Assessment of random-phase approximation and second order M{\o}ller-Plesset perturbation theory for many-body interactions in solid ethane, ethylene, and acetylene

The relative energies of different phases or polymorphs of molecular solids can be small, less than a kiloJoule/mol. Reliable description of such energy differences requires high quality treatment of electron correlations, typically beyond that achievable by routinely applicable density functional theory approximations (DFT). At the same time, high-level wave function theory is currently too computationally expensive. Methods employing intermediate level of approximations, such as M{\o}ller-Plesset (MP) perturbation theory and the random-phase approximation (RPA) are potentially useful. However, their development and application for molecular solids has been impeded by the scarcity of necessary benchmark data for these systems. In this work we employ the coupled-clusters method with singles, doubles and perturbative triples (CCSD(T)) to obtain a reference-quality many-body expansion of the binding energy of four crystalline hydrocarbons with a varying $\pi$-electron character: ethane, ethene, and cubic and orthorhombic forms of acetylene. The binding energy is resolved into explicit dimer, trimer, and tetramer contributions, which facilitates the analysis of errors in the approximate approaches. With the newly generated benchmark data we test the accuracy of MP2 and non-self-consistent RPA. We find that both of the methods poorly describe the non-additive many-body interactions in closely packed clusters. Using different DFT input states for RPA leads to similar total binding energies, but the many-body components strongly depend on the choice of the exchange-correlation functional.


I. INTRODUCTION
Accurate prediction of structural and energetic properties of molecular solids is an important ingredient for their numerous applications, particularly in pharmaceutical and materials science fields. 1,2][10][11] More approximate methods, such as density functional theory (DFT) approximations are simpler to perform but are less accurate due to, e.g., errors related to self-interaction or to description of correlation. 12,13][19][20][21][22] While less accurate than CCSD(T), 10 their errors are more consistent or predictable when compared to those of standard DFT approximations. 22,23For example, RPA with so-called singles corrections was shown to offer a) Electronic mail: klimes@karlov.mff.cuni.czbetter and more consistent results than dispersion corrected hybrids for a set of binding energies of molecular solids. 15,22,24Moreover, the central processing unit (CPU) time required to obtain the RPA energy nonself-consistently with a DFT input from generalizedgradient approximation (GGA) functionals is similar to the time needed for the hybrid calculations within periodic settings. 25,26It is important to identify systems for which the approximations in the simpler schemes lead to reduced accuracy to understand the limits of their applicability.9][40][41][42] The work to improve the accuracy has proceeded in several directions.4][55][56][57][58][59][60][61][62][63] However, some of the modifications mentioned above increase substantially the computational demands of RPA or are not yet available within periodic boundary conditions (PBC).For the application to molecular solids, it is therefore useful to understand how the accuracy of the simple and affordable scheme, RPA with renormalized singles corrections (RSE), 24 is affected by the input DFT states. 17,23,64or molecular solids, the accuracy of different methods is usually tested by comparing the binding energies to reference data. 12,15,65This, however, is partially a limitation as the binding energy is only a single number and it is thus difficult to understand in detail the deviations of approximate methods from the reference.A much more detailed understanding can be obtained from the manybody expansion (MBE) of the binding energy. 4,97][68] This reduces the computational requirements of a single energy evaluation so that the use of the reference methods such as CCSD(T) is feasible. 69The accuracy of the simpler scheme, such as RPA or MP2, can be then tested on each of the individual MBE contributions to obtain the errors of the individual n-body fragments.This elucidates the origin of the error and the extent of error cancellation as well as uncovers problems specific to the methods tested. 4,9,23Such an analysis is especially useful for the RPA binding energies of molecular solids where RPA with singles corrections was shown to give accurate results. 10,22he convergence of MBE can be slow, especially for systems with important electrostatic contributions. 7,702][73][74] For example, periodic Hartree-Fock calculations can be combined with MBE of the correlation energy. 6,73,75,76In another example a fitted empirical force-field is used as the simpler method. 71,77In any case, if the subtractive embedding is to be efficient, the simpler scheme should be such that the number of individual MBE contributions that need to be calculated explicitly is as small as possible.The MP2 and RPA approaches are possible choices for the simpler scheme and we have already shown for a methane clathrate cluster that 4-body terms could be well approximated by RPA. 23Compared to MP2, the benefit of RPA within PBC is its more favorable scaling with the number of k-points sampling the reciprocal space as well as with the number of occupied and virtual states and also a substantially lower cost of diagonalisation if a GGA functional is used. 26,78,79The scaling with the system size is also more favorable for RPA (O(N 4 )) than for MP2 (O(N 5 )) when localised basis sets are used.
In this work we obtain MBE reference n-body energies for a set of molecular solids and use the data to understand the origin of the low errors of RPA-based methods observed before, 10,22 and to assess the suitability of MP2 and RPA for the subtractive embedding scheme.We use four molecular crystals of simple hydrocarbons to obtain reference CCSD(T) energies up to the fourth order of MBE.The distance cut-offs that we use for MBE are sufficient to obtain a diverse set of fragments, with molecules both in contact and separated for the 2-and 3-body contributions.We analyse the basis-set convergence of the different methods and its dependence on the fragment size and on the spatial separation of the molecules of the fragment.This also allows us to identify contributions for which large basis-set sizes are not necessary.We use the reference to assess the predictions of MP2 and RPA with and without the RSE corrections.We use the Perdew-Burke-Ernzerhof (PBE) 80 and the strongly constrained and appropriately normed (SCAN) 81 functionals to provide the input states for RPA as these are readily available within PBC settings with an affordable computational cost.

II. COMPUTATIONAL AND THEORETICAL DETAILS
We selected four crystals for our study: monoclinic ethane 82 and ethylene 83 and cubic and orthorhombic forms of acetylene. 84These molecules are small enough to allow for reference CCSD(T) calculations in high-quality basis sets.The importance of electrostatic contributions increases from ethane to acetylene which we expect to have an effect the relative importance of the different MBE terms or on their convergence with the number of fragments considered.To differentiate between the two forms of acetylene, we denote the cubic form as acetylene/c and the orthorhombic as acetylene/o.
The initial structures of the crystals were taken from the Cambridge Structural Database (CSD), 85 see Table S1 for the CSD codes.7][88][89] We kept the lattice parameters at their experimental values.The geometries of isolated molecules were extracted from the optimized crystal structures and used without further optimization to build the clusters for MBE.All the crystal structures and additional information are provided in the Supporting Information (SI) and data repository. 90he binding energy, E b , of a molecular solid is where E sol and E mol are the energies of the solid per unit cell and isolated molecule, respectively.Z is the number of molecules in the unit cell.There are two main ways to obtain E b : a direct evaluation using periodic boundary conditions and MBE.We use the MBE approach in this work and discuss it in the following.The basic idea of MBE is to decompose a calculation of a large (or infinite) system into many smaller subsystem (fragment) calculations.If all the molecules in a crystal are symmetry equivalent, we can select one of them as a reference molecule (ref).The binding energy of the solid, E b , is then evaluated from interaction energies of dimers ∆ 2 E and non-additive three-, four-, and higherbody energies ∆ 3 E, ∆ 4 E, . . .as follows where i, j, and k are indices of molecules other than the reference one.The summations run over all the molecules in the crystal in principle, but in practice cut-offs are introduced.Here we use cut-offs based on the distance between the molecules in the fragments.For dimer we define the distance as the average Cartesian distance of all the pairs of atoms of the two molecules.For trimers and tetramers the distance is the sum of the distances of all the dimers contained in the cluster.Note that in general, the structure of reference gas phase molecule E mol differs from the one in solid, and in that case a monomer deformation energy should be also included in Eq. 2. However, our main aim here is to compare different theoretical methods, we keep the gas phase structure identical to the one in solid and thus the monomer term is zero.
The two-body interaction energies ∆ 2 E ref,j are obtained from the dimer energies E ref,j and monomer energies E ref and E j as The non-additive 3-body contributions ∆ 3 E ref,j,k are evaluated from the trimer energies E ref,j,k using Finally, the non-additive tetramer contributions ∆ 4 E ref,j,k,l are obtained as We obtained MBE contributions up to the 4-body term for all the considered systems.Higher-order terms are likely to contribute marginally 7,69 and their precise evaluation can be difficult due to numerical errors. 3The structures of the fragments were generated by an in-house library 7 and symmetry equivalent clusters were identified using the approach suggested in Ref. 91.
The MBE calculations were performed for CCSD(T), MP2, and RPA.The Molpro program 92 was used for the MP2 and CCSD(T) calculations.The RPA calculations were performed non-selfconsistently, as it is the current practice for molecular solids and other solid state systems.An in-house code using a canonical-orbital variant of the algorithm described in Ref. 64 was used and the input states were obtained by the PBE and SCAN functionals.All the correlation energies were obtained within the frozen-core approximation.Care was taken to obtain the many-body contributions with a high precision.Specifically, the frequency integration grids are optimized separately for each interacting complex as described in Ref. 64.4][95] Our tests show that this is also the case for RPA based on the SCAN input states, see Table S2.To reduce the numerical errors related to the DFT integration grid, we used a dense molecular grid with 150 radial and 590 spherical points.These settings guarantee a precision of a few percent for the three-body interactions (Table S2 and S3) which is sufficient for the tests presented here.
In all the calculations Dunning's augmented correlation-consistent basis sets, 96 shortened as AVXZ (X = D, T, Q, 5) were used.The energies needed to evaluate each of the individual n-body contributions, i.e., ∆ 2 E ref,j , ∆ 3 E ref,j,k , and ∆ 4 E ref,j,k,l were obtained using the basis set of the whole n-body fragment.To reduce the basis-set incompleteness errors we extrapolate the correlation component of the interaction energies to the complete basis-set (CBS) limit using the formula of Halkier et al. 97 where E X is the energy in the AVXZ basis set.We set n = 3 for the canonical versions of CCSD(T) and MP2 as well as for the RPA calculations. 97n the case of MP2 and CCSD, we used also the explicitly correlated (F12) versions of the methods to reduce the basis-set dependence of the correlation energies. 98,99he correlation energies obtained with the F12 methods have a smaller dependence on the basis-set size and allow thus an independent validation of the CBS limit.While they are often close to the CBS limit when AVQZ basis set is used, we also extrapolated them using Eq.6 with n = 5. 7,100 The triples (T) contribution was scaled for the two-body term 101 and unscaled for the three-and four-body contributions, similar to the approach used in Ref. 23.Finally, the complete auxiliary basis set singles corrections (CABS) 102,103 to the HF energy were also calculated and included where appropriate.

A. Reference CCSD(T) binding energies
In this section we discuss the set-up used for the reference CCSD(T) energies.The cut-offs used to obtain the n-body terms are listed in Table I together with the number of symmetry inequivalent fragments that are within the cut-off for each of the crystals.The assumed values of cut-off distances allow enough configurations to be sampled to reliably assess the accuracy of MP2 and RPA.While the n-body contributions are not completely converged with the finite cut-offs, increasing the cut-offs distances would add to the numerical noise. 3,7The total n-body terms depend also on the basis-set size used for the calculations.In the following we discuss each of the n-body terms separately, focusing first on the dependence on the basis set size and then on the dependence on the cut-off distance.Note that in the tables and text the energies are given to three decimal digits, this is primarily to be able to show also small changes between energies.We now discuss the two-body terms, starting with their basis-set convergence.We expect that the convergence will be similar for all the systems.We thus use ethylene to analyse the convergence in detail.We then use the ethylene data to find a reliable settings that we use to obtain the CCSD(T) reference energies for all the systems.Specifically for ethylene, we obtained the 2-body energies of CCSD(T) and its components using the AVDZ, AVTZ, and AVQZ basis sets.Moreover, we have also calculated the 2-body MP2 energies using AVDZ, AVTZ, AVQZ, and AV5Z basis sets.This allows us to compare the MP2 basis-set convergence behavior to that of CCSD(T) and also to test composite schemes that estimate the CCSD(T) basis-set incompleteness error based on the MP2 data. 104he dimer interaction energies typically depend strongly on the basis-set size.However, one could argue that the importance of using a large basis set might be smaller for dimers where the molecules are far from each other.In general, for large separations the interactions become smaller and also the perturbing potential of the other molecule becomes more homogeneous.We therefore first ask what is the distance dependence of the basis-set error of the dimer energies.To assess this we consider the two-body energies obtained with AVDZ and AVQZ basis sets, taking AVQZ as the reference values.We then calculate the error that occurs when contributions above some distance, called separation distance, are obtained with the less precise AVDZ basis set instead of the AVQZ basis.The resulting error is plotted in Fig. 1 for the different contributions to the CCSD(T) energy.
One can see that the use of a large basis is indeed critical for the nearest neighbors, that is molecules within a distance smaller than ∼ 5 Å.Using AVDZ for all the other dimers leads to errors well below 0.1 kJ/mol for each of the energy component.The data show the error made in the 2-body energy when calculations for dimers above the separation distance are made using the AVDZ basis set instead of the AVQZ basis set.
As the basis-set convergence of the two-body energies depends on the intermolecular distance r, we divided the dimers into two groups: a proximate group (r < 10 Å) and a distant group (10 < r < r cut ).The separation distance of 10 Å is based on the data shown in Fig. 1.For ethylene there are 64 dimers in the proximate group and 364 in the distant group.The basis-set convergence of different energy components for the proximate and distant dimers is shown in Table II and Table III, and we discuss them in the following.
The 2-body HF energy converges quickly with the basis-set size, the value for proximate dimers obtained with the AVTZ basis set differs by less than 0.01 kJ/mol from the energy calculated with the AV5Z basis set (Table II).This small error is further reduced to around 0.002 kJ/mol when the CABS corrections are used.The distant dimers, separated by more than 10 Å, contribute by only 0.061 kJ/mol to the 2-body energy of ethylene (Table III).The value changes only marginally, by 0.002 kJ/mol, when going from the AVDZ to the AV5Z basis set.
The components of the correlation energy depend more strongly on the basis-set size, as expected.As noted before, the dependence is larger for the proximate dimers than for the distant dimers.For example, for the proximate dimers the 2-body CCSD/AVDZ energy differs by almost 10 % from the value obtained with the AVQZ basis.In the case of distant dimers, the difference is only around 2.5 %.The difference is also much more significant in absolute numbers.The error is close to 2.6 kJ/mol for the proximate dimers and around 0.01 kJ/mol for the distant dimers.Clearly, small basis sets are sufficient to obtain the interaction energies of the distant dimers.
Larger basis sets and extrapolations to the CBS limit are required for the proximate dimers and we discuss our findings and the resulting set-up in the following paragraphs.
As the 2-body CCSD energy of the proximate dimers depends strongly on the basis-set size, it is more difficult to obtain the reference data at the CBS limit.There are several ways to obtain values close to the CBS limit and we compared extrapolation, use of the F12 corrections, and the so-called ∆MP2 correction where CCSD is combined with MP2 energies obtained in a larger basis set. 104Extrapolation of the canonical CCSD energies obtained with AVTZ and AVQZ basis sets leads to a value which is close to the CCSD-F12b/AVQZ data, the difference is smaller than 0.1 kJ/mol, see Table II.When the CCSD-F12b values are also extrapolated, the difference to extrapolated CCSD decreases to 0.06 kJ/mol.It is not clear which of the two numbers is more precise without going to even larger basis sets.The MP2 data, which we obtained also with the AV5Z basis set, do not help to identify which of the extrapolated values is closer to the CBS.In fact, the change between AVTZ→AVQZ and AVQZ→AV5Z extrapolated values is similar for MP2 and MP2-F12 so that neither of them can be considered more precise than the other.Keeping these uncertainties in mind, we use the CCSD-F12b/AVTZ→AVQZ extrapolated values as the reference.
In the ∆MP2 approach the 2-body CCSD energy ob-tained with a basis set X, E CCSD X , is corrected with the basis set incompleteness error of MP2, E MP2 CBS − E MP2 X , estimated for the same basis set.Interestingly, for the ethylene dimers, the ∆MP2 scheme is less accurate even when the largest basis sets are used, see Table II.This is caused by the different convergence rate of the MP2 and CCSD energies.Performing the ∆ correction with the F12 methods leads to more consistent results.However, this is more likely due to the fact that the energy differences between different basis sets are smaller when the F12 corrections are used.
The last component of the CCSD(T) energy is the triples (T) contribution.Note that for triples we use the scaling procedure proposed by Knizia and co-workers to reduce its basis-set size dependence. 101There is only a small basis-set dependence of the (T) energy both for proximate and distant dimers of ethylene.The (T) contribution of the proximate dimers obtained with the AVTZ and AVQZ basis sets differ only by around 0.01 kJ/mol (Table II).The distant dimers contribute by less than 0.1 kJ/mol to the 2-body (T) energy and even the small AVDZ basis set is sufficient to obtain the contribution with an error less than 0.01 kJ/mol, see Table III.
We observe similar basis-set convergence trends also for the other systems.The largest uncertainty comes from the evaluation of the 2-body CCSD energy of proximate dimers.Using AVTZ→AVQZ extrapolation for canonical CCSD as well as CCSD-F12b leads to values that are within 0.05 kJ/mol of each other.The uncertainty due to HF is almost an order of magnitude smaller, the change of the 2-body HF+CABS energies is below 0.01 kJ/mol upon going from the AVTZ to AVQZ basis set.All the energy components have a negligible basis-set dependence for the distant dimers.
While analysing the data of acetylene dimers, we noted that the F12 and CABS corrections introduce numerical errors into the 2-body energies when the AVQZ basis set is used.The magnitude of the errors is on the order of few tenths of kJ/mol for F12 and one or two orders less for CABS.Specifically, for the distant dimers of cubic acetylene we find a two-body CCSD-F12b/AVTZ energy of −0.40 kJ/mol and the same value for CCSD in either AVTZ or AVQZ basis set.However, the CCSD-F12b/AVQZ contribution is −0.54 kJ/mol.Similar issues were observed before and they likely stem from finite precision errors and the need to sum contributions of a large number of fragments. 3,7The numerical errors in the AVQZ basis are marginal for ethane and ethylene.
Our final reference two-body energies are obtained with the set-up that follows.For the proximate dimers we use CABS-corrected HF in AVQZ basis set together with AVTZ→AVQZ extrapolated CCSD-F12b and scaled (T) contribution obtained with AVQZ basis set.For distant dimers we take the values obtained with the AVTZ basis set, without extrapolation, but with the use of the CABS and F12b corrections.The magnitude of the CABS and F12b corrections is, however, small in the AVTZ basis set, close to 0.001 of kJ/mol for CABS and below 0.005 kJ/mol for F12b.The final 2-body energies obtained with the aforementioned set-up for the different CCSD(T) energy components are summarized in Table IV.We now turn to the convergence of the 2-body energies with the cut-off distance.The convergence of the HF energies is shown in Fig. 2. The convergence is very fast for ethane while we observe an oscillatory convergence of the energy for the two forms of acetylene.The oscillatory behavior is caused by the different electrostatic moments of the molecules, especially the quadrupole moment: ethane has a zero moment while for acetylene it is around 4 a.u. 105,106Similar behavior of the cut-off dependence of the two-body energies was observed for other systems. 7he contributions to the two-body HF energies of ethane are dominated by Pauli repulsion which is short ranged (decaying exponentially) and repulsive.Pauli repulsion dominates initially for ethylene as well, but attractive electrostatic interactions start to dominate above 6.5 Å and they somewhat reduce the repulsive terms, by ≈1 kJ/mol.For acetylene, the repulsive and attractive interactions almost cancel each other so that the total 2-body HF energy is close to zero, see also Table IV.The 2-body CCSD and (T) correlation energies show the same convergence trend for all the systems, see Fig. 3, Fig. S1, and Table IV.The contributions of molecules at small cut-off distance dominate and there are minimal or no oscillations for larger cut-offs.This is expected as the correlation interactions decay proportionally to −r −6 with the intermolecular distance r.However, the contributions of the distant dimers to the 2-body CCSD energies are around −0.4 kJ/mol and can not be thus neglected.Due to the −r −6 decay, the convergence of the 2body energy with the cut-off distance r cut is proportional to −r −3 cut .This can be used to extrapolate the 2-body energy to the infinite cut-off.The extrapolation is not much sensitive to the choice of the fitting interval.For example, extrapolating the energies using data between 10 and 15 Å or between 12 and 18 Å leads to a difference of around 0.03 kJ/mol for the CCSD-F12b/AVTZ energy of ethylene.The extrapolated 2-body CCSD correlation energies differ from those obtained for a finite cutoff by 0.08 kJ/mol for ethane and ethylene and around 0.03 kJ/mol for the two forms of acetylene.Therefore, they would be still relevant when energies converged with the cut-off distance were sought.
The (T) contribution for distant dimers is close to −0.1 kJ/mol for all the systems, which is also not negligible.In fact, the (T) terms are around 1/5 to 1/6 of the 2-body CCSD correlation energy and this ratio is similar to that obtained for the proximate dimers.Therefore, the relative importance of the (T) contribution does not decay with distance.The possible reason for this is that the (T) terms change the response properties of the molecules.This can be thought of as a change of the effective C 6 coefficients.Overall, when the 2-body HF and CCSD(T) correlation energies are added together, they are rather similar for all the systems, with values between −23.49kJ/mol for ethylene and −27.66 kJ/mol for acetylene/c, see Table V.One can see that the binding at the HF level increases when going from ethane to acetylene/o in Table VI while it decreases for CCSD and (T).The similar binding energies for the different systems are thus a result of a compensation between the mean-field and correlation contributions.

Three-body terms
We now turn to the 3-body terms for which we first analyze the basis set errors taking, again, ethylene as a representative case.The total 3-body contributions obtained for ethylene using a cut-off distance of 26.3 Å and different methods and basis sets are collected in Table VII and we discuss the main findings in the following.
The 3-body HF+CABS contribution shows very little dependence on the basis-set size, changing by 0.004 kJ/mol between the AVDZ and AVQZ basis sets (Table VII).This is consistent with previous results obtained for other systems. 23,69Interestingly, the CABS corrections are essentially negligible.Due to the fast convergence with the basis-set size, we use the AVTZ basis set to evaluate the 3-body HF energies for the other systems as well.To obtain the reference 3-body HF energy we include the CABS corrections.
The non-additive correlation energies converge also faster with the basis-set size than the 2-body terms.For example, the 3-body MP2 energies change only by ∼0.1 kJ/mol when going from the AVDZ to the AVQZ basis set.Extrapolating the AVTZ and AVQZ values for MP2 leads to a value which differs by only ∼0.01 kJ/mol from the AVQZ data and thus extrapolation is hardly necessary.The 3-body MP2-F12 energy shows even a smaller dependence on the basis-set size, the data could be considered converged already in the AVDZ basis set.Note, however, that this does not necessarily guarantee that the convergence of the CCSD or (T) energies would be also fast as MP2 is missing three-body correlation.In any case, the CCSD and CCSD-F12b energies converge also quickly with the basis set size, although with a different rate compared to their MP2 equivalents.Unexpectedly, the F12b-corrected CCSD energy depends more strongly on the basis-set size than the canonical CCSD variant.The changes between AVDZ and AVTZ are, however, only some hundredths of kJ/mol.The (T) terms, which we evaluate without scaling, 23,101 change by even a smaller amount, around 0.004 kJ/mol.We therefore use the AVTZ basis set to evaluate the 3-body CCSD and (T) energies for all the systems and we include the F12b corrections.
We now discuss the convergence of the 3-body energies with the cut-off distance for all the systems.The data are shown in Fig. 4 for HF+CABS, Fig. 5 for CCSD-F12b, and in Fig. S2 for the (T) contribution.We note that even though the values are not completely converged with cut-off, the set contains compact trimers formed by molecules in contact as well as trimers with molecules separated by almost 10 Å.It therefore contains sufficient data for assessing the accuracy of other methods.Cut-off distance convergence of the 3-body HF+CABS/AVTZ energies for all the considered systems.
The convergence of the 3-body HF energies clearly depends on the magnitude of the electrostatic moments, as was the case for the 2-body energy.The convergence is very fast for ethane, with terms above r cut > 16 Å contributing by less than 0.05 kJ/mol to the final value of −0.62 kJ/mol.In contrast, one can see a slow convergence for both forms of acetylene.There are large negative and positive terms for distances below 20 Å that lead to changes of several kJ/mol.Beyond that distance, the 3-body energies are converged to within a few tenths of kJ/mol for all the systems including acetylene.Interestingly, despite the very different convergence with the cut-off, the final 3-body HF energy is between 0 and −1 kJ/mol for all the systems.The 3-body CCSD correlation energies of all the systems are dominated by contributions of trimers with distances below ∼20 Å, the values then change by only ∼0.2 kJ/mol between 20 Å and the cut-off used.The convergence is again affected by the magnitude of the electrostatic moments: it is almost monotonic for ethane while there are significant positive and negative contributions for acetylene.The final 3-body CCSD energies (within the cut-offs) are between 1 and 2 kJ/mol.One can also note that their magnitude decreases when going from ethane to acetylene/o, the ordering is thus the same as for the 2-body energies (Table VI).The 3-body (T) energies are rather small, around 0.2 kJ/mol for all the systems and show a similar convergence as the 3-body CCSD energies (Fig. S2).
Overall, we find that the total 3-body contributions are repulsive for all the systems, with values close to 1 kJ/mol, see Table V.As with the 2-body terms the similar final values are due to a partial cancellation between the HF and correlation contributions.

Four-body terms
We now turn to the 4-body terms starting with their basis-set convergence.As for the 2-and 3-body energies, we assessed the convergence in more detail for ethylene.Previous works have shown that the basis-set convergence of the 4-body terms is fast. 23,69Indeed, we observe that for ethylene the HF and MP2 values change by at most a few thousandths of kJ/mol when going from the AVDZ to the AVTZ basis set (Table VIII).We were only able to perform the CCSD(T) calculations in the AVDZ basis set.Based on the MP2 data and the convergence of CCSD(T) for the 3-body energy, we expect that the basis-set error for AVDZ is negligible, below 0.01 kJ/mol.This is also supported by the small effect of the F12b corrections.We use the HF+CABS/AVTZ and CCSD(T)-F12b/AVDZ values as the reference data.Cut-off distance convergence of the 4-body HF+CABS/AVTZ energy for all the considered systems, note the small scale on the y axis.Note that the y axis scale is approximately one half compared to the HF case in Fig. 6.
We compare the total 4-body CCSD(T) energies for all the systems in Table V and their components in Table VI.Clearly, all the components of the 4-body CCSD(T) energy have, within the cut-offs used, a small magnitude.The HF and correlation contributions have opposite signs and partially cancel each other for all the systems.The final 4-body CCSD(T) energy is close to 0.1 kJ/mol for all the systems but for acetylene/o where it is −0.03 kJ/mol.
The convergence of the 4-body energies with distance is shown in Fig. 6 and Fig. 7 for HF+CABS and CCSD-F12b, respectively.As with the 2-and 3-body energies, the distance convergence of the 4-body HF+CABS energy varies depending on the electrostatic moments of the molecule: For ethane the values stay between 0 and 0.05 kJ/mol, for acetylene/c the energy first reaches a value of almost 0.5 kJ/mol at a cut-off of around 30 Å, and then it returns to zero.In contrast, the 4-body correlation energies show a similar convergence pattern for all the systems.
Note that the 4-body energies are not completely converged with the cut-off distance.We have tested that extending the cut-off by around 2 Å changes the total 4body CCSD(T) energies by −0.03 to −0.08 kJ/mol, but we do not include these data due to possible numerical issues.We note that regardless of the cutoff convergence issues, the CCSD(T) results form a valid reference data set for assessing the performance of approximate methods.

Summary
The n-body CCSD(T) contributions and their sum are summarized in Table V.We note that due to the use of finite cut-offs and basis sets, the n-body energies are not completely converged.We estimate that the deviations from the converged values are only few tenths of kJ/mol.A part of the difference comes from the basis-set incompleteness error.This is most likely significant only for the 2-body energies where the difference between extrapolated CCSD and CCSD-F12b was around 0.05 kJ/mol, the basis-set errors are negligible (close to 0.01 kJ/mol) for the non-additive terms.
The error due to the finite distance cut-off can be almost avoided by extrapolation for the 2-body energies.This leads to an uncertainty of the 2-body terms well below 0.1 kJ/mol.The convergence with the cut-off distance is more problematic for the 3-and 4-body energies, especially for systems with strong electrostatic interactions.However, their convergence with the distance cut-off suggests that they are converged to some tenths of kJ/mol.

B. Accuracy of RPA and MP2
We now use the CCSD(T) energies as a reference to examine the accuracy of the RPA and MP2 methods for predicting the total n-body energies as well as the contributions of the individual fragments.We reiterate that our aim here is to understand the good accuracy of RPA with singles corrections observed for molecular solids within periodic boundary conditions, 22 gain more insight into the difference between PBE and SCAN input states for non-additive energies, 64 and test the suitability of RPA for the subtractive embedding scheme.Before that we briefly comment on the basis-set convergence of RPA and MP2 and the numerical set-up used to perform the calculations.
The basis-set convergence of MP2 is similar to that of CCSD(T), as discussed in the previous part.We therefore use identical set-up to that used for CCSD(T) for the 2-and 3-body terms.For the 4-body MP2 terms we use the AVTZ basis set instead of the AVDZ that we used for CCSD(T).However, the results obtained with AVDZ and AVTZ basis sets differ only marginally (around 0.001 kJ/mol for ethylene), see Table VIII.
The RPA calculations used the same set-up for all the n-body terms.The EXX and RSE energies were obtained with the AVQZ basis set.All the RPA correlation energies were evaluated by AVTZ→AVQZ extrapolation.The extrapolation is performed also for the 3-and 4body energies as they show a stronger dependence on the basis-set size compared to the HF-based methods, as discussed below and shown in Tables S5-S10.

Two-body terms
We discuss first the mean field (or single determinant) contributions, that is HF, EXX, and RSE.The data are shown in Tables IX and X for the proximate and distant dimers, respectively.One can see that EXX gives more repulsive 2-body energies than HF for both PBE and SCAN input states.This is consistent with previous observations. 21,24The repulsion is, however, much smaller for EXX based on SCAN, which agrees with previous calculations for molecular clusters. 23,64When the RSE corrections are added to EXX, the difference to HF is reduced to ∼1-2 kJ/mol for both inputs with the SCAN-based values still closer to HF than the PBE-based data.The mean-field contributions of the distant dimers are small, below 0.2 kJ/mol (Table X).Interestingly, there is a close agreement between the HF values and EXX(SCAN) values for all the systems and the RSE corrections for EXX(SCAN) are negligible.The EXX(PBE) values differ from EXX(SCAN) and HF for both forms of acetylene but the differences are reduced upon addition of the RSE correction.We now add the 2-body correlation energies to the mean-field data for the proximate dimers to compare the methods.Note that this is necessary due to the different 2-body mean-field energies for the post-HF methods and RPA for the proximate dimers.For MP2 we find the expected behavior: the difference to CCSD(T) increases significantly when going from the aliphatic ethane to the molecules with delocalized π-electron systems (Table XI. 27,107The errors are ca. 1 kJ/mol for ethane and as large as 5 kJ/mol for acetylene/c. The RPA correlation energies based on the PBE and SCAN states differ by around 1 kJ/mol from each other with RPA(PBE) giving a stronger binding (Table S4).The differences in the correlation energies then partly cancel the differences in the mean-field EXX energies (Table IX) so that EXX+RPA based on SCAN binds around 0.2 to 2 kJ/mol more strongly than EXX+RPA with PBE input.The situation is reversed when the RSE corrections are added, see Table XI.Overall, the total RPA 2-body energies with the RSE corrections underestimate the CCSD(T) reference by around 1 to 3 kJ/mol, with PBE states giving smaller errors compared to the SCAN input.
For distant dimers, the EXX+RSE are almost identical regardless of the input states and agree with HF.The differences in binding are then entirely due to the correlation part.For all systems considered, using SCAN states as input for RPA leads to a close agreement with CCSD for the long-range interactions, the differences are within 0.01 kJ/mol, see Table XII.The differences are somewhat larger for RPA(PBE).The close agreement of CCSD and RPA can be also seen in Fig. 8(a) for ethane and Fig. 8(b) for acetylene/c which shows the cut-off dependence of the binding energy.Note that the graph shows the sum of contributions of dimers with distance larger than the cut-off on the x axis.As expected, the 2-body MP2 correlation energies of distant dimers do not show a consistent behavior.MP2 is very close to CCSD(T) for ethane (Fig. 8(a)), but the binding is overestimated for acetylene/c (Fig. 8(b)).
The total 2-body energies of MP2 and the two RPA variants are compared to the CCSD(T) values for all the systems in Fig. 9.One see that there is little difference between RPA with RSE based on SCAN and PBE.Using SCAN states produces stronger binding from EXX and RSE, but weaker correlation compared to PBE-based RPA.RPA as well as MP2 predict similar 2-body energies for ethane.However, when going to ethylene and acetylene the MP2 binding gets too strong while either of the RPA variants produce underestimated binding.b) acetylene/c.The energy, in kJ/mol, is a sum of two-body contributions of dimers with a distance larger than the cut-off given on the x axis and smaller than the largest cut-off used.The calculations used the AVTZ basis set.

Three-body terms
We discuss first the mean field energies, that is HF and EXX without and with RSE, shown in Table XIII.The PBE-and SCAN-based EXX energies show a different behavior with respect to the HF values, the first gives a too strong binding, the latter is more repulsive.As with the 2-body terms, the EXX energies based on SCAN states are closer to the HF data than when PBE states are used.Consequently, also the RSE corrections are larger for PBE states than for the SCAN ones.Nevertheless, the 3-body EXX+RSE energies based on PBE differ by around 0.8 kJ/mol from the HF values, the difference is below 0.25 kJ/mol for EXX+RSE based on SCAN.This again hints at smaller many-body errors of SCAN. 108 analyze the distance dependence of the PBE-and SCAN-based EXX and RSE components in Fig. 10.Clearly, all of the methods (i.e., including HF) tend to predict similar energies for larger distances but differ at small separations.The 3-body HF energies at larger distances are especially well reproduced by EXX+RSE based on the SCAN states.Specifically, the difference between 3-body EXX+RSE energy and 3-body HF energy is converged to within 0.01 kJ/mol at a cut-off distance of around 17 Å.When PBE states are used, the difference between EXX+RSE and HF converges more slowly, the difference still changes by around 0.1 kJ/mol above 17 Å.We observe the same trends also for the other systems.Therefore, if one considers the 3-body HF energies as a mean-field reference, the 3-body EXX and EXX+RSE energies based on SCAN are superior compared to the PBE-based 3-body energies.This is again consistent with the behavior observed for methane clathrate.We now turn to the 3-body correlation energies, listed in Table XIV, and the total 3-body energies, shown in Fig. 11.The 3-body MP2 energies are close to 50% of the CCSD or CCSD(T) values for all the systems, less than that for ethane and more than that for both forms of acetylene.This is likely due to the missing 3-body correlations in MP2.
The 3-body mean field (EXX and EXX+RSE) energies gave respectively stronger and weaker binding than the 3-body HF energies for all the systems.Therefore, if the total RPA energy was to recover the total CCSD or even the 3-body CCSD(T) energy, the RPA(SCAN) correlation energies would need to be close to the CCSD correlation energies and the RPA(PBE) values even larger in magnitude (more repulsive).However, we observe nei-ther.The RPA(PBE) correlation energy is only similar to the CCSD correlation for ethane.In all the other cases both RPA(PBE) and RPA(SCAN) correlation energies are much smaller than the CCSD correlation energies.The total 3-body energies are therefore underestimated for either of the RPA methods, see Fig. 11.
We again plot the convergence of the 3-body correlation energies with the cut-off distance to understand possible origins of the differences.The convergence shows similar trends for all the systems and we therefore show only the convergence for ethylene in Fig. 12(a).For small distances, below ≈ 20 Å the MP2 energies are very close to one half of the CCSD energies.However, for larger cut-offs the MP2 data show much smaller variations compared to CCSD.This is most likely due to missing 3-body correlations and the resulting error could likely be re-  duced by including a 3-body correlation correction. 9,109he convergence of the RPA correlation energy with the cut-off distance looks similar to that of CCSD(T) on the first sight, however, there are two possible issues with the behavior of RPA.First, the 3-body energies show a larger dependence on the basis-set size and second, the contributions of the 3-body fragments show different asymptotic behavior compared to the CCSD(T) data.The first issue is illustrated in Fig. 12(a) where one can see that the difference between the 3-body energies obtained from AVTZ and CBS is close to 0.5 kJ/mol for both PBE-and SCAN-based RPA.We note that the difference to the CBS limit is still around 0.2 kJ/mol for the AVQZ basis set.This is likely a consequence of using input states based on DFT, we observe that PBE and SCAN have also a larger basis-set dependence than HF.
The second issue is illustrated in Fig. 12(b) which shows the differences between the distance dependent 3body correlation energies obtained for the various methods and CCSD(T).Clearly, while the difference converges for CCSD and even MP2 within some tenths of kJ/mol above ∼20 Å, the differences obtained for RPA show a much slower convergence.The slower convergence is not caused by some basis-set errors, the basis-set size is mostly relevant for trimers with small distances.We have also checked that the different convergence is not caused by numerical issues by comparing the values obtained by Molpro and the in-house code.Finally, the trend is also present when the total energies, and not only the correlation energies, are compared, so it is not a consequence of a different mean-field reference for RPA and CCSD(T).
Because of the potential use of RPA and MP2 in subtractive embedding schemes, it is important to understand the performance of those methods as a function of the separation of molecules in a cluster.To this end, we divided the ethylene trimers into four groups according to the number of contacts in the fragment.Two molecules are in contact when their intermolecular distance is below 6 Å in the case of ethylene.One can see in Table XV that the as the number of contact decreases the 3-body contributions tend to decrease in magnitude.In the case of CCSD(T), the contributions are reduced by a factor of two when one contact is lost between the molecules.Interestingly, the (T) terms are important only for the compact trimers, their effect is minor already for the group with two contacts.MP2 correlation energy is only around 50% of the CCSD(T) correlation energy for the group with three contacts and below 0.05 kJ/mol in the other groups.Consequently, it underestimates the 3-body contributions for all the groups.Part of the error can be attributed to the missing three body correlations in MP2.
The RPA energies show considerable differences from the reference for the groups with two and one contact and the errors are small only for the group with zero contacts.Part of the error likely stems from the manybody errors of the DFT input states, for example, the 3-body energies for the group with two contacts are 2.33 and −1.22 kJ/mol for PBE and SCAN, respectively. 23It is possible that these errors could be alleviated by going beyond our set-up based on non-self-consistent RPA with (meta-)GGA input states, such as including exchangecorrelation kernels, performing self-consistency, or utilising HF input states.However, these approaches are currently either not available or more computationally demanding within periodic boundary conditions and thus less efficient for the subtractive embedding.In fact, the compact trimer groups with three and two contacts are finite and the erroneous RPA contributions can be replaced by CCSD(T) within the subtractive embedding approach.

Four-body terms
Finally, we compare the 4-body MP2 and RPA energies with the CCSD(T) reference.The 4-body CCSD(T) contributions obtained with our set of tetramers are almost negligible, close to −0.1 kJ/mol for all the systems (Table V).Considering first MP2, we find that the MP2 correlation energies are close to one half of the CCSD(T) correlation for all the systems, see Table XVI.Note that with our tetramer cut-offs all the clusters can be still considered as compact, without fully isolated molecules.Therefore when a tetramer forms, the single particle HF states change due to overlap of the monomers.This changes the MP2 energy compared to the isolated monomers so that the 4-body MP2 energy is non-zero despite there being no 4-body correlation terms in MP2.Moving to RPA, we see a stark difference between the PBE-and SCAN-based RPA energies, both in their mean field and correlation components.The total 4-body energies are small (below 0.2 kJ/mol) and close to the reference values for RPA based on SCAN when RSE are included, see Fig. 13.In contrast, RPA based on the PBE states gives too positive values with errors up to 1.0 kJ/mol for acetylene/c.These differences come both from the mean-field and correlated contributions.The 4body EXX(PBE) component is more repulsive than HF while EXX(SCAN) is more attractive, see Table XVII The magnitude of the 4-body contributions is reduced upon addition of RSE.The 4-body correlation energies are small in magnitude for RPA based on SCAN, in fact, they are even smaller than the MP2 values (Table XVI).In contrast, the RPA correlation energy based on PBE is close to zero only for ethane, the values are few tenths of kJ/mol for the other systems.Finally, we summarise this section by comparing the total RPA and MP2 binding energies to the CCSD(T) reference.The binding energies are listed in Table XVIII and their relative deviations from the reference are shown in Fig. 14.
We start with MP2 for which we find that it predicts well the binding energy of ethane but overestimates the binding for the other systems by at least 14%.For ethane, the small error is a consequence of an error cancellation between small error in 2-body interactions (≈1 kJ/mol) and error in the 3-body interactions (≈ −1.4 kJ/mol) which is partly caused by the missing 3-body correlations.For the other systems, the 3-body errors are similar to those obtained for ethane, but the errors in the 2-body interactions are several kJ/mol.The negative error in both 2-and 3-body terms leads to a substantially overestimated total binding energy.The 2body error comes from inaccurate description of systems with delocalised or π bonds within the second-order perturbation theory and similar issues can be expected for related systems.The errors in 4-body energies are below 0.1 kJ/mol for all the systems and they thus play only a minor role in the final deviation.
The binding energies obtained for SCAN-and PBEbased RPA are rather similar, the relative deviations differ by only a few percent and exhibit essentially the same trends (Fig. 14).As expected, RPA without singles corrections gives binding energies that underestimate the reference data.The range of errors is approximately −20 to −15% for RPA(PBE) and larger, around −22 to −12%, for RPA(SCAN).For either method, the largest error occurs for ethane and the smallest for acetylene/o.When the singles are included we find that RPA(PBE)+RSE performs somewhat better than RPA(SCAN)+RSE for all the considered systems.Specifically, the average difference to the reference data is 5.7% for the first method and 7.5% for the latter.These errors are consistent with those observed for RPA(PBE)+RSE for molecular solids bound dominantly by dispersion in Ref. 22.
While the total binding energies are similar for PBEand SCAN-based RPA, they show significant differences in their many-body components, as discussed in the previous parts.For RPA based on PBE the 3-body error is negative and the 4-body error is positive, there is therefore a partial error cancellation between the 3-and 4body errors.As the 3-body errors are between −1 to −2 kJ/mol and the 4-body errors around 1/2 of that and positive, the sum of the 3-and 4-body errors is around −0.5 to −1 kJ/mol, i.e., too strong binding.The overestimated 3-and 4-body contributions then partly cancel the underestimated 2-body terms leading to the observed underbinding.
The effect of error cancellation between n-body energies is smaller for the SCAN-based RPA.First of all, the mean-field and correlation contributions have opposite differences from HF and CCSD(T) correlation for both the 3-and 4-body terms.Thus these deviations partly cancel for SCAN-based RPA and do not add up as with RPA based on the PBE states.The total 4-body energies are then close to the reference, while the positive 3-body terms are again underestimated.As with the PBE-based RPA, this negative error in the 3-body terms compensates part of the error of the 2-body terms resulting in underestimated binding energies.

IV. CONCLUSIONS
In the present work, we obtained MBE contributions at the CCSD(T) level for four molecular solids and used them to assess the accuracy of MP2 and RPA.The CCSD(T) energies were obtained up to the fourth order of MBE with a finite distance cut-off and we thoroughly tested their convergence with the basis-set size.In doing so we identified strategies that can be used to save computational time without sacrificing significantly the precision of the results.First, a large basis set is required to obtain 2-body energies at the CBS limit, but it is only necessary for dimers with a small intermolecular distance.
The contributions of dimers with a large separation can be obtained using a small basis set such as AVDZ.Moreover, the correlation contributions of the distant dimers can be obtained by extrapolation of the correlation energy with the cut-off distance.The (T) terms need to be considered even for the distant contributions to the 2body energy as they affect the response properties.They can be evaluated using a small basis set (AVDZ) for the 3-body contributions and can only be neglected in the 4-body energy.
We have assessed the suitability of MP2 and RPA approaches for the subtractive embedding schemes for the computation of the total binding energy.The comparison against reference CCSD(T) data supports the following observations.
1.The performance of MP2 for distant 2-body dimers is relatively good for aliphatic systems, but deteriorates significantly for π-electron systems with errors .
2. The largest difference between MP2 and CCSD(T) occurs for the three-body interactions, around 1 kJ/mol.][111] 3. MP2 recovers about 50 % of the correlated contribution to the non-additive 4-body energies.This is enough for accurate total binding energies as for all tested systems the magnitude of the 4-body contribution is small (below 0.1 kJ/mol).
4. Compared to MP2, the performance of RPA does not deteriorate as strongly for π-electron systems.
5. RPA includes three-body correlation terms, but those contributions are of poor quality if there are close contacts in the trimer of molecules.
6.The MBE errors of RPA are clearly affected by the DFT states used to evaluate the RPA energy.While PBE-based RPA with RSE corrections leads to the smallest overall errors, it relies considerably on error cancellation between the different MBE terms.Using SCAN instead of PBE increases the error of the binding energies by a few percent; however, the many-body errors are substantially reduced for RPA(SCAN).Therefore, the SCAN-based RPA is more suitable for the subtractive embedding strategy compared to using the PBE states.
7. On the technical side, the basis-set convergence of the 3-and 4-body energies at the RPA level is slower than for the HF-based methods, which increases the computational cost.
Overall, we conclude that dispersion-dominated systems remain a challenge for approximate electronicstructure methods.The many-body resolved binding energies allow to obtain a much detailed information about the origin of errors for a given method compared to assessment based only on the total binding energies.Moreover, the data presented in this work are intended as a benchmark for the development of novel low-scaling approaches.

Figure 1 .
Figure 1.Difference of the 2-body CCSD(T) energy components of ethylene between the AVDZ and AVQZ basis sets.The data show the error made in the 2-body energy when calculations for dimers above the separation distance are made using the AVDZ basis set instead of the AVQZ basis set.

Figure 2 .
Figure 2.Cut-off distance convergence of the 2-body HF+CABS energy obtained with the AVQZ basis set.

Figure 3 .
Figure 3. Cut-off distance convergence of the 2-body CCSD-F12b correlation energy obtained with the AVQZ basis set.

Figure 4 .
Figure 4.Cut-off distance convergence of the 3-body HF+CABS/AVTZ energies for all the considered systems.

Figure 5 .
Figure 5. Cut-off distance convergence of the 3-body CCSD-F12b/AVTZ correlation energies for all the considered systems.

Figure 7 .
Figure 7. Cut-off distance convergence of the 4-body CCSD-F12b/AVDZ correlation energy for all the considered systems.Note that the y axis scale is approximately one half compared to the HF case in Fig.6.

Figure 8 .
Figure 8.The 2-body correlation energy calculated by different methods for (a) ethane and (b) acetylene/c.The energy, in kJ/mol, is a sum of two-body contributions of dimers with a distance larger than the cut-off given on the x axis and smaller than the largest cut-off used.The calculations used the AVTZ basis set.

Figure 9 .
Figure 9.The 2-body contributions to the total RPA and MP2 binding energies compared to the CCSD(T) reference data.

Figure 10 .
Figure 10.(a) The 3-body energies of ethylene obtained by HF and EXX without and with RSE corrections using both PBE and SCAN orbitals.(b) The differences of the EXX and EXX+RSE data with respect to HF.

Figure 11 .
Figure 11.The 3-body contributions to the total RPA and MP2 binding energies compared to the CCSD(T) reference data.

Figure 12 .
Figure 12.(a) The distance cut-off convergence of the 3body correlation energies of ethylene obtained for different methods.Panel (b) shows the difference with respect to the CCSD(T) curve, i.e.E(rcut) − E CCSD(T) (rcut).

Figure 13 .
Figure13.The 4-body contributions to the total RPA and MP2 binding energies compared to the CCSD(T) reference data.

Figure 14 .
Figure 14.Relative difference of the binding energies with the RPA and MP2 methods with respect to the reference data.

FigureFigure S2 .
Figure S1.Cut-off distance convergence of the 2-body (T) energy obtained with the AVQZ basis set

Table II .
Basis set convergence of the 2-body HF, MP2, CCSD, and (T) contributions for proximate dimers of ethylene in kJ/mol.We also show data obtained with the ∆MP2 procedure.The proximate dimers have intermolecular distance smaller than 10 Å.

Table III .
Basis set convergence of the 2-body term of the HF, MP2, CCSD, and (T) energies for distant dimers of ethylene in kJ/mol.The intermolecular distance is between 10 Å and the total cut-off 18.6 Å for the distant dimers.

Table IV .
Contributions of proximate and distant dimers to 2-body HF, CCSD, and (T) energies.Data in kJ/mol.

Table VII .
Basis set convergence of the 3-body term in the MP2 and CCSD(T) calculations for ethylene, data in kJ/mol.

Table VIII .
Basis set convergence of the 4-body term in the MP2 and CCSD(T) calculations for ethylene, data in kJ/mol.

Table IX .
The 2-body mean-field contributions of proximate dimers (with intermolecular distance below 10 Å), data in kJ/mol.

Table X .
The 2-body mean-field contributions of dimers with intermolecular distance larger than 10 Å (distant dimers), data in kJ/mol.

Table XI .
The 2-body total energy contributions of proximate dimers in kJ/mol.

Table XIII .
The 3-body mean-field energies in kJ/mol.HF values do not include CABS corrections and were obtained with AVTZ basis set, for EXX and RSE the values are based on the PBE and SCAN states and were obtained in the AVQZ basis.

Table XIV .
The 3-body correlation energies in kJ/mol.The MP2 and CCSD energies include F12 corrections and are in the AVTZ basis set, the (T) contribution is "unscaled" and also in the AVTZ basis set.The RPA values were obtained by AVTZ→AVQZ extrapolation.

Table XV .
The 3-body energies of ethylene obtained with CCSD(T), CCSD, MP2, HF, and two variants of RPA.The trimers divided into groups according to number of contacts between the molecules in the trimer.Data are in kJ/mol.

Table XVI .
The 4-body correlation energies in kJ/mol.The AVTZ and AVDZ basis sets were used for the MP2 and CCSD(T) calculations, respectively.The RPA energies are extrapolated based on AVTZ and AVQZ data.

Table XVII .
The 4-body mean-field energies in kJ/mol.HF values do not include CABS corrections and were obtained with AVDZ basis set, for EXX and RSE the values are based on the PBE and SCAN states and obtained in the AVQZ basis.