Performance of Localized-Orbital Coupled Cluster Approaches for the Conformational Energies of Longer n-alkane Chains

We report an update and enhancement of the ACONFL (conformer energies of large alkanes [Ehlert, S.; Grimme, S.; Hansen, A. J. Phys. Chem. A 2022, 126, 3521-3535]) dataset. For the ACONF12 (n-dodecane) subset, we report basis set limit canonical CCSD(T) reference data obtained from MP2-F12/cc-pV{T,Q}Z-F12 extrapolation, [CCSD(F12*)-MP2-F12]/aug-cc-pVTZ-F12, and a (T) correction from conventional CCSD(T)/aug-cc-pV{D,T}Z calculations. Then we explored the performance of a variety of single and composite localized-orbital CCSD(T) approximations, ultimately finding an affordable LNO-CCSD(T)-based post-MP2 correction that agrees to 0.008 kcal/mol MAD (mean absolute deviation) with the revised canonical reference data. In tandem with canonical MP2-F12/CBS extrapolation, this was then used to re-evaluate the ACONF16 and ACONF20 subsets for n-hexadecane and n-icosane, respectively. A revised ACONFL set was thus obtained. It was then used to assess the performance of different localized-orbital coupled cluster approaches, such as PNO-LCCSD(T) as implemented in MOLPRO, DLPNO-CCSD (T1) as implemented in ORCA, and LNO-CCSD(T) as implemented in MRCC, at their various accuracy settings. A three-tier LNO-CCSD(T)-based composite scheme disagrees by only 0.02 kcal/mol from the revised ACONFL reference data. When extrapolated to the complete PNO space limit, DLPNO-CCSD(T1, Tight) and a composite method are the best picks among all the localized coupled cluster methods tested for the dodecane conformers. Dispersion-corrected dRPA-based double hybrids perform remarkably well for the ACONFL set. While the revised reference data do not affect any conclusions on the less accurate methods, they may upend orderings for more accurate methods with error statistics on the same order as the difference between reference datasets.

data do not affect any conclusions on the less accurate methods, they may upend orderings for more accurate methods with error statistics on the same order as the difference between reference datasets.
According to the IUPAC gold book, 1 a conformer is "one of a set of stereoisomers (i.e., isomers that possess identical constitution, but which differ in the arrangement of their atoms in space 1 ), each of which is characterized by a conformation corresponding to a distinct potential energy minimum." A comprehensive understanding of the conformers of organic and biomolecules is critical because they often show distinct chemical and biological activities. 2,3 Hence, accurate estimation of their structural properties and conformational energies (i.e., relative energies of other conformers with respect to the most stable one) is essential for a better understanding of biological phenomena, like protein folding, substrate binding, catalysis by enzymes, and many more (see refs. 4,5 and references therein). Multiple conformers of a certain molecule often span a relatively narrow conformational energy range. Hence, these structures exist as a thermally populated mixture at room or physiological temperature. Therefore, all relevant conformers must be considered to evaluate molecular properties accurately. 6 Due to the high flexibility of open-chain molecules in conformational space, 7 these are often used for modeling flexible biological systems, e.g., in drug discovery and related applications. 8 One class of such systems are linear n-alkanes (i.e., CnH2n+2); simple as they may be, they constitute key building blocks of organic chemistry, of fossil fuels, and of polymers, lipids, and biomembranes.
Since the pioneering work of Pitzer, 9 the existence of different conformers of n-alkane is well known. Over the years, conformational enthalpies and low-lying conformers of shorter unbranched alkane chains have been studied experimentally. [10][11][12] On the other hand, theoretically, the torsional space and equilibrium conformational energies of short n-alkanes have also been explored theoretically. [13][14][15][16][17] Previous theoretical studies 18-21 on the longer n-alkane chains (n > 10) often investigated a handful of the lowest energy conformers to find the balance between repulsive hydrogen contacts and attractive London dispersion and finally predict the energy gap between the linear zigzag and hairpin-like conformers.
In 2009, Gruzman et al. 14 published comprehensive theoretical studies on conformational equilibrium energies of five unbranched n-alkanes isomers: n-butane, n-pentane, n-hexane, nheptane, and n-octane (i.e., CnH2n+2; n=4-8). Later, to elucidate the correct treatment of dispersion, a detailed study of the entire conformer surface of n-pentane was performed by Martin. 13 Both experimental 19 and theoretical 22,23 studies have already established that due to the much stronger attractive London dispersion, the conformational preferences of longer unbranched n-alkanes (both in the gas and liquid phases) are very different from what we generally see for the short n-alkanes.
Hence, a comprehensive study of longer n-alkane conformers beyond the n-octane is vital from chemical and biological perspective.
Recently, Ehlert et al. 24 have proposed a dataset, ACONFL, composed of the thirteen, seventeen, and twenty-one lowest-energy conformers of n-dodecane, n-hexadecane, and nicosane, respectively. That is, the complete ACONFL set consists of 12 n-dodecane conformer energies (i.e., ACONF12), 16 n-hexadecane conformer energies (i.e., ACONF16), and 20 nicosane conformer energies (i.e., ACONF20). In ref. 24 , the reference conformer energies of ACONF12 were calculated at the DLPNO-CCSD(T1)/AV{T,Q}Z level with VeryTightPNO setting. As the CBS extrapolation for the ACONF16 and ACONF20 conformers are still too expensive, the authors in ref. 24  TightPNO, and VeryTightPNO (see Table 1 in Ref. 32 for definitions). The available accuracy thresholds for LNO-CCSD(T) are Normal, Tight, vTight, or vvTight (see Table 1 in Ref. 31 for details). In PNO-LCCSD(T), Default and Tight (see Tables 1-4 in Ref. 26 for more information) are the standard settings. Examples of recent use of these localized orbital coupled cluster methods include the energetics of the (H2O)20 cages using PNO-LCCSD(T)-F12b, 33 (the F12b suffix refers to explicit correlation 34 ), the noncovalent interaction energies of seven large dimers (L7 set 35 ) with LNO-CCSD(T), 36 the main group thermochemistry, barrier heights, intra-, and intermolecular interaction energies of GMTKN55 37 using DLPNO-CCSD(T), 38 and benchmark studies on the Ru(II) complexes involved in hydroarylation, 39  Hence, the main objectives of the present study are: (a) improving the reference conformer energies of longer n-alkane chains and (b) evaluating different pure and composite localized orbital coupled cluster methods relative to the new reference conformer energies. As a by-product, some conclusions about DFT and other approximate methods can also be drawn.

III.
Results and Discussion.
(a) Revised reference conformer energies of n-dodecane: As the first step, we calculated the canonical explicitly correlated RI-MP2-F12 energies of the ndodecane conformers with cc-pVTZ-F12 and cc-pVQZ-F12 basis sets and extrapolated them to Hence, the revised reference data for the ACONF12 set can be considered a "silver" standard.      Table 1 for the extrapolation details]. A detailed inspection of the signed deviations for the twelve n-dodecane conformer energies reveals that DLPNO-CCSD(T1)/CBS overestimates, but DF-CCSD(T) underestimates, the conformer energies across the board (see Table 1).
Owing to their CPU time and resource requirements (over 3 weeks wall clock each, 512GB of RAM, and 3.3 TB of SSD scratch disk), we were only able to calculate five CCSD(T)-(F12*)/VTZ-F12 level conformer energies. We note that unlike CCSD, F12 approaches do not benefit the connected quasiperturbative triples, 76  see Table S1 in the Supporting Information). However, if we assess the performance of RI-MP2 and RI-MP2-F12 against the "new" reference data of n-dodecane conformers, we found that RI-MP2/AV{T,Q}Z is the best performer (MAD=0.18 kcal/mol). Considering the fact that the mean absolute difference between RI-MP2/AV{T,Q}Z and RI-MP2-F12/AV{T,Q}Z-F12 is 0.15 kcal/mol, we can safely say that the former method gets a better answer for the wrong reasons.
Canonical RI-MP2 and explicitly correlated RI-MP2-F12 systematically underestimate the conformer energies with respect to the "new" revised reference (see Table S1 in the Supporting Next, in order to eliminate basis set incompleteness as a 'confounding factor', we use the aug-cc-pVTZ basis set throughout to assess the performance for ACONF12 of LNO-CCSD(T), PNO-LCCSD(T), and DLPNO-CCSD(T1) relative to canonical DF-CCSD(T) (see Table 2). As expected, tightening the accuracy threshold for LNO-CCSD(T) improves its accuracy. Proposed by Nagy et al. 31  Tight] offers the best accuracy among all the individual, and composite LNO-CCSD(T) tested (see Table 2). Instead of using a fixed 0.5 prefactor in the composite schemes, when we optimized the prefactors with respect to canonical CCSD(T)/AVTZ conformer energies, we obtained slightly   Table 2, it is clear that standard and composite LNO-CCSD(T) methods underestimate the conformer energies relative to canonical DF-CCSD(T).
Standard PNO-LCCSD(T) with Default and Tight settings significantly overestimate the ACONF12 energies (see Table 2). The PNO-based composite scheme, Tight + 0. Refitting of the DLPNO-based composite methods with respect to DF-CCSD(T) level ACONF12 energies offered significantly better accuracy but with unphysical negative coefficients (see Table 2). Composite methods proposed in Ref. 31 Revised reference conformer energies of n-hexadecane and n-icosane conformers (i.e., the ACONF16 and ACONF20 subsets): In the nomenclature of Hansen and coworkers, 24 the conformer 0 is "all trans" and 00 is "hairpin"like (see Figures 2-3 in ref. 24 for illustration). As the chain grows longer, eventually dispersion forces will favor a folded over a linear structure. The "all trans" conformer is the lowest in energy for n-dodecane, while the "hairpin" is clearly the global minimum for n-icosane; n-hexadecane lies near the transition point. 19 Table 3 The high-level correction (HLC) we have used to calculate the revised n-dodecane conformer energies is even for them computationally quite expensive, and would become intractable for the larger species. Hence, linear scaling localized coupled-cluster methods would be attractive alternatives for HLC on top of the RI-MP2-F12/CBS level n-hexadecane and nicosane conformer energies. The present study considers 22 such alternative HLCs (see Table 3).
Among these five alternative HLCs, the remarkable accuracy of HLC9, HLC7, and HLC15 can be attributed to fortuitous error compensation between [LCCSD-LMP2] and (T) contributions.
Two low-cost alternatives to HLC14 are HLC13 and HLC8. With CBS extrapolation, DLPNO-CCSD(T1)-based high-level corrections (i.e., HLC4 and HLC5) have the largest deviations relative to the canonical reference data.   Table S2 in the Supporting information).
We have finally selected [RI-MP2-F12/CBS + HLC14] for the revised reference conformer energies of n-hexadecane and n-icosane. (For the high-level correction energies of individual species, see Table S3 in the Supporting Information.) Our best estimates of the ACONFL conformer energies are listed in Table 4 For the ACONFL conformer energies employing HLC13 and HLC8, see  of n-alkane also increases the mean absolute deviation from the reference conformer energies accordingly (see Table 5) While employing the AVTZ basis set for calculation, tightening the accuracy cutoffs from "vTight" to "vvTight" has no additional advantage.
The expression CPS{X,Y} refers to the extrapolation of TCutPNO to the complete PNO space limit using TCutPNO=10 -X and 10 -Y , where Y=X+1 With the "Default" accuracy threshold and AVTZ basis set, PNO-LCCSD(T) performs remarkably well (0.05 kcal/mol) owing to fortunate error compensation between LO error and basis set incompleteness error. Increasing the basis set size from AVTZ to AVQZ adversely affects the accuracy (MAD increases from 0.05 to 0.11 kcal/mol) as LO error dominates with a larger basis set. With the "Tight" setting, increasing the basis set size from AVTZ to AVQZ has no significant influence on PNO-LCCSD(T) performance when the full ACONFL is considered.
Closer scrutiny suggests using a larger basis set degrades the performance of ACONF12 and ACONF16 but marginally improves the accuracy of ACONF20 (see Table 5). Due to the substantial computational cost, we were only able to calculate PNO-LCCSD(T)/AV5Z level conformer energies for ACONF12 and ACONF16 subsets with "Default" and ACONF12 subset with "Tight" cutoffs. Either with the "Default" or "Tight" threshold, Turning to the DLPNO-CCSD(T1) with the "NormalPNO" setting, increasing the basis set size improves accuracy as it should (see Table 5 and DLPNO-CCSD(T1)/AVQZ levels with "TightPNO" and "VeryTightPNO" settings, respectively. With the default "TightPNO" (i.e., TCutPNO=10 -7 ) DLPNO-CCSD(T1)/AV{T,Q}Z and DLPNO-CCSD(T1)/AVQZ offer similar accuracy, which is due to a significant reduction of MAD for ACONF20 while using the prior method (see Table 5). With the default TightPNO and TightPNO at the complete PNO space limit (i.e., TCutPNO=10 -{6,7} ), increasing the basis set size from AVQZ to AV5Z marginally improves the accuracy for n-dodecane conformers. Using a {T,Q} CBS extrapolation and CPS extrapolation from the TCutPNO=10 -6 and 10 -7 energies, DLPNO-CCSD(T1)/TightPNO can achieve 0.14 kcal/mol accuracy, which is in the territory of LNO-CCSD(T, Tight)/AV5Z and LNO-CCSD(T, vTight)/AVQZ. The CPS{6,7} extrapolation with a two-point CBS extrapolation can significantly improve the performance of the ACONF20 subset.
From the MAD values listed in Table 5, it is clear that for the twelve n-dodecane conformers, the performance improvement from "TightPNO" to "VeryTightPNO" is not  Table 5 and Table S6 in the Supporting Information). As earlier, the anomalous values of the coefficients seem hard to justify.
Additionally, we also considered benchmarking the LNO, PNO, and DLPNO-based composite methods optimized for the counterpoise uncorrected (or "raw") S66x8 noncovalent interactions (see Ref. 50 for further details). The RMSD, MAD, and MSD statistics of these methods for ACONFL are listed in Table S6. While using the coefficients from Ref. 50 Table S6 in the Supporting Information).
Hence, we are reluctant to recommend their use.
As a parenthetical remark, for the RMSD, MAD, and MSD statistics of different localized orbital methods relative to the reference conformer energies using HLC8 instead of HLC14 for the ACONF16 and ACONF20 subsets, see Table S7 in the Supporting Information.

(d) Explicitly correlated localized orbital coupled cluster methods:
In this section, we assess the performance of explicitly correlated PNO-LCCSD(T)-F12b and DLPNO-CCSD(T1)-F12 methods in combination with different accuracy thresholds and basis sets.
The mean absolute deviations (MADs), mean signed deviations (MSDs), and root-mean-square deviations (RMSDs) of these methods are listed in Table 6. Following the recommendation of Peterson et al. 75 Table 6). With the "Tight" setting, using the VTZ-F12 basis set, we obtained marginally better performance than VDZ-F12. However, (T)-scaling closed that gap between PNO-LCCSD(Ts)- when "Tight" cutoffs are used. Interestingly enough, tightening the threshold from "Default" to "Tight" does not offer any noticeable advantage for PNO-LCCSD(Ts)-F12b/VDZ-F12. The "Default" and "Tight" settings yield results of comparable quality for ACONF16 with the VDZ-F12 basis set. However, for the ACONF20 set, we get better performance with a tighter threshold.
Increasing the basis set size from VDZ-F12 to VTZ-F12 does more harm than good for DLPNO-CCSD(T1)-F12 when using the "NormalPNO" setting, which is an indication of fortuitous error compensation between errors due to the use of loose accuracy cutoffs and basis set incompleteness. As expected, tightening the accuracy threshold from NormalPNO to TightPNO improves the accuracy of DLPNO-CCSD(T1)-F12/VDZ-F12, which gets even better by using the "VeryTightPNO" setting (see Table 6). Increasing the basis set size from VDZ-F12 to VTZ-F12 significantly worsens the performance of DLPNO-CCSD(T1)-F12 when the "NormalPNO" threshold is used. However, with "TightPNO" cutoffs, the MAD difference between VDZ-F12 and VTZ-F12 basis sets is statistically insignificant. From the mean absolute deviations listed in Table   6, it is clear that VDZ-F12 reaches the basis set limit with the "TightPNO" setting.  75   In this section, we compare how costly different localized coupled cluster methods are relative to DF-CCSD(T). We have considered one n-dodecane conformer (to be more specific, conformer 0).
For the standard and explicitly correlated calculations, the aug-cc-pVTZ basis set was employed throughout. All the calculations were carried out on 16 cores of a node with two Intel(R) Xeon(R) Gold 5320 CPUs (2.20GHz).   Table 7. For PNO-LCCSD(T), the total wall time ratio for "Default" and "Tight" thresholds is   The mean absolute error of the widely used molecular mechanics force fields, UFF 81 and MMFF94, 82,83 was marginally reduced from 2.91 and 3.41 to 2.87 and 3.37 kcal/mol, respectively.
Considering the fact that the average conformer energy of ACONFL is 4.56 kcal/mol, these errors remain unacceptable. With MAD=0.40 kcal/mol, GFN-FF 84 emerges as the best performer among all the force fields, followed by OpenFF-1.0.0 85 (MAD=0.58 kcal/mol). However, it should be mentioned that the surprisingly accurate performance of simple HF with D4 dispersion correction, 0.14 kcal/mol, is now reduced to merely good one, 0.41 kcal/mol. Now, among the SQM methods, the performance of the best pick, PM6-DH4, 86 gets marginally better when we employ the revised reference energies (MAD goes down from 0.55 to 0.48 kcal/mol). However, for PM7, 87 this improvement is a bit more prominent.
Finally, we will focus on the performance of DFT methods in light of the presently revised reference data. The mean absolute errors of PBE-D4, 88 Table 8). As we saw for the GMTKN55 benchmark, 68 Head-Gordon's 16-paramter ωB97M(2) marginally outperforms our six-parameter revDSD-PBEP86-D3BJ for longer n-alkane conformer energies.
Let us switch our focus to the dRPA-based double hybrid functionals. As all the systems of the ACONFL set are closed-shell, Kállay's dRPA75 71  the double hybrid DFT methods tested. With a mean absolute error of 0.04 kcal/mol, these two double hybrids are marginally better than DSD-PBEP86dRPA75-D3BJ. 72 For DSD-PBEP86dRPA75-D4, due to fortunate error compensation, we obtain a marginally better MAD with def2-TZVPP than using a larger def2-QZVPP basis set. Unlike what we observed for the GMTKN55 benchmark, 72 a complete basis set extrapolation from the def2-TZVPP, 108 and def2-QZVPP 108 conformer energies is detrimental to the performance of the dRRA-based double hybrids (see Table 8).
The errors in DFT calculations can be grouped into two camps: imperfections in the functional itself and errors arising from the self-consistent density evaluated with an imperfect functional. In recent years, Sim, Burke, and coworkers 109   The conformer energies are taken from the ESI of ref. 24 In a previous study, 112 we found that for the unbranched n-alkane conformers (n=2-7), 14 using HF densities instead of self-consistent KS densities can significantly improve the performance of pure and hybrid PBE-D4 functionals. However, without dispersion correction, the trend is the opposite. Just for the sake of completeness, with the present dataset in hand, we have tested whether the same is true for the longer n-alkane conformers (see Table 9). Without a dispersion correction, we find that self-consistent KS densities are preferred over HF ones. On the other hand, with the D4 dispersion correction, the performance of HF-PBE-D4 112 and HF-PBE0-D4 112 is statistically indistinguishable from PBE-D4 and PBE0-D4, respectively.

IV. Conclusions.
We have successfully calculated the "silver" standard conformer energies of the ACONF12 set, Finally, from an extensive survey of different pure and composite localized coupled cluster methods for conformational energies of longer n-alkane chains, we can conclude the following: • Increasing the basis set size and/or tightening the accuracy threshold improves the accuracy of the pure LNO-CCSD(T) methods. With the "Normal" setting, the AV{T,Q}Z extrapolation performs better than the more expensive AV{Q,5}Z, but the trend is the opposite when we use the "Tight" or "vTight' setting. With the "Tight" setting, LNO-CCSD(T)/AV{Q,5}Z is the best performer among the LNO-based methods tested; hence the composite methods based on Tight{Q,5} have no additional advantage over the pure method.
• For a certain threshold, increasing the basis set seize from AVTZ to AVQZ helps improve When analyzing the ∆MAD values between old and new reference data, it should be kept in mind that the MAD between the two reference sets themselves is 0.31 kcal/mol. (The latter will also be an upper limit for ∆MAD.) Hence, for methods where the MAD already was several times larger than 0.31 kcal/mol, the choice of reference data will not affect any conclusions as to the suitability of the said methodsbut for approaches with an MAD comparable to or smaller than 0.31 kcal/mol, the choice of reference set may upend some conclusions.
For all dispersion-uncorrected DFT functionals tested in the present study, the MAD value uniformly increases by 0.30 kcal/molbut as they already perform so poorly, this is not an issue.
That being said, for HF-D4, MAD increases nontrivially from 0.14 to 0.41 kcal/mol when substituting the present reference data, while r 2 SCAN-VV10, r 2 SCANh-VV10, 113  Using the AVTZ basis set, we found that localized orbital coupled cluster methods are two orders of magnitude cheaper than the density-fitted canonical CCSD(T) for the n-dodecane conformers. With the "VeryTightPNO" accuracy setting, DLPNO-CCSD(T1) is twice as expensive as LNO-CCSD(T,vvTight). However, for larger systems, one will start witnessing the formation of domains that may make the DLPNO-based methods cheaper than both LNO-CCSD(T) and PNO-CCSD(T).

Acknowledgments:
GS acknowledges a doctoral fellowship from the Feinberg Graduate School (WIS). We gratefully