Molecular Orbital Theory

2.1 Introduction: MO Theory vs Valence Bond Theory

Molecular orbital (MO) theory and valence bond (VB) theory represent two complementary frameworks for understanding chemical bonding. While VB theory, pioneered by Heitler and London in 1927, describes bonds as arising from the overlap of localized atomic orbitals on individual atoms, MO theory takes a fundamentally different approach: electrons are delocalized over the entire molecule, occupying molecular orbitals that extend across all nuclei.

In VB theory, the wavefunction for H$_2$ is constructed by placing one electron on each atom and forming a singlet spin pairing. The VB wavefunction takes the form:

In contrast, MO theory first constructs molecular orbitals as linear combinations of atomic orbitals (LCAO), then fills these orbitals with electrons according to the aufbau principle, Pauli exclusion, and Hund's rules. Each electron moves in the average field of all nuclei and other electrons. The MO wavefunction for H$_2$ is:

where $\sigma_g = \frac{1}{\sqrt{2(1+S)}}(\phi_A + \phi_B)$ is the bonding molecular orbital. The key advantage of MO theory is its natural treatment of delocalization: electrons are not restricted to localized bonds but can spread over the entire molecular framework. This makes MO theory particularly powerful for describing conjugated systems, metallic bonding, and molecules where VB theory requires many resonance structures.

The triumph of MO theory includes its correct prediction of the paramagnetism of O$_2$ (which VB theory in its simplest form fails to predict), its straightforward extension to polyatomic molecules through symmetry-adapted linear combinations, and its natural connection to computational methods such as Hartree–Fock and density functional theory.

2.2 Derivation: LCAO-MO for H$_2^+$

The simplest molecular system is the hydrogen molecular ion H$_2^+$, which has two protons and one electron. This one-electron problem can be solved exactly in confocal elliptic coordinates, but the LCAO approach provides essential physical insight and generalizes to polyatomic systems. We approximate the molecular orbital as a linear combination of 1s atomic orbitals centered on each proton:

where $\phi_A$ and $\phi_B$ are hydrogen 1s orbitals centered on nuclei A and B respectively. We apply the variational principle: minimize the energy expectation value with respect to the coefficients $c_1$ and $c_2$:

Here we define the fundamental integrals. The overlap integral measures the spatial overlap between atomic orbitals on different centers:

The Coulomb integral represents the energy of an electron in orbital $\phi_A$ interacting with both nuclei:

The resonance integral (or exchange integral) captures the stabilization from electron delocalization between the two centers:

By symmetry, $H_{AA} = H_{BB}$ and $S_{AA} = S_{BB} = 1$. Minimizing $E$ with respect to $c_1$ and $c_2$ (setting $\partial E/\partial c_1 = 0$ and$\partial E/\partial c_2 = 0$) yields the secular equations:

For non-trivial solutions ($c_1, c_2 \neq 0$), the secular determinant must vanish:

Expanding this determinant:

Taking square roots and solving for $E$ gives the bonding and antibonding energies:

The bonding orbital $E_+$ corresponds to $c_1 = c_2$ (in-phase combination), giving:

The antibonding orbital $E_-$ corresponds to $c_1 = -c_2$ (out-of-phase combination), giving:

2.3 Derivation: Hückel Theory for π Systems

Hückel molecular orbital (HMO) theory is a remarkably effective simplification of the LCAO-MO method for conjugated π systems. Erich Hückel introduced three key approximations in 1931 that make the secular determinant analytically tractable:

The parameter $\alpha$ is the energy of an electron in an isolated carbon 2p$_z$ orbital (negative, typically around $-11.4$ eV), and $\beta$ is the resonance integral between adjacent p$_z$ orbitals (negative, typically around $-3.0$ eV). With the substitution$x = (\alpha - E)/\beta$, the secular determinant simplifies considerably.

Butadiene (CH$_2$=CH–CH=CH$_2$)

For butadiene, a linear chain of four carbon atoms, the Hückel secular determinant with $x = (\alpha - E)/\beta$ is:

Expanding this determinant by cofactors along the first row:

This is a quadratic in $x^2$. Solving with the quadratic formula:

Since $E = \alpha - x\beta$ and $\beta < 0$, the four π MO energies in order of increasing energy are:

The four π electrons fill $E_1$ and $E_2$. The total π-electron energy is:

For two isolated ethylene double bonds, $E_\pi = 4\alpha + 4\beta$. The delocalization energy is:

Benzene (C$_6$H$_6$)

For benzene, the cyclic topology introduces additional symmetry. The $6 \times 6$ Hückel secular determinant for a cyclic system can be solved using the general formula for cyclic polyenes. With $N = 6$ atoms in a ring, the eigenvalues are:

For benzene ($N = 6$), this yields:

With six π electrons filling the three lowest levels ($E_1$ and the degenerate pair $E_{2,3}$):

Three isolated ethylene bonds would give $E_\pi = 6\alpha + 6\beta$. The delocalization energy is:

2.4 Derivation: MO Diagrams and Bond Order

The bond order provides a quantitative measure of the net bonding character in a molecule. It is defined as:

This definition follows from the fact that each pair of electrons in a bonding orbital contributes one unit of bond strength, while each pair in an antibonding orbital removes one unit. To apply this formula to homonuclear diatomics, we need the MO energy level ordering.

For second-row homonuclear diatomics, the molecular orbitals formed from 2s and 2p atomic orbitals are (in approximate order of increasing energy):

For the lighter diatomics (Li$_2$ through N$_2$), s-p mixing reverses the ordering of$\sigma_{2p}$ and $\pi_{2p}$:

Application to Key Diatomics

Paramagnetism of O$_2$

The paramagnetism of oxygen was one of the great early triumphs of MO theory. In O$_2$, the$\pi_{2p}^*$ level is doubly degenerate. By Hund's rule, the two electrons in this level occupy separate degenerate orbitals with parallel spins, giving a triplet ground state ($^3\Sigma_g^-$) with two unpaired electrons:

2.5 Derivation: Walsh Diagrams for AH$_2$ Molecules

Walsh diagrams, introduced by A. D. Walsh in 1953, show how molecular orbital energies vary continuously with molecular geometry. For AH$_2$ molecules (e.g., H$_2$O, BeH$_2$), the key geometric parameter is the H–A–H bond angle $\theta$, which ranges from 90° to 180°.

At 180° (linear geometry, D$_{\infty h}$ symmetry), the molecular orbitals are classified by the irreducible representations of this point group. As the molecule bends to C$_{2v}$ symmetry, the orbital energies and symmetry labels change. The key orbital energy correlations are:

The critical orbital is the $1\pi_u$ orbital, which is non-bonding in the linear geometry. Upon bending, one component ($3a_1$) gains s-orbital character through mixing and is strongly stabilized, while the other ($1b_1$) remains non-bonding. The energy change of the $3a_1$ orbital as a function of bond angle can be derived from the overlap between the central atom's s and p orbitals with the hydrogen s orbitals:

Why H$_2$O is Bent

Water has 8 valence electrons occupying the $1a_1$, $1b_2$, $2a_1$, and$3a_1$/$1b_1$ levels. The $3a_1$ orbital is occupied and is strongly stabilized upon bending. The net energy change upon bending from 180° to the equilibrium angle is:

The large stabilization of the occupied $3a_1$ orbital upon bending more than compensates for the slight destabilization of $1b_2$, driving H$_2$O to a bent geometry with$\theta = 104.5°$.

Why BeH$_2$ is Linear

BeH$_2$ has only 4 valence electrons, filling $1a_1$ and $1b_2$ (or equivalently,$1\sigma_g$ and $1\sigma_u$ in the linear geometry). Crucially, the $3a_1$ orbital is empty:

Since the $3a_1$ orbital is unoccupied, there is no stabilization from bending. The slight destabilization of $1b_2$ upon bending, combined with the loss of the favorable linear$1\sigma_u$ orbital, means that BeH$_2$ prefers the linear 180° geometry.

2.6 Applications of MO Theory

Computational Chemistry

MO theory is the foundation of virtually all modern computational chemistry methods. The Hartree–Fock method constructs molecular orbitals as self-consistent solutions to the one-electron Fock equations. Post-Hartree–Fock methods (configuration interaction, coupled cluster, Møller–Plesset perturbation theory) and density functional theory all build upon the MO framework. The LCAO-MO approach naturally leads to the Roothaan–Hall equations, which cast the Hartree–Fock problem as a matrix eigenvalue problem solvable by standard linear algebra:

Conjugated Polymers and Organic Electronics

The Hückel model, extended to long conjugated chains, predicts that the π orbital energies form a band structure as $N \to \infty$. The bandwidth is $4|\beta|$, and the HOMO–LUMO gap decreases as $1/N$. This explains the optical properties of conjugated polymers like polyacetylene, polythiophene, and poly(p-phenylene vinylene), which are used in organic LEDs, solar cells, and field-effect transistors. The band gap of polyacetylene is approximately $2|\beta|\sin(\pi/N)$, which for large $N$ approaches zero — predicting metallic behavior, as confirmed experimentally by Shirakawa, MacDiarmid, and Heeger (Nobel Prize, 2000).

MO Theory in Spectroscopy

Electronic transitions observed in UV-Vis spectroscopy correspond to promotions of electrons between molecular orbitals. The absorption wavelength is determined by the MO energy gap. For conjugated systems, Hückel theory predicts the HOMO–LUMO gap:

This explains the bathochromic shift (red shift) observed with increasing conjugation length: as the chain grows, the energy gap decreases and absorption moves to longer wavelengths. Carotenoids, for example, absorb blue light and appear orange/yellow because their extended conjugation produces a HOMO–LUMO gap in the visible range.

Aromatic Stability

The remarkable stability of aromatic compounds is quantitatively explained by MO theory. The Hückel delocalization energy for benzene ($2\beta \approx 150$ kJ/mol) accounts for the experimentally observed thermodynamic stability relative to cyclohexatriene. The$4n+2$ rule emerges naturally: for a cyclic system with $4n+2$ π electrons, all bonding orbitals (including doubly degenerate pairs) are completely filled, leading to a closed-shell configuration with maximum delocalization energy.

2.7 Historical Development

The development of molecular orbital theory stands as one of the great intellectual achievements of twentieth-century chemistry. Its history involves several key figures whose contributions built upon one another over the course of two decades.