The LCAO Approximation

Every production quantum chemistry code — Gaussian, ORCA, Psi4, NWChem, Q-Chem — represents its molecular orbitals as linear combinations of atomic-centered basis functions (Gaussian-type orbitals approximating Slater/atomic orbitals), so the Roothaan–Hall equations solved millions of times a day in academic and industrial computing are, structurally, exactly the secular-determinant problem derived here, just at enormous matrix dimension. Drug discovery pipelines use LCAO-based density functional theory to compute binding energies, reaction barriers, and spectroscopic properties of candidate molecules before a single gram is synthesized. Materials scientists use LCAO-derived tight-binding models (a direct descendant of the Hückel-style simplified secular equations in this article) to compute band structures of semiconductors, the electronic structure of graphene and other 2D materials, and the properties of molecular electronics components. Photovoltaic and OLED design leans on LCAO-computed HOMO/LUMO energies (the two roots of a secular determinant, generalized to many orbitals) to predict absorption and emission colors and charge-transfer efficiency. In spectroscopy, Mulliken's original correlation diagrams — built entirely from LCAO reasoning — are still the standard way to rationalize UV-visible and photoelectron spectra of small molecules in undergraduate and graduate instruction alike. Even qualitative organic chemistry, when explaining conjugation, aromaticity, or the reactivity of enolates and dienes via frontier molecular orbital theory, is implicitly invoking the LCAO expansion to justify why $\pi$ systems can be described by delocalized orbitals built from atomic $p_z$ orbitals.

Ask the tutor: How do the resonance and overlap integrals $H_{AB}$ and $S_{AB}$ actually depend on the internuclear distance $R$ for two hydrogen 1s orbitals?

Picture an electron shared between two nuclei, A and B, separated by some internuclear distance $R$. If the electron happens to be found very close to nucleus A, it experiences almost exclusively the Coulomb field of A, and the wavefunction in that region must reduce to (approximately) a hydrogen-like atomic orbital centered on A — a 1s orbital, a 2p orbital, whatever atomic state is energetically accessible. The same logic applies near nucleus B. And if the electron wanders far from both nuclei, the molecular orbital must decay the way an atomic orbital decays, since a bound electron cannot ignore the exponential falloff dictated by its binding energy. These are not vague hand-waving statements; they are boundary conditions on the exact solution of the molecular Schrödinger equation, and any trial wavefunction that violates them badly will give a poor variational energy. The simplest function satisfying all three conditions simultaneously is a linear combination:

Here $\phi_A$ is a hydrogen-like orbital centered on nucleus A and $\phi_B$ one centered on nucleus B, and $c_A, c_B$ are real, adjustable, dimensionless coefficients. Near A, where $\phi_A$ is large and $\phi_B$ is small (since B's orbital has decayed by the time you reach A, for large enough $R$), $\psi \approx c_A \phi_A$ — an atomic orbital, as required. The same reasoning holds near B. This is the physical heart of LCAO: it is the minimal ansatz consistent with known limits, and everything else — bonding/antibonding character, orbital energies, overlap effects — falls out of optimizing $c_A$ and $c_B$. The coefficients themselves carry chemical meaning. Squaring and integrating, $|c_A|^2$ is (approximately, ignoring overlap) the probability of finding the electron in atomic orbital A's territory, and $|c_B|^2$ in B's. If the two atoms are identical (homonuclear diatomic), symmetry forces $|c_A| = |c_B|$, but the relative sign of $c_A$ and $c_B$ is not fixed by symmetry alone — it is fixed by the variational principle, and that sign is exactly what distinguishes a bonding combination from an antibonding one.

Once the ansatz is fixed, the question becomes purely a matter of numbers: what values of $c_A$ and $c_B$ give the lowest possible energy for $\psi$? This is exactly the setting for the Variational Principle, which guarantees that any normalized trial wavefunction gives an energy expectation value no lower than the true ground-state energy. Minimizing over the coefficients is therefore guaranteed to get us as close as our chosen atomic-orbital basis allows.

The LCAO trial wavefunction for a diatomic one-electron system, in the most general single-basis-function-per-atom form, is

where: $\phi_A, \phi_B$ are normalized (but not mutually orthogonal) atomic orbitals centered on nuclei A and B, and $c_A, c_B$ are real variational coefficients to be determined.

Reading: this single line is the entirety of the LCAO ansatz; all subsequent machinery exists only to pin down $c_A$ and $c_B$. The variational energy of this trial function is the expectation value of the molecular electronic Hamiltonian $\hat H$,

where: the denominator is required because $\psi$ is not automatically normalized once $c_A, c_B$ are free parameters, and $\hat H$ is the full electronic Hamiltonian (kinetic energy plus attraction to both nuclei, plus electron–electron repulsion for many-electron cases).

Reading: this is just the Rayleigh quotient from the Variational Principle; any choice of $c_A, c_B$ gives an $E$ that is an upper bound to the true ground-state energy. Expanding numerator and denominator in terms of the atomic orbitals introduces three families of integrals. Define

where: $H_{ii}$ is the Coulomb integral (the energy of an electron sitting in atomic orbital $i$, feeling the full molecular Hamiltonian), $H_{ij}$ is the resonance (or exchange/hopping) integral coupling orbitals $i$ and $j$, and $S_{ij}$ is the overlap integral measuring how much $\phi_i$ and $\phi_j$ spatially coincide ($S_{ii}=1$ if normalized).

Reading: these three numbers — one Coulomb integral per atom, one resonance integral per bonded pair, one overlap integral per pair — are the entire numerical content of an LCAO calculation; the coefficients $c_A, c_B$ are obtained purely by combining them algebraically. Substituting the expansion into the Rayleigh quotient and minimizing with respect to $c_A$ and $c_B$ (setting $\partial E/\partial c_A = \partial E/\partial c_B = 0$) produces the secular equations:

where: these are two linear homogeneous equations in $c_A, c_B$, valid at whatever energy $E$ makes them simultaneously satisfiable.

Reading: a nontrivial solution ($c_A, c_B$ not both zero) exists only for special values of $E$ — this is precisely a generalized eigenvalue problem, and the special $E$'s are the molecular orbital energies. The condition for a nontrivial solution is that the determinant of the coefficient matrix vanish — the secular determinant:

where: this $2\times2$ determinant, set to zero, is a quadratic equation in $E$ whose two roots are the bonding and antibonding molecular orbital energies.

Reading: solving a Schrödinger-type problem by LCAO always ends here — at a matrix eigenvalue problem whose size equals the number of atomic orbitals in the basis; two atomic orbitals give a $2\times2$ determinant, $N$ atomic orbitals give an $N\times N$ determinant (or, in extended systems, a Hückel-type matrix diagonalization).

Example 1 — Foundational: solving a fully numerical secular determinant. Suppose a (schematic, energy-in-eV) two-orbital LCAO problem has been set up with

Note these are deliberately unequal Coulomb integrals (a heteronuclear-type case) so that no accidental symmetry hides an error. The secular determinant is

Expand each piece. First,

Next,

Subtracting,

Divide through by $0.9559$:

Apply the quadratic formula:

giving two roots,

Since $-15.211 < -13.556$, actually $E_2=-15.211$ eV is the lower root and $E_1=-13.556$ eV the upper one. Both lie close to, but shifted from, the original $H_{AA}$ and $H_{BB}$ values — exactly the expected behavior: mixing pushes the lower orbital energy down slightly and the upper one up slightly relative to the unperturbed atomic levels, a hallmark of any two-level quantum mixing problem (compare non-degenerate Perturbation Theory). Each root, plugged back into a secular equation, fixes the corresponding $c_A/c_B$ ratio and hence the bonding character of that molecular orbital.

Limiting case check: if $H_{AB}\to 0$ and $S_{AB}\to 0$ (atoms infinitely far apart, no interaction), the determinant reduces to $(H_{AA}-E)(H_{BB}-E)=0$, giving back exactly the two isolated atomic energies $E=H_{AA}$ and $E=H_{BB}$ with no mixing — precisely the correct atomic limit, confirming the algebra is consistent.

Example 2 — Applied: LCAO for a heteronuclear diatomic bond (HF-type polarity). Consider a simplified LCAO treatment of a polar bond, such as the σ framework in HF, where the bonding orbital is built from an H 1s orbital ($\phi_H$) and an F 2p$_z$ orbital ($\phi_F$) pointed along the bond axis. Because fluorine is far more electronegative, its atomic orbital energy is much lower: take $H_{HH} = -13.6$ eV, $H_{FF} = -18.7$ eV, with resonance integral $H_{HF} = -2.4$ eV and overlap $S_{HF} = 0.30$. The secular determinant is the same structural equation as before:

Expanding,

Subtracting and collecting:

Divide by $0.91$:

giving $E_{\text{bonding}} = -13.17$ eV and $E_{\text{antibonding}} = -20.74$ eV. Wait: check which is lower — $-20.74$ eV is more negative, so it is actually the lower (bonding) root, and $-13.17$ eV is the antibonding root. This ordering makes physical sense once inspected: the bonding MO sits below the lower atomic level $H_{FF}=-18.7$ eV, consistent with stabilization from constructive overlap, while the antibonding level sits above the higher atomic level $H_{HH}=-13.6$ eV, consistent with destabilization. Because $H_{FF}$ is much lower than $H_{HH}$, the bonding MO coefficient on fluorine will be substantially larger than on hydrogen — recovering, purely from the LCAO secular equations and without invoking electronegativity as a separate postulate, the qualitative picture of a polarized bond with the electron density skewed toward fluorine. This is the essential mechanism by which LCAO theory reproduces bond polarity: unequal Coulomb integrals directly translate into unequal coefficients.

Example 3 — Exam/competition level: three-orbital LCAO and the effect of adding basis flexibility. A common exam-style question asks: does adding a third atomic orbital to the LCAO basis always lower the ground-state energy, and by how much in a specific case? Consider extending Example 1's two-orbital heteronuclear problem by adding a third orbital $\phi_C$ on a nearby atom, with

(keeping $H_{AA}, H_{BB}, H_{AB}, S_{AB}$ from Example 1). The secular determinant is now $3\times3$:

Rather than expand the full cubic (which a student could do numerically), the qualitative examination question is the important part: the variational theorem guarantees that the lowest root of this $3\times3$ determinant must be less than or equal to the lowest root found in the $2\times2$ problem, $E_2 = -15.211$ eV from Example 1. This is because any solution reachable in the smaller basis (simply setting the coefficient on $\phi_C$ to zero) is still an allowed trial function in the larger basis, so the variational minimum over the larger space cannot be worse. Numerically solving the cubic (by expanding along the first row and using the quadratic formula on the resulting depressed cubic, or by Newton's method starting near $-15.2$ eV) gives a lowest root near $E \approx -15.27$ eV — a small additional stabilization of about $0.06$ eV from including $\phi_C$, consistent with the weak coupling integrals $H_{AC}, H_{BC}$ chosen. The lesson generalizes completely: enlarging the LCAO basis can only lower (or leave unchanged) the computed ground-state energy, never raise it, which is exactly the content of the variational theorem applied to nested basis sets, and is the same logic that underlies why modern quantum chemistry pushes toward larger and larger atomic-orbital (or Gaussian) basis sets.

The LCAO method as a systematic tool for molecules is usually credited to the German physicist Friedrich Hund and the American physical chemist Robert S. Mulliken, working largely independently in the late 1920s to interpret molecular spectra in terms of orbitals delocalized over the whole molecule rather than localized bonds. Mulliken, trained originally as a chemist before turning to molecular spectroscopy and quantum theory, was especially insistent that molecular orbitals — built as combinations of atomic orbitals — gave a cleaner and more predictive account of diatomic spectra, ionization energies, and bond orders than the competing valence-bond picture championed by Linus Pauling and others. The mathematical shortcut of expanding an unknown molecular orbital in a known atomic-orbital basis, essentially the content of this article, is often specifically attributed to a 1928 paper by John Lennard-Jones, who first wrote down the LCAO trial function for diatomic molecules in the form used ever since. Mulliken spent decades refining the physical interpretation of LCAO coefficients — introducing the language of bonding, antibonding, and nonbonding orbitals, and the notion of orbital correlation diagrams that connect molecular orbitals smoothly to their separated-atom and united-atom limits. His insistence on population analysis (partitioning electron density among atoms using the LCAO coefficients) remains a standard tool, still called Mulliken population analysis, in every modern electronic structure package.