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Molecular Orbitals for N2
Jmol models of calculated wavefunctions
To view a model, click on a molecular orbital in the energy level correlation diagram shown
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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 twodimensional 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
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calculating contours for an orbital takes about 5–10 seconds
in a good browser in a moderately fast PC (in 2018)
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'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
 N_{2} is a very stable 10valenceelectron molecule, isoelectronic with
CO and with [CN]^{–}
 The formal bond order of N_{2} is 3, from about one σbond and
two πbonds
 Its most important property is its lack of reactivity, so that, as the principal
diluent, it can mitigate the dangerous properties of O_{2} in air
MOs and Natural Atomic Orbitals (NAOs)
Table of Coefficients and of % of each NAO used,
for each σ–MO 
NAO: 
Atom 1 
Atom 2 
N_{2s} 
N_{2pz} 
N_{2s} 
N_{2pz} 
MO 
coeff.  % used 
coeff.  % used 
coeff.  % used 
coeff.  % used 
σN(2s)N(2s) 
0.5815  34 
0.3806  14 
0.5815  34 
0.3806  14 
σ^{*}N(2s)N(2s) 
0.5491  30 
0.4380  19 
0.5491  30 
0.4380  19 
σN(2p)N(2p) 
0.3885  15 
0.5820  34 
0.3885  15 
0.5820  34 
σ^{*}N(2p)N(2p) 
0.2476  6 
0.2605  7 
0.2476  6 
0.2605  7 
 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 viceversa, 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
These notes may be displayed with or without an explanation of the
values of the LCAO coefficients shown in the Table of Coefficients.
Explanation is currently
off
Table of Relative Contributions of
Overlaps to Bonding 
MO 
Overlapping AOs 
Overlap integral S 
Contribution c_{1}c_{2}S 
Total for MO 
Atom 1  Atom 2 
σN(2s)N(2s) 
N_{2s} 
N_{2s} 
0.4795 
0.1621 
σN(2s)N(2s) 
N_{2s} 
N_{2pz} 
0.5075 
0.1123 
σN(2s)N(2s) 
N_{2pz} 
N_{2s} 
0.5075 
0.1123 
σN(2s)N(2s) 
N_{2pz} 
N_{2pz} 
0.1797 
0.0260 
0.4127 
    
σ^{*}N(2s)N(2s) 
N_{2s} 
N_{2s} 
0.4795 
0.1446 
σ^{*}N(2s)N(2s) 
N_{2s} 
N_{2pz} 
0.5075 
0.1221 
σ^{*}N(2s)N(2s) 
N_{2pz} 
N_{2s} 
0.5075 
0.1221 
σ^{*}N(2s)N(2s) 
N_{2pz} 
N_{2pz} 
0.1797 
0.0345 
0.0651 
    
σN(2p)N(2p) 
N_{2s} 
N_{2s} 
0.4795 
0.0724 
σN(2p)N(2p) 
N_{2s} 
N_{2pz} 
0.5075 
0.1147 
σN(2p)N(2p) 
N_{2pz} 
N_{2s} 
0.5075 
0.1147 
σN(2p)N(2p) 
N_{2pz} 
N_{2pz} 
0.1797 
0.0609 
0.0961 
 For the three occupied σ orbitals, for each
of the four pairs of
N—N NAO overlaps, their contribution to bonding
c_{1}c_{2}S_{12} is shown in the Table of Relative Contributions of
Overlaps to Bonding.
S_{12} is the overlap integral between them calculated in the NBO
analysis and c_{1} and
c_{2} 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 N_{2} 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 N_{2}:
 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