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Molecular Orbitals for N2

Jmol models of calculated wavefunctions

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

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

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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 yz plane.   This calculation 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 2018)
• 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

• N₂ is a very stable 10-valence-electron molecule, isoelectronic with CO and with [CN]^–
• The formal bond order of N₂ is 3, from about one σ-bond and two π-bonds
• Its most important property is its lack of reactivity, so that, as the principal diluent, it can mitigate the dangerous properties of O₂ in air

MOs and Natural Atomic Orbitals (NAOs)

• The MO models shown on this web page were obtained at the RMPW1PW91/6–311g(2df) 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 square of the coefficient, of a NAO in a MO, is the fraction of the NAO used in that MO
• Most of these squares are shown as percentages against the correlation lines of the Energy Level Correlation Diagram
• All of the valence shell NAO contributions to the σ–MOs are shown in the Table of Coefficients
• Expressed as percentages, all of the AO contributions to a MO should add up to 100%, and all of the uses of an AO should sum to 100%, since either an AO or a MO represents exactly one electron
• However, some of the Gaussian atomic basis maps into core (1s) or higher (3s or 3p) NAOs, rather than into valence shell (2s or 2p) NAOs
• While the total of valence shell NAO contributions to e.g. π_y, amounts to 99%, or to π^*_y amounts to 95%, the higher energy, empty, 'virtual' MOs are less well accounted for, and consequently are less likely to be realistic.   Thus, only 26% of σ^*N(2p)N(2p) maps to valence shell NAOs, and the rest to n=3 or n=4 NAOs
• Besides the shortfalls in the total contributions to MOs, the Table shows also that each NAO is not wholly accounted for.   This is because the rest maps to even higher virtual MOs

Orthogonality of MOs

• The σ orbitals (black in the Energy Level Diagram) lie symmetrically across the π nodes of the π_x or π_y orbitals (red), so σ and π MOs do not mix
• Similarly, the π_x MO lies symmetrically across the π node of the π_y MO and vice-versa, so the π orbitals are orthogonal to each other and form a doubly degenerate set
• In contrast, the nodal planes of the 2p_z AOs do not correspond to an element of symmetry of the molecule, so they do mix with 2s AOs
• All four of the σ MOs contain both 2s and 2p_z contributions from both atoms

The σ System and sp Mixing

• For the three occupied σ orbitals, for each of the four pairs of N—N NAO overlaps, their contribution to bonding c₁c₂S₁₂ is shown in the Table of Relative Contributions of Overlaps to Bonding.   S₁₂ is the overlap integral between them calculated in the NBO analysis and c₁ and c₂ are their LCAO coefficients given in the first Table


• The total contribution to bonding of the three occupied σ orbitals together is 0.3817, or 47.9% of the total bonding including π bonding


• sp mixing puts p character into σN(2s)N(2s), making it even more stable.   It is now by far the biggest contributor to bond strength at 51.8% of the total bonding
• Without sp mixing, σ^*N(2s)N(2s) would be an entirely antibonding combination of 2s orbitals, and σN(2p)N(2p) would be an entirely bonding combination of 2p_z orbitals
• With sp mixing, σ^*N(2s)N(2s) becomes more stable, with a 38% N_2pz component
• It is still antibonding with respect to its N(2s) - N(2s) overlap, but bonding with respect to its N(2s) - N(2p_z) overlaps
• Overall it is now slightly bonding, and contributes 17.1% of the σ bonding, or 8.2% of the total bonding (including π bonding)
• σN(2p)N(2p) acquires antibonding N(2s) - N(2p_z) overlaps, making it overall slightly antibonding (a contribution of -12.1% to total bonding)
• The effects of the weakly bonding σ^*N(2s)N(2s) and the weakly antibonding σN(2p)N(2p) practically cancel out, leaving the bond order at 3

π Bonding

• Each π or π^* orbital should have only a 2p component from each atom, so each of the four coefficients should be (½)^½, i.e. 0.7071
• In the present wavemechanical calculation, the values found were 0.7053 for the π orbitals and ±0.6890 for the π^* orbitals

• The overlap integral of 2p_y orbitals on atoms 1 and 2 is 0.4178, so the Contribution to Bonding of this π orbital is 0.2078
• The total contribution of both π orbitals together is 0.4156, or 52.1% of total bonding, including σ bonding
• The π overlap integral is smaller than those for N(2s) - N(2p_z) or N(2s) - N(2s) overlaps in the σ system (see table), but bigger in magnitude than the N(2p_z) - N(2p_z) overlap integral
• Because the bonding in N₂ is dominated by σ bonding involving s orbital overlaps and by the π bonding, the equilibrium bond length is such as to favour those overlaps, but is too short for optimum N(2p_z) - N(2p_z) overlap, as shown in the following contour plot of overlapping 2p_z NAOs for N₂:

• Each 2p_z NAO extends beyond the other nucleus and hence the nodal plane of the other 2p_z NAO
• Products of the two functions in that region are positive, making the integral less negative

HOMO and LUMOs

• The HOMO of dinitrogen is σN(2p)N(2p) because the antibonding contribution from sp mixing pushes it above the π–bonding orbitals in energy


• The LUMOs are the doubly degenerate pair of π^* orbitals
• The antibonding nodal surface (approximately the xy plane) is clearly recognisable in the rotatable model