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Molecular Orbitals for Peroxide Ion

Jmol models of calculated wavefunctions

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Notes

• This web page presents evidence from an ab initio modelling calculation, which may be useful to those learning or teaching about molecular orbitals for simple inorganic species
• It does not set out to teach
• If you would like a tutorial aimed at students following a beginners' course about atomic orbitals and their linear combination to make molecular orbitals, you could try Tutorial using the program Orbital

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 Species

• Peroxide ion, [O₂]^2–, has 14 valence-shell electrons and is isoelectronic with fluorine, F₂
• The formal bond order of [O₂]^2– is one, since the effects of the bonding σO(2s)O(2s) orbital and the two bonding π orbitals are notionally cancelled out by the filled antibonding σ^*O(2s)O(2s) orbital and the two filled π^* orbitals respectively (but see below)
• The bond order of one corresponds to there being one empty antibonding valence shell orbital, σ^*O(2p)O(2p)
• As its sodium salt, peroxide ion is much more readily handled than is F₂, but it is also a very strong oxidising agent
• Its protonated derivative H₂O₂, hydrogen peroxide, is manufactured on a very large scale for making bleaching and anti-microbial products

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
• A rationalisation of the calculated coefficients for a homonuclear diatomic species may be seen for N₂ in a separate window.   The same method could be used for peroxide ion

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

• The extent of sp mixing depends upon the closeness in energy of the 2s and 2p NAOs
• They become further apart in energy with increasing effective nuclear charge, e.g. in going from nitrogen to oxygen in the Periodic Table
• In the current NBO analysis for [O₂]^2–, the 2s NAO was calculated to be 0.6814 H (18.543 eV) more stable than 2p_z, compared with 0.4603 H (12.525 eV) for N₂
• There is therefore much less sp mixing in [O₂]^2– than in N₂:   the σO(2s)O(2s) MO of peroxide ion contains 5.6% p character, compared with 29% for N₂, and the σ^*O(2s)O(2s) orbital contains 8.7% p character compared with 38%
• Correspondingly, in peroxide there was 5.8% s character in σO(2p)O(2p) compared with 30% for N₂
• On the corresponding web page for N₂ the coefficients and percentage contributions of NAOs to MOs are presented in the same form as the Table of Coefficients here for peroxide



Table of Relative Contributions of Overlaps to Bonding
MO	Overlapping AOs		Overlap integral S	Contribution c1c2S	Total for MO
	Atom 1	Atom 2
σO(2s)O(2s)	O2s	O2s	0.1830	0.0857
σO(2s)O(2s)	O2s	O2pz	0.3028	0.0348
σO(2s)O(2s)	O2pz	O2s	-0.3028	0.0348
σO(2s)O(2s)	O2pz	O2pz	-0.3045	0.0086	0.1639
σ*O(2s)O(2s)	O2s	O2s	0.1830	-0.0832
σ*O(2s)O(2s)	O2s	O2pz	0.3028	0.0425
σ*O(2s)O(2s)	O2pz	O2s	-0.3028	0.0425
σ*O(2s)O(2s)	O2pz	O2pz	-0.3045	-0.0132	-0.0114
σO(2p)O(2p)	O2s	O2s	0.1830	0.0053
σO(2p)O(2p)	O2s	O2pz	0.3028	-0.0352
σO(2p)O(2p)	O2pz	O2s	-0.3028	-0.0352
σO(2p)O(2p)	O2pz	O2pz	-0.3045	0.1425	0.0774

• For the three occupied σ orbitals, for each of the four pairs of O—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


• Although there is less sp mixing in peroxide, compared with N₂, this does not mean that it is unimportant here
• Because there is no net π bonding, the optimum bond length is longer:  it was calculated for gas-phase peroxide ion in this study as 1.5838 Å, compared with 1.0884 Å for N₂
• The bond length in peroxide is now too long for good 2s — 2s overlap, but 2s — 2p_z overlap is less affected, and 2p_z — 2p_z overlap is considerably stronger
• As shown in the Table of Relative Contributions of Overlaps to Bonding, the pair of 2s — 2p_z overlaps still provide 42% of the bonding contribution of the σO(2s)O(2s) MO, compared with 54% in N₂
• The 2s — 2p_z overlaps in σ^*O(2s)O(2s) are insufficient to make this MO bonding as they do in N₂, so it is net antibonding, as the name used here implies
• Likewise, they are not antibonding enough in σO(2p)O(2p) to prevent that MO from being bonding, in contrast to N₂
• In peroxide ion, the bonding σO(2p)O(2p) orbital is calculated to be below the π bonding orbitals in energy, as is the corresponding MO in F₂, whereas with antibonding character it appears above the π bonding levels in N₂
• The idea that bonding by the σO(2s)O(2s) orbital is cancelled out by antibonding by the σ^*O(2s)O(2s) orbital, leaving the σO(2p)O(2p) orbital to contribute a bond order of 1, is convenient and similar to arguments presented in some elementary textbooks, but according to the present NBO study it is far from the 'truth'.   Because of sp mixing, the σO(2s)O(2s) orbital here contributes 71.3% of σ bonding (or total bonding), while σO(2p)O(2p) contributes only 33.7%.   σ^*O(2s)O(2s) has an antibonding effect of only -5.0%

π Bonding

• Because of the long bond, the overlap integral between 2p_y orbitals on atoms 1 and 2 is 0.1709, compared with 0.4178 for N₂
• The Contribution to Bonding of the π_y orbital of peroxide is 0.0854, compared with 0.2078 for N₂, where strong π bonding is the dominant feature of the MO description and the equilibrium bond length is short so as to allow good π overlap.   (See the figures showing NAO overlaps in the π Bonding section of the N₂ web page)
• However, the smallness of the π bonding contribution in peroxide is of little concern because it is exactly cancelled out, to within the precision quoted, by the antibonding contribution of the corresponding filled π^* orbital, and this would be so, whatever the bond length.   As chemists, we think of this situation as no π bonding

HOMO and LUMOs

• The HOMOs of peroxide ion are the doubly degenerate pair of π^* orbitals
• Attacking Lewis acids, e.g. H⁺, would be expected to bond to opposite ends of these two orbitals, rationalising the known structure of H₂O₂
• Removal of electron density from the antibonding π^* HOMOs, towards attached Lewis acid species, might be expected to strengthen and shorten the O—O bond, just as donation of electron density into the π^* LUMOs of CO is often given as an explanation of weakening of the C—O bond during the formation of carbonyl complexes
• In fact, the O—O bond is shortened to about 1.499 Å in Na₂O₂, when the theoretical Lewis acids are sodium ions, and to 1.475 Å in H₂O₂ in the gas phase, when they are protons


• The LUMO is the σ^*O(2p)O(2p) orbital