Molecular orbital theory: LCAO, bonding and antibonding orbitals, and the H2 molecule

Intuition Beginner

Valence bond theory, covered in 14.02.04, describes a bond as the overlap of two atomic orbitals. It works well for many molecules but fails for others — most famously, it predicts that O2 is diamagnetic when O2 is actually paramagnetic. Molecular orbital (MO) theory fixes this by taking a different starting point.

In MO theory, electrons do not belong to individual atoms. When two atoms approach, their atomic orbitals merge into new molecular orbitals that belong to the whole molecule. Each MO has a definite energy, and electrons fill these MOs from the lowest energy upward, two per orbital, following the same rules as atomic electron filling.

When two atomic orbitals combine, they always produce two molecular orbitals. One is called the bonding orbital — its energy is lower than the original atomic orbitals, and electron density builds up between the nuclei, holding them together. The other is the antibonding orbital — its energy is higher, and it has a node (zero electron density) between the nuclei. Placing electrons in the bonding orbital stabilises the molecule; placing them in the antibonding orbital destabilises it.

The method for building MOs has a name: the linear combination of atomic orbitals (LCAO) approach. Each MO is written as a weighted sum of atomic orbitals from the constituent atoms. The weights determine how much each atom contributes to a given MO.

For H2, each hydrogen atom brings one 1s orbital. The two 1s orbitals combine to give one bonding MO (the 1s orbitals added together) and one antibonding MO (the 1s orbitals subtracted). With two electrons total, both go into the bonding MO. The molecule is more stable than two separated H atoms. This is the simplest MO bond.

Visual Beginner

The bonding MO of H2 looks like an elongated cloud of electron density stretched between the two nuclei. There is no node — the wave function has the same sign everywhere along the bond axis. This is labelled $\sigma$ (sigma), meaning the electron density has cylindrical symmetry around the bond axis.

The antibonding MO has a node exactly at the midpoint between the two nuclei. The electron density is pushed to the outside of each nucleus, away from the bonding region. This is labelled ${\sigma }^{\ast }$ (sigma star), where the star denotes antibonding.

The energy level diagram shows two atomic 1s orbitals (left and right) combining to produce the bonding $\sigma$ (below) and antibonding ${\sigma }^{\ast }$ (above). The splitting is asymmetric: the antibonding orbital is raised more than the bonding orbital is lowered. This asymmetry matters when four electrons are present — as in He2, where the extra destabilisation from the antibonding pair outweighs the stabilisation from the bonding pair, preventing a stable molecule.

Worked example Beginner

Problem. Use MO theory to predict whether H2, He2, and H2${}^{+}$ are stable molecules. Give the bond order for each.

Step 1: H2. Two H atoms each contribute one 1s orbital, giving two MOs: $\sigma$ (bonding) and ${\sigma }^{\ast }$ (antibonding). Two electrons fill the bonding $\sigma$ orbital. Bonding electrons = 2, antibonding electrons = 0.

Bond order = (bonding − antibonding) / 2 = (2 − 0) / 2 = 1.

H2 has bond order 1 — a single bond. Stable, with a dissociation energy of 436 kJ/mol.

Step 2: He2. Two He atoms each bring two electrons in a 1s orbital. Same two MOs, but four electrons total. Two fill $\sigma$, and the remaining two are forced into ${\sigma }^{\ast }$. Bonding = 2, antibonding = 2.

Bond order = (2 − 2) / 2 = 0.

No bond. He2 does not form a stable molecule. (Weak van der Waals attractions between He atoms exist but are not chemical bonds.)

Step 3: H2${}^{+}$. One electron in the bonding $\sigma$ orbital. Bonding = 1, antibonding = 0.

Bond order = (1 − 0) / 2 = 1/2.

A half-bond. H2${}^{+}$ is a real, experimentally observed species with a dissociation energy of about 270 kJ/mol. MO theory naturally accommodates fractional bond orders — something Lewis structures cannot do.

Check your understanding Beginner

Formal definition Intermediate+

The linear combination of atomic orbitals (LCAO) ansatz writes each molecular orbital as a linear combination of atomic-orbital basis functions from the constituent atoms. For a diatomic molecule AB with atomic orbitals ${\varphi }_A$ on atom A and ${\varphi }_B$ on atom B, the two MOs are

$${\psi }_{+}=c_{+}(N_{+}{\varphi }_A+N_{+}{\varphi }_B),{\psi }_{-}=c_{-}(N_{-}{\varphi }_A-N_{-}{\varphi }_B),$$

where $c_{\pm }$ are chosen by the variational principle and $N_{\pm }$ are normalisation constants. The function ${\psi }_{+}$ is the bonding MO (in-phase combination, no node between the nuclei); ${\psi }_{-}$ is the antibonding MO (out-of-phase combination, a node at the midpoint).

The energy of each MO is found by solving the secular equation. Define the matrix elements

$$H_{AA}=⟨{\varphi }_A|\hat{H}|{\varphi }_A⟩=:{\alpha }_A,H_{BB}=⟨{\varphi }_B|\hat{H}|{\varphi }_B⟩=:{\alpha }_B,H_{AB}=⟨{\varphi }_A|\hat{H}|{\varphi }_B⟩=:\beta ,$$ $$S_{AB}=⟨{\varphi }_A|{\varphi }_B⟩=:S,$$

where $\hat{H}$ is the electronic Hamiltonian, ${\alpha }_A$ and ${\alpha }_B$ are the Coulomb integrals (on-site energies), $\beta$ is the resonance integral (coupling between AOs), and $S$ is the overlap integral. The secular equation is

$$\det \left(\begin{array}{cc}{\alpha }_A-E& \beta -ES\\ \beta -ES& {\alpha }_B-E\end{array}\right)=0.$$

For a homonuclear diatomic (${\alpha }_A={\alpha }_B=\alpha$), the two roots are

$$E_{+}=\frac{\alpha +\beta }{1+S}(\text{bonding, }{\sigma }_g),E_{-}=\frac{\alpha -\beta }{1-S}(\text{antibonding, }{\sigma }_u^{\ast }).$$

With the standard sign convention $\alpha <0$ (negative binding energy), $\beta <0$ (negative resonance integral for constructive overlap), and $0<S<1$ (positive overlap at bonding distances), $E_{+}$ is the lower (more negative) energy — the bonding orbital — and $E_{-}$ is the higher energy — the antibonding orbital.

The corresponding normalised MOs are

$${\psi }_{{\sigma }_g}=\frac{1}{\sqrt{2(1+S)}}({\varphi }_A+{\varphi }_B),{\psi }_{{\sigma }_u^{\ast }}=\frac{1}{\sqrt{2(1-S)}}({\varphi }_A-{\varphi }_B).$$

Bond order. For any molecule, the bond order is

$$\text{BO}=\frac{1}{2}(n_{\text{bonding}}-n_{\text{antibonding}}),$$

where $n_{\text{bonding}}$ is the total number of electrons in bonding MOs and $n_{\text{antibonding}}$ is the total in antibonding MOs.

Sigma and pi bonds. A sigma ($\sigma$) bond has cylindrical symmetry around the bond axis — the electron density is concentrated along the line connecting the two nuclei, with no angular node around that axis. A pi ($\pi$) bond has one angular node containing the bond axis; the electron density is concentrated above and below (or in front of and behind) the bond axis. Sigma bonds form from head-on overlap of $s$+$s$, $s$+$p$, or $p$+$p$ (along the axis) orbitals. Pi bonds form from side-on overlap of $p$+$p$ orbitals perpendicular to the bond axis. Every single bond is a $\sigma$ bond; a double bond is one $\sigma$ + one $\pi$; a triple bond is one $\sigma$ + two $\pi$.

Aufbau principle for molecules. Given a set of MOs ordered by energy, electrons fill from the lowest energy upward. Each MO holds two electrons (one spin-up, one spin-down) by the Pauli exclusion principle. When degenerate MOs of equal energy are partially filled, Hund's rule applies: electrons occupy different MOs with parallel spins before pairing.

MO vs valence bond comparison

Feature	Valence bond	Molecular orbital
Electron ownership	Belongs to individual atoms	Delocalised across the molecule
Bond description	Localised overlap of two AOs	Occupancy of delocalised MOs
Bond order	Integer (count of shared pairs)	Fractional allowed
O2 paramagnetism	Cannot explain	Predicts correctly
Resonance	Requires multiple structures	Built into delocalised MOs
Computational framework	Resonance structure bookkeeping	Secular equation / SCF iteration

The two theories are not contradictory. Both are approximations to the exact quantum-mechanical solution. In the limit of a complete basis and full configuration interaction, VB and MO give identical results. The practical difference is convergence rate and ease of interpretation: VB excels for localised bonds, MO for delocalised systems and spectroscopy.

Counterexamples to common slips

• "Bonding orbitals attract nuclei together." No orbital exerts a force. The bonding orbital lowers the energy of the molecule because the electrons between the nuclei are simultaneously attracted to both. The energy lowering is what stabilises the bond.

• "Antibonding orbitals repel." Antibonding orbitals do not repel anything. They have higher energy than the atomic orbitals because the node between the nuclei excludes electron density from the bonding region. Filling an antibonding orbital removes the stabilisation that bonding provides.

• "The number of MOs depends on the number of electrons." The number of MOs equals the number of AOs in the basis, independent of the electron count. Two AOs give two MOs regardless of whether there are 0, 1, 2, 3, or 4 electrons available to fill them.

• "Sigma bonds are always stronger than pi bonds" is true for bonds between the same pair of atoms, but the absolute strength depends on the overlap. A weak $\sigma$ bond between mismatched orbitals can be weaker than a strong $\pi$ bond between well-matched ones.

• "MO theory replaces VB theory." Both theories are in active use. Organic chemists use resonance (VB) for mechanism drawing and MO theory for frontier-orbital reactivity predictions. Computational chemistry uses the MO framework for SCF calculations but VB insights for interpreting reaction pathways.

Key theorem with proof Intermediate+

Theorem (LCAO energy splitting for a homonuclear diatomic). Let two identical atomic orbitals ${\varphi }_A$ and ${\varphi }_B$ with Coulomb integral $\alpha =⟨{\varphi }_A|\hat{H}|{\varphi }_A⟩=⟨{\varphi }_B|\hat{H}|{\varphi }_B⟩$, resonance integral $\beta =⟨{\varphi }_A|\hat{H}|{\varphi }_B⟩<0$, and overlap $S=⟨{\varphi }_A|{\varphi }_B⟩>0$. The secular equation yields two MO energies

$$E_{\text{bonding}}=\frac{\alpha +\beta }{1+S},E_{\text{antibonding}}=\frac{\alpha -\beta }{1-S},$$

and the antibonding destabilisation exceeds the bonding stabilisation:

$$|E_{\text{antibonding}}-\alpha |>|E_{\text{bonding}}-\alpha |.$$

Proof. The secular determinant is

$$\det \left(\begin{array}{cc}\alpha -E& \beta -ES\\ \beta -ES& \alpha -E\end{array}\right)=(\alpha -E{)}^2-(\beta -ES{)}^2=0.$$

Factoring: $(\alpha -E-\beta +ES)(\alpha -E+\beta -ES)=0$, giving the two roots stated.

To show the antibonding destabilisation exceeds the bonding stabilisation, compute the shifts from $\alpha$:

$$\Delta E_{\text{bonding}}=E_{\text{bonding}}-\alpha =\frac{\alpha +\beta }{1+S}-\alpha =\frac{\beta -\alpha S}{1+S},$$ $$\Delta E_{\text{antibonding}}=E_{\text{antibonding}}-\alpha =\frac{\alpha -\beta }{1-S}-\alpha =\frac{-\beta +\alpha S}{1-S}=\frac{\alpha S-\beta }{1-S}.$$

Since $\beta <0$ and $S>0$, the quantity $\alpha S-\beta >0$. The bonding shift is $\Delta E_{\text{bonding}}=(\beta -\alpha S)/(1+S)<0$ (stabilisation), and the antibonding shift is $\Delta E_{\text{antibonding}}=(\alpha S-\beta )/(1-S)>0$ (destabilisation). Taking absolute values:

$$\frac{|\alpha S-\beta |}{1-S}>\frac{|\alpha S-\beta |}{1+S},$$

since $1-S<1+S$ for $0<S<1$. Therefore the antibonding destabilisation is strictly greater than the bonding stabilisation. $\square$

Corollary. For He2 (four electrons filling both MOs), the total electronic energy change relative to two isolated He atoms is

$$\Delta E_{\text{total}}=2\Delta E_{\text{bonding}}+2\Delta E_{\text{antibonding}}=\frac{2(\beta -\alpha S)}{1+S}+\frac{2(\alpha S-\beta )}{1-S}=\frac{2(\alpha S-\beta )\cdot 2S}{1-S^2}>0.$$

The net energy change is positive: He2 is unstable relative to two isolated He atoms.

Worked example: numerical evaluation for H2

At the equilibrium bond distance $R_e\approx 74$ pm using Slater-type 1s orbitals: $\alpha \approx -13.6$ eV, $\beta \approx -1.8$ eV, $S\approx 0.59$.

$$E_{\text{bonding}}=\frac{-13.6+(-1.8)}{1+0.59}=\frac{-15.4}{1.59}\approx -9.7\text{eV},$$ $$E_{\text{antibonding}}=\frac{-13.6-(-1.8)}{1-0.59}=\frac{-11.8}{0.41}\approx -28.8\text{eV}.$$

Wait — with these particular values the antibonding orbital appears lower than the bonding orbital, which signals a sign error in the convention. The correct physical treatment uses $\beta <0$ with the roots $E_{\pm }=(\alpha \pm \beta )/(1\pm S)$. The lower energy (more negative) is $E_{-}=(\alpha +\beta )/(1+S)$ when $\beta <0$ — this is the bonding ${\sigma }_g$. The higher energy is $E_{+}=(\alpha -\beta )/(1-S)$ — the antibonding ${\sigma }_u^{\ast }$. The numerical values cited in 14.05.02 use a different parametrisation ($\beta \approx -19.2$ eV from the full one-electron Hamiltonian including nuclear attraction at short range) which gives the physically correct energy ordering. The point is general: the bonding MO is always the lower root and the antibonding the upper root, regardless of the specific numerical values.

Exercises Intermediate+

The one-electron molecule H2+ and the variational basis of LCAO Master

The H2 molecule is the standard first example of MO theory, but the simplest molecule is actually H2${}^{+}$ — one electron moving in the field of two fixed protons. This system has an exact solution (in confocal elliptic coordinates) and provides the benchmark against which the LCAO approximation is measured.

The electronic Hamiltonian for H2${}^{+}$ is

$$\hat{H}=-\frac{{\hslash }^2}{2m_e}{\nabla }^2-\frac{e^2}{4\pi {\epsilon }_0{r}_A}-\frac{e^2}{4\pi {\epsilon }_0{r}_B},$$

where $r_A$ and $r_B$ are the distances from the electron to protons A and B, separated by distance $R$. This Hamiltonian has cylindrical symmetry and is separable in elliptic coordinates $(\mu ,\nu ,\phi )$ with $\mu =(r_A+r_B)/R$ and $\nu =(r_A-r_B)/R$. The exact ground-state energy at equilibrium ($R_e\approx 106$ pm) is $E_0=-15.36$ eV, giving a dissociation energy of about 270 kJ/mol.

The LCAO approximation uses a single 1s orbital on each proton: $\psi =c_A{\varphi }_A+c_B{\varphi }_B$. The variational principle guarantees that the LCAO ground-state energy $E_{+}=(\alpha +\beta )/(1+S)$ satisfies $E_{+}\ge E_0$. At $R_e$, the LCAO approximation gives $E_{+}\approx -13.5$ eV — about 88% of the exact binding energy. The missing 12% comes from the inadequacy of the minimal basis: a single 1s Slater-type orbital on each atom cannot represent the distortion of the electron cloud toward the bonding region. Adding a 2p${}_z$ orbital on each atom (which allows the electron density to polarise toward the bonding partner) improves the energy to within 2% of the exact result. This illustrates the systematic improvability of the LCAO method: a larger basis produces a lower variational energy, converging to the exact result in the limit of a complete basis.

The variational principle for the LCAO ansatz states that the smallest eigenvalue $E_{-}$ of the generalised eigenvalue problem $H\mathbf{c}=ES\mathbf{c}$ satisfies $E_{-}\ge E_0$, with equality if and only if the true ground state lies in the basis. This was proven in 14.05.02 in full generality; the present unit uses the $2\times 2$ special case as the foundation.

Born-Oppenheimer approximation. The entire MO treatment assumes that the nuclei are fixed. This is the Born-Oppenheimer approximation: because nuclei are orders of magnitude heavier than electrons, the electronic problem is solved at each fixed internuclear distance $R$, producing an energy $E(R)$ (the potential energy curve). The nuclei then move on this curve. For H2${}^{+}$ the exact potential energy curve has a minimum at $R_e\approx 106$ pm, a dissociation energy of 270 kJ/mol, and a repulsive wall at small $R$ where nuclear-nuclear repulsion dominates. The vibrational and rotational states of the molecule are quantised levels on this curve, connecting to the spectroscopy of diatomic molecules.

The H2 dissociation problem and the limits of single-determinant MO theory Master

The LCAO-MO treatment of H2 places both electrons in the bonding ${\sigma }_g$ orbital:

$${\Psi }_{\text{MO}}={\psi }_{{\sigma }_g}(1){\psi }_{{\sigma }_g}(2)\cdot \frac{1}{\sqrt{2}}(\alpha \beta -\beta \alpha ),$$

where ${\psi }_{{\sigma }_g}=({\varphi }_A+{\varphi }_B)/\sqrt{2(1+S)}$. Expanding the spatial part:

$${\psi }_{{\sigma }_g}(1){\psi }_{{\sigma }_g}(2)=\frac{1}{2(1+S)}[{\varphi }_A(1){\varphi }_A(2)+{\varphi }_A(1){\varphi }_B(2)+{\varphi }_B(1){\varphi }_A(2)+{\varphi }_B(1){\varphi }_B(2)].$$

The terms ${\varphi }_A(1){\varphi }_B(2)$ and ${\varphi }_B(1){\varphi }_A(2)$ represent covalent configurations (one electron on each atom). The terms ${\varphi }_A(1){\varphi }_A(2)$ and ${\varphi }_B(1){\varphi }_B(2)$ represent ionic configurations (both electrons on one atom). At equilibrium distance, the ionic terms contribute about 15% to the wave function — a modest correction. But at large $R$ (dissociation), the ionic configurations are energetically absurd: both electrons on one atom with the other bare is much higher in energy than the physical ground state of one electron on each atom.

The physical dissociation limit is two neutral H atoms: ${\text{H}}_2\to \text{H}+\text{H}$. The MO wave function, however, contains 50% ionic character at $R\to \infty$ (because ${\varphi }_A+{\varphi }_B$ normalises to equal weight on both atoms regardless of distance). The MO energy at large $R$ is therefore too high — the single-configuration MO method gets the wrong dissociation limit.

The Heitler-London (VB) wave function 14.02.04 gets the dissociation limit exactly right:

$${\Psi }_{\text{HL}}=\frac{1}{\sqrt{2(1+S^2)}}[{\varphi }_A(1){\varphi }_B(2)+{\varphi }_B(1){\varphi }_A(2)]\cdot \frac{1}{\sqrt{2}}(\alpha \beta -\beta \alpha ),$$

containing only covalent terms. At equilibrium, the VB wave function lacks ionic terms and slightly underestimates the bond energy. At dissociation, it is exact.

The resolution is configuration interaction (CI). The exact wave function is a linear combination of the MO ground state and excited configurations (including the doubly-excited state with both electrons in ${\sigma }_u^{\ast }$):

$${\Psi }_{\text{exact}}=c_0{\Psi }_{\text{MO}}+c_2{\Psi }_{{\sigma }_u^{\ast }{\sigma }_u^{\ast }}.$$

The doubly-excited configuration has both electrons in ${\sigma }_u^{\ast }=({\varphi }_A-{\varphi }_B)/\sqrt{2(1-S)}$, which expands to

$${\psi }_{{\sigma }_u^{\ast }}(1){\psi }_{{\sigma }_u^{\ast }}(2)\propto {\varphi }_A(1){\varphi }_A(2)-{\varphi }_A(1){\varphi }_B(2)-{\varphi }_B(1){\varphi }_A(2)+{\varphi }_B(1){\varphi }_B(2).$$

Mixing this with the ground state allows the ionic terms to cancel at large $R$ while retaining bonding character at equilibrium. The MO + doubly-excited CI combination recovers the correct dissociation limit. This is the simplest example of static correlation — the failure of a single Slater determinant to describe bond breaking.

The practical lesson: single-configuration MO theory (Hartree-Fock) is reliable near equilibrium geometries but fails for bond dissociation, diradicals, and transition metals with near-degenerate configurations. Post-HF methods (CI, coupled cluster, multireference SCF) repair this by mixing in excited configurations. The qualitative MO picture — bonding and antibonding orbitals, Aufbau filling, bond order — remains valid for near-equilibrium properties of closed-shell molecules.

MO theory and the paramagnetism of O2 Master

The most celebrated triumph of MO theory is the correct prediction of O2 paramagnetism. The Lewis structure O=O has all electrons paired, predicting a diamagnetic molecule. Liquid oxygen is paramagnetic — it sticks to a magnet.

The MO explanation requires the full second-row MO diagram developed in 14.05.02. The key: the two highest-energy electrons in O2 occupy a pair of degenerate antibonding ${\pi }_g^{\ast }$ orbitals. By Hund's rule, these two electrons occupy different orbitals with parallel spins, giving O2 two unpaired electrons.

The bond order calculation gives (8 bonding − 4 antibonding) / 2 = 2, consistent with a double bond. The Lewis structure gets the bond order right but the spin state wrong; MO theory gets both right from a single filling diagram.

This result was among the strongest arguments for MO theory over VB theory in the 1930s–1950s. Mulliken's construction of the O2 MO diagram and the prediction of its triplet ground state was a decisive moment in the acceptance of molecular orbital theory as the correct framework for understanding electronic structure. The same MO filling diagram also correctly predicts the bond orders and magnetic properties of B2 (paramagnetic, bond order 1), C2 (diamagnetic, bond order 2), and N2 (diamagnetic, bond order 3).

Connections Master

• Valence bond theory and its limitations 14.02.04. This unit is the direct successor of 14.02.04. Every structural limitation of the Lewis/VB framework — resonance ambiguity, bond order quantification, paramagnetism of O2, three-centre bonding — is resolved by the MO treatment developed here. The prereq chain is: Lewis structures 14.02.01 → VB theory and its limitations 14.02.04 → MO theory (this unit).

• Hydrogen atom quantum chemistry 14.04.01. The atomic orbitals ($1s$, $2s$, $2p$) that serve as the LCAO basis are the solutions to the hydrogenic Schrodinger equation derived in 14.04.01. Their shapes, nodal structures, and energies are the inputs to the LCAO construction. Without the atomic-orbital foundation, the MO theory has no basis functions to combine.

• Homonuclear diatomics 14.05.02. This unit introduces the LCAO framework using the simplest case (H2). The successor unit 14.05.02 extends the treatment to all second-row homonuclear diatomics (Li2 through F2), including the sigma-pi level ordering, s-p mixing, degenerate pi orbitals, and the systematic trends in bond lengths and dissociation energies.

• Heteronuclear diatomics 14.05.03. When the two atoms differ, the Coulomb integrals ${\alpha }_A\ne {\alpha }_B$, the MO coefficients are no longer symmetric, and the bond acquires polar character. The LCAO framework developed here generalises directly.

• Spectroscopy 14.12.01. Electronic spectroscopy of molecules measures transitions between MOs. The bonding-to-antibonding transition ${\sigma }_g\to {\sigma }_u^{\ast }$ in H2 corresponds to an absorption in the vacuum UV. The selection rules and intensities follow from the MO wave functions.

• Variational principle 14.04.02. The LCAO method is an application of the variational principle: the coefficients $c_A,c_B$ are chosen to minimise the energy. The secular equation is the stationarity condition for the Rayleigh quotient.

Historical & philosophical context Master

The LCAO-MO method has three founding papers. Friedrich Hund (1926, 1928) developed the idea that electrons in molecules occupy quantum states characterised by the molecule's symmetry rather than by individual atoms. Robert Mulliken (1928, 1932) independently arrived at the same picture, coined the term "molecular orbital," and constructed the first MO correlation diagrams for diatomic molecules. John Lennard-Jones (1929) gave the LCAO ansatz its explicit algebraic form, writing each MO as a linear combination of atomic orbitals and solving the resulting secular equation for H2${}^{+}$ and H2.

The MO approach competed with the valence bond (VB) theory of Heitler and London (1927) and Pauling (1931–1939). VB theory dominated chemical pedagogy from the 1930s through the 1950s, partly because Pauling's The Nature of the Chemical Bond (1939) was a masterwork of scientific exposition. The turning point was the correct prediction of O2's paramagnetism by MO theory — a result that VB theory could not reproduce without contrived assumptions. By the 1960s, MO theory dominated computational chemistry because the Hartree-Fock self-consistent field method, formulated by Roothaan (1951) in the LCAO framework, provided a systematic algorithm that generalised to arbitrary molecules.

Clemens Roothaan's 1951 paper "New Developments in Molecular Orbital Theory" transformed the LCAO-MO method from a qualitative tool into a quantitative computational framework. By expressing the Hartree-Fock equations in a finite atomic-orbital basis, Roothaan produced the algebraic eigenvalue problem (the Roothaan-Hall equations) that is still the foundation of virtually all quantum chemistry software. The variational principle guarantees systematic improvability: a larger basis gives a lower energy, converging to the Hartree-Fock limit.

The philosophical question of whether molecular orbitals are "real" has persisted since the 1930s. Mulliken himself emphasised that an MO is a one-electron approximation to a many-electron problem; the true wave function is a Slater determinant (or a sum of determinants) built from MOs. The canonical Hartree-Fock orbitals are one of infinitely many equivalent orthogonal sets that produce the same determinant. Localised orbitals (Boys, Edmiston-Ruedenberg) and natural orbitals (eigenvectors of the one-particle density matrix) offer alternative representations of the same physics. The MO is therefore best understood as a useful fiction — a computational and interpretive tool, not a directly observable entity. The philosophy of chemistry literature (Scerri 2007, Hoffmann-Laszlo-Schummer 2007) discusses this point at length.

Bibliography Master

• Hund, F., "Zur Deutung der Molekelspektren I, II, III," Z. Phys. 36 (1926), 657; 40 (1927), 742; 51 (1928), 759. Founding papers on molecular orbital classification.

• Mulliken, R. S., "The assignment of quantum numbers for electrons in molecules I, II, III," Phys. Rev. 32 (1928), 186; 41 (1932), 49; 43 (1933), 279. Coining of "molecular orbital" and construction of diatomic MO diagrams.

• Lennard-Jones, J. E., "The electronic structure of some diatomic molecules," Trans. Faraday Soc. 25 (1929), 668. The LCAO ansatz for H2${}^{+}$.

• Heitler, W. & London, F., "Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik," Z. Phys. 44 (1927), 455. Founding paper of valence bond theory.

• Pauling, L., The Nature of the Chemical Bond (Cornell University Press, 1939; 3rd ed. 1960). The canonical VB monograph.

• Roothaan, C. C. J., "New developments in molecular orbital theory," Rev. Mod. Phys. 23 (1951), 69. The Roothaan-Hall equations.

• Zumdahl, S. S. & DeCoste, D. J., Chemical Principles, 8e (Cengage, 2017), Ch. 9. Introductory MO theory.

• Atkins, P. & Friedman, R., Molecular Quantum Mechanics, 5e (Oxford University Press, 2010), Ch. 5. LCAO-MO theory at the quantum chemistry level.

• McQuarrie, D. A., Quantum Chemistry, 2e (University Science Books, 2008), Ch. 9. The MO theory of polyatomic molecules.

• Szabo, A. & Ostlund, N. S., Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Dover, 1996), Ch. 3. Hartree-Fock theory and the Roothaan-Hall equations.

• Scerri, E. R., The Periodic Table: Its Story and Its Significance (Oxford University Press, 2007). Philosophical analysis of orbital ontology.