Analysis of the E – E Bond in Group-13 Complexes [ ( PMe 3 ) 2 ( E 2 H n ) ] ( E = B – In , n = 4 , 2 , 0 ) †

Quantum chemical calculations at the BP86/def2-TZVPP level have been carried out for the donor-acceptor complexes [(PMe3)2(E2Hn)] for n = 4, 2, 0. The focus of this works lies on the E–E bonding situation. The electronic structure of the molecules was analyzed with the EDA-NOCV method and with NBO calculations. The EDA-NOCV analysis of the E–E interactions in [(PMe3)2(E2Hn)] (n = 4, 2, 0) provide deep insights into the nature and the strength of the bonds. The calculated intrinsic interactions ΔEint suggest that the trend for the bond strength of the E–E single bond [(PMe3)(H)2E– E(H)2(PMe3)] has the order B > Ga > Al > In. The orbital interactions ΔEorb which exhibit the same trend as ΔEint have one dominant contribution which comes from the coupling of the singly occupied orbitals in the (PMe3)(H)2E fragments. A slightly different trend B > Ga ~ In > Al is found for the interaction energy ΔEint of the E–E bonds in [(PMe3)(H)E–E(H)(PMe3)]. The orbital term ΔEorb which has the order B > Ga > In > Al has one major and one minor component which in case of the boron compound may be identified with a σ and a π bond. The heavier homologues [(PMe3)(H)E–E(H)(PMe3)] (E = Al – In) have pyramidally coordinated atoms E. The dominant orbital interactions in the latter species come from the formation of a “slipped” π bond while the minor component comes from the formation of the σ bond. This can be explained with the change in the hybridization of the orbitals at atom E along the formation of the E–E bond. The compounds [(PMe3)E–E(PMe3)] exhibit three different types of bonding situations depending on atoms E. The boron system [(PMe3)B≡B(PMe3)] has a classical triple bond which consist of a σ bond that provides 56 % to the orbital interactions and two degenerate π bonds which contribute 40 % to the covalent bonding. The aluminium and gallium complexes [(PMe3)E–E(PMe3)] (E = Al, Ga) are also triply bonded species where the covalent bonding has one strong and two weaker components. The strong component comes from the “slipped” π bond while the minor components come from the formation of the σ and π bonds. The indium complex [(PMe3)In–In(PMe3)] has only an In–In single bond and two electron lone pairs at the indium atoms. The charge donation Me3P → E2(Hn) ← PMe3 has for all atoms E the trend for n 4 > 2 > 0.


INTRODUCTION
Recently, we reported a structure and bonding analysis of the donor-acceptor bonds in the group-13 complexes [(L) 2 (E 2 H n )] with n = 4, 2, 0 and ligand L being an N-heterocyclic carbene (NHC) or a phosphine ligand PMe 3 . 1,2Substituted derivatives of low-valent and low oxidation state hydrides such as [(NHC Dipp ) 2 (B 2 H 4 )], 3 [(NHC Dipp ) 2 (Al 2 H 4 )], 4 [(NHC Dipp ) 2 (B 2 H 2 )] 5 or [(PPh 3 ) 2 (B 2 H 4 )] 6 are experimentally known.Lately, the complex [(NHC Dipp ) 2 (B 2 )] which is the first group-13 dimer E 2 stabilized in a complex L → E 2 ← L was isolated 7 which has a B≡B triple bond. 8The calculations showed that the structural motives of the complexes [(L) 2 (E 2 H n )] differ significantly with the nature of the ligand L and the group-13 atom E. 1,2 While all analyzed complexes [(L 2 )(E 2 H 4 )] adopt C s symmetry with trans oriented ligands, the heteroatoms E in the [(L 2 )(E 2 H 2 )] compounds are planar coordinated only for the boron complexes and for the NHC ligated aluminum and gallium dimers.The remaining complexes [(L 2 )(E 2 H 2 )] show pyramidal coordination at atom E. For the complexes [(L 2 )(E 2 )] three structure motives were found: The stabilized boron dimers show a linear L-B-B-L arrangement, the heavier NHC complexes and the aluminum and gallium PMe 3 dimers have antiperiplanar Croat.Chem.Acta 87 (2014) 413.coordinated ligands while in [(PMe 3 ) 2 (In 2 )] the phosphine ligands are gauche oriented with an P-In-In-P dihedral angle of 123.0°. 1,2he theoretical findings pose the question about the nature of the E-E bonding in the complexes [(L) 2 (E 2 H n )] and the electronic state of the interacting fragments which could help to explain the different structures of the adducts (Scheme 1).In our previous studies we focused on the donor-acceptor interactions L → E 2 (H n ) ← L. 1,2 The nature of the E-E bonding was not discussed in this work.Therefore, we analyzed the nature of the E-E bonding with the EDA (Energy Decomposition Analysis) method in conjunction with the NOCV (Natural Orbitals for Chemical Valence) scheme. 9Since the nature of the E-E bonding does not change when L = PR 3 or NHC, we restricted our work on the phosphine model compounds where L = PMe 3 .

METHODS
Geometry optimizations have been carried out using TurboMole 6.1 optimizer 10 and gradients at the BP86 11 /def2-TZVPP 12 level of theory.Stationary points were characterized as minima by calculating the Hessian matrix analytically at this level of theory.For all calculations the resolution-of-identity method has been applied. 13or the bonding analyses we optimized the molecules with the program package ADF2009.01. 14BP86 was chosen applying uncontracted Slater-type orbitals (STOs) as basis functions. 15The latter basis sets for all elements have triple-ζ quality augmented by two sets of polarization functions (ADF-basis set TZ2P).This level of theory is denoted BP86/TZ2P.An auxiliary set of s, p, d, f, and g STOs was used to fit the molecular densities and to represent the Coulomb and exchange potentials accurately in each SCF cycle. 16Scalar relativistic effects have been incorporated by applying the zeroth-order regular approximation (ZORA) in all ADF calculations. 17he interatomic interactions were investigated by means of an energy decomposition analysis (EDA, also known as Extended Transition State method -ETS) developed independently by Morokuma 18 and by Ziegler and Rauk. 19The bonding analysis focuses on the instantaneous interaction energy ΔE int of a bond A-B between two fragments A and B in the particular electronic reference state and in the frozen geometry of AB.This interaction energy is divided into three main components [Equation ( 1 The term ΔE elstat corresponds to the quasiclassical electrostatic interaction between the unperturbed charge distributions of the prepared atoms and is usually attractive.The Pauli repulsion ΔE Pauli is the energy change associated with the transformation from the superposition of the unperturbed electron densities  A +  B of the isolated fragments to the wavefunction Ψ° = N Â[Ψ A Ψ B ], which properly obeys the Pauli principle through explicit antisymmetrization (Â operator) and renormalization (N = constant) of the product wavefunction.ΔE Pauli comprises the destabilizing interactions between electrons of the same spin on either fragment.The orbital interaction ΔE orb accounts for charge transfer and polarization effects.The ΔE orb term can be decomposed into contributions from each irreducible representation of the point group of the interacting system.Further details on the EDA/ETS method 14 and its application to the analysis of the chemical bond 20 can be found in the literature.
The EDA-NOCV 21 method combines charge (NOCV) and energy (EDA) decomposition schemes to decompose the deformation density which is associated with the bond formation, Δρ, into different components of the chemical bond.The EDA-NOCV calculations provide pair wise energy contributions for each pair of interacting orbitals to the total bond energy.NOCV (Natural Orbital for Chemical Valence) 22,23 is defined as the eigenvector of the valence operator, V , given by Equation (2): In the EDA-NOCV scheme the orbital interaction term, ΔE orb , is given by Equation (3): provides both qualitative (Δρ orb ) and quantitative (ΔE orb ) information about the strength of orbital interactions in chemical bonds, even in molecules with C 1 symmetry. 24iberg bond indices (WBI), partial charges and Lewis structures were obtained using NBO 3.1 25 as implementted in Gaussian 09 Rev. C.01. 26 NBO electron densities were generated from a single-point calculation of the molecules with BP86/def2-TZVPP basis sets in Gaussian09 from the TURBOMOLE 6.1 optimized geometries.

RESULTS AND DISCUSSION
The optimized geometries of the phosphine complexes 1E, 2E and 3E at def2-TZVPP are shown in Figure 1.The geometrical features have been discussed in our previous studies 1,2 and therefore, we directly proceed to the bonding analysis.
A pivotal question of the EDA-NOCV calculations of complexes [(L) 2 (E 2 H n )] is the choice of the electronic states of the interacting fragments for the analysis of the E−E bonds.The present calculations were carried out using doublet (D) states for (PMe 3 )EH 2 , triplet (T) states for (PMe 3 )EH and quartet states for most of the fragments (PMe 3 )E in the EDA-NOCV calculations of 1E, 2E and 3E respectively.The only exception is (PMe 3 )In where the inspection of the geometry and the shape of the highest lying MOs indicate that the (PMe 3 )In−In(PMe 3 ) bond is a single bond where the interacting fragments are (PMe 3 )In in the electronic doublet state.Generally, the choice of the electronic states was made in the light of our earlier results about the electronic structure of the complexes where we analyzed the donor-acceptor bonds L → E 2 (H n ) ← L. 1,2 The results of the EDA-NOCV calculations are given in Table 1.
The equivalent energy contributions of the α and β electron densities in the EDA-NOCV calculations of the open shell fragments are summed up to give the orbital energy terms ΔE 1 to ΔE 3 .Table 1 shows that the orbital interactions ΔE orb in 1E come mainly from ΔE 1 while the orbital interactions in 2E have two (ΔE 1 and ΔE 2 ) and those in 3E possess three (ΔE 1 − ΔE 3 ) major contributions which come from the interactions of the singly occupied orbitals.The remaining orbital interactions ΔE rest are negligible.
Table 1 shows that the interaction energy ΔE int and thus the strength of the E-E bond in [(PMe 3 ) 2 (E 2 H n )] increases with decreasing number of hydrogen atoms n 4 < 2 < 0 (Table 1).The only exception is complex 3In which possesses a very small interaction energy of ΔE int = −23.6 kcal mol −1 .Inspection of the equilibrium geometries of 3E reveals (Figure 1) that 3In possesses a gauche conformation of the phosphine ligands while the other homologues have either a linear (3B) or a transbent (3Al − 3Ga) form.It will be shown below that 3In has a different electronic reference state of the (PMe 3 )In fragments than the (PMe 3 )Al and (PMe 3 )Ga moieties in 3Al − 3Ga.
The trend of the E−E interaction energy for different atoms B − In along the series of 1E−3E is very interesting.The calculated data for ΔE int suggest that the Al−Al bonds are always significantly weaker than the B−B bonds which is expected while the Ga−Ga bonds are predicted to be clearly stronger than the Al−Al bonds for all compounds 1E − 3E.The In−In bond in 1In is weaker (−54.2 kcal mol −1 ) than the Ga−Ga bond in 1Ga (−61.3 kcal mol −1 ) but the ΔE int values suggest that the In−In bond in 2In (−97.1 kcal mol −1 ) has nearly the same strength as the Ga−Ga bond in 2Ga  1 show that the trend of the orbital interaction term ΔE orb has the same order B > Ga > Al > In as the total interaction energy ΔE int .Note that this does not hold for the electrostatic attraction ΔE elstat where the values for 1B (−113.8kcal mol −1 ) and 1Ga (−113.3kcal mol −1 ) are nearly the same and where 1In (−108.3kcal mol −1 ) has a much larger electrostatic attraction than 1Al (−75.3 kcal mol −1 ).Very extensive investigations of chemical bonds across the periodic table have shown that the trend of the overall strength of a chemical bond * is in most cases the same as the trend of the orbital interactions. 20This explains the great success of the model of orbital interactions in chemistry but it must be pointed out that there are exceptions where Pauli repulsion ΔE Pauli or electrostatic interactions ΔE elstat determine the trend of the bond strength. 28able 1 shows that the orbital interactions in 1E come mainly from one single pair of orbitals.The contribution of ΔE 1 to ΔE orb is always > 90 %.This comes from the pairing of the singly occupied orbitals in (PMe 3 )EH 2 which yield the HOMO of [(PMe 3 ) 2 (E 2 H 4 )].The shape of the latter orbitals of 1Al is shown in Figure 2. The orbitals of the other homologues 1B, 1Ga and 1In look very similar and therefore, they are not shown here.Figure 2 gives also the values of ΔE 1 and the eigenvalues v 1 for all systems along with the plot of the deformation density Δρ 1 which is associated with the orbital interactions ΔE 1 in 1Al.Note that the colour coding of the deformation density Δρ 1 * Note that the strength of a chemical bond is often not related to the bond dissociation energy (BDE), because the BDE values include also the relaxation of the fragments.
indicates the charge flow which has the direction red → blue.The shape of Δρ 1 nicely illustrates the charge accumulation in the bonding region of the E−E bond.The EDA-NOCV results suggest a single E−E σ bond for all molecules 1E which comes from the sharing of the unpaired electrons of the (PMe 3 )EH 2 fragments.This view is supported by the calculated Wiberg bond orders P(E−E) which are shown in Table 2.The calculations give rather uniform values between 0.93 (1B) − 0.89 (1In).Note that the hybridization of the E−E bonding orbital is nearly constant for the different atoms E. The values are 30.4% for B, 30.3 % for Al, 31.2 % s for Ga and 30.2 % s for In.This is remarkable, because sp-hybridization of first octal-row atoms is usually much stronger than for heavier atoms due to the different radii of the ns and np valence AOs. 29The calculated charge distribution shows (Table 2) that the charge donation Me 3 P → (E 2 H 4 ) ← PMe 3 for the boron compounds is very strong (Δq = −1.42e) and that the boron atoms carry all of the negative partial charges while the hydrogen atoms have a small positive charge.This means that there is an umpolung of the B−H bonds with respect to normal boranes and that the hydrogen atoms at boron become slightly acidic.The charge donation in the heavier homologues Me 3 P → E 2 (H 4 ) ← PMe 3 (E = Al -In) has smaller and rather uniform values between −0.64 (Al) and −0.72 (Ga) where the negative charge resides exclusively on the hydrogen atoms while the group-13 atoms carry positive partial charges.
The EDA-NOCV data for [(Me 3 P) 2 (E 2 H 2 )] (2E) in Table 1 show that the strength of ΔE orb has the same order B > Ga > In > Al as the total interaction energy ΔE int but the ΔE int values for the Ga−Ga and In−In bonds are not very different from each other.The orbital interactions in 2E have two large contributions ΔE 1 and ΔE 2 where the first one is always much larger than the latter.The orbitals of the interacting species of the boron hydride (PMe 3 )BH yielding 2B look very different from the MOs of the heavier homologues (PMe 3 )EH (E = Al − In; Figure 3).This is not surprising, because the equilibrium geometry of 2B substantially deviates  from the structures of 2Al − 2In (Figure 1).The latter structures have pyramidally coordinated atoms E while 2B possesses planar coordinated boron atoms.
Figures 3(a) and 3 (b) show the shape and eigenvalues of the relevant orbitals of (PMe 3 )BH and 2B and the plot of the deformation densities Δρ 1 and Δρ 2 along with the associated values of ΔE 1/2 and the eigenvalues v 1/2 .The largest contribution ΔE 1 comes from the formation of the B−B σ bond while ΔE 2 comes from the π bond.Note that the associated eigenvalues have the opposite order v 1 < v 2 , which means that the formation of the weaker π bond involves a larger charge flow than the formation of the σ bond.This can be explained with the shape of the SOMO of LBH where the lobe of the non-bonded part of the orbital is already pointing into the direction of the expected B−B σ bond which requires less distortion for the bond formation than for the π bond.
Figures 3(c) and 3(d) exhibit the shape and eigenvalues of the relevant orbitals of (PMe 3 )AlH and 2Al as well as the associated deformation densities.The shapes of the orbitals and deformations densities of the heavier homologues 2Ga and 2In are very similar to those of the aluminum compound and therefore, they are not shown here.The numerical results of the EDA-NOCV calculations are somewhat surprising.The calculated values for ΔE 1 and ΔE 2 suggest that the formation of the σ-type bond which is associated with ΔE 2 (Figure 4d) provides less stabilization energy than the second bond, which can be considered as "slipped" π bond which is the HOMO of the molecules (Figure 3c).The shape of the deformation densities indicates that Δρ 1 is associated with a significant reshaping of the π-type lone pair orbitals at (PMe 3 )AlH yielding the "slipped" π bond in 2Al while Δρ 2 is associated with the smaller charge accumulation in the σ-bonding region which agrees with the eigenvalues v 1/2 .The large stabilization which is associated with the formation of the HOMO in 2E (E = Al -In) can be explained with the change in the hybridization of the orbital at atom E which is a pure p AO in (PMe 3 )EH but it has between 39.2 % (Al) -36.0 % (In) s-character in the "slipped" π bond.Table 2 gives the NBO results for the E−E bonds which indicate the polarization and the hybridization of the bonds.The calculated Wiberg bond orders suggest a significant double-bond character for 2Al (1.58) and 2Ga (1.49) but less so for 2In (1.16).In contrast, the EDA-NOCV calculations suggest that the orbital interactions in 2Al (−78.2 kcal mol −1 ) are clearly weaker than in 2Ga (−104.6 kcal mol −1 ) and 2In (−92.9 kcal mol −1 ).Bond orders are related to the numbers of bonding orbitals, but they do not provide information about the strength of the associated stabilization energy.We want to point out that the pyramidal coordination at atom E in 2Al − 2In makes it impossible to clearly distinguish between σ and π bonds, because there is no mirror plane in the molecule.The calculated charge distribution shows that the donation Me 3 P → E 2 (H 2 ) ← PMe 3 for the boron compounds is rather strong (Δq = −1.26e) and that the boron atoms carry all of the negative partial charges while the hydrogen atoms are neutral.The charge donation in the heavier homologues 2Al -2In is weaker than in 2B and the negative charge is mainly on the hydrogen atoms.For all systems holds that the charge donation Me The bonding analysis of the complexes [(PMe 3 ) 2 (E 2 )] (3E) shall be divided into three parts, because the nature of the interacting orbitals is different for different atoms E. This becomes obvious from the inspection of the equilibrium geometries of 3E (Figure 1).The linear arrangement (PMe 3 )B−B(PMe 3 ) which is also found in the NHC homologue (NHC Dipp )B≡B(NHC Dipp ) that was recently isolated 7 suggests that the interacting fragment (PMe 3 )B has a quartet spin state as reference electronic state.Figure 4 shows the shape and eigenvalues of the relevant orbitals of the boron species 3B and (PMe 3 )B as well as the associated deformation densities.The figures make it clear that Δρ 1 refers to the formation of the B−B σ bond while Δρ 2 and Δρ 3 are associated with the formation of the degenerate B−B π bond.Note that the larger stabilization of the σ bond (−98.8 kcal mol −1 ) requires less charge flow (ν 1 = 0.370) than the π bond which provides a stabilization of −35.7 kcal mol −1 for each component with a larger charge deformation (ν 2/3 = 0.583).Table 1 shows that the B≡B triple bond has the strongest interaction energy ΔE int = −166.7 kcal mol −1 and also the biggest orbital interaction energy of ΔE orb = −177.2kcal mol −1 of all compounds.According to the The EDA-NOCV results for the aluminum and gallium analogues 3Al and 3Ga are very similar to each other and therefore, we show in Figure 5 only the shape and eigenvalues of the relevant orbitals of 3Al and (PMe 3 )Al with the associated deformation densities, together with the associated numerical values for both compounds.The planar trans-arrangement of the phosphine ligands and the E2 moieties in 3Al and 3Ga (Figure 1) suggest that the electronic reference state of the fragments (PMe 3 )Al and (PMe 3 )Ga is a quartet state.Table 1 shows that the total interaction energies ΔE int and the orbital interaction energies ΔE orb of 3Al and 3Ga using the quartet state of (PMe 3 )Al and (PMe 3 )Ga are still rather large.The largest contribution to ΔE orb comes from the formation of the "slipped" π bond HO-MO-1 which is associated with the charge deformation Δρ 1 .In contrast, the formation of the σ and π bond (HOMO-2 and HOMO in 3Al and 3Ga) stabilizes the molecules much less than the HOMO-1.
Table 1 shows that the In−In bond in 3In is the weakest E−E bond of all investigated species.It is significantly weaker than the In−In bonds in 1In and 2In which deviates from the trend of the other group-13 compounds which have the order 1E < 2E < 3E for the strength of the E−E bond.This can be explained with the electronic state of the fragment (PMe 3 )In which is a doublet rather than a quartet.There is thus only one unpaired electron at the fragments which can only form a single bond.Table 1 shows that the stabilizing orbital interactions ΔE orb in 3In have indeed only one major component ΔE 1 while the lighter homologues have three important components.The dominant orbital interaction is displayed in Figure 6 which shows the shape and eigenvalues of the relevant orbitals of 3In and (PMe 3 )In with the associated deformation density.A comparison of Figure 6 with the results for the aluminum analogue which are shown in Figure 5 reveals interesting differences.The unpaired electron in (PMe 3 )In resides in a p(π) AO at In which upon pairing gives the In−In σ bond (Figure 6).In contrast the unpaired p(π) electron in (PMe 3 )Al gives upon pairing the "slipped" π bond (Figure 5) whereas the Al−Al σ bond comes mainly from the sp(σ) AO at Al. Note, however, that the doubly occupied MOs in 3Al and 3Ga come from a mixture of the MOs of the fragments (PMe 3 )E where the SOMOs which are displayed give only the major component.
The NBO results for compounds 3E (Table 2) show also distinctively different result for 3B, 3Al / Ga and 3In.The bond order for the B−B bond in 3B is 2.41

1 B: 2 B: 2 Al:Figure 3 .
Figure 3. Plot of deformation densities Δρ of the pair wise orbital interactions between (a) and (b) two [(PMe3)(BH)] fragments and (c) and (d) two [(PMe3)(AlH)] fragments in their triplet states, associated energies ΔE in kcal mol −1 and eigenvalues v. Shape of the most important interacting occupied and vacant orbitals of [(PMe3)(EH)] and resulting molecular orbitals.The deformation densities have the direction red → blue.

Figure 4 . 1 Al: 2 Al: 3 Al:
Figure 4. Plot of deformation densities Δρ of the pairwise orbital interactions between two [(PMe3)(B)] fragments in their quartet state, associated energies ΔE in kcal mol −1 and eigenvalues v. Shape of the most important interacting occupied and vacant orbitals of [(PMe3)(B)] and resulting molecular orbitals.The deformation densities have the direction red → blue.

Figure 5 . 1 In:Figure 6 .
Figure 5. Plot of deformation densities Δρ of the pairwise orbital interactions between two [(PMe3)(Al)] fragments in their quartet state, associated energies ΔE in kcal mol −1 and eigenvalues v. Shape of the most important interacting occupied and vacant orbitals of [(PMe3)(Al)] and resulting molecular orbitals.The deformation densities have the direction red → blue.

Table 2 .
NBO results of the E-E bonds at BP86/def2-TZVPP.Partial charges q, bond orders P, occupation and hybridization of the E−E bond orbitals Croat.Chem.Acta 87 (2014) 413.