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Molecular Orbitals for CO

Jmol models of wavefunctions calculated at the RHF/3–21G* level

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

The results displayed may be switched between those from a low level of calculation and those from a high level.

• There is rather little difference, either in the orbital models or in the numbers describing them in the text below, between the two levels of calculation
• Use the low level for comparison of CO, carbon monoxide, with CO₂, carbon dioxide
• Use the high level for comparison with N₂, nitrogen

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

• CO is a very stable 10-valence-electron molecule, isoelectronic with [CN]^– and with N₂, which has a slightly lower bond dissociation energy than CO
• The formal bond order of CO is 3, from about one σ-bond and two π-bonds
• Its most important property is burning in air to give CO₂, in the combustion of fossil fuels
• Chemically, its most important reactivity is as a π–acceptor ligand in metal carbonyl complexes

Polarisation

• More electronegative elements attract their valence shell electrons more strongly, so their AOs lie at lower energy than those of less electronegative elements
• In a simple overlap of two AOs, the resulting bonding and antibonding MOs each resembles most the constituent AO nearest to it in energy
• This means that the bonding MO most resembles the AO of the more electronegative partner
• Bonding MOs are said to be polarised towards the more electronegative partner atom:   since bonding MOs are filled and antibonding MOs tend to be empty, this amounts to the classical definition of electronegativity
• An antibonding MO in a simple overlap is composed of those amounts of AOs not used in making the bonding orbital, so the antibonding MO most resembles the AO of the less electronegative partner.   We say that antibonding MOs are polarised towards the less electronegative partner

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
• This 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
• Some 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 94%, the higher energy, empty, 'virtual' MOs are less well accounted for, and consequently are less likely to be realistic.   Thus, only 51% of σ^*C(2p)O(2p) maps to valence shell NAOs, and the rest to n=3 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

sp Mixing

• 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
• This is in contrast to the 2s and 2p_z AOs of carbon in carbon dioxide, which do not mix because there the nodal plane of 2p_z is an element of symmetry of the molecule.   (2s and 2p_z AOs on oxygen in CO₂ are like those on carbon or oxygen in CO, and do mix)
• Without sp mixing, σ^*C(2s)O(2s) would be an entirely antibonding combination of 2s orbitals, and σC(2p)O(2p) would be an entirely bonding combination of 2p_z orbitals.   The formal bond order would clearly be 3, since bonding due to σC(2s)O(2s) would be cancelled out by the antibonding effect of σ^*C(2s)O(2s), leaving a net of one σ bond to go with the two π bonds
• For simplicity, the unmixed MO labels are retained on this CO web page, even though each σ MO is a mixture of four AOs and in some cases the AOs in the label are not the most important two

• With sp mixing, σ^*C(2s)O(2s) becomes more stable, with a 64% O_2pz component, which is now its main constituent
• It is still antibonding with respect to its C(2s) - O(2s) overlap, but bonding with respect to its C(2s) - O(2p_z) overlap
• Overall it is now slightly bonding, and contributes 11% of the σ bonding, or 5.9% of the total bonding (including π bonding)
• For the three occupied σ orbitals, for each pair of C—O 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


• sp mixing puts p character into σC(2s)O(2s), making it even more stable.   It is now by far the biggest contributor to bond strength at 55% of the total bonding
• In return, σC(2p)O(2p) acquires antibonding s character, making it overall slightly antibonding (a contribution of -9.1% to total bonding)
• The effects of the weakly bonding σ^*C(2s)O(2s) and the weakly antibonding σC(2p)O(2p) practically cancel out, leaving the bond order at 3

HOMO and LUMOs

• The HOMO of carbon monoxide is σC(2p)O(2p) because the antibonding contribution from sp mixing pushes it above the π–bonding orbitals in energy
• Its main components are C_2s and C_2pz, so it is strongly polarised towards carbon, and will bond to σ–acceptor species through carbon, providing that the CO ligand is also acting as a π–acceptor (the 'synergic effect')
• In the rotatable model, you may see that the oxygen contribution is just the p orbital, and there is a curved nodal surface between it and the large C_2s part


• The LUMOs are the doubly degenerate pair of π^* orbitals
• At 72% C_2p, they are clearly polarised towards carbon, and the antibonding nodal surface (approximately the xy plane) is clearly recognisable in the rotatable model
• The bonding of carbon monoxide as a π–acceptor ligand to a π–donor transition metal is often thought of in terms of the reactivity of the CO π^* orbitals of the free molecule