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For a series of comparable metal carbonyls, greater metal-to-CO pi backbonding populates CO antibonding orbitals, weakens its internal bond and lowers its infrared stretching frequency. Coupled normal modes and changing geometry must be considered when comparing observed bands.

At fixed $+2$ oxidation state, increasing effective nuclear charge across the first transition series generally contracts the ionic radius. The first expectation is therefore shorter average metal–nitrogen and metal–sulfur distances toward copper. This is a trend in comparable coordination environments, rather than a requirement that every individual contact shorten monotonically. The occupation of metal–coordination ligand antibonding orbitals, donor identity and lattice distortion all affect the observed distances.

The high-spin $d^5$ and $d^6$ ions both have two electrons in the strongly sigma-antibonding $e_g$ set, so adding the sixth electron to $t_{2g}$ does not itself introduce another sigma-antibonding electron. Copper(II), however, has $e_g^3$ occupation and is subject to the Jahn-Teller effect. A tetragonal Jahn-Teller distortion commonly gives four shorter and two longer contacts, so an elongated copper contact can oppose the overall size trend. Here the nitrogen and sulfur donors are already chemically inequivalent; the supplied schematic alone does not determine which trans pair becomes longest. One should compare actual M–N and M–S sets separately, rather than infer a unique axis or a numerical distance from the magnetic moments. The experimentally determined structures illustrate why the donor geometry and individual contacts matter.^[ref]

Take the M–C–O direction as $z$. carbon monoxide binds through carbon because its occupied $5\sigma$ donor orbital has a substantial carbon-end lobe. End-on overlap with an empty metal orbital of sigma symmetry gives CO-to-metal sigma donation. At the same time, occupied metal $d_{xz}$ and $d_{yz}$ orbitals overlap sideways with the two corresponding empty CO $2{\pi }^{\ast }$ orbitals, giving metal-to-CO pi backbonding. Both channels strengthen the metal–carbon bond, while backbonding lowers the internal C–O bond order. The processes are synergic: sigma donation helps supply metal electron density for backbonding, while backbonding relieves electron accumulation at the metal.

The original orbital sketch shows one of the two perpendicular pi channels. It is the occupied metal orbital, rather than an empty metal orbital, that supplies electrons to CO antibonding orbitals.

In a metallocene, several orbitals with substantial metal $d$ character are nonbonding molecular orbitals or only weakly bonding or antibonding with respect to the strong Cp–metal framework. Adding or removing a small number of electrons from them can therefore change oxidation state or spin without destroying the principal ring–metal bonds. This allows stable compounds on either side of the 18-electron count: for example, the usual neutral counts for chromocene, ferrocene, cobaltocene and nickelocene are respectively 16, 18, 19 and 20. The 18-electron rule remains a useful reference, but occupation of a weakly antibonding orbital is much less disruptive than occupation of a strongly antibonding orbital. Redox stability also depends on geometry, substituents and reaction conditions, so this is a bonding explanation of the observed flexibility rather than a guarantee of equal stability for all counts.

In a boroxine, each boron has three sigma bonds and is approximately trigonal planar, with an $s{p}^2$ description for its sigma framework and an available perpendicular $2p$ orbital. oxygen can be described approximately as $s{p}^2$: two in-plane hybrids form B–O sigma bonds, one contains an in-plane lone pair, and a perpendicular $2p$ lone pair overlaps with neighbouring boron $2p$ orbitals. These three oxygen donor pairs supply a six-electron pi system around the alternating ring. The combination of good same-shell overlap and a low-strain six-membered ring favours planarity and partial B–O multiple-bond character. The bonds remain polar; a six-electron diagram alone does not establish benzene-like aromatic stabilization quantitatively.

In a siloxane, each silicon instead has four sigma bonds, two to oxygen and two to its organic groups, giving an approximately tetrahedral $s{p}^3$ sigma framework. oxygen has two sigma bonds and two lone pairs, so an approximately $s{p}^3$ localized description is a useful starting point, although actual Si–O–Si angles and lone-pair hybrids vary considerably. There is no comparable ring of vacant silicon $2p$ orbitals. A description based on strong oxygen-$2p$ to silicon-$3d$ pi bonds is also misleading: those orbitals have poor energy and radial matching. Strongly polar Si–O sigma bonding, supplemented by negative hyperconjugation from oxygen lone pairs into appropriate Si–R antibonding orbitals, gives a better account. The ring can therefore pucker rather than requiring continuous planar pi overlap. This follows the broader conclusion that strong main-group bonds and apparently expanded valence do not imply dominant valence $d$-orbital participation.^[ref]

The anomeric effect is the preference, in suitable cyclic acetals, for an electronegative substituent at an anomeric carbon to occupy an axial position even when simple steric reasoning favours equatorial. An important orbital contribution is donation from an endocyclic oxygen lone pair into the antibonding orbital of the adjacent exocyclic C–O bond, $n_{\mathrm{O}}\to {\sigma }_{\mathrm{C}-\mathrm{O}}^{\ast }$. The chair's axial arrangement supplies favourable antiperiplanar overlap. Electrostatic and solvation contributions can also affect the observed conformational preference.

The reacting Homo is the tertiary-amine nitrogen lone pair. The relevant LUMO is the C-I antibonding orbital, ${\sigma }_{\mathrm{C}-\mathrm{I}}^{\ast }$, of MeI. Backside lone-pair donation into that orbital forms N-C while weakening C-I. The neutral OH group is not first converted to an alkoxide in the stated reaction.

Strong ligand-to-metal pi donation is favoured by an occupied ligand p orbital and a suitable empty metal d orbital of matching symmetry, with reasonably similar energies and good spatial overlap. Small donor atoms such as O or N can supply short bonds and efficient overlap; oxo, imido and nitrido ligands are familiar examples. Early transition metals in high oxidation states, often $d^0$, are particularly favourable because the metal acceptor orbitals are vacant and relatively low in energy. More spatially extended $4d$ and $5d$ orbitals can strengthen the overlap.

Metal-to-ligand pi backbonding is the complementary case: electron-rich, low-oxidation-state metals donate occupied d orbitals into low-lying ligand antibonding orbitals, as for CO. In both directions, filling a metal–ligand pi-antibonding orbital offsets the multiple-bond stabilization. The relevant prediction is orbital occupancy and overlap, rather than the presence of a formal double-bond line in a drawing.

For oxygen, eight valence electrons occupy bonding orbitals and four occupy antibonding orbitals. Its bond order isThe additional electron in superoxide enters a ${\pi }_g^{\ast }(2p)$ antibonding orbital, givingReduced bond order weakens and lengthens the bond, consistent with the longer bond in superoxide. The diagram explains the direction of the change; it does not predict the two numerical bond lengths without a quantitative electronic-structure calculation.

Two atomic orbitals. In the linear combination of atomic orbitals approximation, use $|\psi ⟩=c_A|A⟩+c_B|B⟩$, where the two basis functions are localized atomic orbitals. Projecting $\hat{H}|\psi ⟩=E|\psi ⟩$ onto each orbital gives the generalized eigenvalue problemHere the scalar overlap in the matrix is $S=⟨A|B⟩$, while $H_{ab}=⟨a|\hat{H}|b⟩$. Normalization requires $c^{†}Sc=1$, with $S$ now denoting the full overlap matrix. For equivalent real orbitals with diagonal energy $E_a$, real hopping $t$, and real overlap $s$, the eigenstates and energies areFor the usual negative hopping convention, the symmetric combination is the lower bonding orbital and the antisymmetric combination is the higher antibonding orbital. In an orthonormal approximation, $s=0$ and the result reduces to a two-by-two Hermitian matrix eigenproblem.

Hybridized bands. At each $\mathbf{k}$, take the two Bloch basis states to be orthonormal, as implicit in the stated band model, and write ${ϵ}_k={\hslash }^2{k}^2/(2m)$. The Hamiltonian isFor $|{\psi }_{\mathbf{k}}⟩=a_{\mathbf{k}}|{\varphi }_{\mathbf{k}}⟩+b_{\mathbf{k}}|{\chi }_{\mathbf{k}}⟩$, the eigenvalue condition isso $(E-{ϵ}_k)(E-E_d)-|V{|}^2=0$. Thus band hybridization givesFor nonzero $V$, a convenient explicit normalized pair of eigenstates isA phase convention can make $V$ real, but the eigenvalues depend only on its modulus. These are exact within the truncated two-band model, and approximate as a description of a material.

Dispersions and limits. For $V=0$, the parabola crosses the flat band atFor $V\ne 0$ the crossing is avoided, with a direct hybridization gap $2|V|$ at $k_0$. The lower and upper branches exchange orbital character through this avoided crossing. At the origin their energies areso for $|V|\ll E_d$ the small-$k$ expansions areAt large $k$, where ${ϵ}_k-E_d\gg |V|$,The lower branch approaches $E_d$ from below and becomes flat; the upper branch approaches the free-electron parabola from above. At zero coupling the unsorted orbital bands remain exactly flat and parabolic. Sorting them by energy instead gives $E_{-}=\min ({ϵ}_k,E_d)$ and $E_{+}=\max ({ϵ}_k,E_d)$, which cross at $k_0$ without mixing.

Fermi wavevector and velocity. A partially occupied perfectly flat band has chemical potential $E_d$. Thus before mixing, the occupied broad band has radius $k_0$ defined above. For spin degeneracy two, its particle density is $n_1=k_0^3/(3{\pi }^2)$; equal population of the flat sector gives total density $n=2n_1$.

For any nonzero mixing, $E_{-}(k)<E_d$ at finite $k$, while $E_{+}(k)>E_d$. Moreover $E_{-}(k)$ increases monotonically. At zero temperature and this fixed density, a single occupied sphere in the lower branch therefore accommodates all particles, and the upper branch is empty. Fermi-volume conservation in two hybridized electron bands givesThe new radius is about $26\%$ larger than the old broad-band radius. The strictly flat uncoupled band has no uniquely determined Fermi radius of its own; its prescribed population is what enters the count.

The radial group velocity is $v_{-}(k)={\hslash }^{-1}d{E}_{-}/dk$. Differentiating the exact dispersion givesAt $k_F$, put $\delta =2^{2/3}-1$, so ${ϵ}_{k_F}-E_d=\delta E_d$. Relative to the original broad-band Fermi velocity $v_F^{(1)}=\hslash k_0/m$,Expanding for $|V|\ll E_d$ gives the flat-band hybridization Fermi-velocity suppressionThe coefficient is about $3.65$. The enlarged occupied volume coexists with a small velocity because the new Fermi surface lies on the almost-flat part of the lower branch. This result assumes fixed particle density, sufficient available states before any Brillouin zone boundary, and temperature low enough to resolve energy differences of order $|V{|}^2/E_d$. The limit $V\to 0$ at a fixed nonzero temperature need not give this sharp redistribution. Giving the narrow band a finite dispersion can also leave both pockets occupied, so the result is specific to the ideal model.