Conformational energies of reference organic molecules: benchmarking of common efficient computational methods against coupled cluster theory

We selected 145 reference organic molecules that include model fragments used in computer-aided drug design. We calculated 158 conformational energies and barriers using force fields, with wide applicability in commercial and free softwares and extensive application on the calculation of conformational energies of organic molecules, e.g. the UFF and DREIDING force fields, the Allinger’s force fields MM3-96, MM3-00, MM4-8, the MM2-91 clones MMX and MM+, the MMFF94 force field, MM4, ab initio Hartree–Fock (HF) theory with different basis sets, the standard density functional theory B3LYP, the second-order post-HF MP2 theory and the Domain-based Local Pair Natural Orbital Coupled Cluster DLPNO-CCSD(T) theory, with the latter used for accurate reference values. The data set of the organic molecules includes hydrocarbons, haloalkanes, conjugated compounds, and oxygen-, nitrogen-, phosphorus- and sulphur-containing compounds. We reviewed in detail the conformational aspects of these model organic molecules providing the current understanding of the steric and electronic factors that determine the stability of low energy conformers and the literature including previous experimental observations and calculated findings. While progress on the computer hardware allows the calculations of thousands of conformations for later use in drug design projects, this study is an update from previous classical studies that used, as reference values, experimental ones using a variety of methods and different environments. The lowest mean error against the DLPNO-CCSD(T) reference was calculated for MP2 (0.35 kcal mol−1), followed by B3LYP (0.69 kcal mol−1) and the HF theories (0.81–1.0 kcal mol−1). As regards the force fields, the lowest errors were observed for the Allinger’s force fields MM3-00 (1.28 kcal mol−1), ΜΜ3-96 (1.40 kcal mol−1) and the Halgren’s MMFF94 force field (1.30 kcal mol−1) and then for the MM2-91 clones MMX (1.77 kcal mol−1) and MM+ (2.01 kcal mol−1) and MM4 (2.05 kcal mol−1). The DREIDING (3.63 kcal mol−1) and UFF (3.77 kcal mol−1) force fields have the lowest performance. These model organic molecules we used are often present as fragments in drug-like molecules. The values calculated using DLPNO-CCSD(T) make up a valuable data set for further comparisons and for improved force field parameterization. Graphical abstract Supplementary Information The online version contains supplementary material available at 10.1007/s10822-023-00513-5.

Conformational sampling using force fields of the drug molecules in SBDD problems can be accomplished using fragment-based approaches [15,16], in which the candidate drug molecule is divided into fragments and the smaller organic molecule conformations are sampled before carrying out the docking calculations.Candidate conformations of the entire molecule are computed by recombining favorable fragment conformations.A database of minimized conformations of the fragments allowed re-using them during conformer generation of other molecules, including drugs and large bioactive molecules, which improves the time-efficiency of sampling as, for example, in the open-access Quantum-Mechanical Properties of Drug-like Molecules (QMugs) data set; the QMugs collection comprises QM properties on optimized molecular geometries using ωB97X-D/def2-SVP, e.g., QM wave functions, including DFT density and orbital matrices, of more than 665 k biologically and pharmacologically relevant molecules extracted from the ChEMBL database, totaling ~ 2 M conformers [17,18].QM7-X [19] is another comprehensive dataset of 42 physicochemical properties for ~ 4.2 million equilibrium and non-equilibrium structures of small organic molecules with up to seven nonhydrogen optimized with PBE0 + MBD, i.e., a third-order self-consistent charge density functional tight binding (DFTB3) supplemented with a treatment of many-body dispersion (MBD) interactions.These methods use ensembles of conformations that capture the bioactive conformation as one of a diverse set of energetically accessible conformations [20][21][22][23].Another approach involves using pre-existing knowledge of small-molecule conformations to further restrict the space of the conformational search to likely torsion angles and other combinations.Such knowledge-based methods [24,25] derive torsion angle preferences from molecular mechanics or QM simulations of small molecules or structural databases such as the Cambridge Structure Database (CSD) [26] or the Protein Data Bank (PDB) [27].Such datasets combined with ML can lead to the calculation up to 20 million off-equilibrium conformers of organic molecules [28].
Benchmarks of conformer generation tools have been performed, not based on low-energy/geometry [23], but also by comparing the geometry of an experimental crystal structure against an ensemble (e.g., 50 to 200+) of conformers [22,[29][30][31][32][33][34][35]; given a reasonable tool, one might guess that generating enough conformers should produce something close to the experimental geometry.Thus, finding a metric, such as energy, to filter, score or rank conformers is critical.Also, Bayesian optimization for conformer generation [36] or Graph Neural Network [37] have been applied to find the lowest energy conformer rather than an ensemble of conformers.Although most methods to score conformations, i.e., calculate conformational energies and identify low energy conformers, use a molecular mechanics energy function, e.g., the MMFF94 force field [38] implemented in MOE [39], in OMEGA program [9,29], in RDKit [40], the OPLS2001 force field [41] implemented in the CONF-GEN [42], the reliability of classical force fields and other quantum mechanical methods needs to be validated [43].Other studies explored conformational search algorithms to regenerate bioactive conformations from protein-ligandcomplexes.It was found that in 73% of the studied molecules in protein complexes from Protein Data Bank (PDB) structures the "bioactive" conformation was within 3k B T from the most-stable conformation in solution as determined by density functional theory (DFT) calculations [22].
Despite the enormous amount of energy values that can be produced with the current computing resources, systematic comparisons of force fields and QM methods provide always helpful results [44] for the calculation of the conformational energies of organic molecules [36,45].
Given that drug molecules consist of a few tens of atoms connected by covalent bonds and that the possible organic small molecules number hundreds of billions [70], hardware progress can lead to the accurate coupled cluster method with single-, double-, and perturbative triple excitations [CCSD(T)] chemical energies for 133,000 molecules with less than 10 carbons [71], or B3LYP/6-31G* optimized geometries for 2.6 million molecules [72].
Continued development of deep learning molecular potentials generated from QM data sets can provide high accuracy predictions of QM reference calculations, while maintaining a computational cost comparable to classical force field.ANI-2x potential provided chemically accurate energy predictions for molecules containing seven atomic elements (H, C, N, O, S, F, Cl) of interest to computational drug design (CADD) and showed similar accuracy to DFT methods, while outperformed MMFF94 and PM6 for conformer scoring.The ANI-2x potential retained the same computational scaling as classical force fields, providing a 10 6 speedup over the DFT level it has been trained against [73].It has been tested across a wide range of applications relevant to drug development on diverse test sets.
For drug design purposes the accurate description of the diverse local minima is needed for each drug-like molecule [74], and datasets have been developed towards that aim, e.g. the platinum dataset of 2912 protein-bound ligand conformations extracted from the PDB for which the chemical space was shown to be representative of the chemical space of approved drugs [75].
In the present work we seek to revisit results of calculations for model organic molecules of standard conformational interest rather than drug-like compounds and focuses on assessing the accuracy of force fields, frequently used in conformer search applications, but also standard ab initio and DFT methods.Compared to previous works, e.g. in Refs.[48,50] we increased the number of tested molecules and the number of methods applied, which include many force field methods but also Hartree-Fock (HF) theory, the post-HF second-order Møller-Plesset perturbation theory (MP2) and the standard DFT functional B3LYP.Accurate reference values for evaluating all these methods were obtained with the gold-standard basis-set extrapolated DLPNO-CCSD(T) method [76], in contrast to previous works which often used inconsistent experimental values or low theory levels as reference values.The suitable energies for comparison with CC-calculated conformational energies are energies measured in the gas phase with spestroscopic methods.Compared to previous works, this manuscript also reports the gas phase experimental and calculated with various theories conformational energies and describes the molecular basis of these conformational preferences for the organic molecules tested.These model organic molecules are often present as fragments in drug-like molecules.The values calculated using DLPNO-CCSD(T) make up a valuable data set for further comparisons and for improved force field parameterization.

Test set
A data set consisting of 158 small molecules consisting solely of carbon, hydrogen, nitrogen, and oxygen (CHON) atoms, was the same as that used before for validation of the accuracy of MMFF94 force field and subset of these has been previously used [48,50].The three dimension (3D) structures of the compounds used are available following the link found in the Supporting Information.

Details for the calculations
Molecular mechanics calculations with the MM+ force field were performed using the Hyperchem program (Hypercube Inc.) which provided enthalpic values.For each molecule an initial structure was constructed and minimized first using a first derivative algorithm (conjugate gradient or steepest descents) and then Newton-Raphson algorithms and an energy gradient tolerance of 0.001 kcal mol −1 Å −1 .HyperChem provides a variety of tools for the convenient manipulation of 3D structures like changing chirality, reflection through planes, easy insertion of a variety of torsional restraints etc.When a constraint was required the force constant used was as high as 150 kcal mol −1 degree −2 .MMX, MM3-96 [77][78][79] and MMFF94 [32] were implemented with PCMODEL9/Windows software (Serena Software) and calculated enthalpies.3D structures were minimized first using steepest descents and then Newton-Raphson algorithms and an energy gradient tolerance of 0.0001 kcal mol −1 Å −1 .MM3-00 and MM4-08 [80][81][82] were performed using the commercial programs developed by Allinger and we calculated enthalpies [83,84].Calculations were carried out in the gas phase using a dielectric value of 1.5 and no cutoff for the nonbonded interactions.In the case of MM+ [85] available with Hyperchem program and MMX, which are clones of MM2-91 [86] force field, lone pair (lp) pseudo-atoms were added where needed for proper description of the molecular structure.Calculations of enthalpic values with UFF [87], Dreiding [88] force fields were carried out with Gaussian-03 software [89].MM+, UFF and Dreiding are universal force fields and can be used for the calculations of any structure since, in cases for which no parameters are available, empirical rules are applied.Structure manipulations, restraints, etc. were applied using the software tools.In a few cases, structure manipulation was performed using HyperChem because this software is user friendly.The structures were then saved in PDB format and opened with other software pieces.B3LYP/6-31G(d,p) and MP2/6-31G(d,p) electron energy calculations were performed using Gaussian-09 [89] and the energies were calculated in geometry-optimized structures.−  The B3LYP/6-31G(d,p) geometry-optimized structures were used for the DLPNO-CCSD(T) electronic energies calculations.Reference values for the electronic energies of all species were obtained at the CCSD(T) level [76] with separate extrapolation of the Hartree-Fock (HF) and electron correlation energies to their complete basis set (CBS) limits: The domain-based local pair natural orbital (DLPNO) methodology enabled the use of large correlation-consistent basis sets of polarized triple-zeta quality in the coupledcluster calculations for all molecules included in this study.
CBS extrapolation used correlation-consistent basis sets cc-pVnZ with successive cardinal values n = 2 and 3 (cc-pVDZ and cc-pVTZ) [90][91][92].DLPNO-CCSD(T) calculations with the cc-pVQZ basis sets were not possible for all compounds due to memory limitations; therefore, for consistency, only cc-pV[D/T]Z extrapolated results will be discussed.Using a two-point exponential extrapolation [93,94], the HF energy has been suggested to converge as: where X and Y represent the successive cardinal numbers of the basis sets.E (X) HF and E (Y) HF are the SCF energies obtained with the two basis sets.The parameter α takes the value of 4.42 for the cc-pV[D/T]Z extrapolation [94].On the other hand, the CBS limit for the CCSD(T) correlation energy was obtained as: Here X and Y are the cardinal numbers as above and the optimal value for the parameter β for the cc-pV[D/T]Z combination of basis sets was shown to be 2.46 [94,95].The SCF component of the calculations employed the RI-JK approach in conjunction with cc-pVTZ/JK basis sets, while the cc-pVDZ/C and cc-pVTZ/C basis sets were used in the correlation treatment [96].The ORCA program package was used for all DLPNO-CCSD(T) calculations [97].TightSCF and TightPNO settings as defined in ORCA were used for all calculations.PNO extrapolation using different TCutPNO values was also tested [98][99][100][101], but it made no difference compared to the standard TightPNO computed energies for the molecules in the test set, confirming that the DLPNO values are converged with respect to the PNO space.
In the relevant tables we also included experimental values previously obtained and used for some of these molecules for comparative purposes.We note, however, that these are not always directly comparable because the computed energies reported here are electronic gas-phase values and do not incorporate thermodynamic or solvent contributions [48].In the tables, when a method performs with an error larger than 1.5 kcal mol −1 , the result is indicated in boldface and bold underlined when the deviation is larger than 3 kcal mol −1 .When a conformational energy difference with a tested theory has opposite sign compared to the reference theory but the energy value differs by less than ~ 0.1 kcal mol−1, we considered this case as a correct prediction with the tested theory.

Hydrocarbons
The results of the calculations and experimental data for the studied molecules appear in Table 1.For n-butane, the anti conformation is stabilized with respect to the gauche conformation by experimentally determined energies in the gas phase of 0.69 kcal mol −1 [113], 0.67 kcal mol −1 [114], 0.71 kcal mol −1 [115].The rotational barriers and conformational energies in the gas phase have been measured [116] and it has been proposed that in the lower-energy trans conformer the hyperconjugative orbital interaction between antiparallel C-H bonds, σ C-H → σ* C-Η (Fig. 1) contributes to the stabilization of the anti conformer [117].The hyperconjugative phenomenon simply suggests that the Lewis structures for organic molecule representation is an approximation.The importance of hyperconjugative interactions in the     conformational analysis of organic molecules was reported in 2001 [118], where it was suggested that the lower energy of staggered compared to the eclipsed ethane results, not from smaller steric repulsions, but from hyperconjugative stabilization (Fig. 2), which is equivalent to the formation of more bonds that lowers the energy.After some rebuttal [119,120], it has been suggested that both steric effects and hyperconjugative interactions play important roles in stabilizing the staggered conformation in ethane.While steric effects make the dominant contribution [121][122][123],

Νο. wrong con-formers
hyperconjugation interactions contribute about one third of the total torsional barrier in ethane.In butane, the calculated potential energy surfaces and the Natural Bond Orbital (NBO) analysis suggested that the gauche conformer is destabilized because of the steric repulsions between the gauche methyl groups while hyperconjugative interactions play an important, but not prelevant role for the relative conformational energies [121][122][123].All the tested theories calculate the anti conformation as the global minimum in accordance to the DLPNO-CCSD(T) theory and experimental values in the gas phase (Table 1) [113][114][115].Compared to the DLPNO-CCSD(T) value, the Dreiding, MMFF94, MM+, MMX, MM3-96, MM3-00 force fields have a deviation of ~ + 0.2 kcal mol −1 , UFF force field and HF theories perform with a deviation of ~ + 0.4 to 0.5 kcal mol −1 and B3LYP functional with a deviation of + 0.3 kcal mol −1 .The MM4-08 force field (+ 0.08 kcal mol −1 ) and MP2 theory (+ 0.03 kcal mol −1 ) have the smallest deviation.According to 13 C-NMR at very low temperatures in 2,3-dimethylbutane the preference for anti conformation, like E (Fig. 3) is small compared to the n-butane, despite the common observation that the anti conformation has only two gauche interactions versus three in the gauche conformation F [124,125].This is also confirmed by the DLPNO-CCSD(T) calculations (Table 1).The increase in gauche conformation population can be stabilized because steric forces between vicinal methyl groups are reduced through opening up of the Me-C-Me bond angles and steric interactions may be further eased by rotation about the central bond resulted in structure H [126].In contrast, in anti conformation E there is no option for steric strain relief because opening up of the Me-C-Me bond angles forces the vicinal methyl groups together, as shown in structure G [126].Compared to DLPNO-CCSD(T) theory, the Dreiding, UFF and MMFF94 force fields calculate erroneously the anti conformation as the global minimum with an energy deviation in the range of 0.28-1.20 kcal mol −1 while HF theories provide energy deviations in the range of 0.10-0.20 kcal mol −1 .In contrast, the Allinger force fields, B3LYP and MP2 calculate accurately this small conformational energy difference.
In 1-butene (Fig. 4), the skew conformation has lower energy than the cis conformation, in good accord with the experimental data [127] while DPLNO-CCSD(T) theory    calculates the cis with a slight lower energy (0.01 kcal mol −1 ) relative to the skew conformation (Table 1).Compared to the DPLNO-CCSD(T) calculations, all theories [128], except UFF, stabilize or overstabilize (> 2 kcal mol −1 with the Dreiding force field) the skew conformation as the global energy minimum.In this case MP2 performed the best.
The UFF force field overstabilizes the cis conformation by > 3 kcal mol −1 .Dispersion (attractive van der Waals) forces act at distances longer than the sum of van der Waals radii [129,130].We evaluated the ability of the tested methods to calculate the contribution of dispersion interactions in conformational preferences by studying a few relevant molecules, e.g., the 1,2-diphenylethane, tetraethylmethane, tetramethylhexane, tri-neopentyl-benzene and tetra-benzyl-ethene (Fig. 5).
In 1,2-diphenylethane the gauche conformations is stabilized by π-π interactions compared to the steric relief in the anti conformation.For several decades the results were non-conclusive [131][132][133][134] and an experimental energy difference of 1.19 kcal mol −1 [134] or 0.57 kcal mol −1 [132] in favor of the anti conformation was suggested.However, a recent computational chemistry and spectroscopic investigation showed that 1,2-diphenylethane exists as a mixture of gauche and anti conformations, with the gauche being the global minimum [135].Our DPLNO-CCSD(T) calculations suggest the stabilization of the gauche conformation compared to the anti conformation albeit only by 0.32 kcal mol −1 .Compared to the DPLNO-CCSD(T) calculations all the Allinger force fields, the HF theories and the conventional B3LYP functional stabilize or overstabilize the anti conformation while low-order post-HF (MP2) approaches strongly favor the gauche conformation [135].
However, inclusion of semiempirical dispersion effects in density functionals or coupled cluster post-HF models agree in forecasting the simultaneous presence of both conformers in the gas phase with a slightly larger stability (0.32 kcal mol −1 ) of the gauche conformation.Surprisingly, Dreiding and UFF predict gauche conformer as the global minimum for 1,2-diphenylethane with Dreiding performing with an error > 2 kcal mol −1 .
In tetraethylmethane [136] the T1 conformation is lower in energy than the T3 conformation, according to our DPLNO-CCSD(T) calculations, in good agreement with the dynamic NMR data [136].In T1 conformation compared to T3 conformation the two methyl groups are in a syn position (as shown in the upper and right part of the T1 conformation) where dispersion forces act stabilizing more T1 compared to the T3 conformation.Except for UFF, which calculates both T1 and T3 conformations with equal stability, all the other theories calculate the right global minimum (T1 conformation) for tetraethylmethane.The DPLNO-CCSD(T) calculations show that in tetramethylhexane [137] the C 2h conformation has almost equal energy-slightly lowercompared to C 2 conformation.In the C 2h conformation the two ethyl groups are in a syn instead of an anti orientation, respectively, and attractive London dispersion (LD) forces antagonize Pauli repulsion (steric hindrance) forces leading to equal energies of C 2h and C 2 conformations according to the DPLNO-CCSD(T) calculations, in contrast to dynamic NMR in solution where C 2h is prevailed in the ~ 60:40 mixture with C 2 (ΔG = 0.22 kcal mol −1  In tetrabenzylethene [103] or 1,3,5-trineopentyl benzene, the benzyl or t-butylmethyl substituents form π-π or alkyl-alkyl dispersive interactions when they are in a syn orientation. In tetrabenzylethene, in the D 2 conformation, which is also observed in the solid state for the benzene dimer, the phenyl groups, each linked through a methylene to the same unsaturated carbon, are in anti orientation while in C 2h all benzene rings are in syn orientation.In C 2h conformation the phenyl groups are in syn orientation and dispersion attraction antagonize Pauli repulsion.In tetrabenzylethene the D 2 conforation is clearly more stable by 4.5 kcal mol −1 than C 2h conformation according to our DPLNO-CCSD(T) calculations suggesting that the repulsive interactions prevail.Compared to the DPLNO-CCSD(T) calculation the MM+ and UFF calculate the wrong global minimum.Dreiding, MMX, MMFF94 and particularly the MM3-96, MM3-00, MM4-08 force fields underestimate the energy difference, and the HF theories overestimate the energy difference, with Allinger force fields and HF theories performing with deviation > 3 kcal mol −1 .The B3LYP (− 1.20 kcal mol −1 ) and MP2 (− 0.67 kcal mol −1 ) perform the best with this model molecule with the latter having a smaller deviation.
For cyclohexane or cyclohexene and cyclooctane it has been shown experimentally in the gas phase or with dynamic NMR in solution, respectively, that the chair cyclohexane is more stable over twist-boat [112] by 5.5 kcal mol −1 [139], the half-chair cyclohexene is more stable than the boat by 5.5 kcal mol −1 [140,141], and for cyclooctane the boat-chair (BC) is more stable than twist-chair-chair (TCC) by 1.9 kcal mol −1 [142] (Fig. 6).Τhe DPLNO-CCSD(T) calculations calculate these conformational energies 5.97 kcal mol −1 , 5.54 kcal mol −1 and 1.97 kcal mol −1 .All theories calculate the chair cyclohexane as more stable than twist-boat while UFF and Dreiding force fields overestimating the energy by 2.85 and 1.70 kcal mol −1 , respectively.In the case of cyclohexene all theories calculate the halfchair cyclohexene conformation as the global minimum but MMFF94 (2.06 kcal mol −1 ) and MM4-08 (1.73 kcal mol −1 ) deviate most from the DPLNO-CCSD(T) calculations.As regards cyclooctane, Dreiding and UFF force fields perform with the largest errors with the first force field calculating TCC and BC conformations with same energy and the second force field calculating TCC as more stable than the BC conformation by 2.41 kcal mol −1 .From the other theories, the HF and B3LYP theories perform with the largest error (> 1.5 kcal mol −1 ) but the MMFF94 and Allinger's force field perform better.For all the three molecules MP2 performed with the smallest deviation.
The gauche conformation is preferred over anti in 1,2-difluoroethane as showed by experiments and DLPNO-CCSD calculations (Table 2) [150,151].This preference, which is also observed for other electronegative substituents, is known as the gauche effect.An explanation for this effect has been proposed on the basis of MP4/6-311++G(d,p) level calculations [152,153] according to which: (a) the anti rotamer is destabilized [152] because in this position the trans electronegative substituents cause the C-C bond orbitals to bent in opposite directions resulting in bending geometry of the C-C bond (see left part of Fig. 7) and a weaker bond, whereas in the gauche rotamer the C-C bond orbitals bend in the same direction or/and (b) the gauche conformation is stabilized over competing electrostatic interactions between the fluorine atoms because of favouring hyperconjugative interactions σ(C-H) → σ*(C-F) [117] being possible due to the gauche position of fluorine substituents (see right part of Fig. 7).The opposite preference is observed experimentally in the gas phase in 1,2-dichloroethane [154] where gauche conformer is destabilized over anti due to the repulsion of bond C-Cl dipoles as showed by MP2/6-311++G** calculations [153].While the cause of gauche conformation stability was also suggested as due to 1,3 C•••F electrostatic polarization interactions that stabilize nearby carbon atoms [155] or similarly to electrostatic and exchange-correlation interactions [156] using state-of-the-art DFT calculations at theory level ZORA-BP86-D3(BJ)/QZ4P the rotational isomerism of 1,2-dihaloethanes XCH 2 CH 2 X (X = F, Cl, Br, I) was investigated as the interplay of hyperconjugation with Pauli repulsion between lone-pair-type orbitals on the halogen substituents that constitutes the causal mechanism for the gauche effect.Only in the case of the relatively small fluorine atoms, steric Pauli repulsion is too weak to overrule the gauche preference of the hyperconjugative orbital interactions.For the larger halogens, X⋅⋅⋅X steric Pauli repulsion

Cyclohexane derivatives
In monosubstituted cyclohexanes [112] (R = Me or i-Pr or t-Bu [165], Ph [166], Me 3 Si [167,168], NH 2 [169], OH or OMe [83], CO 2 Me or COMe [170], SH [171], PH 2 [172], F [173,174], Cl [175,176], the equatorial orientation is lower in energy with citations included for the different substituents [177].The stereoelectronic reasons for the higher stability of the equatorial over the axial (ax) conformations in monosubstituted cyclohexanes are still under investigation.The traditional model of 1,3-diaxial steric interactions between the axial substituent and the axial C3-H and C5-H bonds (steric gauche butane interaction between the axial substituent and carbons C3, C5) [165] provide a model adequate for most cases.However, compared to the synaxial repulsive interactions [165] model which destabilized the axial conformation compared to the equatorial (eq) conformation, it has been also proposed that the equatorial orientation is more stable than the axial orientation because of the stabilizing hyperconjugative σ C-Hax → σ* C-Hax interactions [178].These include in the equatorial conformation the axial C-H bond of the carbon bearing the equatorial group and the axial C-H bond of the adjacent carbon [178] (Fig. 8).
For groups with heteroatoms, X = N, O, F, Cl, electrostatic interactions stabilizing the gauche conformation in 1-fluoropropane or 1-propanol (Figs.7,12) are expected to stabilize also the axial conformation over equatorial [179].Since the experimental data show that the equatorial conformer is the most stable in these cases [143], the previous effects dominate.In a selected group of substituted cyclohexanes the ΔE ax-eq , of monosubstituted cyclohexanes with OR (R = Me, Et, i-Pr and t-Bu) and R substituents (R = Me, Et, i-Pr and t-Bu) was calculated with HF, MP2 and QCISD theories with the 6-311G* and 6-311+G* basis sets [180].
The natural bond orbital method was applied to quantify the hyperconjugative contribution, ΔE hyp , to the relative stability of conformers.From the calculated values of ΔE ax-eq and ΔE hyp an estimate of the differential steric effect, ΔE ster , of substituents in cyclohexane was obtained.The values of ΔE hyp and ΔE ster show that they have a similar magnitude for OR substituents, while for R substituents the values of ΔE ster are greater.The shift in the conformational equilibrium towards the axial conformer, the so-called anomeric effect, takes place when, within a series of substituents, hyperconjugative interactions and steric interactions balance in favour of the stability of this conformer.After our suggestion that axial substituents in cyclohexanes exert not only Pauli repulsion but also attractive LD interactions [129,181,182] and that DFT potential including the Grimme correction for LD interaction can be included for a more accurate description of ΔE ax-eq systematic study using DLPNO-CCSD(T)/aug-cc-pVQ//B3LYP/def2-TZVP led to A-value scale that is can no longer be considered purely to arise from steric factors.Even for groups that do not participate in charge transfer or electrostatic interactions, the A-value includes Pauli repulsion and attractive LD interactions [183].It has been observed with DNMR in solution an increase in population of axial conformer when passing from Me 3 SiO to the bulkier Ph 3 SiO group.An explanation was suggested for this effect, i.e. that is due to the increase in the attractive van der Waals interactions between SiR 3 and axial CH bonds in the axial conformation; the number of these stabilizing interactions is larger in Ph 3 SiO-cyclohexane compared to the Me 3 SiO derivative [184].Actually DLPNO-CCSD(T) calculations show that in the gas phase the axial conformation is more stable for Me 3 SiO while when this group is changed to Ph 3 SiO the axial and equatorial conformations become equal in energy which is the reversed from what is observed in solution [184].MMFF94 fails in five compounds, Dreiding in three compounds, UFF, MM3-00, and strikinly also MP2 in two compounds, while MM+, MMX, MM3-93, MM4-08, HF/ cc-pDVZ, HF/CBS, B3LYP only in one case.All force fields failed to calculate the axial conformer as the most stable one for Me 3 SiO group.Interestingly, all force fields, except Dreiding, as well as MP2 calculate fairly the increase in population of axial conformer when passing from Me 3 SiO to the bulkier Ph 3 SiO group.UFF have the largest deviations being > 1.5 kcal mol −1 in 3 cases and > 1 kcal mol −1 in 1 case.Interestingly all HF theories have a deviation > 1 kcal mol −1 for phenylcyclohexane.Compared to DLPNO-CCSD(T) reference energy values the experimental results disagree for the Me 3 SiO group.
In trans-1,2-dihalogen cyclohexanes, the di-equatorial is destabilized because of the repulsive interactions between the C-X dipoles compared to the di-axial conformation, while the di-axial conformation is destabilized because of the Pauli repulsion between axial C-X and axial C-H bonds which is particularly important in the trans-1,2-dichloro and trans-1,2-dibromo derivatives compared to the trans-1,2-difluoro because of the bigger size of bromine and chlorine over the not significant size of fluorine [185].However, in the diaxial conformation also attractive interactions exist between axial C-X dipoles and between axial C-X dipoles and axial C-H bond (Fig. 9).The electrostatic repulsion between the C-F dipoles is larger in the di-equatorial trans-1,2-difluoro cyclohexane and the equilibrium is more shifted to the di-axial conformation which has a 54% population as was shown by experimental measurements in the gas phase with electron diffraction [186] and QCISD/6-311+G(2df,p) calculations while in solution the diequatorial predominates for the trans-1,2-dihalogen cyclohexanes [187,188].Thus, the conformational preference is not the same as in trans-1,2-difluoroethane where the gauche conformer is preferred over the anti as previously discussed in haloalkanes [150,151].
In the trans-1,2-dichloro the experimental measurements in the gas phase [186] and the CCSD/6-311+G(2df,p) calculations [186] show that the diaxial has an 60% population as the Pauli repulsion between axial C-X and axial X-H bonds cannot destabilize the diaxial over the diequatorial conformation.
The diequatorial substitution is also observed in 1,2-dimethylcyclohexane [190], albeit less pronounced because of the steric repulsion between the gauche methyl groups.In 1,3-dimethylcyclohexane [191] the preference for the diequatorial conformer equilibrium returns to the common value since the two methyl groups are now apart enough to interact seriously.In the trans-1,2-bis(trimethylsilyl) cyclohexane the diaxial conformer is more stable than the diequatorial conformer because of the severe steric repulsion   in the last and also due to the LD attractive interaction of axial SiMe 3 groups [192].For the trans-1,2-dimethylcyclohexane and cis-1,3-dimethylcyclohexane all theories calculated the right global minimum, i.e. the diequatorial over diaxial conformation with UFF and HF theories showing deviation > 1.5 kcal mol −1 .Dreiding and MMFF94 performed with deviation > 3 kcal mol −1 and HF theories with a deviation > 1.5 kcal mol −1 .As regards the four trans-dihalogen cyclohexanes, Dreiding and UFF calculate the wrong global minimum in all four molecules examined with strong deviation for three out of the four cases, MM+, MM4-08, HF/CBS failed to predict the right global minimum in two cases while MM3-00, HF/cc-pVTZ, and B3LYP in just one case.

Oxygen-containing compounds
We performed calculations in important categories of oxygen-containing compounds, i.e., carboxylic acids, esters, aldehydes, ketones, alcohols, ethers, acetals (Table 4).In formic acid [193][194][195], carboxyl group adopts two distinct planar geometries in rare gas matrices at low temperature and prefers a Z-or syn-conformation in which the C=O and O-H or O-CH 3 bonds are in eclipsed orientation.In the formic acid the O-H group is oriented at ∼ 60° with respect to the C═O in the gas phase and in the E-or anti-conformation the O-H is antiparallel to the C═O.The Z(syn) is more stable by 3.90 kcal mol −1 in formic acid [196] according to microwave spectroscopy.The Z conformation of methyl formate has been found to be 4.8 kcal mol −1 more stable than the E form, and with methyl acetate the energy difference was found to increase to 8.5 ± 1 kcal mol −1 [197].Methyl formate has been also studied with IR and by DNMR and the free energy difference with the latter method has been determined to be 2.15 kcal mol −1 in an apolar solvent [198,199].Using femtosecond 2D-IR spectroscopy [200] it was demonstrated that formic acid adopts the two distinct, long-living conformations syn and anti in deuterated acetonitrile and heavy water solutions, The fractions of the anti-conformation and the syn-conformation are 20-30% and 80-70%, respectively, both in deuterated acetonitrile and in heavy water solutions.The distinct conformers of the carboxylic acid and their slow exchange at room temperature shows that these conformers are separated by high energy barriers.As a result, the presence of these conformers can have a large effect on the structure and dynamics of (bio) molecular systems.Similar conformational behaviour exist for methyl formate [201] or methyl acetate studied also in the gas phase [197,202].In solution formate species have been studied by DNMR [203].The considerably higher energy content of 8.5 kcal mol −1 [197] in E(anti) conformation in methyl acetate is due to proximity of methyl groups.In ethyl acetate in the E(anti) conformation around (O=)Csp 2 -OCH 2 CH 3 rotor the eclipsed and skew con formations depending if the methyl or C-H groups of ethyl groups are eclipsed as regards the C=O bond.
The experimental results for propanal [204] or 2-butanone [205] show that the global minimum corresponds to an eclipsed orientation of carbonyl bond and 3-or 4-methyl groups, respectively, to avoid steric repulsion between methyl groups in the skew conformation with relative conformational energies 0.95 or 2.0 kcal mol −1 .MP2/6-311G(d,p) calculations [206] suggest that for 2-butanone or propanal, the Pauli repulsive and the bond dipole interactions are primarily responsible for the conformational preference of the skew (gauche in Ref. [205]) by 1.81 kcal mol −1 or 1.22 kcal mol −1 , respectively, over the eclipsed in good agreement with experimental [205] and other computational results [207,208].
UFF, Dreiding force fields failed to predict the right global minimum in all cases and B3LYP6-31G(d,p) in one case.MM+ force field deviate in two cases by more than 3 kcal mol −1 while MMX force field deviate in one case by more than 3 kcal mol −1 and one case more than 1.5 kcal mol −1 while HF/cc-pVDZ theory deviate in one case by more than 1.5 kcal mol −1 and MP2 theory perform best.
Ethanol [232,233] and ethyl methyl ether [234] have been studied in the gas phase as mixture of anti and gauche conformations.For ethanol, calculations have been performed at the MP2/aug-cc-pVTZ, CCSD(T)/ aug-cc-pVTZ or aug-cc-pVQZ theories [235] along with various other theories; the anti conformation was calculated theory to be 0.13 kcal mol −1 more stable compared to the gauche conformation using CCSD(T)/aug-cc-pVQZ theory [235] in excellent agreement with the experimental value of 0.129 kcal mol −1 [232] which is close to our 0.16 kcal mol −1 using DLPNO-CCSD(T) calculations.For ethyl methyl ether also various levels of theory have been applied, e.g.MP2 [236,237] or CCSD, QCISD or CCSDT, QCISDT [237] with various basis sets which prodided energy values 1.38 or 1.36, 1.34 kcal mol −1 or 1.30, 1.30 kcal mol −1 using 6-31G(d) basis set [237] which are also very close to our calculated 1.30 kcal mol −1 with DLPNO-CCSD(T) theory.
According to HF/6-31+G(d) calculations the stabilization (by 0.3 kcal mol −1 ) [143,245] of the gauche conformation over anti can be attributed to the attractive electrostatical interaction, shown in the first line in the left-hand part of Fig. 12, since the C δ+ -Ο δ− dipole induces an excess positive charge at the hydrogen atoms of methyl C-H bonds resulting in attractive interactions.This attractive interaction counterbalances the steric repulsion between OH and CH 3 groups in the gauche conformer.Additionally, in the gauche conformation the hyperconjugative interaction σ(C2-H) → σ*(C 1 -O) with a second-order perturbation energy 4.42 kcal mol −1 contribute to the stabilization of gauche conformation (Fig. 10, first line, right hand part).
All theories predict correctly the correct global minimum for ethanol, methyl ethyl ether, 2-propanol, i.e. the anti conformation, is more stable than the gauche conformation.The MP2 and B3LYP theories have the smallest deviation from DLPNO-CCSD(T) following by HF/cc-pVTZ, HF/CBS, MM4-08, MMFF94.The biggest deviation was observed for UFF and then MM3-96, MM3-00 and the MM2 analogs MM+ and MMX.
As regards 1-propanol, all the force fields fail to predict the correct global minimum, i.e. the gauche conformation.Ηowever, using the MM4 force field the correct conformational preference in 1-propanol is calculated since MM4 includes terms to account for the induced dipoles [80] compared to MM3 [246].More accurate parameters relative to the C-C-O angle bending and the barrier of the C-O bond rotation were included in MM4 compared to MM3 [246].Additionally, HF/cc-pVDZ and MP2 calculate correcty the gauche conformation as global minimum while B3LYP and HF/cc-pVDZ predict gauche and anti conformations with equal energy.
According to experimental 3 J ( 1 H-1 H) values the same conformational preference of gauche relative to anti conformation, as regards the O-C1-C exo -C dihedral angle, is observed in the C-and O-glycosides (see second and third lines in Fig. 12).The calculations in Table 4 for the O-Et glycoside, as regards the comdormations that generate as regards torsion O-C 1 -C exo -C, show that all theories tested calculate the correct conformation, with UFF and MMFF94 deviating the most, followed by HF (Table 4).
Dimethoxymethane, which is the dimethyl acetal of formaldehyde, prefers the gauche conformation around central The steric repulsive gauche interaction is compensated by the two anomeric interactions.The anomeric interaction is defined as the increased stabilization resulting if a nonbonding electrons pair of heteroatom has an antiparallel orientation with respect to a polarized C-O bond.In dimethoxymethane, there are two such anomeric interactions, each including a non-bonding electrons pair in one oxygen with an antiparallel orientation with a polarized C-Ο bond.It has been explained that this preference is observed as the result of minimization of the repulsive interactions between C-Ο dipoles and electron pairs.The anomeric g − g − conformation is known to be the global energy minimum form of dimethoxymethane, according to a number of experiments employing electron diffraction [249,250], nuclear magnetic resonance [251], X-ray diffraction [252], infrared spectroscopy in argon matrices [253], or rotational spectroscopy [254,255] and ab initio or DFT calculations [253,256,257] with the more recent at CCSD(T)/aug-cc-pVDZ//B3LYP/aug-cc-pVTZ level [257].This conformational preference is due to the hyperconjugative interaction n(O) → σ*(C-Ο).In terms of resonance structures this lone pair electrons donation can be described with the structures C-Ο-C-Ο-C ↔ C-Ο + =C − Ο-C [244,258].Additional experiments and ab-initio calculations suggested that the preference for the anomeric g − g − conformation is due to attractive C-H⋯O interactions [129,182,[259][260][261], e.g. in g − g − conformation there are two gauche attractive interactions between oxygen lone pairs and C-H bond but in ga and aa conformations there is only one [259].The anomeric conformation is the global minimum also for acetaldehyde dimethylacetal according to Cambridge Crystallographic Database and of molecular mechanics calculations, and by NMR measurements of simple model acetals [262,263].Results based on coupling constants 1 J C-H , 3 J C-H showed that for acetals R 1 CH(OMe) 2 the common anomeric conformation is quickly destabilised as R 1 increases in size [263].The steric gauche interaction between groups R 1 and OR 2 forces group OR 2 to eclipse C-H bond (Fig. 12) through rotation by ~ 180° since in the new g,eclipsed conformation the two anomeric (hyperconjugative) interactions are maintained.Thus, while the formaldehyde dimethylacetal, i.e. the dimethoxyethane (R 1 = H, R 2 = R 3 = Me), adopts the standard anomeric conformation g − g − , the g,eclipsed conformation is considerably populated in acetaldehyde dimethylacetal (R 1 = R 2 = R 3 = Me) and is the global minimum for bigger alkyl groups, e.g. when R 1 = i-Pr, t-Bu.
For formaldehyde and acetaldehyde dimethyl acetal UFF and Dreiding force fields failed to calculate correctly the anomeric effect and both calculate as global minimum the aa conformation for the former compounds and UFF the g eclipsed for the latter.MM+, MMX force fields perform with significant deviations.Most accurate are B3LYP and MP2 theories with next the HF/cc-pVDZ theory then the MM4-08 force field following by the other theories.
The conformational energy of 2-methyltetrahydropyran is higher than that of methylcyclohexane because the smaller length of C-O bond forces the axial methyl to be in closer distance with the axial C-6 hydrogen.In 4-methyltetrahydropyran the conformational energy value is similar to that of methylcyclohexane since the smaller in length C-O bond does not affect the distance between axial Me and axial H in 1,3-positions.In 3-methyltetrahydropyran the destabilization of the axial conformer is smaller compared to the 4-methyl analogue, since the synaxial 1,3-Me⋯H is replaced by the synaxial Me⋯Lp [178] for which Pauli repulsion is less.All theories calculate these preferences.Between all theories tested MP2 shows values closer to the DLPNO-CCSD(T) calculations, followed by MM3-96, MM3-00, MM+, MMX.
Among the tested compounds having two functional groups (Table 4) ethanediol has two vicinal hydroxyl groups (Fig. 15).A large number of ab initio studies have been carried out in the gas-phase for ethanediol, ranging from HF calculations and partially optimized geometries to G2(MP2) calculations with fully optimized MP2/6-31+G* geometries [274][275][276].All of these investigations found that the relative energies of all 10 rotamers lie within 3.49 kcal mol −1 , with the g − g + a isomer being the lowest in energy (Fig. 15).These theoretical results are in good agreement with experimental results [277,278].According to οur reference DLPNO-CCSD(T) calculations the g − g + a conformation is the global minimum stabilized by the formation of a hydrogen bond between the hydroxyl groups.UFF and Dreiding are the only theories that calculate the aaa isomer instead of the g − g + a conformation while MMX calculate g − g + a and aaa conformations with equal energy.MM + is the next worst with deviation − 1.81 kcal mol −1 then HF/cc-pVTZ and HF/ CBS with deviation − 0.88 kcal mol −1 and − 1 kcal mol −1 , MM3-00 and HF/cc-pVTZ with deviations − 0.53 and − 0.49 kcal mol −1 , respectively.MM4-08, B3LYP and MP2 have the best performance with deviation − 0.3, + 0.07 and + 0.27 kcal mol −1 , respectively.
An interesting case arises when the substituent at 2-position of N-methylpiperidine is a bulky secondary or tertiary alkyl group, like 2-or 1-adamantyl [293].In both two molecules the chair conformation N-Me(ax), C-Ad(eq) is by far more stable than the other minima.The interaction between adamantyl and methyl is much more important than the axial preference over equatorial just for the N-methyl group that determines the conformational preferences for the 2-alkyl-N-methylpiperidines where alkyl is small.Furthermore, in 2-(1-adamantyl)-N-methylpiperidine and 2-(2-adamantyl)-N-methylpiperidine while the chair conformation N-Me(ax), C-Ad(eq) is the global minimum the second more stable conformation is different between the two molecules.In 2-(1-adamantyl)-N-methylpiperidine the N-Me(eq), C-Ad(eq) conformation is the second more stable conformation but in 2-(2-adamantyl)-N-methylpiperidine the diaxial conformation is the second more stable!In 2-SnBu 3 -N-Me-piperidine the conformations N-Me(eq), C-SnBu 3 (eq) and N-Me(eq), C-SnBu 3 (ax) are almost isoenergetic.The major reason appears to be a distortion of the conformation in which the C-2-Sn bond is synclinal to the nitrogen lone pair [294].
For the trimethyphosphate the global minimum corresponds to a ggg orientation of P=O with the three OMe groups by rotation around the P-O single bond [340][341][342], as shown by matrix isolation IR and DFT computations, which have been expanded to the higher analogues like tri-n-butyl phosphate [343].

Conjugated compounds
For 1,3-butadiene the energy difference between the gauche (cis) form and ground state trans form was determined in the gas phase using Raman [347] or microwave [348] or UV [349] spectroscopy to be 2.94 kcal mol −1 which agrees well with the ~ 3.01 kcal mol −1 from MP2/ aug-cc-pVTZ [350] or very accurate CCSD(T)(FC)/ CBS + CCSD(T)(CV)/cc-pwCVQZ + scalar relativistic effects correction + CCSDT(Q)(FC)/cc-pVDZ correction [351] (Table 7).The first microwave spectrum of "cis" butadiene unambiguously shows that it possesses a nonplanar gauche structure [348].Acrolein has been studied experimentally in the gas phase [352] and with ab initio calculations at the CCSD(T)/CBS level of theory [353] providing energies 2.20 kcal mol −1 and 2.06 kcal mol −1 , respectively.Methacrolein has been studied experimentally in the gas phase [354] and with ab initio calculations at the CCSD(T)/CBS [355] providing energies 3.02 kcal mol −1 and 3.47 kcal mol −1 , respectively.Methyl vinyl ketone has been studied experimentally in the gas phase [356] and with ab initio calculations at the CCSD(T)/CBS [355] providing energies 0.80 kcal mol −1 and 0.61 kcal mol −1 , respectively.Our results are in good agreement with values provided by CCSD(T)/CBS.
In the case of methyl vinyl ketone, UFF, Dreiding and HF/cc-pVDZ failed to calculate the trans conformation as the ground state.In all other cases the theories tested calculated the correct global minimum.MP2 is the most accurate following B3LYP for 1,3-butadiene, B3LYP and MMFF94 for acrolein, MMFF94 for methyl vinyl ketone.Among force fields, MMFF94 is the best performer while from Allinger's force fields it is MM4-08, which is better parameterized for conjugated compounds [357].
The transition states for ring and nitrogen inversion were investigated for some systems.The structure of the transition state for the ring inversion in cyclohexane [77,376], cyclohexene [141], and N-methylpiperidine [377] are shown in Fig. 23.For N-methylpiperidine, N-methylpyrrolidine and 3,3-dimethyl-N-pyrrolidine [378] nitrogen inversion transition has a planar nitrogen configuration.
MP2 best performed with calculated values close to the reference with exception of C-P bond rotation in dimethyl phosphine.It has been reported that semilocal DFT potentials including the DFT-HF hydrid methods such as B3LYP can perform well for rotational of conformational barriers [379].B3LYP performed fairly but in many cases other theories performed more accurately, e.g., MMX, MM+, MM3-96 [79][80][81], MM3-00 or HF/cc-pVDZ; even Dreiding and UFF performed well in a few cases.Among the force fields, the MM3-00 and MM4-08 deviate the least from the reference values.

Conclusions
In the present work we revisited previous works assessing the accuracy of force fields as regards the conformational preferences and energies of reference organic molecules.We calculated 158 conformational energies and barriers from 145 organic molecules, including hydrocarbons, haloalkanes, conjugated compounds, and oxygen-, nitrogen-, phosphorus-and sulphur-containing compounds.We reviewed in detail the conformational aspects of these model organic molecules providing the current understanding of the steric and electronic factors that determine the stability of low energy conformers and the literature including previous experimental observations and calculated findings.The suitable energies for comparison with CC-calculated conformational energies are energies measured in the gas phase with spestroscopic methods.Compared to previous work [48,50], we increased the number of tested molecules and the number of methods applied.We used the UFF and DREIDING force fields, the Allinger's force fields MM3-96, MM3-00, MM4-80, the MM2-91 clones MMX and MM+, the MMFF94 force field, ab initio theories, e.g.HF, the low-order post-HF MP2 method and the standard DFT model B3LYP.As reference conformational energy values to test the accuracy of these theories we performed basis-set extrapolated DLPNO-CCSD(T) calculations.This enabled us to have a common high-level reference for all compounds and all energetic quantities used in this work, compared to previous studies which often used inconsistent experimental values or low theory levels as reference values.
As shown in Fig. 24, the lowest mean error value was calculated for MP2 (0.35 kcal mol −1 ), followed by B3LYP (0.69 kcal mol −1 ) and the HF theories (0.81-1.0 kcal mol −1 ).As regards the force fields the lowest errors were observed for the Allinger's force fields MM3-00 (1.28 kcal mol −1 ), ΜΜ3-96 (1.40 kcal mol −1 ) and The MM4-08 force field's lower performance is of some interest but is likely consistent with the effort of Allinger and colleagues to build a set of parameters that might be more useful for vibrational data.The DREIDING (3.63 kcal mol −1 ) and UFF (3.77 kcal mol −1 ) force fields have the lowest performance.At this point, it is necessary to point out the dramatically different computational cost of the methods compared in this study.The present work considers three distinct categories of computational methods: force-field based molecular mechanics approaches, self-consistent-field QM methods (HF and DFT), and correlated wave-function methods (MP2 and DLPNO-CCSD(T)).Although the results and the errors are presented and discussed in common, it is important to keep in mind that the computational cost of each class of method differs by approximately an order of magnitude or more.Thus, molecular mechanics calculations for even the largest molecules in the present work are completed in time scale of seconds, HF and DFT calculations within several minutes, while the most expensive DLPNO-CCSD(T) calculations may require tens of minutes to a few hours to complete for the largest compounds.The heightened sensitivity of the ab initio quantum chemical methods and their non-linear scaling with respect to the basis set size is an additional consideration that does not apply to molecular mechanics methods.Moreover, in addition to increased time requirements, the correlated wave function methods have much steeper scaling memory/storage requirements with increasing size of the molecule or basis set.These considerations make it impossible to establish cost/error relationships for the whole variety of methods examined herein.Although the abovementioned order-of-magnitude cost comparison should always be considered, each class of method has its own scope, and often a combination of methods with different accuracy/cost profile can be beneficial in practice.Therefore, the choice of method in actual applications should consider not simply the average error and expected accuracy of any given method, but also the substantially different time and memory requirements, the total number of calculations required (e.g., a small set of compounds or a library of thousands of compounds), and the purpose of the study (e.g., rapid screening or benchmark-quality results).
Overall, the current study reviewed and commented on the current state of the art as regards the conformational energies of model organic molecules often present in druglike molecules and provides a new data set with DLPNO-CCSD(T) calculated values that can be used in future evaluation of approximate computationally efficient methodologies, or even in the training and parameterization of refined force fields.

6 1 )
)/aug-cc-pVTZ//MP2/aug-cc-pVTZ c ΔG, gas phase d ΔG, solution, DNMR e CASSCF/MS-CASPT2//B3LYP/6-31 + G** b,e Previous reported highest level theory f Refers to dihedral lp-N-C-C g The first chapter letter refers to C-N-C-C dihedral and the second letter refers to lp-N-C-C dihedral h Carbon C-2 is outside of the plane of the other atoms i,j,k See Figs.17, 18, and 19 Table Conformational energies of some compounds containing sulfur and phosphorus (kcal mol −, low temp.NMR d ΔH, solution, low temp.NMR e Rotation around C-S bond (dihedral C-C-S-C or C-C-S-H)

Fig. 7 A
Fig. 7 A Shows the stabilization of the gauche conformation by rotation about C2-C3 bond in 1-fluoropropane through attractive electrostatic interactions (left) and/or via hyperconjugative interaction (right).B Shows destabilisation of the anti conformer because

Fig
Fig.8Top: equilibrium between low energy conformers in methylcyclohexane and the C-H bonds which participated (in bold) in the most important hyperconjugative interactions.Bottom: C-H bond participates in two hyperconjugative interactions in axial methylcyclohexane and in four hyperconjugative interactions in equatorial conformer

Fig. 12
Fig. 12 Top: stabilization of gauche conformation of 1-propanol over anti conformation by rotation of C2-C3 bond is likely due to electrostatical interactions (left) and hyperconjugative interactions (right).Middle and bottom: the same conformational preferences are observed for the O-C1-C exo -C dihedral in C-and O-glycosides

Fig. 15
Fig. 15 Conformations of 1,2-ethanediol and description of the conformarional space of trehalose by rotation around dihedral angles φ and φ′

Fig. 16
Fig.16 Conformations of hexahydropyrimidine and of its 3-OH analogue

Fig. 22
Fig. 22 Possible structures for the transition state by rotation around CO-N bond

Table 1
Relative conformational energies of few hydrocarbon molecules (kcal mol −1 ) a ΔΗ, gas phase b ΔH, solution, Raman c ΔG, gas phase d ΔG, solution, low temp.NMR e Refers

Table 3
Relative conformational energies of some cyclohexane derivatives (kcal mol−1

Table 8 (
). UFF, Dreiding, MMFF94, MM3-96 calculate clearly C 2h as the global minimum while MM3-00 and MM4-08 predicted clearly the C 2 as the global minimum for tetramethylhexane.MM+, MMX, Hyperconjugative interactions in trans and gauche n-butane.There are four hyperconjugative σ C-Η → σ* C-H interactions in trans conformation but two σ C-H → σ* C-Η and two σ C-Η → σ* C-C in gauche conformation B3LYP and MP2 theories calculate correctly that C 2h and C 2 conformations are equal in energy.