug-cc-pVTZ orbital basis set. For 12 isogyric reactions involving these molecules, combining the aug-cc-pVTZ correlation energies with the aug-cc-pVQZ Hartree-Fock energies produces the electronic reaction energies with a mean absolute deviation of 1.4 kJ mol(-1) from the experimental values. The method has the same number of optimized parameters as the corresponding CCSD(T) model, does not require any modification of the coupled-cluster computer program, and only needs a small triple-zeta basis to match the precision of the considerably more expensive standard quintuple-zeta CCSD(T) computation.     

Perturbation theory corrections to the two-particle reduced density matrix variational method

In the variational 2-particle-reduced-density-matrix (2-RDM) method, the ground-state energy is minimized with respect to the 2-particle reduced density matrix, constrained by N-representability conditions. Consider the N-electron Hamiltonian H(lambda) as a function of the parameter lambda where we recover the Fock Hamiltonian at lambda=0 and we recover the fully correlated Hamiltonian at lambda=1. We explore using the accuracy of perturbation theory at small lambda to correct the 2-RDM variational energies at lambda=1 where the Hamiltonian represents correlated atoms and molecules. A key assumption in the correction is that the 2-RDM method will capture a fairly constant percentage of the correlation energy for lambda in (0,1] because the nonperturbative 2-RDM approach depends more significantly upon the nature rather than the strength of the two-body Hamiltonian interaction. For a variety of molecules we observe that this correction improves the 2-RDM energies in the equilibrium bonding region, while the 2-RDM energies at stretched or nearly dissociated geometries, already highly accurate, are not significantly changed. At equilibrium geometries the corrected 2-RDM energies are similar in accuracy to those from coupled-cluster singles and doubles (CCSD), but at nonequilibrium geometries the 2-RDM energies are often dramatically more accurate as shown in the bond stretching and dissociation data for water and nitrogen.     

Toward a less costly but accurate calculation of the CCSD(T)/CBS noncovalent interaction energy

The popular method of calculating the noncovalent interaction energies at the coupled-cluster single-, double-, and perturbative triple-excitations [CCSD(T)] theory level in the complete basis set (CBS) limit was to add a CCSD(T) correction term to the CBS second-order Møller-Plesset perturbation theory (MP2). The CCSD(T) correction term is the difference between the CCSD(T) and MP2 interaction energies evaluated in a medium basis set. However, the CCSD(T) calculations with the medium basis sets are still very expensive for systems with more than 30 atoms. Comparatively, the domain-based local pair natural orbital coupled-cluster method [DLPNO-CCSD(T)] can be applied to large systems with over 1,000 atoms. Considering both the computational accuracy and efficiency, in this work, we propose a new scheme to calculate the CCSD(T)/CBS interaction energies. In this scheme, the MP2/CBS term keeps intact and the CCSD(T) correction term is replaced by a DLPNO-CCSD(T) correction term which is the difference between the DLPNO-CCSD(T) and DLPNO-MP2 interaction energies evaluated in a medium basis set. The interaction energies of the noncovalent systems in the S22, HSG, HBC6, NBC10, and S66 databases were recalculated employing this new scheme. The consistent and tight settings of the truncation parameters for DLPNO-CCSD(T) and DLPNO-MP2 in this noncanonical CCSD(T)/CBS calculations lead to the maximum absolute deviation and root-mean-square deviation from the canonical CCSD(T)/CBS interaction energies of less than or equal to 0.28 kcal/mol and 0.09 kcal/mol, respectively. The high accuracy and low cost of this new computational scheme make it an excellent candidate for the study of large noncovalent systems.     

Implementation of the CCSD(T)-F12 method using cusp conditions

The explicitly-correlated coupled-cluster singles and doubles with perturbative triples method (CCSD(T)-F12) is implemented using the cusp conditions. Numerical tests for a set of 16 molecules have shown agreement of correlation energies within 1 mE(h) between the cusp-condition and fully-optimized CCSD(T)-F12 methods. Benchmark calculations on 13 chemical reactions with the cusp-condition CCSD(T)-F12 method reproduce experimental enthalpies within 2 kJ mol(-1). It is also shown that regular unitary-invariant ansatz cannot exactly satisfy singlet and triplet cusp conditions in open-shell situations. We present an extended ansatz which can handle both conditions exactly.     

Explicit correlation and basis set superposition error: the structure and energy of carbon dioxide dimer

We have investigated the slipped parallel and t-shaped structures of carbon dioxide dimer [(CO(2))(2)] using both conventional and explicitly correlated coupled cluster methods, inclusive and exclusive of counterpoise (CP) correction. We have determined the geometry of both structures with conventional coupled cluster singles doubles and perturbative triples theory [CCSD(T)] and explicitly correlated cluster singles doubles and perturbative triples theory [CCSD(T)-F12b] at the complete basis set (CBS) limits using custom optimization routines. Consistent with previous investigations, we find that the slipped parallel structure corresponds to the global minimum and is 1.09 kJ mol(-1) lower in energy. For a given cardinal number, the optimized geometries and interaction energies of (CO(2))(2) obtained with the explicitly correlated CCSD(T)-F12b method are closer to the CBS limit than the corresponding conventional CCSD(T) results. Furthermore, the magnitude of basis set superposition error (BSSE) in the CCSD(T)-F12b optimized geometries and interaction energies is appreciably smaller than the magnitude of BSSE in the conventional CCSD(T) results. We decompose the CCSD(T) and CCSD(T)-F12b interaction energies into the constituent HF or HF CABS, CCSD or CCSD-F12b, and (T) contributions. We find that the complementary auxiliary basis set (CABS) singles correction and the F12b approximation significantly reduce the magnitude of BSSE at the HF and CCSD levels of theory, respectively. For a given cardinal number, we find that non-CP corrected, unscaled triples CCSD(T)-F12b/VXZ-F12 interaction energies are in overall best agreement with the CBS limit.     

Low-lying electronic states of M(3)O(9)(-) and M(3)O(9)(2-) (M = Mo, W)

Multiple low-lying electronic states of M(3)O(9)(-) and M(3)O(9)(2-) (M = Mo, W) arise from the occupation of the near-degenerate low-lying virtual orbitals in the neutral clusters. We used density functional theory (DFT) and coupled cluster theory (CCSD(T)) with correlation consistent basis sets to study the structures and energetics of the electronic states of these anions. The adiabatic and vertical electron detachment energies (ADEs and VDEs) of the anionic clusters were calculated with 27 exchange-correlation functionals including one local spin density approximation functional, 13 generalized gradient approximation (GGA) functionals, and 13 hybrid GGA functionals, as well as the CCSD(T) method. For M(3)O(9)(-), CCSD(T) and nearly all of the DFT exchange-correlation functionals studied predict the (2)A(1) state arising from the Jahn-Teller distortion due to singly occupying the degenerate e' orbital to be lower in energy than the (2)A(1)' state arising from singly occupying the nondegenerate a(1)' orbital. For W(3)O(9)(-), the (2)A(1) state was predicted to have essentially the same energy as the (2)A(1)' state at the CCSD(T) level with core-valence correlation corrections included and to be higher in energy or essentially isoenergetic with most DFT methods. The calculated VDEs from the CCSD(T) method are in reasonable agreement with the experimental values for both electronic states if estimates for the corrections due to basis set incompleteness are included. For M(3)O(9)(2-), the singlet state arising from doubly occupying the nondegenerate a(1)' orbital was predicted to be the most stable state for both M = Mo and W. However, whereas M(3)O(9)(2-) was predicted to be less stable than M(3)O(9)(-), W(3)O(9)(2-) was predicted to be more stable than W(3)O(9)(-).     

Explicit correlation and intermolecular interactions: investigating carbon dioxide complexes with the CCSD(T)-F12 method

We have optimized the lowest energy structures and calculated interaction energies for the CO(2)-Ar, CO(2)-N(2), CO(2)-CO, CO(2)-H(2)O, and CO(2)-NH(3) dimers with the recently developed explicitly correlated coupled cluster singles doubles and perturbative triples [CCSD(T)]-F12 methods and the associated VXZ-F12 (where X = D,T,Q) basis sets. For a given cardinal number, we find that results obtained with the CCSD(T)-F12 methods are much closer to the CCSD(T) complete basis set limit than the conventional CCSD(T) results. The relatively modest increase in the computational cost between explicit and conventional CCSD(T) is more than compensated for by the impressive accuracy of the CCSD(T)-F12 method. We recommend use of the CCSD(T)-F12 methods in combination with the VXZ-F12 basis sets for the accurate determination of equilibrium geometries and interaction energies of weakly bound electron donor acceptor complexes.     

Highly Accurate CCSD(T) and DFT–SAPT Stabilization Energies of H‐Bonded and Stacked Structures of the Uracil Dimer

The CCSD(T) interaction energies for the H‐bonded and stacked structures of the uracil dimer are determined at the aug‐cc‐pVDZ and aug‐cc‐pVTZ levels. On the basis of these calculations we can construct the CCSD(T) interaction energies at the complete basis set (CBS) limit. The most accurate energies, based either on direct extrapolation of the CCSD(T) correlation energies obtained with the aug‐cc‐pVDZ and aug‐cc‐pVTZ basis sets or on the sum of extrapolated MP2 interaction energies (from aug‐cc‐pVTZ and aug‐cc‐pVQZ basis sets) and extrapolated ΔCCSD(T) correction terms [difference between CCSD(T) and MP2 interaction energies] differ only slightly, which demonstrates the reliability and robustness of both techniques. The latter values, which represent new standards for the H‐bonding and stacking structures of the uracil dimer, differ from the previously published data for the S22 set by a small amount. This suggests that interaction energies of the S22 set are generated with chemical accuracy. The most accurate CCSD(T)/CBS interaction energies are compared with interaction energies obtained from various computational procedures, namely the SCS–MP2 (SCS: spin‐component‐scaled), SCS(MI)–MP2 (MI: molecular interaction), MP3, dispersion‐augmented DFT (DFT–D), M06–2X, and DFT–SAPT (SAPT: symmetry‐adapted perturbation theory) methods. Among these techniques, the best results are obtained with the SCS(MI)–MP2 method. Remarkably good binding energies are also obtained with the DFT–SAPT method. Both DFT techniques tested yield similarly good interaction energies. The large magnitude of the stacking energy for the uracil dimer, compared to that of the benzene dimer, is explained by attractive electrostatic interactions present in the stacked uracil dimer. These interactions force both subsystems to approach each other and the dispersion energy benefits from a shorter intersystem separation.     

Binding in transition metal complexes: Reduced multireference coupled-cluster study of the MCH2+ (M=Sc to Cu) compounds

The recently developed reduced multireference coupled-cluster method with singles and doubles (RMR CCSD), which is perturbatively corrected for triples [RMR CCSD(T)], is employed to compute binding energies of nine transition metal ions with CH2. Unlike analogous compounds involving main-group elements, the MCH2+ (M=Sc to Cu) transition metal complexes often exhibit a non-negligible multireference character. The authors thus employ the RMR CCSD(T) method, which represents an extension of the standard single-reference (SR) CCSD(T) method and can account for multireference effects, while employing only small reference spaces. In this way the role of quasidegeneracy effects on the binding energies of these complexes can be assessed at a higher SD(T) level than is possible with the widely used ab initio methods, namely, with the standard SR CCSD(T) approach, and provide a new benchmark for these quantities. The difference between the RMR and the standard CCSD(T) methods becomes particularly evident when considering nonequilibrium geometries.     

Accurate ab initio binding energies of alkaline earth metal clusters

The effects of basis set superposition error (BSSE) and core-correlation on the electronic binding energies of alkaline earth metal clusters Y(n) (Y = Be, Mg, Ca; n = 2-4) at the Moller-Plesset second-order perturbation theory (MP2) and the single and double coupled cluster method with perturbative triples correction (CCSD(T)) levels are examined using the correlation consistent basis sets cc-pVXZ and cc-pCVXZ (X = D, T, Q, 5). It is found that, while BSSE has a negligible effect for valence-electron-only-correlated calculations for most basis sets, its magnitude becomes more pronounced for all-electron-correlated calculations, including core electrons. By utilizing the negligible effect of BSSE on the binding energies for valence-electron-only-correlated calculations, in combination with the negligible core-correlation effect at the CCSD(T) level, accurate binding energies of these clusters up to pentamers (octamers in the case of the Be clusters) are estimated via the basis set extrapolation of ab initio CCSD(T) correlation energies of the monomer and cluster with only the cc-pVDZ and cc-pVTZ sets, using the basis set and correlation-dependent extrapolation formula recently devised. A comparison between the CCSD(T) and density functional theory (DFT) binding energies is made to identify the most appropriate DFT method for the study of these clusters.     

Simple coupled-cluster singles and doubles method with perturbative inclusion of triples and explicitly correlated geminals: The CCSD(T)R12 model

To approach the complete basis set limit of the "gold-standard" coupled-cluster singles and doubles plus perturbative triples [CCSD(T)] method, we extend the recently proposed perturbative explicitly correlated coupled-cluster singles and doubles method, CCSD(2)(R12) [E. F. Valeev, Phys. Chem. Chem. Phys. 8, 106 (2008)], to account for the effect of connected three-electron correlations. The natural choice of the zeroth-order Hamiltonian produces a perturbation expansion with rigorously separable second-order energy corrections due to the explicitly correlated geminals and conventional triple and higher excitations. The resulting CCSD(T)(R12) energy is defined as a sum of the standard CCSD(T) energy and an amplitude-dependent geminal correction. The method is technically very simple: Its implementation requires no modification of the standard CCSD(T) program and the formal cost of the geminal correction is small. We investigate the performance of the open-shell version of the CCSD(T)(R12) method as a possible replacement of the standard complete-basis-set CCSD(T) energies in the high accuracy extrapolated ab initio thermochemistry model of Stanton et al. [J. Chem. Phys. 121, 11599 (2004)]. Correlation contributions to the heat of formation computed with the new method in an aug-cc-pCVXZ basis set have mean absolute basis set errors of 2.8 and 1.0 kJmol when X is T and Q, respectively. The corresponding errors of the standard CCSD(T) method are 9.1, 4.0, and 2.1 kJmol when X=T, Q, and 5. Simple two-point basis set extrapolations of standard CCSD(T) energies perform better than the explicitly correlated method for absolute correlation energies and atomization energies, but no such advantage found when computing heats of formation. A simple Schwenke-type two-point extrapolation of the CCSD(T)(R12)aug-cc-pCVXZ energies with X=T,Q yields the most accurate heats of formation found in this work, in error on average by 0.5 kJmol and at most by 1.7 kJmol.     

General-model-space state-universal coupled-cluster methods for excited states: diagonal noniterative triple corrections

The recently developed multireference, general-model-space, state-universal coupled-cluster approach considering singles and doubles (GMS SU CCSD) has been extended to account perturbatively for triples, similar to the ubiquitous single-reference CCSD(T) method. The effectiveness of this extension in handling of excited states and its ability to account for the static and nondynamic correlation effects when considering spin- and/or space-symmetry degenerate levels within the spin-orbital formalism is examined on the example of low-lying excitation energies of the C2, N2, and CO molecules and a comparison is made with the (N,N)-CCSD method used for the same purpose. It is shown that while the triple corrections are very effective in improving the absolute energies, they have only a modest effect on the corresponding excitation energies, which may be even detrimental if both the ground- and excited-state levels cannot be given a balanced treatment. While the triple corrections help to avoid the symmetry-breaking effects arising due to the use of the spin-orbital formalism, they are much less effective in this regard than the (N,N)-CCSD approach.     

The convergence of complete active space self-consistent-field energies to the complete basis set limit

Examination of the convergence of full valence complete active space self-consistent-field energies with expansion of the one-electron basis set reveals a pattern very similar to the convergence of single determinant Hartree-Fock energies. Calculations on 26 molecular examples with the sequence of ntuple-zeta augmented polarized (nZaP) basis sets (n=2, 3, 4, 5, and 6) are used to evaluate complete basis set extrapolation schemes. The most effective extrapolation reduces the rms one-electron basis set truncation errors from 3.03, 0.58, and 0.12 mhartree to 0.23, 0.05, and 0.014 mhartree for the 3ZaP, 4ZaP, and 5ZaP basis sets, respectively.     

Approximations to complete basis set-extrapolated, highly correlated non-covalent interaction energies

The first-principles calculation of non-covalent (particularly dispersion) interactions between molecules is a considerable challenge. In this work we studied the binding energies for ten small non-covalently bonded dimers with several combinations of correlation methods (MP2, coupled-cluster single double, coupled-cluster single double (triple) (CCSD(T))), correlation-consistent basis sets (aug-cc-pVXZ, X = D, T, Q), two-point complete basis set energy extrapolations, and counterpoise corrections. For this work, complete basis set results were estimated from averaged counterpoise and non-counterpoise-corrected CCSD(T) binding energies obtained from extrapolations with aug-cc-pVQZ and aug-cc-pVTZ basis sets. It is demonstrated that, in almost all cases, binding energies converge more rapidly to the basis set limit by averaging the counterpoise and non-counterpoise corrected values than by using either counterpoise or non-counterpoise methods alone. Examination of the effect of basis set size and electron correlation shows that the triples contribution to the CCSD(T) binding energies is fairly constant with the basis set size, with a slight underestimation with CCSD(T)∕aug-cc-pVDZ compared to the value at the (estimated) complete basis set limit, and that contributions to the binding energies obtained by MP2 generally overestimate the analogous CCSD(T) contributions. Taking these factors together, we conclude that the binding energies for non-covalently bonded systems can be accurately determined using a composite method that combines CCSD(T)∕aug-cc-pVDZ with energy corrections obtained using basis set extrapolated MP2 (utilizing aug-cc-pVQZ and aug-cc-pVTZ basis sets), if all of the components are obtained by averaging the counterpoise and non-counterpoise energies. With such an approach, binding energies for the set of ten dimers are predicted with a mean absolute deviation of 0.02 kcal/mol, a maximum absolute deviation of 0.05 kcal/mol, and a mean percent absolute deviation of only 1.7%, relative to the (estimated) complete basis set CCSD(T) results. Use of this composite approach to an additional set of eight dimers gave binding energies to within 1% of previously published high-level data. It is also shown that binding within parallel and parallel-crossed conformations of naphthalene dimer is predicted by the composite approach to be 9% greater than that previously reported in the literature. The ability of some recently developed dispersion-corrected density-functional theory methods to predict the binding energies of the set of ten small dimers was also examined.