Molecular Orbital Theory and LCAO

Why does He₂ not exist, while H₂ is one of the strongest bonds known per electron pair? Valence Bond Theory is built from overlapping atomic orbitals paired into bonds. It cannot answer this cleanly. It treats bonding electrons as localized between two nuclei. It struggles to explain why adding two more electrons to a perfectly good bonding overlap in He₂ produces no bond at all. Molecular Orbital (MO) theory answers it in one line: those two extra electrons go into an antibonding orbital that cancels the bonding stabilization exactly. This single idea — that combining atomic orbitals produces as many new orbitals as you started with, some lower in energy (bonding) and some higher (antibonding) — is the conceptual engine behind essentially all of modern inorganic and organometallic bonding theory. MO theory matters because it is quantitative. Lewis structures and simple orbital-overlap pictures are not. It explains paramagnetism of O₂. This is impossible to rationalize with a simple Lewis structure. It explains the bond-order trends across the second-period diatomics. It explains the color and reactivity of transition-metal complexes. It also explains the electronic structure of everything from graphene to metal carbonyls. The mathematical machinery — the Linear Combination of Atomic Orbitals (LCAO) — is deceptively simple. Molecular orbitals are just weighted sums of atomic orbitals, $\psi = c_A\phi_A + c_B\phi_B$ . Yet this simple ansatz generates much more. Combined with the variational principle and symmetry, it generates the entire qualitative and semi-quantitative structure of chemical bonding. The big picture is this: atomic orbitals on different atoms are not eigenstates of the molecular Hamiltonian. When two atoms approach, their orbitals must be replaced by new one-electron wavefunctions. These are molecular orbitals. They are eigenstates (approximately) of the full molecular problem. LCAO is the practical recipe for building these new orbitals. It builds them out of the atomic orbitals you already know. It uses overlap integrals, symmetry-matching, and energy-matching. These three factors are the gatekeepers. They decide how strongly any pair of atomic orbitals combines. Once you have the MOs and their energies, you fill them with electrons. You use the same aufbau and Pauli rules you already know from atoms. This article builds the LCAO formalism from first principles. It starts from the two-orbital problem. It derives the bonding/antibonding energy split. It quantifies why overlap and energy matching control bond strength. It sets up, but deliberately does not exhaust, the diatomic MO diagrams. It also sets up the bonding/antibonding orbital classification that follows directly from this machinery. Those are best treated as their own topics. This is true once the LCAO engine here is fully understood.

Key takeaways

Molecular orbitals are built as linear combinations of atomic orbitals (LCAO): $\psi_\pm = c_A\phi_A \pm c_B\phi_B$ .
Combining $n$ atomic orbitals always gives exactly $n$ molecular orbitals — orbitals are never created or destroyed, only recombined.
In-phase combination lowers energy (bonding MO); out-of-phase combination raises energy (antibonding MO), and the antibonding rise is larger than the bonding drop.
Three matching rules control how strongly orbitals combine: symmetry match, energy match, and spatial overlap match.
The overlap integral $S$ and resonance integral $\beta$ (or $H_{AB}$ ) set the magnitude of the bonding/antibonding splitting.
Filling MOs with electrons via the aufbau principle gives bond order, and explains magnetism and bond strength beyond what Lewis structures can.
Key idea: Atomic orbitals on interacting atoms mix into new bonding (lower-energy, constructive) and antibonding (higher-energy, destructive) molecular orbitals, and the number and filling of these orbitals — not the drawing of localized 2-electron bonds — determines a molecule's stability, bond order, and magnetism.

Where this lives today

LCAO-MO reasoning is the backbone of essentially every quantitative electronic-structure method used in modern chemistry, from semi-empirical Hückel and extended-Hückel calculations taught alongside Homonuclear Diatomic MOs all the way to full ab initio Hartree–Fock and density-functional-theory codes, which represent molecular orbitals as linear combinations of Gaussian-type atomic-orbital basis functions — the LCAO ansatz, just with far more basis functions per atom and numerically fitted coefficients instead of the hand-derived $c_A, c_B$ above. Every commercial and academic quantum chemistry package (Gaussian, ORCA, Q-Chem, VASP for solids) solves a generalized version of exactly the secular equation derived here, just expanded to hundreds or thousands of basis orbitals. In transition-metal and organometallic chemistry, LCAO thinking underlies crystal-field and ligand-field theory extensions that explain d-orbital splitting patterns, back-bonding in metal carbonyls (CO's empty π* orbital mixing with filled metal d orbitals — a direct LCAO bonding interaction between orbitals of matching symmetry and reasonably matched energy), and the electronic spectra used to assign oxidation states and coordination geometries in catalysis research. In materials science, band theory in solids is literally the LCAO idea taken to the thermodynamic limit: combining $N\to\infty$ atomic orbitals across a periodic lattice produces not two levels but a continuous band of $N$ closely spaced MOs, and the band gap of semiconductors (silicon, gallium arsenide) is the solid-state descendant of the bonding/antibonding gap derived above for a two-atom system. Graphene's electronic structure, described by simple Hückel-type LCAO models of overlapping carbon 2p $_z$ orbitals, underlies its famous linear ("Dirac cone") electronic dispersion exploited in next-generation electronics research. In biochemistry and drug design, molecular orbital calculations (frontier orbital, HOMO/LUMO reasoning building directly on the bonding/antibonding language introduced here) predict sites of nucleophilic and electrophilic attack in enzyme mechanisms and guide rational design of reactive pharmacophores. Spectroscopically, MO energy-level diagrams directly predict UV-visible absorption energies (HOMO→LUMO-type transitions) used routinely in identifying conjugated organic chromophores and charge-transfer bands in coordination compounds — an experimental fingerprint of the very $E_\pm$ splitting derived algebraically above.

Ask the tutor: How do the bonding/antibonding MOs built here combine into the full σ/π energy-level diagrams for real homonuclear diatomics like N₂ and O₂?

The physical picture

Start with two hydrogen atoms far apart. Each has one electron in a 1s orbital, $\phi_A$ and $\phi_B$ , each an exact eigenstate of its own isolated atomic Hamiltonian. As the atoms approach, the electron on atom A begins to feel the nucleus of atom B (and vice versa), and $\phi_A$ , $\phi_B$ are no longer eigenstates of the true two-center Hamiltonian. The correct one-electron wavefunctions for the molecule must be found some other way — and the simplest useful way is to guess that the true molecular orbital is some mixture of the two atomic orbitals we already know:

\psi = c_A \phi_A + c_B \phi_B

This is the LCAO ansatz. It is a variational trial function: we don't know $\psi$ exactly, but we know it should look like $\phi_A$ near nucleus A and like $\phi_B$ near nucleus B, and a linear combination is the simplest function with that property. The coefficients $c_A, c_B$ are then chosen to minimize the energy (the variational principle guarantees this gives the best possible answer of this functional form). For two identical atomic orbitals (same atom, same energy, e.g. two H 1s orbitals), symmetry forces the coefficients to be equal in magnitude, giving exactly two solutions: the in-phase (symmetric) combination and the out-of-phase (antisymmetric) combination.

Figure

LCAO of two 1s orbitals: bonding and antibonding combinations

The energy consequence of this is the central diagram of MO theory: the bonding combination lies below the average energy of the isolated atomic orbitals, and the antibonding combination lies above it, with the antibonding rise larger in magnitude than the bonding drop (because overlap itself is destabilizing in the denominator — see the derivation below).

Figure

Energy-level diagram: two atomic orbitals combine into bonding + antibonding MOs

The number of MOs produced always equals the number of AOs combined — two atomic orbitals give exactly two molecular orbitals, never one and never three. This conservation of orbital count is a strict bookkeeping rule and a useful sanity check whenever you build an MO diagram: count your basis orbitals, and count your resulting MOs; they must match. Three matching conditions determine how much two atomic orbitals mix, and are worth stating explicitly because they generalize far beyond the two-orbital toy problem to polyatomic and transition-metal MO diagrams: 1. Symmetry match — the orbitals must belong to the same irreducible representation of the molecular point group (equivalently, for diatomics: same nodal symmetry about the internuclear axis — σ mixes with σ, π mixes with π, not with each other). 2. Energy match — orbitals close in energy mix strongly; orbitals far apart in energy mix weakly, because the bonding/antibonding splitting for dissimilar energies is suppressed (this is why core orbitals essentially never participate in bonding). 3. Spatial overlap match — the orbitals must have appreciable spatial overlap ( $S \neq 0$ ); orbitals that point away from each other or are too diffuse/contracted relative to the internuclear distance barely interact. These three rules are exactly the qualitative content that the quantitative treatment below (the secular equations) reproduces automatically.

The core ideas

The overlap integral quantifies rule 3 above:

S_{AB} = \int \phi_A^* \phi_B \, d\tau

where: $\phi_A,\phi_B$ are normalized atomic orbitals, $d\tau$ is the volume element, and $S_{AB}$ ranges from 0 (no overlap) toward 1 (orbitals coincide) as the atoms approach; $S_{AB}=0$ for symmetry-mismatched orbitals at any distance.

Reading: $S$ measures the degree to which two atomic wavefunctions occupy the same region of space with matching phase. It is the single number that tells you whether "rule 3" is satisfied, and it appears explicitly in the energy formulas below — nonzero overlap is a precondition for bonding, not a guarantee of it (energy mismatch can still suppress bonding even when $S\neq0$ ). The Hamiltonian matrix elements needed are the Coulomb integral and resonance (exchange) integral:

\alpha = H_{AA} = \int \phi_A^* \hat{H} \phi_A \, d\tau, \qquad \beta = H_{AB} = \int \phi_A^* \hat{H} \phi_B \, d\tau

where: $\hat H$ is the one-electron molecular Hamiltonian, $\alpha$ (the Coulomb integral) is close to the energy of an electron in the isolated atomic orbital, and $\beta$ (the resonance integral) is negative and grows in magnitude with increasing overlap; it is the quantity directly responsible for bonding.

Reading: $\alpha$ is "what energy would this orbital have if the other atom weren't there" (roughly the atomic orbital energy, corrected for the field of the other nucleus); $\beta$ is "how much energy do we gain by letting the electron delocalize between both centers" — it is always negative for a bonding-type overlap of orbitals with the same sign convention, and it is the direct microscopic origin of the covalent bond. Solving the two-orbital secular problem (derived in full below) gives the bonding and antibonding MO energies:

E_\pm = \frac{\alpha \pm \beta}{1 \pm S}

where: $E_-$ (lower sign combo throughout, using the convention that $\beta<0$ ) is the bonding MO energy and $E_+$ is the antibonding MO energy; $S$ is the overlap integral defined above.

Reading: because $S>0$ and $\beta<0$ , the bonding root $E_- = (\alpha+\beta)/(1+S)$ lies below $\alpha$ , and the antibonding root $E_+=(\alpha-\beta)/(1-S)$ lies above $\alpha$ by a larger amount — the antibonding orbital is always more destabilized than the bonding orbital is stabilized, because the $(1-S)$ denominator is smaller than $(1+S)$ . This asymmetry is why filling both bonding and antibonding MOs (as in He₂) gives net repulsion, not a null result. The corresponding coefficients, from normalization plus the symmetry constraint for two identical orbitals, are:

c_A = \frac{1}{\sqrt{2(1+S)}}, \qquad c_B = \pm\, c_A

where: the $+$ sign gives the bonding MO ( $\psi_- = c_A(\phi_A+\phi_B)$ ) and the $-$ sign gives the antibonding MO ( $\psi_+ = c_A(\phi_A-\phi_B)$ ), with the extra $(1\pm S)^{-1/2}$ ensuring $\int\psi^2 d\tau = 1$ even though $\phi_A,\phi_B$ are not orthogonal.

Derivation ASecular equations for a two-orbital LCAO problem

Goal. Starting only from the variational principle and the LCAO trial function $\psi = c_A\phi_A + c_B\phi_B$ , derive the bonding/antibonding energies $E_\pm$ and show explicitly why the antibonding level rises more than the bonding level falls.

Step 1 — Variational energy functional. For any trial wavefunction $\psi$ , the variational principle states that the expectation energy

E[\psi] = \frac{\int \psi^* \hat H \psi \, d\tau}{\int \psi^* \psi \, d\tau}

is an upper bound to the true ground-state energy, and is minimized by the best choice of the adjustable parameters — here $c_A$ and $c_B$ — within the trial form. Substituting $\psi = c_A\phi_A+c_B\phi_B$ , the numerator (using $H_{AA}=H_{BB}=\alpha$ by symmetry of a homonuclear pair, and $H_{AB}=H_{BA}=\beta$ ) becomes

\int \psi^*\hat H\psi\,d\tau = c_A^2\alpha + c_B^2\alpha + 2c_Ac_B\beta

and the denominator, using $\int\phi_A^2=\int\phi_B^2=1$ and $\int\phi_A\phi_B = S$ , becomes

\int \psi^*\psi\,d\tau = c_A^2 + c_B^2 + 2c_Ac_BS

Step 2 — Minimize with respect to the coefficients. Setting $\partial E/\partial c_A = 0$ and $\partial E/\partial c_B = 0$ (treating $E$ as the ratio above and using the quotient rule) produces the secular equations

c_A(\alpha - E) + c_B(\beta - ES) = 0

c_A(\beta - ES) + c_B(\alpha - E) = 0

These are two linear homogeneous equations in $c_A, c_B$ ; a nontrivial solution exists only if the determinant of coefficients vanishes.

Step 3 — Secular determinant. Requiring

\begin{vmatrix} \alpha - E & \beta - ES \\ \beta - ES & \alpha - E \end{vmatrix} = 0

expands to

(\alpha - E)^2 = (\beta - ES)^2

Taking the square root gives two cases, $(\alpha-E) = \pm(\beta - ES)$ , which rearrange to

E(1\mp S) = \alpha \mp \beta

so that

E_\pm = \frac{\alpha \pm \beta}{1 \pm S}

recovering the boxed result above, with the top signs together giving the bonding root and the bottom signs together giving the antibonding root.

Step 4 — Coefficients from the secular equations plus normalization. Substituting $E=E_-=(\alpha+\beta)/(1+S)$ back into the first secular equation gives $c_A(\alpha - E_-) = -c_B(\beta-E_-S)$ ; direct substitution shows both sides reduce to demanding $c_A = c_B$ . Substituting $E_+$ instead forces $c_A = -c_B$ . Normalizing $\int\psi^2 d\tau = c_A^2+c_B^2+2c_Ac_BS=1$ with $c_A=\pm c_B$ gives $2c_A^2(1\pm S)=1$ , i.e. $c_A = [2(1\pm S)]^{-1/2}$ , matching the coefficients quoted above.

Step 5 — Why antibonding destabilizes more than bonding stabilizes. Compare the two shifts from $\alpha$ :

E_- - \alpha = \frac{\alpha+\beta}{1+S} - \alpha = \frac{\beta - \alpha S}{1+S}, \qquad E_+ - \alpha = \frac{\alpha-\beta}{1-S} - \alpha = \frac{-\beta - \alpha S}{1-S}

Go deeper · Derivation B · 2 minBond order and the He₂ non-bond from MO filling

Derivation BBond order and the He₂ non-bond from MO filling

Goal. Use the aufbau filling of the bonding/antibonding pair derived above to define bond order quantitatively, and show why H₂ is stable but He₂ is not — the paradigm case MO theory gets right and simple orbital-overlap pictures get wrong.

Step 1 — Define bond order from MO occupation. Once the bonding ( $\sigma$ ) and antibonding ( $\sigma^*$ ) MO energies are known from Step 3 above, electrons are assigned to them via the aufbau principle and Pauli exclusion, exactly as for atomic orbitals. Bond order is defined as

\text{Bond order} = \frac{n_b - n_a}{2}

where $n_b$ is the number of electrons in bonding MOs and $n_a$ the number in antibonding MOs. The factor of $\tfrac12$ reflects that a conventional single (Lewis) bond corresponds to a net of two bonding electrons.

Step 2 — Apply to H₂. Two H atoms contribute two electrons total (one 1s electron each). Both go into the lower-energy $\sigma_{1s}$ bonding MO (opposite spins, satisfying Pauli exclusion), so $n_b=2, n_a=0$ , giving

\text{Bond order (H}_2\text{)} = \frac{2-0}{2} = 1

consistent with the single H–H bond and with the large binding energy (measured well depth $D_e \approx 458$ kJ/mol for H₂, arising because both electrons occupy the stabilized $\sigma$ level with energy $E_-<\alpha$ ).

Step 3 — Apply to He₂. Two He atoms contribute four electrons total (two 1s electrons each). Filling in order of increasing energy: two electrons fill $\sigma_{1s}$ (bonding), and the remaining two are forced by Pauli exclusion into $\sigma^*_{1s}$ (antibonding), since only two electrons fit in any one spatial orbital. Thus $n_b=2, n_a=2$ , giving

\text{Bond order (He}_2\text{)} = \frac{2-2}{2} = 0

Step 4 — Why bond order zero means "no bond," quantitatively. Using the result of Step 5 in the first derivation, the antibonding destabilization $|E_+-\alpha|$ exceeds the bonding stabilization $|E_--\alpha|$ for any $S>0$ . The net energy of He₂ relative to two separate He atoms is

\Delta E = 2(E_- - \alpha) + 2(E_+-\alpha) = 2\left[\frac{\beta-\alpha S}{1+S} + \frac{-\beta-\alpha S}{1-S}\right] < 0 \text{ in magnitude of stabilization, net } >0

Working the algebra (common-denominator over $1-S^2$ ) gives a net positive (destabilizing) $\Delta E$ whenever $S>0$ , because the antibonding term's larger destabilization wins. This is why He₂ is unbound (bond order 0, observed only as an extremely weak van der Waals dimer with a dissociation energy of order $0.001$ kJ/mol, orders of magnitude below any covalent bond) — a result that falls straight out of LCAO bookkeeping but requires extra, separate argument in Valence Bond language.

Worked examples

Example 1 — Foundational: energies and coefficients for H₂⁺ with realistic numbers. H₂⁺ (one electron, two protons) is the cleanest test of the two-orbital formulas because there genuinely is only one electron to place, so there's no electron-electron repulsion complicating the picture. Take typical LCAO parameters appropriate near the equilibrium bond length of H₂⁺ ( $R \approx 106$ pm): $\alpha = -13.6$ eV (the H 1s orbital energy), $S = 0.59$ , and $\beta = -11.0$ eV (these are standard illustrative values used to reproduce the correct qualitative and semi-quantitative shape of the H₂⁺ curve). Bonding MO energy:

E_- = \frac{\alpha+\beta}{1+S} = \frac{-13.6 + (-11.0)}{1+0.59} = \frac{-24.6}{1.59} = -15.47\ \text{eV}

Antibonding MO energy:

E_+ = \frac{\alpha-\beta}{1-S} = \frac{-13.6-(-11.0)}{1-0.59} = \frac{-2.6}{0.41} = -6.34\ \text{eV}

Check the asymmetry demanded by the derivation: the bonding drop is $E_- - \alpha = -15.47-(-13.6) = -1.87$ eV, while the antibonding rise is $E_+-\alpha = -6.34-(-13.6)=+7.26$ eV. As required, the antibonding rise (7.26 eV) is much larger in magnitude than the bonding drop (1.87 eV) — a direct numerical confirmation of the $(1-S)$ vs $(1+S)$ asymmetry derived above. The single electron in H₂⁺ occupies $E_-$ , giving one electron in a bonding MO, bond order $\tfrac12$ , consistent with the experimentally observed H₂⁺ bond (weaker than H₂'s, as expected from half the bond order). Coefficient check: $c_A = 1/\sqrt{2(1+0.59)} = 1/\sqrt{3.18} = 0.561$ , so $\psi_- = 0.561(\phi_A+\phi_B)$ ; normalization check $c_A^2+c_B^2+2c_Ac_BS = 2(0.561)^2(1+0.59) = 2(0.3147)(1.59)=1.000$ ✓.

Example 2 — Applied: why F₂ has a weaker bond than expected from naive bond-order counting, using overlap/energy matching. Both N₂ and F₂ formally have MOs filled to give bond orders that follow from the same aufbau logic as Example 1 (deferred in full detail to the homonuclear diatomic MO diagrams built on this framework), but here we use the LCAO matching rules directly to rationalize a puzzle: F₂'s bond enthalpy (154 kJ/mol) is anomalously weak compared to what a naive extrapolation from lighter halogens might suggest, and weaker even than a single P–P or Cl–Cl bond in the naive picture. Apply rule 2 (energy match) and rule 3 (overlap match) directly. Fluorine's 2s and 2p orbitals are highly contracted (F is small, $Z_{\text{eff}}$ is large), so the F–F internuclear distance (142 pm) forces the bonding lone pairs on each F to sit very close together. The non-bonding lone-pair orbitals on adjacent F atoms — while not part of the LCAO bonding/antibonding pair being filled — have significant mutual overlap because the atoms are so close, producing strong lone-pair/lone-pair repulsion that is not "wrong" MO occupancy but an additional destabilization outside the simple bonding/antibonding count. Quantitatively, this shows up as an anomalously small resonance integral $|\beta|$ for the bonding combination relative to what atomic size alone would predict, because much of the favorable core contraction is offset by repulsive close contact of filled non-bonding orbitals. The LCAO framework explains this as an overlap-integral effect: $S$ for the bonding σ orbital is unusually degraded relative to isoelectronic N₂, precisely because short internuclear distance forces destabilizing lone-pair contacts that the pure two-orbital model (Examples above) does not include but that real polyatomic-orbital counting must.

Example 3 — Exam/competition level: overlap-integral sign change and the "does this MO exist at all" question for p-orbitals off-axis. A very common trap question: does a 2p $_z$ orbital on atom A (pointing along the internuclear z-axis) combine with a 2p $_x$ orbital on atom B (perpendicular to the axis) to form a bonding MO? Set up the overlap integral explicitly using the geometry. Place both atoms on the z-axis, separated by $R$ . The 2p $_z$ orbital on A has lobes of opposite sign along $+z$ and $-z$ from A; the 2p $_x$ orbital on B has lobes of opposite sign along $+x$ and $-x$ from B, i.e. perpendicular to the A–B axis. By the mirror symmetry of the molecule through the plane containing the z-axis and perpendicular to x (the $yz$ plane, say), the 2p $_x$ orbital on B is exactly antisymmetric under reflection through that plane (one lobe maps to minus the other lobe), while the 2p $_z$ orbital on A is exactly symmetric under the same reflection (both lobes lie in the plane and are unchanged). Formally,

S = \int \phi_{2p_z}^A(\mathbf r)\, \phi_{2p_x}^B(\mathbf r)\, d\tau

Split the integration into the region $x>0$ and $x<0$ . Because $\phi_{2p_z}^A$ is even in $x$ (unchanged under $x\to-x$ ) while $\phi_{2p_x}^B$ is odd in $x$ (changes sign under $x\to-x$ ), the integrand is odd overall, and the contributions from $x>0$ and $x<0$ are exactly equal in magnitude and opposite in sign:

S = \int_{x>0} + \int_{x<0} = \int_{x>0} \phi^A_{2p_z}\phi^B_{2p_x}\,d\tau - \int_{x>0}\phi^A_{2p_z}\phi^B_{2p_x}\,d\tau = 0

So $S=0$ identically, for any internuclear distance $R$ and any radial form of the orbitals — this is a symmetry-forced zero, not a numerical coincidence, and it is exactly rule 1 (symmetry match) from the qualitative picture, now proved rigorously by direct integration. Since $\beta \propto S$ in the sense that $H_{AB}$ also vanishes by the identical symmetry argument applied to the Hamiltonian matrix element, these two orbitals do not mix at all: a 2p $_z$ (A)/2p $_x$ (B) pair contributes zero bonding and zero antibonding character, and must instead be treated as two independent nonbonding orbitals in the full molecular MO diagram. This is the rigorous justification for the standard rule that only like-symmetry orbitals (σ with σ, π $_x$ with π $_x$ , π $_y$ with π $_y$ ) are connected by lines in a diatomic MO diagram.

Limiting cases, scope & common traps

Regime	What happens	Why
$S \to 0$ (orbitals far apart or symmetry-forbidden)	$E_\pm \to \alpha \pm \beta$ , but $\beta\to 0$ too, so both MOs collapse onto $\alpha$ — no bonding	No spatial overlap means no shared electron density to lower or raise the energy
$S \to 1$ (unphysical, orbitals coincide)	$E_-\to(\alpha+\beta)/2$ , $E_+\to\infty$	Denominator $(1-S)\to 0$ diverges; signals breakdown of the simple two-center model at very short range where orbitals are no longer well-approximated as atomic
Very different atomic orbital energies ( $\alpha_A \neq \alpha_B$ , heteronuclear)	Mixing becomes unequal ( $c_A\neq c_B$ ); MO is polarized toward the lower-energy atomic orbital	Energy-match rule: large $\lvert\alpha_A-\alpha_B\rvert$ suppresses the bonding stabilization and localizes the bonding MO mostly on the more electronegative atom
Core orbitals (e.g. 1s of a second-row atom)	Essentially non-bonding despite being "present"	Core orbitals are far in energy from valence orbitals of the partner atom and spatially contracted close to their own nucleus — both energy-match and overlap-match rules fail
Filling both bonding and antibonding MOs (e.g. He₂, or any "closed shell + closed shell" pair)	Net destabilization, no bond	Antibonding rise exceeds bonding drop for any $S>0$ (proved above)

Common trapMisconception: "antibonding electrons just cancel bonding electrons, giving zero net effect"

It is tempting to think that filling both $\sigma$ and $\sigma^*$ with two electrons each simply returns you to "no interaction," as if the bonding and antibonding effects were mirror images that sum to nothing. They are not mirror images: because the antibonding level is pushed up by more energy than the bonding level is pushed down (a direct consequence of the $(1-S)$ vs. $(1+S)$ denominators derived above), filling both fully gives a net destabilization relative to separated atoms, not a null result. This is precisely why He₂ is not merely "as stable as two free He atoms" but is measurably less stable than the separated atoms at all but the longest (van der Waals) range — the LCAO algebra forces this conclusion.

The chemist behind it

Origins · 1 minRobert S. Mulliken and how this was discovered

Robert S. Mulliken

1896–1986 · American physical chemist, principal architect of molecular orbital theory

Robert Mulliken built molecular orbital theory into a rigorous, quantitative framework in a series of papers from the late 1920s through the 1930s, working in close dialogue with Friedrich Hund (whose name is attached to the closely related Hund–Mulliken approach) and drawing on the group-theoretical and quantum-mechanical foundations laid by Heitler, London, and Slater for valence bond theory. Where Linus Pauling's valence bond framework treated a bond as an overlap of two localized atomic orbitals paired into a two-electron bond — an extension of the Lewis picture — Mulliken insisted on building delocalized one-electron molecular orbitals extending over the whole molecule from the outset, filled according to the same aufbau and Pauli principles used for atoms. This was initially the less popular approach among chemists, who found valence bond theory's connection to familiar Lewis structures more intuitive, but molecular orbital theory ultimately proved indispensable for explaining phenomena — the paramagnetism of O₂, the electronic spectra of diatomics, charge-transfer complexes, and eventually the electronic structure of solids and transition-metal complexes — that resisted simple valence-bond treatment. Mulliken received the Nobel Prize in Chemistry in 1966 "for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method."

History

- 1927 — Walter Heitler and Fritz London give the first quantum-mechanical treatment of the H₂ bond (valence bond approach).

- 1928–1932 — Friedrich Hund and Robert Mulliken independently develop molecular orbital theory, building orbitals delocalized over the whole molecule.

- 1931 — Erich Hückel applies LCAO-MO ideas to π systems in conjugated and aromatic organic molecules.

- 1932 — Mulliken introduces the term "molecular orbital" and systematizes σ/π nomenclature and bonding/antibonding classification.

- 1966 — Mulliken awarded the Nobel Prize in Chemistry for his development of molecular orbital theory.

What is the difference between LCAO and molecular orbital theory itself?

Molecular orbital theory is the broader conceptual framework: electrons in a molecule occupy delocalized one-electron wavefunctions (molecular orbitals) filled by aufbau and Pauli rules, exactly as in atoms. LCAO is the specific mathematical method used to construct approximate molecular orbitals, by writing them as weighted sums of known atomic orbitals. You could in principle have molecular orbital theory without LCAO — for instance, numerically tabulated MOs from a fully numerical solution of the Schrödinger equation — but LCAO is by far the most practical and widely used route, because it lets you reuse known atomic-orbital shapes and energies, and it is the basis of essentially all computational quantum chemistry software in use today.

Why does the bonding MO coefficient formula include a factor of 1/√(2(1+S)) instead of simply 1/√2?

The simple 1/√2 normalization would be correct only if the two atomic orbitals were orthogonal (zero overlap), as basis functions on the same atom are. But $\phi_A$ and $\phi_B$ sit on different atoms and have nonzero overlap $S$ , so the cross term $2c_Ac_BS$ in the normalization integral $\int\psi^2\,d\tau = c_A^2+c_B^2+2c_Ac_BS$ is not zero. Including this term and solving for $c_A=c_B$ gives $2c_A^2(1+S)=1$ , hence the $\sqrt{2(1+S)}$ denominator. Physically, because bonding overlap adds constructively, you need slightly smaller coefficients than 1/√2 to keep the total probability equal to one; the antibonding MO, with destructive overlap, needs a slightly larger coefficient instead ( $\sqrt{2(1-S)}$ in the denominator).

Does LCAO only work for diatomic molecules?

No — LCAO generalizes directly to polyatomic molecules and solids; the two-orbital secular equation derived here is simply the smallest possible case of a general $n\times n$ secular determinant built from all the valence atomic orbitals in the molecule. For polyatomics, symmetry (via group theory and the molecule's point group) is used to sort atomic orbitals into symmetry-adapted linear combinations first, which then combine pairwise (or in small symmetry-matched sets) exactly as in the two-orbital case treated here. Band theory in solids is the same equation taken to the limit of an infinite, periodic array of atomic orbitals, producing continuous bands instead of discrete bonding/antibonding pairs.

How is LCAO-MO theory different from Valence Bond Theory in practice, and which one is "right"?

Both are approximate variational methods, and both converge to the same exact answer if pushed to sufficiently high levels of refinement (configuration interaction on the MO side, or resonance among many VB structures on the VB side). The practical difference is the starting picture: Valence Bond Theory and Orbital Overlap builds a bond as a two-electron overlap between two atomic orbitals, one on each atom, closely tied to the Lewis dot structure and easy to connect to hybridization and geometry (VSEPR). MO theory instead builds delocalized orbitals spanning the whole molecule from the start, filled with all the molecule's electrons via aufbau. MO theory more naturally explains delocalization, resonance, paramagnetism, and spectroscopic transitions; VB theory more naturally explains geometry, hybridization, and localized bond/lone-pair pictures. Modern chemists use whichever is more convenient for the question at hand.

Why is the resonance integral β always negative for a bonding interaction?

β = H_AB is a Hamiltonian matrix element representing the energy associated with an electron delocalizing between the two nuclei rather than sitting on one atom. For orbitals of matching phase in the internuclear region (constructive overlap), the extra electron density placed between the two positively charged nuclei is stabilized by attraction to both nuclei simultaneously, lowering the energy relative to being localized on one atom — this lowering is encoded as a negative β in the standard sign convention (with α also negative, representing a bound atomic orbital energy relative to a free electron at infinity). If you flip the relative phase of one orbital, the sign of β flips too, which is exactly what converts a bonding combination into an antibonding one algebraically in the secular equations.

Can three or more atomic orbitals combine at once, not just two?

Yes, and this is in fact the general case for any polyatomic molecule. If $n$ atomic orbitals of matching symmetry are all close enough in energy to interact simultaneously, the correct treatment builds an $n\times n$ secular determinant (a direct generalization of the 2×2 determinant derived here), yielding $n$ molecular orbitals spanning a range of bonding, nonbonding, and antibonding character. Simple examples include the three p orbitals of oxygen in ozone mixing into a 3-center π system, or the six 2p $_z$ orbitals of benzene mixing into six π molecular orbitals (three bonding, three antibonding) in Hückel theory. The two-orbital case treated in depth here is the essential building block; everything more complex follows the same variational logic with a bigger matrix.

Is overlap (S) ever negative, and what would that mean?

Yes. The sign of $S$ depends on the relative phase convention chosen for the two atomic orbitals and on the specific orbital shapes and orientation. For a fixed phase choice of two p orbitals pointing toward each other along the bond axis, head-to-head overlap of same-signed lobes gives $S>0$ (as assumed throughout this derivation), but if you instead chose the opposite phase convention for one orbital, the labels "bonding" and "antibonding" would simply swap, and $S$ would appear with the opposite sign in the formulas — the physical energies $E_\pm$ and the physical electron density are unchanged, because phase convention is not itself observable. What is physically meaningful is only the relative phase between the two orbitals in the region of overlap, which determines whether the resulting combination is bonding or antibonding.