D2h	E	C2 (z)	C2 (y)	C2 (x)	i	σ (xy)	σ (xz)	σ (yz)
ag	+1	+1	+1	+1	+1	+1	+1	+1
b1g	+1	+1	-1	-1	+1	+1	-1	-1
b2g	+1	-1	+1	-1	+1	-1	+1	-1
b3g	+1	-1	-1	+1	+1	-1	-1	+1
au	+1	+1	+1	+1	-1	-1	-1	-1
b1u	+1	+1	-1	-1	-1	-1	+1	+1
b2u	+1	-1	+1	-1	-1	+1	-1	+1
b3u	+1	-1	-1	+1	-1	+1	+1	-1

For further information on this point group, see
Symmetry at Jacobs University

Select the format in which molecular structures are to be shown:

• JSmol requires HTML 5.0, and can be slow
• Jmol requires Java to be installed on the client machine, but is sometimes much faster

Molecular Orbitals for Diborane, B₂H₆

Jmol models of calculated wavefunctions

To view a model, click in a molecular orbital circle in the energy level correlation diagram shown

• Ignore any popup warning and click on the green Continue button which appears

Mouse Control of Models

Left mouse drag to rotate; Shift Left drag up or down to resize; Shift Right drag or Shift Left drag horizontally to z-rotate;
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Notes

Usage

• The orbitals models are shown in two popup windows, which are reused alternately so that you can compare one orbital with another
• Contours on a two-dimensional plot correspond to surfaces in three dimensions
• The initial view of a model is with surfaces at ψ = ±0.04
• A radio button is provided to 'Switch contours on'. This shows a two-dimensional contour plot in the xy or xz plane, chosen so that you see some contours for each model.   This is the plane of the screen when the model first appears, though you may then turn it around
• The calculation of the contour plot is slow in the default HTML5.0_JSmol format for the structures:   calculating contours for an orbital takes about 5–10 seconds in a good browser in a moderately fast PC (in 2019)
• You may see the contour plot better if you 'Switch surfaces off' and click 'Contours coloured'.  Black and white contours are provided as the default because the coloured ones do not show up very well through the coloured surfaces
• If you 'Switch ball and stick off' you can see contours right up to the nuclei, and where a 2s orbital contributes much you may see its inner lobe with opposite phase

The Molecule

• B₂H₆ may be considered to be a relatively weak acid-base complex between two BH₃ units
• While BF₃ is a strong Lewis acid, the acidity of boron is reduced by π-donation from its fluoride ligands, and BF₃ exists as a monomer
• No such π-donation is possible for BH₃, so even the weakly basic hydride ligand of another BH₃ unit can react to produce the dimer B₂H₆, which has two hydride bridges
• Diborane is often given as an example of electron-deficient bonding
• Almost all stable molecules, larger than diatomic, have filled bonding and non-bonding molecular orbitals, and empty antibonding orbitals
• B₂H₆ has the correct number of electrons to fit this prescription, so is not electron-deficient in this sense, unlike BH₃, which has an empty non-bonding orbital

The Energy Level Correlation Diagram

• This is highly schematic:  I have presented the calculated MO energy levels only qualitatively in the order found by the ab initio calculation used
• The MOs are shown with sequential orbital number and group theory representation names, as brief useful labels
• A character table is provided so that users may check the symmetry properties of the molecular orbital models which are generated
• Longer titles for the MOs are given in their popup model windows, and in the Table of Coefficients shown below in the present page:  these describe the main bonding or antibonding nature of the orbital
• Numbering of MOs starts at #3 because the first two MOs counted by an ab initio package, such as Gaussian, are combinations of the two boron core 1s levels
• MO energy level correlation diagrams in textbooks, for simpler, atom-centred molecules such as BF₃, usually have central atom AOs on one side, ligand AOs on the other, and the MOs in a central column so that correlation lines may be drawn to indicate the contributions of AOs to MOs
• Since the situation of bridging hydrogen atoms and of terminal hydrogen atoms in B₂H₆ are quite different, I have chosen to present their AOs in separate columns, which allows two columns of MOs with correlations lines, one for MOs which involve mainly the hydrogen bridges and one for MOs which involve mainly the terminal hydrogens
• All of the occupied MOs are involved also in some bonding or antibonding directly between the boron atoms, but these are generally less important contributions to bonding than contributions from overlap of boron AOs with hydrogen AOs
• Since hydrogen has only a 1s AO available for bonding, MOs which have a nodal plane lying through a hydrogen atom are completely non-bonding to it.   Thus, MOs with b_1u symmetry (red on diagram) are antisymmetric with respect to σ (xy) so are non-bonding to the terminal hydrogens, which lie in the xy plane.   These MOs may bond only the bridging hydrogens
• Similarly, MOs with b_1g or b_2u symmetry (green) are antisymmetric with respect to σ (xz) so are non-bonding to the bridging hydrogens, which lie in the xz plane.   These MOs may bond only the terminal hydrogens
• MOs with symmetry a_g (black) contain contributions from both bridging and terminal hydrogen AOs.   I have tried to assign them to the column of MOs corresponding to their bigger contribution.   Thus, MO #7 is assigned to the terminal bonding column rather than to bridge bonding

Choice of Axes

• B₂H₆ has D_2h symmetry, having three different C₂ axes, one along each Cartesian axis direction
• Which Cartesian axis is assigned to which C₂ axis, of the physical molecule, determines which group theory representation name is given to each MO
• Thus, the third most stable valence shell MO, labelled b_1u here, where the bridging hydrogen atoms lie along the z–axis and the x–axis runs through the boron atoms, would be labelled b_3u if the x–axis were chosen to lie through the bridging atoms and the z–axis through the borons
• This may be seen in the character table for D_2h, presented here, where representation b_1u would become representation b_3u if the values in the pair of columns labelled 'C₂ (z)' and 'C₂ (x)' were interchanged, and similarly the pair of columns labelled 'σ (xy)' and 'σ (yz)'
• There does not seem to be a consensus on this, for B₂H₆, among textbooks and various web sources
• The energies and appearances of the MOs remain the same:  only the symmetry labelling of the MOs changes when axis assignments are interchanged
• As I have used Gaussian 09 for my ab initio calculation for B₂H₆, I am using the axis convention of its 'Standard orientation'

MOs and Natural Atomic Orbitals (NAOs)

• The MO models shown on this web page were obtained at the RHF/3–21G^* level in a conventional ab initio calculation, using a Gaussian atomic basis set
• The Gaussian atomic basis set is an approximation to Natural Atomic Orbitals, 2s, 2p_z, etc., which are not very amenable to computation
• A Natural Bond Orbital analysis of the resulting MOs produced a set of NAOs and the coefficients of these needed to make the calculated MOs
• The valence shell NAO contributions to the MOs are shown in the Table of Coefficients
• The absolute value of a coefficient is given only once, since all NAOs of a given kind, e.g. B (2s), are related by the symmetry of the molecule
• This symmetry means that the coefficients may have the same or different signs.   The array of signs is given in the second row of each MO entry, in the order of atom numbers shown at the head (and foot) of the table
  • The relative signs in each array, for a set of NAOs, is the same for all MOs having a particular group theory representation name, though the array may be multiplied by –1, so as to generate the antibonding nodes which the make the MOs of a particular representation orthogonal to each other
  • As an example, MO #5 (b_1u) has signs + + for the B (2p_z) coefficients and + – for the bridging hydrogen coefficients
  • Inspection of the Jmol model for MO #5 shows that this makes it entirely bonding with respect to both boron-boron and boron-hydrogen interactions, since opposite signs on the two hydrogen orbitals are required to match the opposite signs of the two lobes of each p orbital
  • MO #13 (b_1u) has the same + – sign combination for the H orbitals, but has – – for the B (2p_z) coefficients.   This inverts the boron contribution to this MO and produces the clearly visible antibonding nodal surfaces
• The square of the coefficient, of a NAO in a MO, is the fraction of the NAO used in that MO. This, expressed as a percentage, is given in the third row for the MO
• The fourth row is this contribution, before rounding, multiplied by two or four depending on how many symmetry related instances of this NAO there are.   This shows how much of the MO is involved in bonding or antibonding to this kind of NAO
• Across a row, all of the NAO contributions per MO should add up to 100%, and, down a column, all of the uses of a NAO should sum to 100%, since either a NAO or a MO represents exactly one electron
• For the occupied MOs, the sums per MO are in the range 98.0–100.0%, so the analysis in terms of valence-shell NAOs is valid
• The non-bonding LUMO, MO #9, is only 89.9% described in terms of valence shell NAOs:   the remainder maps onto NAOs from higher shells
• The remaining, higher energy, empty, 'virtual' MOs are less well accounted for, and consequently are less likely to be realistic.   The order of their energies also varies more, depending on the level of theory used in the wavemechanical calculation, so may be different in different textbooks etc.
• Down a column, the contributions of each valence shell NAO sums to about 80%. The rest corresponds to its contributions to higher shell MOs

sp Mixing and MOs with Components which are Antibonding by Symmetry

• Of the boron 2p orbitals, only the 2p_x orbitals can give combinations of the same symmetries (a_g + b_3u) as do the boron 2s, and therefore give MOs which are sp mixtures
• For a particular boron atom, B (2p_x) and B (2s) appear the same when viewed from the centre of the molecule, because the nodal planes of B (2p_x) are not elements of symmetry of B₂H₆
• MOs of symmetry b_3u, e.g. MO #4, are antisymmetric with respect to σ (yz), which passes through the bridging hydrogen atoms.   This means not only that they are non-bonding to the bridging hydrogens, but also that they are antibonding with respect both to B (2s)—B (2s) and to B (2p_x)—B (2p_x) overlaps   They may be bonding or antibonding with respect to B (2s)—B (2p_x) overlap, since this is not imposed by the symmetry of the MO
• Similarly, though not involving B (2s) orbitals, MOs #8 (b_1g) and #9 (b_2g) are antisymmetric with respect to the yz plane, so their B (2p)—B (2p) overlaps also are antibonding by symmetry
• Of the H (1s)—H (1s) overlaps, H_bridge—H_bridge is antibonding by symmetry in MO #5 (b_1u), as a result of this being antisymmetric with respect to σ (xy), and gem–H_terminal—H_terminal is antibonding in MO #6 (b_2u) and MO #8 (b_1g), being antisymmetric with respect to σ (xz)

Three-centre, two-electron Localised Bonding versus the MO Approach

• Each of the two bridging hydride atoms may be considered to be involved in symmetric 3c—2e bonding, so in a MO treatment we might expect to find two occupied MOs which are largely bonding to the bridging hydrogens

• In fact we find three: MO #3 (a_g), #5 (b_1u) and #7 (a_g), which contribute to bridge bonding, but MO #7 is nearly non-bonding with respect to bridge bonding and B—B bonding taken together, so the simple 3c—2e bonding model, found traditionally in textbooks of Inorganic Chemistry, retains some validity
• The 3c—2e model implies that there should be two empty, non-bonding, boron-centred orbitals, one for each bridge, accounting for the observed Lewis acidity of B₂H₆
• The LUMO, MO #9 (b_2g), like the non-bonding π orbitals of CO₂, certainly is of this nature, though with an antibonding direct interaction between the boron 2p_z orbitals, while the next most stable empty MO, #10, is clearly antibonding to the bridging H atoms

Contributions of Orbitals to Bonding