Explaining and Fixing DFT Failures for Torsional Barriers
EExplaining and Fixing DFT Failures for Torsional Barriers
Seungsoo Nam, Eunbyol Cho, and Eunji Sim ∗ Department of Chemistry, Yonsei University, 50 Yonsei-ro Seodaemun-gu, Seoul 03722, Korea
Kieron Burke
Departments of Chemistry and Physics, University of California, Irvine, CA 92697, USA
February 16, 2021
Abstract
Most torsional barriers are predicted to high accuracy (about 1 kJ/mol) by standard semilocal functionals, but a smallsubset has been found to have much larger errors. We create a database of almost 300 carbon-carbon torsional barriers,including 12 poorly behaved barriers, all stemming from Y=C-X group, where X is O or S, and Y is a halide. Functionalswith enhanced exchange mixing (about 50%) work well for all barriers. We find that poor actors have delocalization errorscaused by hyperconjugation. These problematic calculations are density sensitive (i.e., DFT predictions change noticeablywith the density), and using HF densities (HF-DFT) fixes these issues. For example, conventional B3LYP performs asaccurately as exchange-enhanced functionals if the HF density is used. For long-chain conjugated molecules, HF-DFT canbe much better than exchange-enhanced functionals. We suggest that HF-PBE0 has the best overall performance.
The accurate prediction of the torsional energy landscapeof a molecule plays a crucial role in chemical and biolog-ical processes, such as estimating the selectivity of chem-ical reactions,[1, 2, 3, 4, 5] protein folding,[6, 7] drugdesign through docking simulations,[8, 9] and molecularelectronics,[10, 11, 12, 13, 14, 15] etc. Density functionaltheory (DFT) calculations play an essential role in estimat-ing the torsional energy profile[1, 11, 12] and also serve asa benchmark for force-field parameterization in molecular dy-namics simulations.[16, 7, 8] Standard DFT calculations usingsemilocal functionals and hybrids such as B3LYP,[17] achieveuseful accuracies for the torsional profile, showing an error ofless than 2 kJ/mol for typical torsional barriers.[18, 19, 20]However, DFT is known to inaccurately estimate some tor-sional energies quantitatively and sometimes even qualitatively,particularly for π -conjugated molecules.[18, 19, 21, 22, 23, 24,25, 26] The delocalization error in DFT that overstabilizes thedelocalized electronic structure[27] is known to be the cause ofthese inaccuracies.[28] This delocalization error occurs becausethe delocalized exact exchange hole in the π -conjugated sys-tem is poorly described by local- or semi-local exchange holemodels.[21] By increasing the exact exchange portion or tuningthe range-separation parameters in range-separated DFT,[29]this problem can be partially solved.[24, 30, 14] Tahchievaet al. could also fix poor barriers from standard DFT withempirical corrections. These quantitative errors sometimesyield qualitatively incorrect results. For example, standardDFT overpredicts the torsional profile of resorcinol, producingincorrect trajectories in molecular dynamics simulations.[31]Other studies showed that relative energies between two tor-sional conformations of methyl vinyl ketone cation predictedusing popular B3LYP[17] or CAM-B3LYP[32] functionals donot match experiment.[33] ∗ [email protected] 𝑠𝑝 - 𝑠𝑝 CTB33-45 𝑠𝑝 - 𝑠𝑝 CTB32-190 𝑠𝑝 - 𝑠𝑝 CTB22di-32 M A E , S b [ k J / m o l ] 𝑠𝑝 - 𝑠𝑝 CTB22ds-12hybridization: subset:
Figure 1:
Barrier mean absolute error (MAE) and barriersensitivity ( S b ) of subsets in CTB-279. (Hatched bars denoteself-consistent DFT, filled bars HF-DFT.) CTB xy - N consists of N torsional barriers with an axis of torsional rotation betweenthe carbon ( sp x )-carbon( sp y ) bond. CTB22ds-12 is a subset ofCTB22-44 composed of barriers with high barrier sensitivity, andCTB22di-32 contains the rest. The triangle denotes average bar-rier sensitivity S b for each functional. Representative moleculesand their single bonds of interest are also depicted. In this study, we explain the source of the DFT torsionalbarrier error and propose a simple method to improve vari-ous carbon–carbon torsional barriers, including long polymers.First, we create a carbon torsional barrier (CTB) dataset con-sisting of 279 torsional barriers with various functional groupsattached to carbon atoms. (See methods for details) This CTB-279 dataset is further classified into CTB xy - N ( x , y =
2, 3 ,and N represents the number of barriers in a subset) subsetsbased on the hybridization of the carbon atoms ( sp or sp )that make up the torsional axis of rotation. For example, a r X i v : . [ phy s i c s . c h e m - ph ] F e b orsional barrier using HF-DFT • TCCL Yonsei Preprint
CTB22-44 consists of 44 torsional barriers of short, conjugatedmolecules with a torsional rotation of C( sp )-C( sp ). Wefurther divided the CTB22-44 subset into CTB22di-32 andCTB22ds-12, where CTB22ds-12 is composed of barriers withhigh barrier sensitivity, and CTB22di-32 is the rest. (Thedefinition of barrier sensitivity and criteria for this division willbe explained later.) The hatched bars in Figure 1 indicate themean absolute errors (MAEs) of various levels of functionalapproximations. Regardless of the level of approximation, theMAE mildly increases with the number of double bonds (i.e., sp - sp → sp - sp → sp - sp ). But the DFT performance issignificantly different in the CTB22di-32 and CTB22ds-12 sub-subsets. In CTB22ds-12 sub-subset, the generalized gradientapproximation and global hybrids perform particularly poorly,whereas M05-2X,[34] with a high portion of exact exchange,and the long-range corrected ω B97 series perform well. (SeeFigure S1 and Table S1 for statistics.) → We also demon-strate a simple remedy for reducing DFT errors: the use ofHF-DFT,[35] The solid bars in Figure 1 show that the HF-DFTversions of PBE[36] and B3YLP[17] (namely, HF-PBE andHF-B3LYP, respectively) now perform similarly to M05-2X (onwhich the application of HF-DFT has almost no effect).[37]Because M05-2X is recommended for a specific purpose,[38]HF-DFT is a more general alternative that can be applied tosystems that include not only torsional barriers but also thetype of interactions for which M05-2X may fail.[39] The rest ofthis paper explains these results and shows their implications.
R= H, CH , NH , OH, FY= CH , NH, O, S X= H, F, Cl, Br R R R Y X 𝐸 r e l [ k J / m o l ] (c) CH /OH (a) CTB33-45 (d) CH /NH/Cltorsional angle 𝜙 [ ° ] 𝜙 𝜙 PBE
B3LYPM05-2X (b)
CTB32-190
Figure 2:
Quality of typical semilocal functionals for typicaltorsional barriers. Schematics of molecules in the (a) CTB33-45and (b) CTB32-190 subsets. Torsional angle φ is defined as theR -C-C-R dihedral angle for (a) and R-C-C-Y dihedral anglefor (b). Typical torsional profiles of molecules in the (c) CTB33-45 and (d) CTB32-190 subsets, with reference marked in black.For each profile, barriers are defined as the energy differencebetween the local minimum and nearest local maximum, depictedwith a gray arrow. As a torsional axis of rotation, molecules in the CTB-279dataset contain two single-bonded carbon atoms. To coverdiverse chemical compositions around the carbon atoms, var-ious functional groups are attached. Figures 2a and b show the schematics of molecules in the CTB33-45 and CTB32-190subset, respectively, and Figures 2c and d depict the corre-sponding torsional profiles. Torsional barriers are defined asthe energy differences between the local minimum and near-est local maximum of each torsional profile, as depicted inFigure 2d.As shown in Figures 2c and d, DFT torsional profiles (orange,green, and purple dashed curves) almost overlap with thereference profiles (black curves). As depicted in Figure 1,the DFT MAEs are extremely small (approximately 1 kJ/molor less), which is consistent with previous studies.[18, 19,20] In the absence of molecular conjugation, such as in theCTB33-45 and CTB32-190 subsets, the torsional energy isprimarily determined by steric hindrance between adjacentsubstituents,[40, 41, 42] indicating that steric hindrance doesnot cause significant DFT torsional errors.
Y= CH , NH, O, S X= H, F, Cl, Br YX Y X 𝐸 r e l [ k J / m o l ] (c) COFCOF (d)
COClCOCl (b) butadiene (a)
CTB22-44 torsional angle 𝜙 [ ° ] 𝜙 PBEB3LYPM05-2X
Figure 3:
Failures of semilocal functionals for CTB22-44: (a)Schematic. Torsional angle φ is defined as the dihedral angle of Y-C-C-Y. Torsional profile of (b) butadiene (CH CHCHCH ), (c)oxalyl fluoride (COFCOF), and (d) oxalyl chloride (COClCOCl)obtained from the reference calculation (black), DFT methods(colored dashed lines), and HF-DFT methods (colored solid lines).All energies are relative with zero for reference at global minimum,approximations match reference maximum in (b) and (c), and φ = ◦ in (d). The gray arrows in (c) and (d) denote somebarriers corresponding to CTB22ds-12. The torsional barrier of a single bond that participates in π -conjugation (i.e., sp - sp ), such as molecules in Figure 3a,is well known to be significantly more problematic in DFTthan those without conjugation,[18, 19, 21, 22, 23, 24, 25, 26]as seen in Figure 1. The use of DFT on these types of tor-sional barriers needs caution. In a π -conjugated molecule, thetorsional barrier is determined by the competition between con-jugation and steric hindrance.[43] When the molecules are pla-nar, such as cis - ( φ = ◦ ) or trans conformation ( φ = ◦ ), π -conjugation is formed and stabilizes the molecules. In con-trast, no π -conjugation is formed for the ortho conformation( φ ≈ ◦ ).Butadiene is one of the simplest molecules that exhibits π -conjugation. Its torsional profiles are depicted in Figure 3b. • TCCL Yonsei Preprint -0.12-0.0600.060.12-12-60612 C / H C / F C / C l C / B r N / H N / F N / C l N / B r O / H O / F O / C l O / B r S / H S / F S / C l S / B r -10-5 Δ 𝐸 o t [ k J / m o l ] 𝑁 𝐿 𝑃 o t [ e ] butadiene COFCOF CSBrCSBr (a) Δ 𝐸 o t [ k J / m o l ] 𝑁𝐿𝑃 ot [e] (b) butadiene COFCOF CSBrCSBr
XHFClBr YCH NHOS
H F Cl BrX:Y: H F Cl Br H F Cl Br H F Cl Br Δ𝐸 ot 𝑁𝐿𝑃 ot CH NH O S
Figure 4:
For B3LYP calculations of CTB22-44 molecules,(a) error of ortho-trans energy differences ( ∆ E ot , bars), andcorresponding non-Lewis population ( NLP ot , a line with mark-ers). Note that NLP ot is drawn in reverse for easy comparisonwith ∆ E ot . (b) Correlation between NLP ot and ∆ E ot . The graydashed line is a guide for the eye. All properties ( χ ) are calculatedas ortho-trans, χ ot = χ ( ortho ) − χ ( trans ) . Relative to the reference (black solid line), all DFT methods(colored dashed lines) in the figure indicate an overestimation oftorsional barriers between the ortho and trans conformations.This has been studied extensively and its physical origin is wellknown: local or semilocal exchange hole models (commonlyused in many DFT methods) underestimate the exchangeenergy density when the exchange hole is delocalized.[44, 21]This results in the overstabilization of the delocalized ( π -conjugated) conformations. In other words, a delocalizedelectronic structure is responsible for the delocalization errorin DFT.However, this is not true for all DFT torsional profiles. Re-cently, Tahchieva et al . showed that glyoxal, oxalyl halides,and their thiocarbonyl derivatives have large DFT errors intheir torsional profiles. These errors are not due to the miss-ing of dispersion interactions in DFT.[26] Moreover, althoughnot explicitly stated, the direction of the DFT error of themolecules covered in that work is the opposite of typical DFTerrors. For example, for COFCOF (oxalyl fluoride) shown inFigure 3c, the shape of the torsional profile is similar to thatof butadiene; semi-local and hybrid functionals with moderate exact exchange predict the energy of the planar form to be rel-atively high. (See Figure S2a) This is contrary to what wouldbe expected in butadiene. Therefore, such a behavior cannotbe explained by the delocalization error of the π -conjugatedform. COClCOCl (oxalyl chloride) suffers a more serious prob-lem. As depicted in Figure 3d, PBE and B3LYP incorrectlypredict ortho as the equilibrium conformation. (Figure S3aand S4a confirms this also happens for semi-local and hybridfunctionals with moderate exact exchange.) Knowledge aboutthe molecules that have the opposite tendency for DFT errorsis lacking; in particular, it is unclear as to why DFT actsatypically on these molecules.The CTB22-44 subset consists of 44 torsional barriers of16 molecules, including 8 molecules discussed in the reference26 which exhibit the opposite DFT error trend. To simplifythe analysis, we focus on the differences between the ortho and trans conformations ( χ ot = χ ( ortho ) − χ ( trans ) fora computable property χ ). The ortho and trans conforma-tions are expected to exhibit the most different physical andchemical properties, regardless of the detailed shape of thewhole torsional profile. Any method that yields accurate E ot (that is, the energy difference between the ortho and trans conformations, which becomes a torsional barrier when ortho and trans conformations are the local maximum and nearestminimum on the torsional profile, respectively) should showgood performance on the whole torsional profile, including anybarriers.In Figure 4a, the B3LYP error trend of the ortho - trans energy difference ( ∆ E ot = E B3LYPot − E Refot ) is consistent withthat in Figure 3b ∼ d; ∆ E ot > (B3LYP overestimates) forbutadiene and ∆ E ot < (B3LYP underestimates) for oxalylhalides. When the Y group changes as CH → NH → O → Sand X group changes as H → F → Cl → Br, ∆ E ot becomesincreasingly negative, indicating the increasing overstabilizationof the ortho conformation (relative to trans ) by B3LYP. Thisis not limited to B3LYP. Other DFT methods follow the sametrend (See Figure S5). The overall size of ∆ E ot in Figure 4ais small on an absolute scale, but is large enough to yieldincorrect equilibrium conformation predictions in some cases,such as COClCOCl in Figure 3d.To investigate the source of the DFT error, we performednatural bond orbital (NBO) analysis[45] on the wavefunctionsof the molecules in CTB22-44. In the NBO scheme, mostof the electrons are strictly localized to the Lewis-type core,bonding, and lone-pair NBOs. The remaining “non-Lewis”electrons, occupying antibonding or Rydberg NBOs, repre-sent the delocalization correction to the idealized local Lewisstructure,[46] so the degree of delocalization can be quantifiedusing the non-Lewis electron population (NLP; the number ofnon-Lewis electrons). In Figure 4a, the markers connected witha line indicate the NLP difference between ortho and trans conformations ( N LP ot = N LP ortho − N LP trans ) obtainedfrom the B3LYP calculations. From this perspective, the signof
N LP ot indicates the relative degree of delocalization be- • TCCL Yonsei Preprint tween ortho and trans conformations. For example, when the trans conformation is more delocalized than the ortho con-formation,
N LP trans > N LP ortho and thus
N LP ot < , and vice versa . (Split sentences) For the molecules in CTB22-44subset, the almost linear relationship in Figure 4c indicates astrong correlation between the delocalization error ( ∆ E ot ) anddegree of electron delocalization ( N LP ot ).For butadiene (top left on Figure 4b), the negative signof N LP ot matches the chemical intuition; the π -conjugated trans conformation is more delocalized when compared withthe ortho conformation. In Figure 4a, when the Y groupchanges as the CH → NH → O → S and X group changes asH → F → Cl → Br, the
N LP ot becomes increasingly positive,indicating that electrons in the ortho conformation becomeincreasingly delocalized than those in the trans conformation.However, for CSBrCSBr (bottom right), N LP ot > impliesthat the ortho conformation exhibits greater electron delo-calization than the π -conjugated trans conformation. Thisobservation is somewhat unexpected and implies that both the π -conjugation in the planar form and the hyperconjugationmaximized in a nonplanar form have a significant influence onthe electron delocalization of the molecules in CTB22-44. (Weuse the term hyperconjugation to denote any type of electrondelocalization induced by the orbital interaction between thefilled and antibonding orbitals.) This observation also indicatesthat the electron delocalization due to the π -conjugation inthe planar conformation may not always have a higher strengththan the hyperconjugation in twisted conformations. The signof N LP ot in Figure 4 is due to the competition between twodelocalization factors. When the delocalization in the trans conformation ( π conjugation) prevails over the delocalizationin the ortho conformation (hyperconjugation), N LP ot < ,and vice versa .Figures 5a and 5b show schematics of E ot . A smaller de-localization error in the ortho conformation than that in the trans conformation of butadiene incorrectly increases E ot from24.4 kJ/mol (reference) to 29 kJ/mol (B3LYP). Conversely, inCOFCOF, the delocalization error of the ortho conformationis greater than that of the trans form, reducing E ot from7.7 kJ/mol (reference) to 5.5 kJ/mol (B3LYP). Figures 5c ∼ fdepict the representative hyperconjugative interactions maxi-mized at ortho conformation for the COFCOF molecule. Sim-ilar types of hyperconjugation exist for other molecules; how-ever, n ( X ) → π ∗ ( C = Y ) exists only for halide molecules. For theCTB22-44 molecules, in which the X group is hydrogen, hy-perconjugative interaction, such as those shown in Figure 5e,does not exist and the π conjugation is relatively more domi-nant than in their halogen counterparts. The absence of thishyperconjugation for X=H results in negative N LP ot (i.e., the ortho conformation is relatively less delocalized), as shown inFigure 4. Some molecules in CTB22-44 are known to exhibithyperconjugation,[47, 48] and these types of hyperconjugationscontribute to electron delocalization in the ortho conforma-tion, possibly increasing the delocalization error in the DFT 𝜎 CO∗ 𝜋 CO 𝜋 CO∗ 𝑛 F 𝜎 CF transortho transortho (a) Butadiene (b)
COFCOF (c) 𝜎 CF → 𝜋 CO∗ (d) 𝜋 CO → 𝜎 CO∗ (e) 𝑛 F → 𝜋 CO∗ (f) 𝜋 CO → 𝜎 CF∗ 𝜋 CO∗ 𝜎 CF∗ 𝜋 CO Figure 5:
Schematics of E ot = E ortho − E trans for (a) buta-diene and (b) oxalyl fluoride (COFCOF). Black (green) energylevels denote the reference (B3LYP), whereas arrows with gra-dation indicate the delocalization error. The size of arrows isnot to scale. (c ∼ f) shows the selected NBO donor-acceptor pairswith maximum energy stabilization in the ortho conformation ofCOFCOF. The isovalue is set to ± . calculations. Therefore, the explanation that “a delocalizedelectronic structure is responsible for the large delocalizationerror of DFT,” which holds for butadiene,[18, 19, 21] alsoholds for molecules in CTB22-44. The difference in the DFTtendency arises from the difference in the more delocalizedconformations.As depicted in Figure 1, a simple solution, called HF-DFT,can improve the performance of conventional functional ap-proximations (especially for the CTB22ds-12 subset) regardlessof the source of the error. The density sensitivity of a givenDFT calculation is defined as [49] S = | ˜ E [ n LDA ] − ˜ E [ n HF ] | , (1) where n LDA and n HF are the density from the local densityapproximation and HF, respectively, and ˜ E [ · ] is an energy ofsome density functional approximation. Depending on theproperty of interest, ˜ E in Eq. 1 can be the energy of a given • TCCL Yonsei Preprint conformation (e.g., conformational sensitivity, S ( φ ) ) or the en-ergy difference between adjacent local maximum and minimumconformations (e.g., barrier sensitivity, S b ). Equation 1 mea-sures the energy difference between two extreme nonempiricaldensities. A small S implies that the use of any reasonably ac-curate density to calculate the DFT energy will not significantlychange its energetic error. A large value of S indicates densitysensitivity and then often the use of HF-DFT improves theresults.[49, 37] In CTB22ds-12, barrier sensitivities are greaterthan 1.5 kJ/mol in B3LYP (or 4 kJ/mol for PBE). Sometimes,only a subset of a molecule’s barriers are density sensitive.Figure 1 depicts that for the density-sensitive CTB22ds-12subset (significantly more than other subsets, see triangles),HF-DFT significantly reduces MAE over conventional DFT.Other subsets are density insensitive, and thus the reductionin error with HF-DFT is small. In this sub-subset, HF-DFTversions of PBE, BP86, TPSS, TPSSH, and PBE0 all out-perform any conventional DFT considered in this work. (SeeFigure S1 and Table S1 for statistics.) Figure 3d depicts that,for COClCOCl molecule, where barrier sensitivites for two ma-jor barriers (gray arrows in Figure 3d) exceed 1.5 kJ/mol withB3LYP calculations, HF-PBE and HF-B3LYP fix the minima inthe torsional profile, as do other functionals (See Figure S3 andFigure S4 for COBrCOBr). Indeed, HF-DFT shows comparableperformance compared to the torsion-corrected atom-centeredpotential correction method introduced in the reference 26.(See Table S2 ) -2 -1 -0.12 -0.06 0 0.06 0.12 𝑆 o t ( k J / m o l ) 𝑁𝐿𝑃 ot [e] XH F ClBr butadiene
CSBrCSBrYCH NH OS Figure 6:
Correlation between
NLP ot and S ot obtained fromB3LYP calculations. Note that S ot can take negative values. Thegray dashed line is a guide for the eye. Figure 6 depicts how the electron delocalization correlateswith density sensitivity. To compare relative amount of sen-sitivity of two conformations, we define S ot = S ( ortho ) − S ( trans ) , the relative conformational sensitivity between ortho and trans conformations. A positive S ot indicatesthat for the given molecule, ortho conformational sensitivityis higher than that of trans and vice versa . In most con-formations, the signs of N LP ot and S ot are the same—forexample, butadiene and CSBrCSBr, the two molecules at the extreme from the perspective of N LP ot . More delocalizedconformations are more density sensitive.Although delocalization due to hyperconjugation is con-siderable on sp - sp molecules, it is also present in sp - sp molecules.[42] Hyperconjugation affects the DFT barrier erroron CTB32-190, and the barrier error (although reduction issmall) is observed to be reduced using HF-DFT, as shown inFigure 1. We note that the procedure in HF-DFT is recom-mended to use HF densities only for density sensitive cases,[37]but Figure 1 shows using HF densities in all cases, showingthis does not harm in density insensitive cases.Our analysis is not limited to small molecules. Figure 7a de-picts the barrier sensitivity of polyacetylene (CH (C H ) m CH )for various lengths ( m ) and various DFT methods. For mostfunctionals, S b increases with the increase in polymer length m .This corresponds to extending the gray solid line in Figure 6 inthe lower-left direction. (Note that S b is always positive while S ot is not.) In other words, a longer polyacetylene exhibitslarger electron delocalization of the π -conjugation type, andthis increases S b for most functionals. An exception is M05-2Xbecause the functional has a significantly high portion of exactexchange.Torsional barriers of polyacetylene molecules computed usingDFT and HF-DFT and their errors are depicted in Figures 7band 7c, respectively. DFT torsional barriers are typicallyoverestimated for the same reasons as for butadiene, and theoverestimation is larger for longer polyacetylenes owing tothe stronger π conjugation (i.e., delocalization error). Again,for torsional barriers with a significant barrier sensitivity, HF-DFT effectively reduces the error of DFT. The increase inerror with the increase in m is smaller for HF-DFT than forconventional DFT, and HF-B3LYP outperforms M05-2X forlong polyacetylene torsional barriers. This trend also appliesto polymers containing triple bonds of sp hybridization. Thelower panel in Figure 7 depicts the torsional barriers of variouspolydiacetylene (CH (CHC CCH) m CH ) molecules. Althoughpolydiacetylenes have different types of hybridization, sp - sp - sp - sp , the physical origin of barrier overestimation byDFT is the same as that for polyacetylenes or butadiene.Figures 7c and f show that for these polymers, the performanceof 100% long-range exact-exchange functionals is not optimal,as is known for aromatic compounds.[51] Also, the excellentperformance for short oligomers with M05-2X disappear as m increases, while the error of HF-PBE0 and HF-B3LYP saturatesor even decreases. The HF-DFT version of hybrid functionalswith a moderate amount of exact exchange (B3LYP, PBE0,and M06, see Figure S6) shows the best performance in thetorsional barrier of the polymer considered in this work.To conclude, in this work, we investigated the carbon–carbontorsional profiles of various molecule types using standard DFT.In general, the DFT error in the torsional barrier is larger for π -conjugated molecules than for nonconjugated molecules.Typically, for π -conjugated molecules, planar conformationsare overstabilized by DFT and so torsional barriers are over- • TCCL Yonsei Preprint 𝑚 (a)(d) 𝐸 b [ k J / m o l ] 𝐸 b [ k J / m o l ] Δ 𝐸 b [ k J / m o l ] Δ 𝐸 b [ k J / m o l ] 𝑆 b [ k J / m o l ] 𝑆 b [ k J / m o l ] 𝑚𝑚 (b) (c)(e) (f) Figure 7: (a) Barrier sensitivity of polyacetylene (CH (C H ) m CH ), where m is the number of repeated units. (b) Comparison ofDFT (dashed lines) and HF-DFT (solid line) barriers with reference (black line) RI-MP2-F12 calculation. (c) E b error relative to thereference. (d ∼ f) The same data but for polydiacetylene (CH (CHC CCH) m CH ). Following reference 24, barriers are defined as E b = E ( ◦ ) − E ( ◦ ) for polyacetylene, and E b = E ( ◦ ) − E ( ◦ ) for polydiacetylene. The curves in (b, c, e, f) are generatedusing the effective conjugation length model fitting in the reference 50. estimated. But there exist some atypical torsional profileswhere non-conjugated conformations are overstabilized by DFT,causing significant barrier errors. We demonstrate that thisatypical error behavior originates from electron delocalizationdue to strong hyperconjugation. The error in DFT torsionalprofiles of conjugated molecules is determined by the compe-tition of two electron-delocalization factors— π -conjugationand hyperconjugation—whose relative strength varies withconformation, and becomes severe when one factor dominatesthe other. For oxalyl or thiocarbonyl halides, hyperconjugationdominates, while for long conjugated molecules, π -conjugationdominates, and both suffer from severe DFT errors. All thesepoor actors are density sensitive, and so HF-DFT significantlyimproves the torsional profiles regardless of the type of electrondelocalization. This is also true for long polymers. For densitysensitive torsional barriers, applying HF-DFT on semi-localfunctionals even outperforms exchange-enhanced or range-separated hybrid functionals. Taken all together, HF-PBE0provides high accuracy for all small molecule torsional barriers,and its errors remain small for long-chain polymers. Computational Details
All ab-initio calculations were performed using the ORCA4.2.1 program package.[52] Most of the reference calcula- tions on the CTB-279 dataset were obtained at the DLPNO-CCSD(T)[53]/TightPNO/cc-pV(T+Q)Z level, and we con-firmed that the use of the NormalPNO setting gives nomeaningful difference. Some exceptions were observed forX=F, Cl, and Br for the CTB22-44 subset. Therefore, weused CCSD(T)/cc-pV(T+Q)Z for those molecules. We usedthe VeryTightSCF keyword for all CC calculations to main-tain high integral accuracy. We confirmed that T1 diag-nostics of all molecules in the CTB-279 dataset are all lessthan 0.02. For polyacetylene and polydiacetylene, RI-MP2-F12/aug-cc-pVDZ was used. All DFT and HF-DFT calcu-lations (SVWN,[54, 55] PBE,[36] BP86,[56, 57] BLYP,[56,58] TPSS,[59] TPSSH,[60] B3LYP,[17] PBE0,[61] M06,[62]BHHLYP,[63] M06-2X,[62] M05-2X,[34] CAM-B3LYP,[32] ω B97, ω B97X,[64] and ω B97M-V[65]) were performed withthe PySCF program package[66]. For the CTB-279 datasetand polymer DFT calculations, Dunning’s[67, 68, 69] cc-pVQZand aug-cc-pVDZ basis sets were used, respectively. Reference35 introduces simple scripts to run HF-DFT calculations. TheNBO analysis was performed using the NBO 3.1 program[45]included in Gaussian 16.[70]To generate the CTB-279 dataset, molecules were generatedautomatically by attaching various functional groups (R, X,and Y in Figure 2 and 3). In this step, the initial conformationof each molecule was selected such that no intramolecular • TCCL Yonsei Preprint hydrogen bond was formed. For example, two OH groupsin glyoxal were forced to move in opposite directions. SinceMP2 yields very accurate geometries for covalently bondedsystems[71] and RI-MP2 gives very close geometry to MP2,[72]we optimized the molecules using RI-MP2/def2-TZVP[73] witha fixed φ with ◦ intervals. After optimization, we manuallyremoved the molecules that formed intramolecular hydrogenbonds at any φ . In this process, 15, 66, and 16 moleculesremained for the CTB33, CTB32, and CTB22 subsets, respec-tively. With these molecules, we performed DLPNO-CCSD(T)or CCSD(T)/cc-pV(T+Q)Z calculations to obtain the refer-ence energies. With the reference torsional profile, we usedcubic spline interpolation to calculate the local maximum andminimum in ◦ precision. Torsional profiles in Figure 2 and 3are also obtained with this procedure. After locating the localmaximum and minimum, geometry optimization with a fixedtorsional angle was performed. On that geometry, referenceand DFT calculations were performed. As such, CTB33 andCTB32 contained 45 and 190 torsional barriers, respectively. Acknowledgement
This study was performed at Yonsei University andwas supported by a grant from the Korean ResearchFoundation (2020R1A2C2007468 and 2020R1A4A1017737).KB acknowledges the NSF for Grant CHE 1856165.
Supporting Information • Figures and tables corresponding to ortho-trans energydifferences, statistics for CTB-279 dataset, comparison withPBE-TCACP and HF-DFT, problematic torsional profiles, andpolymer torsional barrier errors for various functionals. • Spreadsheet containing CTB-279 and polymer data • Geometry files used in this work
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Supporting Information
Table S1 and Figure S1 compares DFT and HF-DFT for tor-sional barriers. For highly density sensitive CTB22ds-12, HF-DFT shows dramatic improvement over conventional DFT,while for other density insensitive subsets, HF-DFT yieldssimilar results with conventional DFT.
CTB33-45 CTB32-190
CTB22-44
CTB22di-32 CTB22ds-12 Both
DFT HF-DFT DFT HF-DFT DFT HF-DFT DFT HF-DFT DFT HF-DFT
SVWN 0.79 0.65 2.16
PBE 0.89 0.99 1.21 1.02 1.33 1.42 6.00
BP86 1.06 1.14 1.18 1.04 1.38 1.43 6.22
BLYP 1.19 1.12 1.21 1.12 2.19 2.15 8.14
TPSS 1.30 1.37 1.02 0.99 1.29 1.26 3.96
TPSSH 0.92 1.00 0.84 0.74 1.16 1.13 2.74
PBE0 0.29 0.30 1.01 0.82 1.10 1.12 2.57
CAM-B3LYP 0.25 0.28 0.64 0.54 1.12 1.19 2.76 2.27 1.57 1.49 ω B97 0.28 0.23 0.73 0.70 1.12 0.98 1.86 1.93 1.32 1.24 ω B97X 0.22 0.23 0.71 0.64 1.08 0.98 1.98 1.87 1.33 1.22 ω B97M - V 0.31 0.26 0.56 0.47 0.89 0.84 1.32 1.72 1.01 1.08
Table S1:
Torsional barrier MAEs (in kJ/mol) of various func-tionals. Red numbers indicate MAE over 1.5 kJ/mol. Bold num-bers indicate that HF-DFT reduces MAE by at least 0.5 kJ/molcompared to DFT. S V W N P B E B P B L Y P T P SS T P SS H B L Y P P B E M B HH L Y P M - X M - X C A M - B L Y P ω B ω B X ω B M - V SC-DFTHF-DFT S V W N P B E B P B L Y P T P SS T P SS H B L Y P P B E M B HH L Y P M - X M - X C A M - B L Y P ω B ω B X ω B M - V SC-DFTHF-DFT01 S V W N P B E B P B L Y P T P SS T P SS H B L Y P P B E M B HH L Y P M - X M - X C A M - B L Y P ω B ω B X ω B M - V SC-DFTHF-DFT 0246810 S V W N P B E B P B L Y P T P SS T P SS H B L Y P P B E M B HH L Y P M - X M - X C A M - B L Y P ω B ω B X ω B M - V SC-DFTHF-DFT (a) CTB33-45 (b) CTB32-190(c) CTB22di-32 (d) CTB22ds-12 M A E [ k J / m o l ] M A E [ k J / m o l ] DFT functionals
Figure S1:
Torsional barrier MAEs of various functionals.Torsional barriers are defined as the energy difference betweenthe local minimum and nearest local maximum in each torsionalprofile.
Figure S2, S3, and S4 shows torsional profiles of COFCOF,COClCOCl, and COBrCOBr, respectively, obtained with con-ventional DFT and HF-DFT. Overall, ortho conformations ofthese molecules are typically overstabilized with local/semi-local/hybrid functionals with moderate amout exchange. Whilesome conventional DFT predicts incorrect global minimum con-formations for COClCOCl and COBrCOBr, HF-DFT capturescorrect global minimums.Figure S5 shows that various DFT methods, not only B3LYP,typically shows increasingly negative ∆ E ot when the Y groupchanges as CH → NH → O → S and X group changes as H 𝐸 r e l [ k J / m o l ] torsional angle 𝜙 [ ° ] (a) SC-DFT (b) HF-DFT Figure S2:
COFCOF (oxalyl fluoride) torsional profile forreference (CCSD(T)/cc-pV(T+Q)Z, black) and various (a) SC-DFT methods and (b) HF-DFT methods. 𝐸 r e l [ k J / m o l ] torsional angle 𝜙 [ ° ] (a) SC-DFT (b) HF-DFT Figure S3:
COClCOCl (oxalyl chloride) torsional profile forreference (CCSD(T)/cc-pV(T+Q)Z, black) and various (a) SC-DFT methods and (b) HF-DFT methods. 𝐸 r e l [ k J / m o l ] torsional angle 𝜙 [ ° ] (a) SC-DFT (b) HF-DFT Figure S4:
COBrCOBr (oxalyl bromide) torsional profile forreference (CCSD(T)/cc-pV(T+Q)Z, black) and various (a) SC-DFT methods and (b) HF-DFT methods. → F → Cl → Br.Table S2 compares performances between DFT, HF-DFTand PBE-TCACP[26] on 12 conformational energy differencesof some CTB22-44 molecules. HF-DFT with standard func-tionals shows comparable performances with empirically pa-rameterized PBE-TCACP.Figure S6 shows DFT and HF-DFT (left and right panel,respectively) ∆ E b for polyacetylene and polydiacetylene (upperand lower panel, respectively). For short polymers, exchange-enhance functionals such as BHHLYP, M06-2X, and M05-2Xshow very small errors, but as m increases, their error increases. • TCCL Yonsei Preprint Δ 𝐸 o t ( k J / m o l ) YX Figure S5: ∆ E ot for CTB22-44 molecules. Squares with dashedlines denotes molecules containing CTB22ds-12 barriers. Dashedlines with circle markers denotes DFT, while solid lines withtriangle markers denotes HF-DFT. Method DFT HF-DFTPBE-TCACP 1.3SVWN 2.5 3.6PBE 5.0 1.2BP86 5.1 1.2BLYP 6.6 2.7TPSS 3.5 1.2TPSSH 2.5 1.1B3LYP 4.3 2.7PBE0 2.3 1.2M06 3.2 2.7BHHLYP 3.5 3.2M06-2X 2.3 2.2M05-2X 2.1 2.1CAM-B3LYP 2.5 2.3 ω B97 2.2 2.0 ω B97X 2.2 1.9 ω B97M-V 1.6 1.5
Table S2:
Comparision of DFT, HF-DFT and PBE-TCACP(taken from 26) results. Shown numbers are MAEs of 12 energydifferences between two torsional conformations (defined in 26)of 8 molecules (Y=O,S with X=H,F,C,Br) used in both 26 andthis work. Reference to determine errors are taken from ourcalculations: DLPNO-CCSD(T)/TightPNO/cc-pV(T+Q)Z forX=H and CCSD(T)/cc-pV(T+Q)Z for others on RI-MP2/def2-TZVP geometries. • TCCL Yonsei Preprint Δ 𝐸 b [ k J / m o l ] Δ 𝐸 b [ k J / m o l ] 𝑚 (a) (b) (c) (d) 𝑚 Figure S6:
DFT and HF-DFT (left and right panel, respectively) ∆ E b for polyacetylene and polydiacetylene (upper and lower panel,respectively)for polyacetylene and polydiacetylene (upper and lower panel,respectively)