1 Introduction

This issue of Structure and Bonding is dedicated to the 100th anniversary of the landmark publication “The Atom and the Molecule” by Gilbert Lewis where he introduced the model of electron-pair bonding into chemistry [1]. It was a bold suggestion that was born from the attempt to explain the wealth of chemical information which was available at that time with a model for molecular structures that was shaped by classical physics. The boldness of the suggestion lies in the fact that Lewis knew about the inability of classical physics to correctly describe chemical bonding in terms of electrostatic attraction as underpinning forces for the model of electron-pair bonding. He speculated about a possible deviation from Coulomb’s law when he wrote that “Electric forces between particles which are very close together do not obey the simple law of inverse squares which holds at greater distances” [2]. Eleven years later, this foresighted postulate was proven by Heitler and London to be correct [3], although Lewis could not foresee the paradigm change which was to come by the quantum theory that was suggested by Schrödinger and Heisenberg.

The work by Heitler and London published in 1927 was the first study which correctly described chemical bonding in terms of modern quantum theory that was introduced by Heisenberg and Schrödinger two years earlier. It was the birth of quantum chemistry which received its first textbook Einführung in die Quantenchemie [Introduction to Quantum Chemistry (in German)] in 1937 by Hans Hellmann [4]. (The book has recently been republished with biographical notes of the son Hans Hellmann Jr. by Andrae [5].) The quantum chemical explanation of chemical bonding was a revolutionary view of the nature of the interatomic interactions. In order to understand the strong attraction between two neutral atoms which form a chemical bond, electrons have to be considered as waves rather than particles, and the covalent bond must be understood as a (translated from German by the authors) “quantum mechanical vibrational phenomena” [3] which comes from the mixing of the wave functions. Thus, chemical bonding is not due to the formation of an electron pair as suggested by Lewis. The tendency of molecules to have electron pairs is rather due to the Pauli principle which allows a maximum of two electrons in the same region of space. A chemical bond can be formed with only one electron such as in H2 +.

It is illuminating to consider the position of Gilbert Lewis, who had an excellent knowledge of modern physics, to quantum theory. In his pioneering publication in 1916, he shortly discusses new models which were suggested to explain the apparent violation of the common laws of electricity in the atomic region. He writes that “The most interesting and suggestive of these theories is the one proposed by Bohr and based upon Planck’s quantum theory” [6]. But then he dismisses Bohr’s atomic model because it “…is not only inconsistent with the accepted laws of electromagnetics but, I may add, is logically objectionable, for that state of motion which produces no physical effect whatsoever may better be called a state of rest”. In spite of his reservations against quantum theory which was only known in the suggestions of Planck and Bohr and prior to the work of Schrödinger and Heisenberg, Lewis devoted more space and thoughts to it in his seminal book “Valence and The Structure of Atoms and Molecules” which was published in 1923 [7]. It was the first publication by Lewis fully devoted to chemical bonding after his now celebrated 1916 study. In the meantime Irving Langmuir had published a series of papers where he further developed the theory of electron-pair bonding [8–10]. This led to the situation that the electron-pair model was widely attributed to Langmuir rather than to Lewis, and still today the name Langmuir–Lewis model is sometimes used. In fact, the term “covalency” and the octet rule are due to Langmuir and not to Lewis [11, 12].

Lewis devoted a section in his book to the topic entitled “The Quantum Theory” where he acknowledges that the discrete nature of matter introduces a fundamentally new understanding of atomic structure and light. He frequently refers to Einstein’s thoughts about quantum theory, and he mentions a remark which Einstein made to Lewis that “…the quantum theory was not really a new theory, but merely a recognition of the falsity of previous theories” [13]. Lewis goes on and points out that quantum theory (which was prior to the Schrödinger/Heisenberg development of the theory) was not really capable to furnish an understanding of interatomic interactions. But then he concludes “Quantum theory has been criticized for furnishing no adequate mechanism, but presumably the root of our present problem lies deeper than this, and it is hardly likely that any mechanism based on our existing modes of thought will suffice for the explanation of the many new phenomena which the study of the atom is disclosing” [13]. Lewis shared the deep-rooted dislike of quantum theory which he refers to as “the entering wedge of scientific bolshevism” [14] with Einstein but sensed at the same time that something new was coming up to explain chemical bonding and other molecular properties. He speculated that some of the abstractions which he used in his book may in the future have to be abandoned while others “…may have to be modified, and my chief purpose in writing the present section is not so much to predict just how these modifications are to occur as it is to emphasize the necessity of maintaining an opening of mind; so that, when the solution of these problems, which now seem so baffling, is ultimately offered, its acceptance will not be retarded by the conventions and the inadequate mental abstractions of the past” [15]. These are the closing remarks in the book which are a challenge and a legacy of Gilbert Lewis to the following generations.

After reading the original works by Gilbert Lewis about chemical bonding and in particular his 1916 paper [1] and the 1923 book [7], it may be recognized that his legacy is not just the suggestion that the chemical bond shall be identified with an electron pair. It is also the appeal to future generations not to hang on to old conventions and traditional models but to continue in developing new models and to be open to new insights which become available when future methods provide more information about chemical bonding.

One of the models which Lewis introduced in his book shall be the topic of this article which is written in the spirit that is expressed in the above-cited closing statement of the author. It is the general definition for acids and bases which now carries his name Lewis acids and Lewis bases. In the chapter “Remnants of the Electrochemical Theory”, he devotes a section to “The Definition of Acids and Bases” where he introduces his model with the statement: “A basis substance is one which has a lone pair of electrons which may be used to complete the stable group of another atoms, and....an acid substance is one which can employ a lone pair from another molecule in completing the stable group of one of its own atoms” [16]. With other words, Lewis distinguishes between two types of electron-pair bonds, i.e. the shared-electron bond A–B and the dative bond A → B (Scheme 1). Fifteen years later in 1938, when quantum chemistry was already blossoming and Linus Pauling was on his way to provide a quantum theoretical underpinning of the electron-pair model in his book “The Nature of the Chemical Bond” that was eventually published in 1939 [17, 18], Lewis elaborated on the topic in his later work entitled “Acids and Bases” [19]. He mentions the model of resonance that was suggested by Pauling for describing the electronic structure which is particularly important for unsaturated species such as Lewis acids. But he points out that the development of quantum chemistry does not really alter the essential definition of acids and bases in terms of lone electron-pair donation.

Scheme 1
scheme 1

Schematic representation of (a) an electron-sharing electron-pair bond and (b) a dative (donor–acceptor) electron-pair bond. Electron lone pairs are represented in this and the other figures by a bar

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The relevance of distinguishing between electron-sharing bonds A–B and dative bonds A → B for understanding molecular structures of main-group compounds has been stressed by Haaland in a review article in 1989 [20]. The model of donor–acceptor interactions is well established in transition metal chemistry since Dewar suggested in 1951 that the structure of Zeise’s salt can be understood in terms of σ donation and π backdonation [21]. The donor–acceptor model was generalized to other transition metal complexes in a series of papers by Chatt together with Duncanson and other co-workers [22, 23] (the contributions of Chatt to the present understanding of chemical bonding in transition metal chemistry have been highlighted in [23]), and therefore, it is now known as Dewar–Chatt–Duncanson (DCD) model (for a discussion of the DCD bonding model in the light of quantum chemical calculations, see [24]). The DCD model which uses dative bonds is the predominant description of chemical bonding in transition metal chemistry [25]. Prior to the works by Dewar and Chatt, similar suggestions were made by Hieber [26] and later by Orgel [27] who pointed towards a synergic bonding in transition metal complexes.

The review by Haaland [20] is a good starting point for the present manuscript. It summarizes the knowledge of classical Lewis acid/base complexes mainly of group 13/15 adduct which were known at that time. It is shown that the discrimination between electron-sharing bonds and dative bonds is very useful for understanding molecular structures and stabilities. But during the last decade, it was realized that there are molecules of main-group atoms which were previously described with electron-sharing bonds that may better become discussed with dative bonds [28–39]. This led to the prediction of new adducts with unusual bonds which could become synthesized and structurally characterized by X-ray analysis [40–44]. Numerous experimental studies particularly in the area of low-valent main-group atoms reported about exotic molecules whose structures were explained with dative bonds (representative examples: [45–56]). The increasing number of molecules that were sketched with dative bonds was not undisputed [57], but it was shown that many features and properties of the newly synthesized compounds are easily understood with the model of dative bonds [58] (for a reply, see [59]). We believe that it is an appropriate contribution to the special issue of Structure and Bonding celebrating the 100th anniversary of Gilbert Lewis’ epochal paper to show that his model of dative bonding is still a powerful tool for finding new molecules and to explain unusual structures.

2 Carbon Dioxide CO2 and Carbon Suboxide C3O2

Carbon dioxide CO2 is a well-known compound while carbon suboxide C3O2 is a more exotic species which has received less attention in the literature, although it has been synthesized already in 1906 [60]. The molecules are usually sketched with electron-sharing double bonds O=C=O and O=C=C=C=O which let one expect linear geometries. CO2 has a linear equilibrium structure while gas-phase studies revealed in 1986 that carbon suboxide has a bent geometry with a bending angle of 156° at the central carbon atom [61, 62]. The bending potential was found to be very flat, and the molecule adopts a linear structure in the solid state [63]. The linear geometry of CO2 and the bent structure of C3O2 can easily be understood when the two types of electron-pair bonding are considered. Figure 1 schematically displays the possible bonding situations in the molecules in terms of electron-sharing bonds and dative bonds. It also shows the electronic reference states of the relevant bonding fragments and the excitation energy which is required for promotion from the electronic ground state. (The experimental values of the atoms have been taken from [64]. The excitation energies for diatomic molecules were taken from [65].) Note that the carbon atom in the excited 1D state would be a σ donor and π acceptor in CO2 but a σ acceptor and π donor in C3O2.

Fig. 1
figure 1

Sketch of electron-pair bonds in CO2 and C3O2 with electron-sharing bonds and dative bonds. Below each structure are the electronic reference states of the atoms and CO for the respective bonding interactions. At the bottom are the excitation energies from the electronic ground state to the excited state which were taken from [61]

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The electron-sharing double bonds of CO2 require oxygen atoms in the 3P state which is the electronic ground state, while the carbon atom requires the excited 5S state which is 96.4 kcal/mol higher than the ground state. Even more promotion energy is necessary to prepare oxygen and carbon for possible dative bonds. The 3P → 1D excitation energies for two oxygen atoms entail 2 × 45.4 = 90.8 kcal/mol and the 3P → 1D excitation of carbon necessitates another 29.1 kcal/mol. Thus, a total of 119.9 kcal/mol is required to promote oxygen and carbon for possible donor–acceptor interactions O⇆C⇄O, while only 96.4 kcal/mol is necessary for electron-sharing bonds O=C=O. Since electron-sharing bonds are stronger than dative bonds between the same atoms, it is clear that CO2 should be written as O=C=O.

The situation looks different for carbon suboxide. Here, the promotion energy for dative bonds is only 29.1 kcal/mol which comes from the 3P → 1D excitation of carbon atom. Possible electron-sharing bonds O=C=C=C=O request excitation of the CO groups from the X1Σ+ ground state to the a3Π excited state which amounts to 2 × 139.3 = 278.6 kcal/mol and excitation of carbon atom to the excited 5S state (96.4 kcal/mol) which leads to a total promotion energy of 375.0 kcal/mol. The difference of 345.9 kcal/mol in promotion energy is easily compensated by the strength of the donor–acceptor interactions (OC)⇆C⇄(CO). The bond dissociation energy for the reaction C3O2 → C + 2 CO is D e = 136.0 kcal/mol which gives a bond energy of 68 kcal/mol for each dative (double) bond, much less than an electron-sharing double bond. The bond dissociation energy of the electron-sharing double bond in ethylene H2C=CH2 is D e = 180 kcal/mol (Hermann M and Frenking G. Calculated at UCCSD(T)/cc-pVTZ. Unpublished). Thus, carbon suboxide should be written as C(CO)2, and the bonding situation should be sketched as (OC)⇆C⇄(CO). The σ donation OC → C ← CO is found to be stronger than the OC ← C → CO π backdonation [30] and leaves some lone-pair character at the central carbon atom. This explains why carbon suboxide has a bent geometry.

The assignment of dative bonds for C3O2 does not mean that the description with double bonds is wrong. Bonding models are not right or wrong; they are more or less useful. It is important to distinguish between the physical reality of the interatomic interactions and the description in terms of a model. Nature does not know electron-pair bonds nor does it discriminate between dative bonds and electron-sharing bonds. Our argument in favour of dative bonds in C3O2 rests on the straightforward explanation of (a) the bend equilibrium geometry of the molecule which is difficult to understanding using electron-sharing bonds,Footnote 1 (b) the rather small bond dissociation energies of the C–C bonds which do not agree with a genuine C=C double bond and (c) the trend of decreasing bond angles L-C-L′ when L becomes a weaker π acceptor (see the following section about carbones). Moreover, the high excitation energy of CO to the triplet state, which is required for building an electron-sharing double bond, makes it physically more reasonable to consider dative bonds. Thus, we find it more useful to discuss the bonding of carbon suboxide in terms of donor–acceptor interactions, because it offers an explanation and not only a mere description of the bonding situation.

The very high excitation energy of CO to the triplet state explains also the finding that the dimer ethylenedione in the singlet state is not even a minimum on the potential energy surface, although it can nicely be written with the Lewis formula O=C=C=O. The excitation energy of 278.6 kcal/mol for the required 3Π state of CO is too high to be compensated by the bonding energy of a C=C double bond. Dative bonding in the linear form is also unfavourable, because the two fragments are both donors. Only lately has linear OCCO in the 3Σg − state been observed as quasi-bound species while singlet states of OCCO were found to be dissociative [66] (see also [67]).

Very recently, the anion [B(CO)2 −] which is isoelectronic to C(CO)2 could become isolated in a low-temperature matrix experiment [68]. The analysis of the vibrational spectra and the comparison with ab initio calculations at the CCSD(T)/aug-cc-pVTZ level indicated that the boron dicarbonyl anion has a linear structure. This could be rationalized with stronger CO π backdonation OC ← B− → CO, but the structure might also be sketched with double bonds [O=C=B=C=O]−. The authors investigated the nature of the bonding using an energy decomposition analysis. They compared the energy change which is associated with the formation of the molecular structure from closed-shell fragments or from proper open-shell fragments which have the frozen geometries of the molecules. The results which are shown in Table 1 provide also a detailed insight into the bonding interactions. The crucial term which indicates the relaxation of the molecular orbitals that comes from the formation of the molecular wave function is ΔE orb. The calculated value when one starts with the singlet fragments which yield dative bonds is ΔE orb = −433.4 kcal/mol, while the open-shell fragments which lead to electron-sharing bonds give ΔE orb = −493.9 kcal/mol. Thus, the anion [B(CO)2]− should be described in terms of dative bonds (OC)⇆B−⇄(CO) although it has a linear geometry.

Table 1 Energy decomposition analysis of [B(CO)2 −] at the BP86/TZ2P+ level
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The energy decomposition analysis makes it possible to visualize the charge migration which is associated with the dative interactions and to provide a quantitative estimate of the relative strength of donation and backdonation. Figure 2a displays schematically the orbitals which are involved in the donor–acceptor interactions. Only one component of the degenerate OC ← B(–) → CO π backdonation is shown. Figure 2(b)-(e) shows the associated charge flows of the orbital interactions. The (+,+) component of the (OC) → B(–) ← (CO) σ-donation involves the 2s AO of boron while the (+,−) component implicates donation into the 2p(σ) AO. The charge flow which is associated with the dative interactions has the colour code red → blue. The strongest contributions ΔE 1 and ΔE 2 (−127.7 kcal/mol each) come from the degenerate π backdonation OC ← B(–) → CO. A slightly weaker stabilization ΔE 3 = −114.2 kcal/mol arises from the out-of-phase (+,−) σ-donation of the CO lone-pair orbitals OC → B(–) ← CO into the vacant p(σ) AO of boron. The in-phase (+,+) σ-donation OC → B(–) ← CO into the vacant 2s AO of boron is much weaker (ΔE 4 = −54.8 kcal/mol) than the out-of-phase (+,−) σ-donation, although the 2s AO is energetically lower lying than the 2p AO. The stronger stabilization of the donation into the latter comes from the larger overlap of the p(σ) AO than the 2s AO of boron with the donor orbitals.

Fig. 2
figure 2

(a) Schematic representation of the orbitals involved in the OC → B(–) ← CO σ-donation and the OC ← B(–) → CO π backdonation. Only one component of the latter is shown. (b–e) Plot of the charge flow which is connected with the pairwise orbital interactions in [B(CO)2]− together with the associated interaction energies ΔE n. The charge flow is red → blue

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The results in Table 1 and Fig. 2 demonstrate the progress which has been made since 1916 when Gilbert Lewis suggested the electron-pair model. At the same time the essential kernel of his model is retained. The development of modern quantum chemical methods did not erase the intuitive proposal of Lewis, which is complemented and can now be quantitatively expressed by advanced models of quantum chemistry.

3 Carbones CL2 and Related Molecules

The finding that carbon suboxide may be understood as donor–acceptor complex suggests that there could be other molecules with the generic formula L → C ← L where L is a σ-donor ligand and where the carbon(0) atom retains its valence electrons as two lone pairs. A literature search shows that the replacement of CO by a phosphine PR3 leads to well-known compounds whose structures and chemical properties agree with the model of dative bonding. Figure 3 shows a decreasing bond angle at the central carbon atom from C(CO)2 (156.0°) to C(CO)(PPh3) (145.6°) and C(PPh3)2 (131.7°) which conforms with the weaker π-acceptor strength of phosphine ligands that is known from transition metal (TM) chemistry [69]. Carbodiphosphorane C(PPh3)2 is known since 1961 [70] and has been extensively studied ever since [71–74]. It easily accepts two protons at the central carbon atoms which are strongly attracted by the lone electron pairs at carbon [75, 76]. The molecule was considered in the past as diylid [70–73], and only recently it was recognized that C(PPh3)2 is better described with donor–acceptor bonds [28].

Fig. 3
figure 3

Experimental bond angles at the central carbon atom of the compounds, C(CO)2, C(PPh3)(CO) and C(PPh3)2

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The model of dative bonds and the known behaviour of ligands in TM complexes helped to predict another type of carbon(0) compound where the ligands are N-heterocyclic carbenes (NHCs) [29]. There is experimental evidence that NHC ligands have similar σ-donor/π-acceptor properties as phosphines PR3 [77]. Calculations showed that carbodicarbenes C(NHC)2 have a bent geometry and a bending angle of 132° which is close to the value of carbodiphosphorane C(PPh3)2.[29, 35]. The theoretical prediction was soon verified by experiment. The first carbodicarbenes (CDCs) were synthesized and structurally characterized by X-ray analysis [78, 79] shortly after they had been calculated [29]. CDCs are now a very active and promising field of experimental research [80–83]. “The root for this very recent development lies in the suggestion of Gilbert Lewis that chemical bonds in molecules may be formed by the attraction between a fragment which has an electron lone pair and an electron deficient species that has a two-electron gap in the valence shell”. The carbon atom in the 1D state has two such gaps and thus, it can accommodate two donors. Compounds CL2 which have a carbon(0) atom with two electron lone pairs are a class of compounds whose chemical reactivity exhibits characteristic differences from carbenes CR2 that have only one lone pair at the carbon(II) atom [84, 85]. The name “carbone” has been coined for divalent C(0) compounds CL2 [86]. Further information about the chemistry of carbones can be found in the literature [87, 88].

Carbodicarbenes are good examples to demonstrate the dichotomy of electron sharing vs. dative bonding. CDCs may also be considered as amino-substituted allenes which can be sketched with electron-sharing bonds. Figure 4 shows schematically the bonding situation in parent allene and in tetraaminoallenes (TAAs) with the two types of electron-pair bonds. Below each structure are the relevant fragments in the required electronic reference state. The crucial factor is the singlet–triplet (S/T) excitation energy of the terminal carbene fragments CR2 (R=H, NMe2, NEt2) and NHC. CH2 has a (3B1) ground state which needs no promotion for electron-sharing bonding with C (5S). There is no doubt that the parent allene has electron-sharing double bonds H2C=C=CH2. The C(NMe2)2 carbene has a S/T gap of 33.5 kcal/mol which requests a promotion energy of 67 kcal/mol of the ligands to engage in electron-sharing bonds. The linear geometry suggests that the molecule may be written as (NMe2)2C=C=C(NMe2)2, but in the absence of EDA calculations, it cannot be ruled out that the molecule has dative bonds (NMe2)2C → C ← C(NMe2)2. The latter bonding situation is clearly more appropriate for the NEt2 derivative (NEt2)2C → C ← C(NEt2)2 which has in spite of the more bulky ethyl groups a calculated bonding angle at the central carbon atom of 169.5° [29]. The S/T gap of C(NEt2)2 is 41.1 kcal/mol which means that the promotion energy of the ligands to engage in electron-sharing bonds is 82.2 kcal/mol which is higher than for C(NMe2)2. An much higher S/T gap of 82.0 kcal/mol is calculated for NHC which leaves no doubt that C(NHC)2 should be written with dative bonds (NHC) → C ← (NHC). There is experimental and theoretical evidence that C[C(NMe2)2] and C[C(NEt2)2] can both be considered as carbones CL2, although the former species has a linear structure. The molecules readily add CO2 and CS2 at the central carbon atom-yielding adducts [89] that were also found for carbodiphosphorane [90]. Calculations showed that TAAs have very large first and second proton affinities which have been found as distinctive difference to carbenes [91].

Fig. 4
figure 4

Sketch of electron-pair bonds in allenes with electron-sharing bonds and dative bonds. Below each structure are the electronic reference states of the carbon atom and the carbene ligands CR2 for the respective bonding interactions. At the bottom are the excitation energies from the electronic ground state to the excited state. The value for the carbon atom was taken from [61], and the values for the carbenes were calculated at BP86+D3(BJ)/def2-TZVPP

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The model of dative bonding proved to be a very powerful tool for related molecules EL2 which are isoelectronic to carbones. Figure 5 shows a survey of group-15 homologues (N+)L2 and group-13 complexes (BH)L2 which were calculated and in part synthesized. Some of them could become synthesized following predictions that were based on the model of dative bonding. The experimental values for the bond angles of isolated molecules are given in parentheses. We want to point out that the trend of the bond angles in EL2 nicely follows the strength of the L ← E → L π backdonation, unless steric repulsion of the bulky substituents prevents smaller angles. For example, the bond angles increases for the isoelectronic species (N+)(N2)2 (111.2°) < (N+)(CO)2 (130.7°) < C(CO)2 (156°).

Fig. 5
figure 5

List of isoelectronic molecules EL2 showing calculated bond angles and partial charges Δq of the central fragments for E=BH, C, N+ [89]. Experimental values of the bond angles of isolated molecules are given in parentheses

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The synthesis of two unusual molecules from the series shown in Fig. 5 which were previously unknown confirms the predictive value of the Lewis model of dative bonding. One molecule is the borylene complex (BH)(CAAC)2 where two cyclic (alkyl)(amino)carbene ligands (see Fig. 5) stabilize a BH fragment in the highly excited 1Δ state where the lone-pair electrons occupy a p(π) AO [39]. Although the excitation energy of BH from the X1Σ+ ground state to the 1Δ reference state is very high (131.5 kcal/mol) [65], the strong CAAC → (BH) ← CAAC σ donation and the CAAC ← (BH) → CAAC π backdonation sufficiently stabilize the molecule that it can be isolated and structurally characterized by X-ray analysis [40]. The molecule was the first example of a tricoordinated boron compound where the boron atom is a Lewis base, and thus, it is isoelectronic with an amine. Figure 6 shows the HOMO of the molecule. The shape nicely shows the extent of the CAAC ← (BH) → CAAC π backdonation. Since the HOMO is energetically rather high lying, it can easily be protonated and can also be ionized to the radical cation which could become isolated and structurally characterized by X-ray analysis. For further details we refer to the original literature [39, 40].

Fig. 6
figure 6

Plot of the highest-lying occupied molecular orbital (HOMO) of the borylene complex (BH)(CAAC)2 [90]

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The second remarkable molecule is a substituted homologue of the borylene complex where the stabilizing donor ligands are carbonyls. The borylene dicarbonyl complex (BR)(CO)2 which has a bulky aryl group R has very recently been synthesized [92, 93]. The X-ray analysis confirms the trigonal planar geometry which was calculated for the parent system (Fig. 5). The bond angle OC-B(R)-CO is 104.0° which is more acute than the calculated value in the parent system OC-B(H)-CO because of steric repulsion of the carbonyl ligands with the substituent R. The borylene dicarbonyl reacts chemically like a transition metal carbonyl complex which opens the door to a new area of the p-block atoms [92, 93].

The model of dative bonding served also as useful guideline for heavy atom homologues of carbones EL2 (E = Si – Pb). It actually seems that dative bonds are even more common in molecules of heavier main-group atoms than for the first octal-row elements. In the years 2003–2005 it was reported that the first silicon and germanium homologues of allenes had been isolated [94, 95]. However, the molecular structure did not exhibit the characteristic features of allenes which have a linear backbone R2C=C=CR2 where the terminal groups are orthogonal to each other. The isolated species have a bent geometry, and the planes of the substituents are twisted (Fig. 7a). The structural features resemble more carbodicarbenes (Fig. 7b), but the cyclic end groups do not have nitrogen atoms in α-position like CDCs.

Fig. 7
figure 7

Schematic representation of the bonding situation in (a) alleged heavy allenes and (b) carbodicarbenes C(NHC)2

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Calculations of NHE and cycE (E = C – Pb) where NHE and cycE are group-14 homologues of NHCs and cyclopentylidene showed that the singlet/triplet gap of NHE decreases but that of cycE increases when E becomes heavier (Table 2). The excitation energy 3P → 1D of the heavier atoms Si - Pb is smaller than for carbon which means that dative bonding for the systems (cycE) → E ← (Ecyc) becomes more favourable. For E = Si, the singlet fragments in Si(cycSi)2 are favoured by (2 × 27.1 kcal/mol) – 18.0 kcal/mol = 36.2 kcal/mol, and for E=Ge the singlet fragments in Ge(cycGe)2 are lower in energy by (2 × 31.0 kcal/mol) – 20.4 kcal/mol = 41.6 kcal/mol. The bonding situation of a genuine allene in E(cycE)2 would only be possible if stronger electron-sharing bonds would compensate for the differences in the excitation energies. It has been shown, however, that for heavier atoms E, the E → E (E = Si – Pb) donor–acceptor interactions between singlet fragments may have a similar strength as E–E electron-sharing interactions between open-shell fragments [96]. Thus, the alleged “trisilallene” and trigermaallene” [94, 95] are rather examples for heavy group-14 homologues of carbones. Table 3 shows the names which have been suggested for tetrylones EL2 in analogy to tetrylenes ER2.

Table 2 Energy differences of the ligands NHE and cycE between different spin multiplicities at BP86/TZVPP and experimental atomic excitation energies
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Table 3 Proposed nomenclature for divalent E(0) compounds
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We close this section with an example where the different bond strength of dative bonds and electron-sharing bonds of the same fragments could be quantitatively estimated. The results provided an explanation for a puzzling experimental result. In 2009, the silylene complex NHC → SiCl2 could become isolated which was the first stable silylene adduct at room temperature [97] (a stable SiBr2 adduct was reported in the same issue: [98]). The complex was later reacted with the CAAC carbene which is a stronger σ donor and stronger π acceptor than NHC [99, 100]. Instead of the expected exchange reaction yielding CAAC → SiCl2, the product had two CAAC ligands attached to the silylene fragment in CAAC-(SiCl2)-CAAC where the Si–C bonds were significantly shorter than in CAAC → SiCl2 [101]. Moreover, the latter molecule was found to have an electronic triplet state.

Figure 8 shows schematically the bonding situation in the complex NHC → SiCl2 and in the triplet species SiCl2(CAAC)2 which has electron-sharing bonds between SiCl2 and the CAAC ligands that accommodate the unpaired electrons.Footnote 2 It shows also calculated energies which are relevant for the system. The bond dissociation energy (BDE) of NHC → SiCl2 is D e = 40.5 kcal/mol. The BDE of the hypothetical adduct CAAC → SiCl2 of D e = 42.5 kcal/mol is as expected slightly bigger. The calculated BDE of the isolated species SiCl2(CAAC)2 is D e = 227.5 kcal/mol. In order to enable electron-sharing bonds, the fragments SiCl2 and CAAC must be promoted to the triplet state which requires 60.1 kcal/mol + (2 × 49.9 kcal/mol) = 159.9 kcal/mol. Subtracting this value from the BDE gives a net gain of ΔE = 67.3 kcal/mol that is larger than the BDE of the hypothetical complex CAAC → SiCl2 (D e = 42.5 kcal/mol). Thus, the formation of the triplet species SiCl2(CAAC)2 which possesses electron-sharing bonds is energetically favoured. In contrast, the formation of the hypothetical triplet molecule SiCl2(NHC)2 is energetically unfavourable in comparison with the complex NHC → SiCl2 although the BDE of SiCl2(NHC)2 (D e = 245.5 kcal/mol) is higher than that of SiCl2(CAAC)2 (D e = 227.5 kcal/mol). This is because the S/T gap of NHC (88.9 kcal/mol) is much higher than for CAAC (49.9 kcal/mol). The net stabilization energy ΔE for the formation of SiCl2(NHC)2 is only 245.2 kcal/mol–60.1 kcal/mol – (2 × 88.9 kcal/mol) = 7.3 kcal/mol which is much less than the BDE of NHC → SiCl2 (D e = 40.5 kcal/mol).

Fig. 8
figure 8

Schematic view of the bonding situation in (a) the complex NHC → SiCl2 and (b) the molecule SiCl2(CAAC)2 in the triplet state. Below are the calculated bond dissociation energies at M05-2x/def2-TZVPP of the complexes SiCl2(NHC) and SiCl2(CAAC) and the compounds in the triplet state SiCl2(NHC)2 and SiCl2(CAAC)2 as well as the singlet–triplet gaps of the fragments. The bottom lines give the net stabilization energies ΔE for the formation of the triplet compounds SiCl2(NHC)2 and SiCl2(CAAC)2

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The data in Table 2 suggest that tetrylones with NHC ligands E(NHC)2 are promising targets for the synthesis of stable divalent E(0) compounds. Very recently, the stable silylone SiL2 and germylone GeL2 have been isolated with two NHC ligands that were bridged by a methylene group [42, 43], and the silylone Si(CAAC)2 could become isolated (Fig. 9) [41].

Fig. 9
figure 9

Schematic representation of the silylones and germylones E(CAAC)2 and E(NHC-NHC) (E = Si, Ge) which have been isolated

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The Lewis bonding model still remains remarkably fruitful when it becomes connected with quantum chemical calculations even 100 years after its first presentation.

4 Dative Bonding in Heavy Homologues of Acetylene HEEH (E = Si – Pb)

The fundamental difference between atoms of the first octal row and heavier homologues to form stable molecule with multiple bonds was well known when Lewis published his works on chemical bonding [1, 7]. He wrote in his book: “…..the ability to form multiple bonds is almost entirely, if not entirely, confined to elements of the first period of eight, and especially to carbon, nitrogen and oxygen” [102]. This statement was based on chemical knowledge which was available at that time. It was only in 1976 and 30 years after Lewis was deceased in 1946 that the first stable molecule whose structure could be measured by X-ray crystallography with a Sn=Sn double bond was isolated [103] followed by the syntheses of the other group-14 homologues R2E=ER2 ((E = Si: [104]), (E = Ge: [105], see also [106]), (E = Pb: [107])). Stable group-14 homologues of alkynes RE≡ER (E = Si – Pb) were reported between 2000 and 2004. Measurement of the X-ray analysis showed that the equilibrium geometries of ditetrylenes R2E=ER2 and ditetrylynes RE≡ER are very different from those of the carbon systems. R2E=ER2 are not planar but possess pyramidally coordinated ER2 groups while ditetrylynes RE≡ER have a trans-bent arrangement of the substituents instead of a linear structure which raised the question whether they have genuine E≡E triple bonds [108–112].

Even more surprising were the results of accurate quantum chemical calculations of the parent systems E2H2 [113–127] (E = Si – Pb) which showed a variety of unusual equilibrium structures for several isomers (Fig. 10) that were later found in low-temperature matrix experiments [128–133]. None of them are the linear form HE≡EH which is an energetically high-lying second-order saddle point on the potential energy surface. The doubly bridged butterfly structure A is the global energy minimum for all heavier systems E = Si – Pb followed by the singly bridged isomer B. The vinylidene form C is the only isomer which is common for all group-14 atoms. Since the atomic connectivity differs from those of the other species, it is not further considered. There are two trans-bent forms D1 and D2 which are important for discussion. Only D1 is an energy minimum while D2 which has a more acute bonding angle is a transition state which is, however, for the lead system energetically lower lying than D1. With bulky substituents R it becomes an energy minimum [134] which could become isolated [135].

Fig. 10
figure 10

Schematic view of the structures E2H2 which are found as energy minima on the potential energy surfaces and calculated relative energies. The values were taken from [96]

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The isomers which are shown in Fig. 10 are difficult to sketch with electron-sharing bonds in an unambiguous way. It has been shown that the reason for the heavier homologues of acetylene to adopt unusual structures and the relative energies of the isomers can be explained with the dichotomy of electron-sharing and dative bonds [96]. The explanation is based on an earlier model that was suggested to rationalize the pyramidal structures of some ditetrylenes where the electron-sharing bonds for carbon R2C=CR2 are substituted by dative bonds R2E⇄ER2 (Fig. 11) [136, 137]. This model proved in conjunction with quantum chemical calculations to be even more helpful for the exotic structures of REER.

Fig. 11
figure 11

Schematic representation of (a) electron-pair bonding in ethylene and planar analogues and (b) dative bonds in heavier group-14 homologues

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The starting point for the discussion of the bonding situation in HEEH is the inspection of the electronic structure of the fragments EH. Figure 12 shows that the electronic ground state 2Π has one σ electron pair and one unpaired electron in the p(π) AO of atom E. The electronic reference state which is required for electron-sharing triple bonds HE≡EH is the first excited 4Σ− state. The excitation energy 2Π → 4Σ− is 16.7 kcal/mol for E = C but it becomes much higher for the heavier atoms E.

Fig. 12
figure 12

Schematic representation of the electron configuration of the 2Π electronic ground state and the a4Σ− excited state of EH (E=C–Pb). Experimental [65] and calculated (BP86/QZ4P; [96]) excitation energies in kcal/mol

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Table 4 shows the calculated bond dissociation energies D e for all linear species HE≡EH with E = C – Pb which exhibit a regular decrease for the heavier atoms. The decrease is particularly large from C to Si. The right column shows the net energy gain of the electron-sharing triple bonds when the promotion energies 2Π → 4Σ− of the fragments EH are subtracted. Note that the 2Π state would lead to a HE-EH electron-sharing single bond which is in competition with the HE≡EH triple bond that can be formed from the excited 4Σ− state. The energy gain for the carbon system to form a triple bond amounts to 240.0 kcal/mol which is much higher than the bond energy of a typical single bond. The situation is clearly different for the heavier homologues. The net energy gain for a HSi≡SiH triple bond is only 44.5 kcal/mol which is less than the value for a typical Si–Si single bond (75–80 kcal/mol) [138]. The same conclusion holds for the remaining species with E = Ge – Pb. It follows that only carbon binds through the excited 4Σ− state of CH while the heavier hydrides EH bind through the 2Π ground state.

Table 4 Calculated bond dissociation energies D e (kcal/mol) of linear HE≡EH → 2 EH (a4Σ−) and X2Π → a4Σ− excitation energies ΔE exc (kcal/mol) of EH at BP86/QZ4P
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Connecting the EH fragments in the 2Π ground state through the unpaired electrons and taking care of the octet rule straightforwardly lead to the energy minima on the potential energy surface of E2H2 that are shown in Fig. 10. Figure 13 shows three different arrangements where the unpaired electrons yield a σ bond HE–EH. The planar syn- and anti-forms F (Fig. 13a) and D2 (Fig. 13c) which have an electron sextet in the valence shell of atom E are transition states with respect to rotation about the E–E bond. As mentioned above, D2 becomes an energy minimum for bulky groups R and it can be isolated when E = Pb, although the lead atom has only six electrons in the valence shell [134, 135]. The octet rule is violated for very heavy main-group atoms due to relativistic effects [139]. Rotation of F or D2 by 90° about the E–E axis enables mutual donation of the E–H bonds into the formally vacant p(π) AO of the other atom E, yielding structure A which fulfils the octet rule (Fig. 13b). The E–H bonds are tilting in order to maximize the E–H donation which straightforwardly leads to the doubly bridged butterfly structure A. Such kind of electron donation was already envisaged by Lewis in his book: “….when there are not enough electrons in a molecule to provide each atom with its stable octet by the process of forming normal bonding pairs, two contiguous atoms may, to some extent, share a second or third pair of electrons, although this sharing is by no means so complete or unambiguous as in the single bond” [140]. Again, a remarkable foresight which, however, was followed by a restriction which is not correct: “….this ability to share a second or third pair is almost entirely limited to carbon, nitrogen and oxygen”. The electron donation from the E–H bond into the empty p(π) orbital can be understood as a variant of the dative bond where the electron pair comes from a bond but not from a lone pair.

Fig. 13
figure 13

Qualitative model for the orbital interactions between two EH molecules in different orientations where the unpaired electrons yield a σ bond

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The doubly bridged butterfly structure A has three electron-pair bonding components for E–E bonding, i.e. one electron-sharing σ bond and two E–H donor–acceptor bonds. It may thus be considered to contain a triple bond where the electron-sharing π bonds of a classical triple bond are replaced by dative bonds. Note that the E–E distances in A (Si = 2.23 Ǻ, Ge = 2.39 Ǻ, Sn = 2.78 Ǻ, Pb = 2.93 Ǻ) are clearly longer than in the linear structure E (Si = 1.98 Ǻ, Ge = 2.05 Ǻ, Sn = 2.41 Ǻ, Pb = 2.48 Ǻ), and yet, the former species are much lower in energy than the latter. This is because the dative bonds in A do not require the large excitation energy 2Π → 4Σ− of the fragments EH. We want to point out that the charge donation from the E–H bonds in A is stronger than the hypothetical sideward donation of the electron lone pairs, because the lone-pair orbitals of the heavy atoms E have mainly s-character and because hydrogen is more electronegative than Si–Pb.

The unpaired electrons of the EH fragments in the 2Π ground state may also couple and form an electron-sharing π bond. Figure 14 displays three different orientations of the fragments with a HE-EH π bond. The arrangement in Fig. 14a shows that the E–H bond of the left fragments may donate into the vacant p(π) AO of the right EH species. In order to maximize the donor–acceptor interactions, the E–H bond and the vacant p(π) AO are tilting which induces a concomitant uplift of the E–H bond of the right fragment. This explains nicely the very unusual geometry of structure B where the terminal E–H bond is at the same side as the bridging hydrogen atom. The donation of the electron lone pair from the right to the left moiety complements the octet shell of this atom E. The octet rule is thus fulfilled. Isomer B also possesses three electron-pair bonds between atoms E which consist of one electron-sharing π bond, one E–H dative bond and one lone-pair dative bond. Rotation of the right E–H fragment by 90° leads to structure G which enables two E–H dative bonds besides the electron-sharing π bond (Fig. 14b). Structure G is therefore lower in energy than isomer B, but it is a transition state for the degenerate wing-flapping motion of the global energy minimum structure A [128]. The opposite tilting of the E–H fragments leads to structure D1 (Fig. 14c) where the dative bonds come from the lone-pair orbitals. The electron donation is therefore weaker in D1 which is higher in energy structures B and G. Steric repulsion enforces a trans-arrangement of bulky substituents in compounds REER which therefore adopt the structures D1 for E=Si–Sn and D2 or E=Pb in agreement with the relative energies of the parent systems (Fig. 10). D1 has three electron-pair bonds which consist of one electron-sharing π bond and two lone-pair dative bonds while D2 has one electron-sharing σ bond. (A similar explanation for the trans-bent geometry of HEEH (E = Si – Pb) has been given in [127]. The trans-bent structure of valence isoelectronic digallium compounds [RGaGaR]2− was discussed in terms of HOMO–LUMO mixing by [141].) For a more detailed discussion, we refer to the original literature [96].

Fig. 14
figure 14

Qualitative model for the orbital interactions between two EH molecules in different orientations where the unpaired electrons yield a π bond

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Is it possible to sketch the bonding situation of the energy minimum structures A, B, D1 and D2 with simple Lewis formulas and possibly resonance forms? The answer is yes if the difference between an electron-sharing bond and a dative bond is depicted. Figure 15 shows a viable way how this can be done. The arrows indicate whether the dative bond comes from a lone-pair orbital or from an E–H bond. The sketches are slightly more complicate than standard Lewis structures, but they convey information about the different type of electron-pair bonding.

Fig. 15
figure 15

Suggested Lewis structures for isomers A, B, D1 and D2 of E2H2 isomers

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5 Concluding Remarks

The electron-pair model of Gilbert Lewis remains 100 years after its introduction a powerful tool for creative chemical research where it continues to serve as a guideline for finding new molecules and for understanding molecular structures if it is combined with quantum chemical calculations. The understanding of chemical bonding which originates through a quantum theoretical resonance phenomenon in terms of electron pairs requires the knowledge of the electronic structure which can be analysed with a variety of modern theoretical methods [142, 143]. The legacy of Gilbert Lewis includes both the electron-pair bonding model with its associated rules and the permanent willingness for “maintaining an opening of mind; so that, when the solution of these problems, which now seem so baffling, is ultimately offered, its acceptance will not be retarded by the conventions and the inadequate mental abstractions of the past” [15]. We do not find better words than the original statement by this pioneer of chemistry.