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Occupying an antibonding molecular orbital opposes a chemical bond. For equivalent $1s$ atomic orbitals of the hydrogen atom, the difference ${\sigma }_u^{\ast }={\sigma }_{-}$ vanishes on the plane halfway between the protons: every point there is equally far from both centres, so the amplitudes cancel. This nodal plane removes the contribution of this molecular orbital to electron density on that plane; it does not make an entire finite region empty. The $1s$-derived antibonding molecular orbital lies above its bonding molecular orbital partner in energy. See Field, MIT 5.61, lecture 25, pp. 1–2.

In the ${\mathrm{He}}_2$ minimal model, $({\sigma }_u^{\ast }{)}^2$ adds $N_a=2$ and cancels the bond order contribution of $({\sigma }_g{)}^2$. Opposite signs on the two lobes describe probability amplitudes, while both lobes have nonnegative probability density after squaring. The star is an orbital label, not an extra electron or an electric charge.

The neutral molecule Be-two. Its simplest filled sigma-two-s/bonding and antibonding description gives zero bond order, but mixing of accessible two-p configurations and electron correlation produce a weak covalent bond. Resolved vibrational levels establish a bound ground-state potential.^[ref]

Occupying a bonding molecular orbital favours a chemical bond. For the two equivalent $1s$ atomic orbitals of the hydrogen atom, the sum ${\sigma }_g={\sigma }_{+}$ has constructive interference between the protons. The resulting electron density in that region helps stabilize the combined system. Its lower energy in this model follows from the Quantum Hamiltonian, not from the plus sign alone; see Field, MIT 5.61, lecture 25, pp. 1–2.

For example, $({\sigma }_g{)}^2$ in molecular hydrogen supplies $N_b=2$, hence bond order one when $N_a=0$. The sum in the molecular orbital theory toy has midpoint probability density $0.303468$, whereas the difference has zero there. The normalization factors must be included in this comparison.

In the elementary molecular orbital theory count, bond order is half the excess of electrons in bonding molecular orbitals over those in antibonding molecular orbitals. Thus adding one electron to a bonding molecular orbital raises $b$ by $1/2$, while adding one to an antibonding molecular orbital lowers it by $1/2$. The configurations drawn above give $1$, $1/2$, and $0$. This count summarizes net covalent bond occupancy; it is neither a measured energy nor a universal test for every kind of binding. See Chemistry 2e, 8.4: molecular orbital bond order.

Andreas Orthaber, Ferdinand Belaj and Rudolf Pietschnig, Comptes Rendus Chimie 13 (2010), 923–928. Section 3.2 analyses the strongly polar P–O sigma bond and oxygen lone-pair negative hyperconjugation; this is not a claim that every orbital-localization scheme assigns the same bond order.

An electron domain is a region of electron density counted around a central chemical atom in the VSEPR model. One lone pair counts as one domain; a bond to one neighbour counts as one domain even when its bond order is two or three. Domain count is therefore different from simply counting electron pairs.

A heteronuclear three-valence-electron diatomic. A simple bonding/antibonding two-orbital description predicts bond order one-half and one unpaired electron. More electronegative beryllium carries the negative end of the bond dipole.

The neutral molecule Li-two. In the elementary valence molecular orbital description, its two valence electrons occupy the bonding sigma-two-s orbital, giving bond order one and a closed-shell ground state.

A metal-metal bond joins two metal centers directly through overlap of their atomic orbitals. A single bond contributes one electron to the local count at each metal in conventional electron counting. Bridging ligands can accompany direct bonding, but a short bridged separation alone does not establish its bond order.

Molecular orbital theory describes the electronic arrangement of a molecule using molecular orbitals: one-electron spatial wavefunctions extending over the combined atomic nuclei. Filling bonding molecular orbitals can stabilize a chemical bond; filling antibonding molecular orbitals opposes that stabilization. A molecular orbital is a building block of a many-electron description, rather than the entire many-electron wavefunction. See Robert W. Field, MIT 5.61, lecture 24, pp. 1–4: molecular orbitals and the LCAO approximation.

First make the interference visible with an exactly normalized one-dimensional toy. Replace the three-dimensional $1s$ atomic orbitals of the hydrogen atom by two Gaussian functions centred at $x=\pm a$, with dimensionless $x$ and $a>0$:The normalized sum and difference are ${\psi }_{\pm }=({\varphi }_A\pm {\varphi }_B)/\sqrt{2(1\pm S)}$, following the same linear combination of atomic orbitals algebra. At $a=1$, the orbital overlap is $S=e^{-1}\approx 0.367879$. At the midpoint, ${\varphi }_A(0)={\varphi }_B(0)\approx 0.455581$, givingBoth probability densities have total integral one: interference redistributes probability, rather than creating an extra electron. These are toy wavefunctions, not actual hydrogen atom orbitals or predictions of molecular hydrogen energies.

Now fill the two $1s$-derived molecular orbitals in the minimal model. The lower bonding molecular orbital is labelled ${\sigma }_g$; the upper antibonding molecular orbital is ${\sigma }_u^{\ast }$. The Pauli exclusion principle permits at most two electrons in each spatial molecular orbital, with opposite spin projections along a chosen axis. For the lowest electron configuration in this model, fill ${\sigma }_g$ first. The subscripts $g$ and $u$ denote even and odd behaviour under inversion through the midpoint; the star marks an antibonding molecular orbital. See Flowers et al., Chemistry 2e, 8.4: occupancy and bond order and Field, MIT 5.61, lecture 25, pp. 1–3: bonding, antibonding, and symmetry labels.

With $N_b$ electrons in bonding molecular orbitals and $N_a$ in antibonding molecular orbitals, the model's bond order is $b=(N_b-N_a)/2$. The three elementary counts are:

The zero bond order says this minimal model gives no net covalent bond for ${\mathrm{He}}_2$. It does not rule out every bound pair of helium chemical atoms: a very weak Van der Waals force can bind a neutral helium dimer, which has been observed experimentally. See Zeller et al., Imaging the He2 quantum halo state using a free electron laser, PNAS 113, 14651–14655 (2016), p. 14651: van der Waals binding and Luo et al., The weakest bond: Experimental observation of helium dimer, Journal of Chemical Physics 98, 3564–3567 (1993). The bond order count does not include that weak binding mechanism.

The useful surprise is that sharing alone does not determine a chemical bond: the relative quantum phase and the occupancy matter together. The same two shapes explain why adding an electron to a bonding molecular orbital increases the model's bond order, while adding one to an antibonding molecular orbital decreases it. Molecular orbital theory also accounts for the unpaired electrons and paramagnetism of molecular oxygen, which its all-paired Lewis structure misses. See Chemistry 2e, 8.4: the molecular orbital description of oxygen.

Eight chloride ions contribute charge $-8$. For an overall $4-$ charge, the two metals together contribute $+4$, so each is formally Mo(II). Molybdenum is in group six, giving four d electrons per centre and eight metal-based electrons in total.

In the ligand-perturbed configuration developed below, they occupy ${\sigma }^2{\pi }^4{\delta }^2$, with no Mo–Mo antibonding electron. Therefore the formal bond order isThis is one sigma bond, two pi bonds and one delta bond. Before including ligands, the two ideal delta channels are degenerate and the allocation of their last two electrons is not determined by the bare energy sketch alone; the ligands remove that ambiguity and select the surviving low delta channel. The eight-electron bonding count still gives formal order four.

The occupied bonding groups are experimentally meaningful: molybdenum X-ray emission resolves three metal–metal bonding energy groups in this ion, alongside a deeper metal–chloride bonding feature.^[ref]

For A, linear nitrogen coordination with predominantly nitrile-like bonding is consistent with all three measurements:The short C–N and long C–S distances correspond approximately to a triple bond and a single bond. An approximately $sp$ nitrogen donor has its donor lone pair directed along the N–C axis, explaining the linear M–N–C angle.

For B, bent nitrogen coordination with increased ${}^{-}\mathrm{N}=\mathrm{C}=\mathrm{S}$ chemical resonance character accounts for the longer C–N and shorter C–S bonds. In this limiting description both internal bonds have substantial double-bond character and nitrogen is approximately $s{p}^2$. One nitrogen lone pair can bind a metal with an M–N–C angle near $120^{\circ }$; alternatively, donation of both nitrogen lone pairs can give a trigonal, nitrogen-bridging unit:The nitrogen-bridging drawing is a compatible proposed bonding mode, but the listed internal distances and one angle alone do not prove the number of attached metals. In particular, B cannot be identified simply as terminal sulfur coordination: the specified angle explicitly contains an M–N bond. resonance structures explain relative bond orders; they do not imply an isolated coordination ligand with permanently localized integer bonds.

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 ionic counting, two ${\eta }^5$ cyclopentadienyl anions donate $12$ electrons. Therefore ferrocene, cobaltocene and nickelocene contain 18, 19 and 20 valence electrons, respectively: Fe(II) is $d^6$, Co(II) $d^7$ and Ni(II) $d^8$. ferrocene fills the bonding and approximately nonbonding levels while leaving the next metal–ring antibonding pair empty. cobaltocene and nickelocene add one and two electrons to that antibonding set.

Increasing antibonding occupation lowers the net metal–ring bond order. It therefore explains the simultaneous increase in M–C distance and decrease in dissociation enthalpy from Fe to Co to Ni. The trend does not follow the shrinking size of the isolated metal ions; the orbital occupation is decisive here. The specified enthalpy separates the entire sandwich into a gaseous metal dication and two gaseous ring anions. It measures that complete ionic dissociation process, not ten independent localized M–C bond energies, so dividing it by ten is not a justified bond-energy determination.

Both coordinated frequencies are substantially lower than that of free N${}_2$. In a local harmonic model,Thus coordination lowers the approximate N–N stretching force constant by about 27%, with B marginally more activated than A by this measure. Donation into N${}_2$ pi-antibonding orbitals provides the natural explanation: the M–N${}_2$ bond has an important backbonding component and N–N loses some multiple-bond character.

The same qualitative red shift occurs for CO when metal-to-CO backbonding populates CO ${\pi }^{\ast }$ orbitals. N${}_2$ is generally a poorer pi acceptor than CO, so low-valent electron-rich metal fragments are particularly helpful for stabilizing its complexes. A numerical claim that N${}_2$ always shifts more or less than CO would require a matched metal/ligand comparison. Neither the two frequencies nor the local-force approximation establish integer N–N bond orders, complete N–N cleavage, or catalytic nitrogen conversion; they establish significant weakening of intact coordinated N${}_2$.

The carbon end of a carbonyl ligand has a filled $5\sigma$ molecular orbital with an outward-facing lobe. Its overlap with a vacant metal orbital of the same axial symmetry gives CO-to-metal sigma donation. Occupied metal $d_{\pi }$ orbitals can simultaneously overlap with the empty CO $2{\pi }^{\ast }$ antibonding molecular orbitals, giving metal-to-CO pi backbonding. Two perpendicular pi channels are possible; the schematic represents one. Orbital signs indicate phase, not electric charge.

The two interactions reinforce one another: sigma donation transfers density toward the metal, and pi backbonding distributes it into the ligand acceptor orbitals. Both strengthen M–C attachment, while backbonding weakens C–O. Occupation of CO ${\pi }^{\ast }$ decreases the internal C–O bond order, generally increasing its bond length and lowering its stretching spectroscopic wavenumber. This is why a strongly backbonded carbonyl ligand can have a strong metal–carbon bond and a weakened internal carbon–oxygen bond at the same time.

Use an ${\eta }^5$ cyclopentadienyl ligand and four terminal carbonyl ligands at each titanium, with a Ti–Ti single metal-metal bond. The middle panel shows this connectivity. Neutral electron counting assigns five electrons to each Cp and givesEquivalently, the two Cp anions require an average oxidation state Ti(I), hence $d^3$; ionic counting gives $3+6+8+1=18$. The formal Ti–Ti bond order is one. The large coordination envelope can make this bond weak and long; reaching 18 electrons is bookkeeping, rather than proof of a short, strong bond. There is no need to invent extra CO bridges or a higher Ti–Ti bond order.

The principal electronic variable is the net population of metal–metal bonding molecular orbitals relative to antibonding molecular orbitals. For otherwise comparable complexes, a larger bond order generally gives a shorter metal-metal bond; removing an antibonding electron can shorten a bond even though the total electron count falls. Different orbital symmetries also matter: sigma, pi and delta overlaps respond differently to distance and rotation of the metal fragments.

metal size, oxidation state and radial orbital extent set the underlying distance scale. Comparing a first-row metal with a larger heavier congener does not isolate bond order. Increasing positive charge often contracts the metal orbitals, but can also change electron occupancy, ligand binding and electrostatic interactions. The ligand field and donor/acceptor properties of the surrounding ligands in coordination chemistry determine which metal orbitals remain available for direct bonding and how much density instead participates in pi backbonding.

Finally, the geometry can enforce or oppose the preferred separation. Bridging ligands impose bond-angle and bond-length constraints; bulky substituents produce steric hindrance; crystal packing can perturb the observed distance. A bond length is evidence for bonding only in its electronic and structural context. In particular, two bridges can hold metals close without proving a high direct bond order, while a sterically congested formal single bond can be unusually long.

For the doubly bridged tetracarbonyl, each iron has one terminal carbonyl ligand, two bridging carbonyl ligands and one cyclopentadienyl ligand. Neutral electron counting gives $8+5+2+2=17$ before direct bonding, so a single Fe–Fe bond gives 18 at each iron. For the triply bridged tricarbonyl, each iron instead has $8+5+3=16$ before direct bonding; formal completion to 18 requires a double Fe–Fe bond. Equivalently, the total fragment counts before metal–metal bonding are 34 and 32 respectively.The increased direct bond order explains the substantial contraction. The pentamethylcyclopentadienyl ligands also alter steric and electronic effects, and bridge geometry changes, so this is not a controlled measurement of bond order alone. Nevertheless, the markedly shorter distance in the less carbonyl-saturated dimer is consistent with stronger direct Fe–Fe bonding. The isolated triply bridged compound was characterized structurally and magnetically in the primary work.^[ref]

A carbonyl ligand binds through carbon, whose occupied $5\sigma$ molecular orbital has a substantial outward-facing carbon lobe. Overlap with an empty metal orbital along the M–C axis gives ligand-to-metal sigma donation. Conversely, an occupied metal $d_{\pi }$ orbital overlaps sideways with an empty CO $2{\pi }^{\ast }$ antibonding molecular orbital, giving metal-to-ligand pi backbonding. There are two mutually perpendicular pi channels. The original orbital schematic below shows one of them; $+$ and $-$ denote orbital phase, not electric charge.

The interactions are synergic: donation makes electron density available at the metal, and backbonding helps accommodate it while strengthening the metal–carbon interaction. Occupying CO ${\pi }^{\ast }$ reduces the C–O bond order, lengthens that bond and lowers its bond force constant. Therefore stronger pi backbonding generally lowers the carbonyl stretching spectroscopic wavenumber even while strengthening the M–C attachment.