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 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

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;
Right click for menu

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

• Because of the symmetry of the homonuclear molecule, for a perfect model based only on hydrogen-like valence shell NAOs, only two independent variables would be required to determine all 16 coefficients
• For example, if the coefficients of the 2s NAO of atom 1 in σN(2s)N(2s) and in σ^*N(2s)N(2s) were known, then the magnitudes of all of the other 14 coefficients could be derived from them because of the requirement that each MO represents exactly one electron, and that each NAO is used exactly once
• Their relative signs are all known because of the requirement that all MOs are orthogonal to each other
• The coefficients in the first table may be understood using the following mental process
• MOs are considered as linear combinations of AOs
• A linear combination produces the same results, irrespectively of the sequence in which the terms are added together, so we can consider the process in two steps

• The first step is entirely imaginary, so it can be perfect: the intermediate coefficients generated in this step are exactly known to any desired precision
• The 2s orbitals from each atom are combined to make a bonding σ_g(s) and an antibonding σ_u(s)^* combination
• The g subscript indicates that the value of the wavefunction at any point is the same as that at a corresponding point on the other side of the centre of symmetry, which is at the middle of the bond.   The u indicates values of the same magnitude as each other, but of opposite signs, at such a pair of points
• As there are only two, equal components in σ_g(s), each must represent half of an electron, so the coefficients of the AOs will each be (½)^½, i.e. 0.7071
• In the order of NAOs shown in the first table, the row of coefficients for σ_g(s) would
  be 0.7071, 0, 0.7071, 0
  while for σ_u(s)^* it would be 0.7071, 0, -0.7071, 0
• Likewise, we can consider bonding, σ_g(p) and antibonding σ_u(p)^* combinations of the 2p_z orbitals
• The row of coefficients for σ_g(p) would be 0, -0.7071, 0, 0.7071
  while for σ_u(p)^* it would be 0, 0.7071, 0, 0.7071
• Note that, in the bonding combination, the coefficients have opposite signs because the negative lobe of one 2p_z orbital overlaps the positive lobe of the other, so the overlap integral is negative.   A bonding orbital has a positive contribution to bonding c₁c₂S₁₂, where S₁₂ is the overlap integral ∫φ₁φ₂dτ and c₁ and c₂ are the coefficients, so if the overlap integral is negative then the product of the coefficients must also be negative
• Now, in the second stage of our imaginary linear combination, since they have the same symmetry σ_g relative to molecular symmetry, the partial combinations σ_g(s) and σ_g(p) may be further combined to make the calculated, more bonding and less bonding, combinations σN(2s)N(2s) and σN(2p)N(2p)
• For simplicity, the unmixed MO labels are retained on this N₂ web page, even though each σ MO is a mixture of four AOs
• For N₂, the names represent the major components in each case
• This time, the amounts of σ_g(s) and of σ_g(p) will not be equal by symmetry, but should add up so that each MO describes one electron, and each partial combination is used once altogether, over the two MOs to which each contributes

• Their values may be deduced by working back from the optimum values of coefficients of NAOs in MOs found by the wavemechanical calculation and presented in the Table of Coefficients
• Thus, the coefficient of the 2s NAO of atom 1 in σN(2s)N(2s) is shown as 0.5815, so the coefficient of σ_g(s) in this MO is 0.5815/0.7071 = 0.8224, or 67.6%
• Similarly, the coefficient of σ_g(p) in this MO comes out as 0.5382, i.e. 29.0%
• The percentages do not quite sum to 100 because of inexact mapping of NAOs to Gaussian basis set orbitals, as explained above
• For σN(2p)N(2p) the coefficient of σ_g(s) is 0.5494 (30.2%) and of σ_g(p) is -0.8231 (67.7%), accounting for the remainder of the σ_g(s) and σ_g(p) partial combinations
• Notice that coefficients of σ_g(p) are of opposite sign in making the pair of MOs σN(2s)N(2s) and σN(2p)N(2p).   Just as for making bonding and antibonding combinations, this is to make the MOs orthogonal to each other
• The same principles apply to putting together σ_u(s)^* and σ_u(p)^* to make the σ_u MOs σ^*N(2s)N(2s) and σ^*N(2p)N(2p)
• Here, the coefficient of σ_u(s)^* in σ^*N(2s)N(2s) is 0.7765 (60.3%), and of σ_u(p)^* is 0.6194 (38.4%)
• Because of the poor mapping of valence shell NAOs to σ^*N(2p)N(2p), it is not worthwhile to work back from its calculated coefficients.   However, we can see that they should account for the rest of σ_u(s)^* and σ_u(p)^*, so, ideally, the coefficients would be about 0.63 (40%) and -0.78 (60%)

• 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