Introduction

We have been interested in a series of topics that we will illustrate by a pair of references: (i) heterocycles, mainly azoles, and often related to their aromaticity [1, 2]; (ii) tautomerism, mainly of heterocyclic compounds [3, 4]; (iii) weak interactions [5, 6]; NMR properties, both chemical shifts [7, 8] and coupling constants [9, 10]; and (iv) crystallography [11, 12]. We will report in this review our results concerning conformational analysis and intramolecular dynamic phenomena, thus excluding intermolecular proton transfers [2, 13]. Obviously, some of these topics overlap.

We have followed Eliel and Wilen book plan [14] with small adaptations to include most of our contributions:

  1. 1.

    Stereochemistry of alkenes and aza analogs

    1. 1.1.

      CC double bonds

    2. 1.2.

      CN double bonds

    3. 1.3.

      NN double bonds

  2. 2.

    Conformation of acyclic molecules

    1. 2.1.

      CC and CN bonds

      1. 2.1.1.

        sp3/sp3

      2. 2.1.2.

        sp3/sp2

      3. 2.1.3.

        sp2/sp2

    2. 2.2.

      Amines, phosphines, and sulfur compounds

    3. 2.3.

      Atranes

  3. 3.

    Configuration and conformation of cyclic molecules

    1. 3.1.

      Six-membered rings

      1. 3.1.1.

        Carbohydrates

      2. 3.1.2.

        Heterorings

    2. 3.2.

      Other than six-membered rings

      1. 3.2.1.

        Three-membered rings

      2. 3.2.2.

        Four-membered rings

      3. 3.2.3.

        Five-membered rings

      4. 3.2.4.

        Rings larger than six-membered

    3. 3.3.

      Stereochemistry of other related systems

  4. 4.

    Chirality in molecules devoid of chiral centers

    1. 4.1.

      Biphenyls, atropisomerism

    2. 4.2.

      Molecular propellers

      1. 4.2.1.

        Methanes

      2. 4.2.2.

        Borates (scorpionates)

    3. 4.3.

      Helicenes

    4. 4.4.

      Truxenes

The selection of these topics is based on our interests in dynamic phenomena but also in those of our coauthors that often wanted to know if theoretical calculations could provide a solid foundation to their experimental observations. Among them are Curt Wentrup (Australia), Artur M. S. Silva (Portugal), Rosa M. Claramunt (Spain), Krzysztof Zborowski (Poland), Patricio F. Provasi (Argentina), Juan Jesús López (Spain), Wolfgang Holzer (Austria), Janet E. Del Bene (USA), José Luis Serrano (Spain), Manuel Yáñez (Spain), Christian Roussel (France), Emilio J. Cocinero (Spain), and others. In this review covering our work from 1989 to 2019, our results will be discussed in rapport with literature results, in general subsequent to our work.

Stereochemistry of alkenes and aza analogs

CC double bonds

The rotational barriers about CC single, double, and triple bonds were studied at the B3LYP/6-311++G(d,p) level [[15]]. Intuitively, one expects that the barrier around a C≡C bond should be higher than that around a C=C bond, but that barrier is not physically observable. The conclusion was the rotational barrier around a CC triple bond could be estimated to be 355 kJ mol−1 (leading to a double bond) or 645 kJ mol−l (leading to a single bond). These calculations were compared with the Cr–Cr quintuple bond [16].

Rotation around the π-C=C double bond, leaving the σ-(C=C) bond intact, is one of the most fundamental processes in chemistry. In the case of the central C=C bond in push–pull ethylenes 1 with R substituents like COR, CO2R, NO2, CN, .... The orthogonal state is not a biradical but a zwitterion [17]. The conformation and barriers of these compounds are determined by the presence (barriers of ~ 125 kJ mol−1) or absence (barriers of ~55 kJ mol−1) of intramolecular hydrogen bonds (IMHBs). For comparison purposes, the ethylene barrier amounts to 272 kJ mol−1. The energy profiles in function of the dihedral angle θ follow an empirical equation related to the Pitzer equation for ethane [14]. For further related studies on this topic, see [18].

figure a

A further example of rotation about central CC bonds was studied in cases related to Feringa’s molecular rotors [19].

figure b

The isomerization barriers of bithioxanthenes 2–4 were calculated at the B3LYP/6-311++G(d,p) level; in the case of bis(4H-thiopyran) (5), a planar molecule, complete active space self-consistent field calculations (CASSCF) were carried out followed by Multi-State CASPT2 calculations on the singlet and triplet transition states [20]. The barriers for the singlet 90° were 106.0 (B3LYP/6-311++G(d,p)) and 92.6 kJ mol−1 (CASPT2). Although there is a difference of 13.4 kJ mol−1, the conclusion was that the excited states do not affect the energy barrier at low temperatures. Therefore, the barriers for the bithioxanthenes 2–4 were calculated at the B3LYP/6-311++G(d,p) level; these compounds present two conformations, one boat-like or up/up (uu) and the other chair-like or up/down (ud), the ud minima being more stable than the uu ones by about 34 kJ mol−1. Their isomerization mechanisms, inversion and rotation, have been studied; the rotation mechanism is always preferred.

CN double bonds

The C=N group is presented in a variety of structures comprising imines, hydrazones, azines, and oximes. Although we have devoted a paper to hydrazones [21], our main interest was on azines. This is due that we studied experimentally these compounds several years ago [22,23,24] (for a recent review, see [25]). Two papers were devoted to computational studies of the structure of aldazines and ketazines, part 1 to simple compounds (6, 7) [26] and part 2 to halogen (8) and α,β-unsaturated derivatives like 9 [27], including cinnamaldazine (10) [24].

figure c

In the case of simple azines, B3LYP/6-311++G(d,p) calculations yield good geometries as well as E/Z ratios and IR spectra in agreement with experimental results; increasing the level of the calculations (MP2, QCISD) does not improve the geometries [26]. In the second paper, only B3LYP/6-311++G(d,p) calculations were carried out. For simple azines and for conjugated azines, EE and EEEE conformations, respectively, were preferred. Two mechanism of isomerization of CN double bonds, inversion and rotation, were studied for simple imines; in the case of azines, the TS has a “inversion-rotation” structure corresponding to barriers between 50 and 100 kJ mol−1. The potential surface of 8 (X = F) was calculated [27].

Two further papers explore the structure of aldazines bearing aromatic substituents (6, R = aryl or heteroaryl). In the first case, the existence of IMHBs in salicylaldazine (11) and Lumogen (12) stabilizes the EE dihydroxy isomers but in the case of 12, the hydroxy-oxo tautomer was observed in solution by NMR [28].

figure d

The structure of aldazines derived from formyl-1H-imidazoles, 13 and 14, some of them double-labeled with 15N, has been studied by NMR (solution and solid-state) and by X-ray crystallography [29]. Computational calculations at the B3LYP/6-311++G(d,p) level were used to determine the conformation about CC bonds and the configuration about C=N ones. Tautomerism and lone pair/lone pair repulsions determined the preferred structures.

figure e

Barriers about C=N bonds in aldimines, oximes, hydrazones, and azines have been computed at the B3LYP/6-311++G(d,p) and G3B3 levels [30]. Bond rotation and nitrogen inversion processes were compared (see Fig. 1) and in all cases the inversion process is preferred save in the case of azines where there is some rotation character. These results were cited in a paper about the photochemistry of imines [31].

Fig. 1
figure 1

Schematic representation of the rotation and inversion processes in imines

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NN double bonds

Somewhat related to azines 13 and 14, the structure of the azo derivative, 3(5),3′(5′)-azopyrazole (15) was studied theoretically [32] because the syn/anti isomerization of azobenzene is the main strategy for modifying the distances in molecular machines [32] and references therein]. Energy calculations, 13C and 15N chemical shifts, 1H–1H coupling constants, and electronic spectra reduced the 20 possible calculated structures to only one, the 3,3′-Z,Z-anti-azopyrazole.

figure f

Conformation of acyclic molecules

CC and CN bonds

sp3/sp3

Interested in the relationships between molecular and supramolecular chemistry (for instance, extending the Cahn-Ingold-Prelog rules to supramolecular structures [33, 34]), we studied the rotational barriers about covalent bonds and hydrogen bonds, in particular, comparing 1,1,1-trifluoroethane (Csp3–Csp3 bond) with the trifluoromethane/ammonia complex (HB) [35]. These studies were extended to the extremely rare 3JHH bonds through two heteroatoms (H–N–N–H and H–N–O–H) [36,37,38] that were compared to the 4hJHH in supramolecular complexes such as [H3N···H···NH3]+, [HOH···NH3] and [HOH···OH2] [39]. Finally, the Karplus relationship was calculated for H3C–(C≡C)n–CH3 systems in function of the number of CC triple bonds until n = 6; by extrapolation, it was estimated that for n = 25, the JHH will amount to 0.1 Hz [40]. These studies have been extended by other authors to 3JHSi spin–spin coupling constants [41] and to F–(C≡C)n–F JFF coupling constants until n = 11 [42].

The conformations of other systems where two heteroatoms form the pivotal bond have also been studied. The most reported are X–O–O–Y related to oxygen peroxide (X = Y = H): racemization process and optical rotatory power (X = H, Y = CCH, CH3, CF3, t-Bu, CN, F, Cl) [43]; chiral discrimination in the hydrogen bonded dimers of X–O–O–H (X = H, CH3, CF3, HCO) [44]; resolution of the optical rotatory power into atomic contributions (X = H, Y = CH3) [45, 46]; NMR of homo- and heterochiral complexes of X–O–O–Y with lithium cation (X, Y = H, CH3) [47, 48]. The central atoms of the X–O–O–Y molecule have been replaced by S and Se and their hydrogen and chalcogen bonds studied [49]. Related to hydrazine are the X2–P–P–Y2 molecules, H2P–PH2 and H2P–PHF; their energy profiles on the rotation about the PP bond have been computed at the MP2/aug′-cc-pvTZ [50].

Somewhat related to this topic was the exploration of the methane surface [51]. Seven stationary points of the methane hypersurface were first explored concerning geometries and energies to check previous data. On these geometries, absolute 1H and 13C NMR shieldings as well as 1JCH and 2JHH coupling constants were calculated. For planar methane, D4h, the relative energies were 604 (CCD/6-311++G(d,p)), 575 (CCSD(T)/aug-cc-pVTZ), and 558 kJ mol−1 (M05-2x/aug-cc-pVTZ). Jackowski and Makulski reported these results in their work towards a 13C scale for MAS NMR spectroscopy [52].

sp3/sp2

Concerning Csp2–Csp3 single bonds, the libration (restricted rotation) of phenyl groups in 2-benzyl-1H-benzimidazole (16) was determined by solid-state NMR (SSNMR) variable temperature experiments, although this phenomenon is not observed by X-ray crystallography. The barrier was calculated to be 59 kJ mol−1, a rather high value compared with the calculated value for the gas phase (10 kJ mol−1, B3LYP/6-311++G(d,p)) [53].

figure g

Our interest in azoles and benzazoles has resulted in several conformational papers related to Nsp2 (pyrrole-like)–Csp3 bonds conformations. Compound 16 presents a very unexplained behavior in the solid state, being an achiral compound in solution (due to CC bonds free rotation) and a chiral compound in the solid state (conglomerate) where always the same enantiomer is present [54, 55].

The double addition of azoles to glyoxal result in meso and d,l isomers like 17 in the case of 1H-benzotriazole [56]. Eight minimum energy conformations were calculated and compared with NMR results in solution in the case of pyrazole.

figure h

We have reported a series of studies of N-benzylazoles and N-benzylbenzazoles. The N-benzyl as well as other N-aryl and N-heteroaryl groups has clear conformational preferences and the CH2 protons eventual anisochrony is very useful for dynamic NMR studies (DNMR). Using this approach, the all-syn conformation of C3-symmetrical N-(Hetero)arylmethyl triindoles, for instance 18 in the case of benzyl, was ascertained [57].

figure i

A group similar to the benzyl one but with fluorine instead of protons is the difluoromethyl that can be used as a chiral probe [58]; this group was never previously been studied. In the case of camphor derivatives 19a and 19b, the 19F NMR spectra correspond to ABX systems of the diastereotopic fluorine atoms (Fig. 2).

Fig. 2
figure 2

The 19F NMR spectra of compounds 19a and 19b

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A systematic experimental and theoretical study of the whole family of N-benzylazoles (20–29) and N-benzazoles (30–37) was carried out and reported in two publications (compound 29 is unknown and compound 31 is very unstable, both were calculated) [59, 60]. Using a combination of X-ray crystallography, and NMR and DFT calculations, the structure and conformation, including rotational barriers of these compounds, were determined and the possibility to observe diastereotopic protons at very low temperatures discussed.

figure j

One of the few papers concerning Nsp3–Csp2(Ar) bonds reports the structure of palbociclib (38) [61] (for a review, see [62]).

figure k

sp2/sp2

Some curcuminoids such as 39 present conformational structures related to rotation about single bonds linking two Csp2 atoms [63], while the conformation of N-heterocyclic carbenes 40 (and related compounds) depends on the rotation about two Nsp2–Csp2(Ar) bonds [64]. In this paper was introduced the use of a tesseract (hypercube) of 16 vertices and 32 edges representation that was used by other authors [65].

figure l

Amines, phosphines, and sulfur compounds

This topic is related to the previous one when the single bond involves an sp3 nitrogen atom; for instance, carbenes 40 have also been studied for imidazolines (no CC double bond) where the N atoms are clearly pyramidal [64]. The nitrogen inversion and its relevance in conformational analysis have been studied in 13 simple amines, 41 to 53, based on calculated chemical shifts [66].

figure m

The position of the N lone pair (LP) of a series of 17 amines and the 13 represented above plus azatetrahedrane, azacubane, 1-azabicyclo[1.1.0]butane, and azirine has been located with the help of Bader’s methodology; besides, a geometrical model based on symmetry was examined [67]. The related P inversion of phosphines was also studied [68,69,70].

The conformation of sulfamide itself, H2N–SO2–NH2 [71], glibenclamide, a sulfamide derivative, R–HN–SO2–Ar [72], and rimonabant, a hydrazide [73], was also studied. These works have been cited several times [74,75,76].

Atranes

When the three legs of NR3 or PR3 are part of a cyclic system, the N inversion (amines) and the related P inversion (phosphines) results in the deformation of a basket (Fig. 3).

Fig. 3
figure 3

The structure of atranes (silatranes, phosphatranes, and related compounds)

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Two theoretical publications were devoted to the study of these compounds, the first one to the study of their geometrical, energetic, and NMR properties [77], and the second one to the modulation of in:out and out:out conformations in [X,X′,X″] phosphatranes by Lewis acids [78]. This field is continuing to be very active [79, 80].

Configuration and conformation of cyclic molecules

Six-membered rings

Carbohydrates

This topic, part of the thesis of Luis Miguel Azofra, was not a usual topic of our research. In the case of d-erythrose (54) and d-threose (55), we characterized at the B3LYP/6-311++G(d,p) level 174 and 170 minima for the open-chain structures of 54 and 55, respectively. G3B3 calculations indicate that the α-furanose configuration is the most stable for both d-erythrose and d-threose. The hydrogen bonds present in these molecules were classified as 1-2, 1-3, or 1-4, based on the number of C–C bonds [81] (Fig. 4). In a subsequent paper, the acid catalysis of the mutarotation mechanism in the two aldotetroses, 54 and 55, was studied at B3LYP/6-311++G(d,p) computational level in gas phase and in solution employing the PCM–water model. The acid catalysis has been studied taking into account the effect of (i) a classical Lewis acid as BH3, (ii) a classical hard-Pearson acid as Na+, (iii) two classical Brønsted acids such as H+ and H3O+, and (iv) the combined strategy using H3O+ and one bridge-H2O molecule as an assistant in the proton transfer [82].

Fig. 4
figure 4

Open-chain and α- and β-furanose configurations of d-erythrose (54) and d-threose (55)

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The preceding works were followed by a study of d-ribose (56) and 2-deoxy-d-ribose (57) carbohydrates. A theoretical DFT (B3LYP and M06-2X) and MP2 study has been undertaken considering the five possible configurations (open-chain, α-furanose, β-furanose, α-pyranose, and β-pyranose) of these two carbohydrates with a comparison of the solvent treatment using only a continuum solvation model (PCM) and the PCM plus one explicit water molecule. In addition, experimental vibrational studies using both nonchiroptical (IR-Raman) and chiroptical (VCD) techniques have been carried out. The theoretical and experimental results show that α- and β-pyranose forms are the dominant configurations for both compounds (Fig. 5) [83]. This was followed by a full exploration of the conformational landscape of 56 and 57 monosaccharides in the gas phase which has been performed using DFT methods (B3LYP and M06-2X) [84]. Up to 954 and 668 stable structures have been obtained for d-ribose and 2-deoxy-d-ribose. For other authors’ contribution to this field, see [85,86,87,88,89,90,91,92].

Fig. 5
figure 5

Left to right and top to bottom: open-chain, furanose, and pyranose configurations of d-ribose and 2-deoxy-d-ribose. The orientation of the hydroxyl group on the anomeric carbon atom (C1) in the cyclic forms gives the α- (exo face) and β- (endo face) diastereoisomers.

Full size image

Heterorings

Some six-membered rings containing elements other than C, N, and O have been studied. Pyrazaboles 58a–58f have four possible conformations, i.e., boat, chair, bent, and planar. The predominance of one of these conformations in the solid state (X-ray crystallography) depends on the R1 and R2 substituents. Theoretically (B3LYP/6-311+G** and MP2/6-31G* computational levels), the boat conformation is an energy minimum, with the planar and chair conformations as transition states in an energy diagram. In solution, the compounds are in a boat conformation (with a boat-to-boat dynamic equilibrium) irrespective of their crystal structures [93] (see also [94]).

figure n

BODIPY 493/503 59 is a pentamethyl derivative of the BODIPY skeleton [95]. Compound 59 is a bright, green fluorescent dye with similar excitation and emission to fluorescein with several uses in biochemistry, for staining lipids, membranes, and other lipophilic compounds. Its conformation was optimized at the B3LYP/6-311++G(d,p) computational level to afford a starting geometry for calculating NMR properties. Finally, three six-membered –N=S–S=N– heterocycles 60–62, all of them non-planar, have been studied theoretically [96].

Other than six-membered rings

Three-membered rings

The conformational landscapes of triaziridine and the water trimer supramolecular system were calculated (Fig. 6) [B3LYP/6-311+G(3df,2p)] and compared; also the monofluoro and monochloro derivatives were calculated [97]. Both systems behave similarly; more importantly, even though in the former the energy required to induce conformational rearrangements is much smaller than that needed for typical molecular systems, there is a fairly good linear relationship between both magnitudes. Theoretical studies on triaziridine have been pursued at higher levels [98].

Fig. 6
figure 6

Triaziridine (62), minima (uud and uuu), and transition states (udp, uup, ppp)

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The rotation/inversion barriers of formamide (63), N-formyl aziridine (64), and N-formyl azirine (65) were calculated [MP2/6-311++G(d,p)]. The results provide a quantitative picture of the influence of ring strain on the properties of amides, with special emphasis on the effects associated with nitrogen pyramidalization [99]. Related to this problem, the chemistry of bridged lactams has been reviewed [100] and the conformational aspects of cyclic peptides derived from aziridine-containing amino acids have been studied [101].

figure o

Diaziridine 66 has two conformations a (1R,2S) and b (1S,2S); the optical rotatory power of 66b has been calculated (B3LYP/6-311++G(2d,2p) and MP2/6-311++G(d,p)) [102] as well as the barrier that separated 66a of 66b by a TS = 149.6 kJ mol−1, conformer b being 25.1 kJ mol−1 more stable than a. Five different topologies of the cluster that present two HB interactions per monomer have been considered (Fig. 7 shows the optimized ones for the hexamer). The preferred conformation 66b is consistent with that of the 1,2,3-trisubstitued diaziridines [103].

Fig. 7
figure 7

Optimized geometries of the five configurations considered for the clusters. The ones shown correspond to the hexamer of 66

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The results show that the clusters with alternated chiral molecules are the preferred ones and that the proton transfer proceeds with low energetic barriers in the protonated systems. Proton transfer along the chains can invert stepwise the chirality of the molecules producing what we have called racemization waves [104].

Four-membered rings

Similar studies than those reported previously on triaziridines and water trimers (section “Three-membered rings”) were carried out for tetrazetidine (67) (Fig. 8), water tetramers [97], and their monofluoro and monomethyl derivatives.

Fig. 8
figure 8

Tetrazetidine 67 stationary points

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Five-membered rings

Molecular dynamics studies of 3-oxohexahydroindolizino[8,7-b]indole derivatives (68) were carried out in order to study their potential as novel β-turn mimetics [105, 106]. While 68a and 68b are able to adopt type II′ β-turn conformations, in the case of 69a–69c, they show extended conformations of non-standard folding.

figure p

Another publication belonging to this section concerns the study of 1,3,5-triphenyl-Δ2-pyrazoline (or 4,5-dihydro-1H-pyrazole, 70). Figure 9 shows that the X-ray structure (this compound crystallizes as a conglomerate) and the optimized one (B3LYP/TZVP) are very similar [107]. Compound 70 has been proposed as organocatalyst via iminium activation [108].

Fig. 9
figure 9

Structures of compound 70

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Rings larger than six-membered

The minimum energy conformation of the seven-membered ring B of colchicine (71), a compound that continued to be much studied [109], has been calculated at the B3LYP/6-311++G(d,p) level showing agreement with the X-ray structure and providing torsion angles consistent with vicinal 1H–1H spin–spin coupling constants (Karplus equation) [110].

figure q

Seven-membered rings containing two nitrogen atoms are called diazepines, the most common are the benzo[e][1,4]benzodiazepines, like chlordiazepoxide, diazepam, lorazepam, and many others. In our case, we have studied the benzo[b][1,4]benzodiazepines. In benzodiazepines 72 [111], 73 [111, 112], and 74 [113] and in diazepines 75a and b [114] and 76 [115], tautomerism and ring inversion (none of these seven-membered rings are planar) have been computed at different levels.

A conformational analysis of of 2,3,6,7-tetrahydroazepines was carried out with MM3 and CHARMm molecular mechanics, and AM1 semi-empirical, as well as Hartree–Fock and local density functional (LDF) ab initio methods [116]. Similar geometrical characteristics were found with all methods although there are important differences in the rank order of the relative energies. The importance of the solvation of these compounds in the affinity for the dopamine D1 receptor was also studied [117].

figure r

The ability of a series of six- 77 and seven-membered ring structures 78 to mimic the properties of an ideal γ-turn has been studied by means of three molecular similarity indices [118]. In general, the compounds with seven-membered rings show good overall molecular similarity indices when compared to an ideal inverse γ-turn while six-membered rings provide good overall similarities with classic and inverse γ-turns.

figure s

5,6,11,12-Tetrahydrodibenzo[a,e]-cyclooctene or -[8]annulene (79) has three conformations: chair, twist-boat, and twist [119]; using GIAO/B3LYP/6-31G* calculations together with 1H and 13C DNMR experiments, the long-standing problem of its conformation has been solved and the interconversion barriers determined. The two most stable conformations, chair and twist-boat, have similar energies and interconvert through processes having activation energies about 42 kJ mol−1 slightly dependent on the solvent. The different conformations have different sizes and that has been used to prepare a thermal contracting polymer [120, 121].

The tetraindoles 80 show ring inversion when R = H, inversion inhibited by N-substitution [122]. These multi-indole structures are part of the PAHs (polycyclic aromatic hydrocarbons) [123]. Conformational analysis of eight-membered rings containing four 81 or two N atoms, 82 and 83, was carried out (B3LYP/6-311++G(d,p)) and compared with 75 [124]. The energy profiles are similar to those determined experimentally.

figure t

A few rings larger than 8-membered have been studied. The 12-membered ring derivative 84 has played an important role in the history of the conformational analysis. B3LYP/6-311++G(d,p) calculations allow to determine four minima and four TSs; the calculated ring inversion barrier, 42.6 kJ mol−1, is in excellent agreement with the measured barrier 41.4 ± 0.8 kJ mol−1 [125]. Sixteen-membered ring 85 and 18-membered rings 86 and 87 were studied (DNMR and crystallography) and their ring barriers measured and calculated with the AM1 Hamiltonian [126]. Compounds related to 86 and 87 but with o-phenyl groups instead of p-phenyl ones have been reported [127, 128].

Chirality in molecules devoid of chiral centers

Biphenyls, atropisomerism

We have studied several of the ten possible situations of Fig. 10 in a paper covering many situations [129] and a review reporting the literature until 2011 [130] (see also [131]). The nomenclature used in Fig. 10 to define each central bond includes the atoms involved in the bond and, between parentheses, the number of atoms in each ring.

Fig. 10
figure 10

6-6, 5-6, and 5-5 atropisomers. The C–Hs can be replaced by Ns and the N–Hs by Os or Ss

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Two papers were devoted to C(6)–C(6) systems, the first one to 1,1′-binaphthalenes bearing at positions 2,2′ CH(OH)CF3 substituents (three stereoisomers, R,S,S; R,S,R; and R,R,R and their enantiomers, combining axial and central chirality). B3LYP/6-31G(d) calculations were used to determine the mechanism of formation of the diols by reduction of the double COCF3 derivatives [132]. The static and dynamic properties of BINOL (1,1′-bi-2-naphthol) and its conjugated acids and bases were studied using mass spectrometry, microwave rotational spectroscopy, NMR in superacid media, and MP2/6-311++G(d,p) calculations [133]. Our calculated BINOL barrier was used to discard a hypothetical mechanism [134].

figure u

Lamotrigine (88), the treatment of choice for epilepsy and bipolar disorder, is a racemate of rapid interconverting enantiomers [135]; this was established by 1H NMR at 600 MHz in the presence of ABTE (Virgili′s [(S,S)-α,α′–bis(trifluoromethyl)-9,10-anthracenedimethanol] chiral auxiliary) and B3LYP/6-311++G(d,p) calculations. The predominance of the diamino structure proved relevant for the study of the hydrolysis of Lamotrigine-N2-glucuronide in wastewater [136].

We have calculated the barrier to the rotation about the C6–C1′ bond through a near-planar structure that corresponds to a racemization process. There are two possibilities that correspond to diastereomers: that the Cl atom at position 20 is on the side of N1 (Fig. 11a) and that it is on the side of the 5-NH2 (Fig. 11b). The barriers are almost the same, 61.9 and 62.1 kJ mol−1, respectively, therefore in the range that can be measured by dynamic NMR (DNMR) in the presence of a chiral additive. Using this technique, we have determined in CDCl3 a value of ΔG‡TC = 62.4 kJ mol−1, the agreement being excellent.

Fig. 11
figure 11

a, b Structures of the two transition states of 88

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Compound J147 (89) is one of the most promising compounds to treat Alzheimer’s disease [12] and it was used to design a series of 1,2,3-triazoles 90 whose conformational preferences were calculated [137].

Ten structures corresponding to two heterocycles C–C bonded, C(5)–C(5) 91, were studied theoretically [MP2/6-311++G(d,p)] to see if chalcogen–chalcogen interactions (conformation 91b) determined the preferred conformation. It appears that they are important although dipole–dipole effects also contribute [138].

figure v

A review was published covering the conformational aspects of meso-tetraarylporphyrines (92) [139], not only with aryl but also with heteroaryl groups like pyridines and C-substituted pyrazoles yielding four atropisomers (α,α,β,β; α,α,α,β; α,β,α,β; and α,α,α,α) that are conformationally stable at room temperature although they interconvert in solution at high temperatures [140]. The conformation of 2,2′-, 3,3′-, and 4,4′-bipyridines as well as their monoprotonated and diprotonated forms was studied with particular emphasis on their NMR chemical shifts and conformational barriers [141] (see also [142]). The rates of enantiomerization of chiral 2,2′-bipyridines with restricted rotation 93a and b were calculated [B3LYP/6-311++G(d,p)] [143].

figure w

The most studied series in this section belong to Het(N)–Ar(C) compounds; only one was a six-membered heterocycle (a pyrimidin-2-thione) [144] cited by Roussel in [145], all the others were N-arylazoles N(5)–C(6). The addition of imidazoles and benzimidazoles to quinones affords mono (94, 95) and disubstituted derivatives (96–99) and related compounds; meso and d,l compounds are formed [146].

figure x

The reaction of hexafluorobenzene with sodium benzimidazolates affords hexakis (benzimidazol-1′-yl) benzenes 100 as the only products. These compounds, called propellenes [147] by analogy with hexaphenylbenzene [148,149,150] (Fig. 12), present eight possible conformations depending on the up/down position of the benzimidazolyl residue; three of them have been characterized c, f, and g [151].

Fig. 12
figure 12

A six-blade propeller

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figure y

All of the above systems have been studied, each one presenting specific situations: 101 is an example of orthogonal interaction between the N atom of the nitro group and the N2 atom of the azole (besides pyrazoles, triazoles, tetrazoles, indazoles, and benzotriazoles have been studied) [152, 153]. N-Arylindazoles, both 1H102 and 2H103, were examined starting from X-ray structures reported in the Cambridge Structural Database [154]; the search includes aza-derivatives (N atoms instead of CH groups in the six-membered ring) [155]. The 1,2,3-triazole 90 present also a conformation about an N–C bond [137]. A series of 2-, 3-, 4-, 2,4- (like 101), 2,6-, and 2,4,6-nitro-pyrazoles and indazoles were examined with special emphasis on the X-ray structures [156]. The chemical shifts of 2-phenyl-2H-benzotriazole (104) in the solid state were calculated with the GIPAW and GIAO-PCM (DMSO) and both methods compared; only for 15N SSNMR chemical shifts GIPAW proved better [157]. Finally, the atropisomerism of 105 was studied showing the influence of hydrogen bonding on the racemization rates that were determined by treatment of the plateau-shape chromatogram during chromatography on chiral support [158] (for similar results, see [159]). An example of the much less frequent case where instead of an aryl ring there is a six-membered heterocycle concerns 2,4,6-tris(1H–pyrazol-1-yl)-1,3,5-triazine, a compound presenting polymorphism, pseudopolymorphism, and co-crystals [160].

Molecular propellers

In this section, we will discuss three-blade propellers (Fig. 13) much more common than the previous six-blade ones [14, 161, 162]. Usually, they are tris-azolyl derivatives related to trisarylmethanes [163,164,165,166]; the tetrakisazolyl derivatives, related to tetrakisarylmethanes [167,168,169], are less common.

Fig. 13
figure 13

A three-blade propeller

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Methanes

In 1994, Breitmaier et al. published a paper where Grignard alkylation of aldehydes occurs with very large enantiomeric excess due to the effect of the static magnetic field of an NMR spectrometer [170]. Since this result seems improbable, we carry out a reaction that we feel will be much more sensitive to chiral induction [164].

figure z

The reaction of 2-methyl-1H-benzimidazole (106) with chloroform in the conditions of PTC affords the tris(2-methylbenzimidazol-1-yl)methane (107) as a racemic that by 1H NMR in the presence of Pirkle’s alcohol was proved to be a 50:50 mixture of both enantiomers. The enantiomers were separated by chromatography on microcrystalline cellulose triacetate [171]; they are stable with a racemization barrier of 120 kJ mol−1 at 343 K. When the reaction was carried out inside of an NMR spectrometer (7.05 T), we obtain again a 50:50 mixture, i.e., no enantioselectivity. We sent our paper to Angewandte Chemie where it was rejected based on “the journal does not published negative results.” We sent it immediately to Heterocyclic Communications [172] where it was received on June 30, 1994. Although Heterocyclic Communications continue to be published, the first issue where our paper appeared was not included in the Web of Science. Shortly afterwards, Breitmaier’s paper was withdrawn [173,174,175,176]. A series of papers were published on this topic in subsequent years, by ours [177, 178] and by other groups [179, 180]. Among the derivatives studied, experimentally (X-ray) and theoretically, was 108 [181]. A comprehensive review on poly(pyrazol-1-yl)methanes up to 2017 was published [182].

The reaction of 3(5)-methyl-1H-pyrazole with CHCl3 affords four tris(pyrazol-1-yl)methanes 109 in proportions that obey a (a + b)3 model (a = 3-methyl; b = 5-methyl) with great accuracy. Isomer 109-555 was prepared from tris pyrazol-1-yl)methane (110) [183].

figure aa

When the reaction was carried out with CCl4, the proportion of tetrakis derivatives does not follow an (a + b)4 polynomial expansion, probably due to steric effects in the last step, going from Mepz3Cl to Mepz4C [184].

Borates (scorpionates)

Following Trofimenko’s seminal work [185, 186] and sometimes collaborating with him [7, 187], we have devoted some publications to the study of scorpionates mainly using NMR data and conformational analyses: fluxional behavior of 111 and 112 [188], multinuclear NMR and space groups 113–115 [189], crystallography and SSNMR of the thallium salts of scorpionates 116–117 with a discussion of the 11 motifs observed for tetrakis scorpionate derivatives [190], and the structure of four thallium tris(1H-pyrazol-1-yl)hydroborates 118a–118d [191].

figure ab

Helicenes

Helicenes are fascinating molecules [14, 162, 192] (for two recent publications related to our work, see [193, 194]) that we have studied three times. Hexaaza[5]helicenes 119 racemize (M/P helix conversion) by a classical mechanism and not by a ring opening 120/ring closing one [195]. The kinetic parameters are ΔH = 17.6 kJ mol−1 and ΔS = 53.8 J mol−1 K−1.

figure ac

The racemization barriers of the 1-aza series 121–124 were calculated at the M05-2x/6-31G(d) level. The [4]helicene 121 is planar; for the three others, the barriers are 57.0 (122), 141.1 (123), and 181.6 kJ mol−1 (124); the experimental value of 123 is 134.7 kJ mol−1 [196]. Dimers linked by alkaline cations (Li+, Na+, K+) were calculated to study the chiral distinction between homochiral and heterochiral dimers.

figure ad

We have in process the calculation of the barriers of helicenes reported in Table 1.

Table 1 Racemization barriers of helicenes in kJ mol-1
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figure ae
figure af

In the process of preparing [7]circulene (130), Yamamoto et al. went through 1,6-dehydro [6]helicenes 129a–129d [201] determining the X-ray structure of 129c and the racemization barriers of the four compounds (Table 1). Contrary to the barriers of helicenes that the calculations (different levels) reproduce with accuracy (Table 1), those of 1,6-dehydro[5]helicenes 129 (except 129a, the less hindered) are grossly overestimated (increasing the level of the calculations, up to MP2 and CCSD does not solve the problem). A possible explanation is that the TS involves an open-shell diradicaloid structure.

The results of Table 1 shows that with a moderate level of calculation (M05-2x/6-31G(d)), the experimental results are well reproduced: Removing 129b to 129d, we obtain, Exp. = (1.012 ± 0.008) B3LYP/6-31G(d), n = 12, R2 = 0.999. The deviation of compounds 129b to 129d can be calculated adding a dummy to be 78 ± 6 kJ mol−1.

Truxenes

The only representative of this section are truxenes 131, an interesting family of concave shape hydrocarbons [202] that have been much studied by their proximity to fullerenes [123, 203, 204]. Truxene derivatives may be considered molecules with three helicene regions.

figure ag

Conclusion

The great diversity of structures, 1–131, added to the different techniques (solution and solid-state NMR, chiral reagents for enantiorecognition, dynamic NMR (DNMR) in the presence of a chiral additive, X-ray crystallography, circular dichroism, vibrational circular dichroism, chiral chromatography, kinetics, equilibria, etc.) were tied together by theoretical calculations. They provided a rationale for the measured values and, at the same time, allow predicting unmeasured properties.

The future of chemistry is tied to the progress in computational chemistry. The number of possible structures, being much larger that the elemental particles in the Universe [205] and the biased distribution of known molecules in the multidimensional space of chemical structures [206, 207], obliges to develop theoretical methods of prediction of properties (physical –materials– and biological –drugs–) to attain these compounds that will protect the humanity in the future.

This review reports many examples of the success of theoretical chemistry in explaining known properties; a step further is necessary to predict properties of unknown compounds. This should be the main conclusion of this review.