LCAO molecular orbitals

Result

LCAO molecular orbitals

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Statement

Building molecular orbitals from atomic ones.

Why it matters

born-oppenheimer already justifies solving the electronic Schrödinger equation at fixed nuclear positions, and hydrogen-atom-solution supplies exact atomic orbitals for a one-electron atom; but no molecule beyond a one-electron ion has an exactly solvable electronic Schrödinger equation. LCAO is the standard, practical way around this: build an approximate molecular orbital directly out of the known atomic orbitals sitting on each nucleus, turning an intractable differential equation into a tractable linear-algebra problem. It underlies huckel-theory's treatment of conjugated pi systems and is the conceptual seed of every larger, more accurate quantum-chemical method (basis-sets, hartree-fock) built later in the programme.

Hypotheses

A molecular orbital \(\psi\) can be approximated as a linear combination of a finite set of atomic-orbital basis functions, \(\psi=\sum_i c_i\phi_i\).This is an ansatz, not an exact statement; particle-in-a-box and hydrogen-atom-solution's own machinery only guarantees exact solutions for specific idealised potentials, and no such exact solution exists for a general molecule. variational-principle justifies searching over the coefficients \(c_i\) to minimise energy, and guarantees the result is an upper bound to the true energy, however large the basis. The basis atomic orbitals retain roughly their free-atom shape and symmetry within the bound molecule.This is reasonable when atoms are only moderately perturbed by bonding; for a highly accurate treatment the atomic orbitals themselves may need to be supplemented with additional, more diffuse or polarised functions not present on the free atom (basis-sets covers this extension explicitly). For a simple two-centre case, only two atomic orbitals of matching symmetry and comparable energy are combined.Orbitals of mismatched symmetry (e.g. an \(s\) orbital combined with a \(p\) orbital oriented perpendicular to the bond axis) have a vanishing overlap integral by symmetry and contribute negligible bonding interaction, so they are excluded from the minimal treatment developed here.

Proof

\psi = c_A\phi_A + c_B\phi_B

For a diatomic case such as \(\text{H}_2^+\), the trial molecular orbital is built from exactly one atomic orbital on each centre, with coefficients \(c_A,c_B\) left free to be optimised. A

\begin{vmatrix}H_{AA}-E & H_{AB}-ES\\ H_{AB}-ES & H_{BB}-E\end{vmatrix}=0

Applying the variational principle (minimising \(E\) with respect to \(c_A,c_B\)) leads to the secular equations, whose non-trivial solution requires this determinant to vanish; \(S\) is the overlap integral between \(\phi_A\) and \(\phi_B\), generally nonzero since the two orbitals occupy overlapping regions of space. B

H_{AA}=H_{BB}=\alpha,\quad H_{AB}=\beta \ \Rightarrow\ E_\pm = \frac{\alpha\pm\beta}{1\pm S}

For two identical atoms, symmetry sets both diagonal integrals equal (\(\alpha\), the atomic energy) and the determinant solves directly for two roots, \(E_+\) (lower) and \(E_-\) (higher), since \(\beta\) (the resonance integral) is negative for a bonding-type interaction. B

\psi_+ = \frac{\phi_A+\phi_B}{\sqrt{2(1+S)}} \ \ (\text{bonding}), \qquad \psi_- = \frac{\phi_A-\phi_B}{\sqrt{2(1-S)}}\ \ (\text{antibonding})

The lower-energy root corresponds to the symmetric (in-phase, constructive) combination of the two atomic orbitals, and the higher-energy root to the antisymmetric (out-of-phase, destructive) combination. A

|\psi_+|^2 \text{ enhanced between nuclei}; \qquad |\psi_-|^2=0 \text{ at the midpoint (a node)}

Constructive interference in \(\psi_+\) builds up electron density directly between the two nuclei, screening their mutual repulsion and lowering energy; destructive interference in \(\psi_-\) creates a node between the nuclei, removing exactly the density that would otherwise stabilise the bond, raising energy above the separated-atom value \(\alpha\). A

Result

E_\pm = \frac{\alpha\pm\beta}{1\pm S}, \qquad \psi_\pm = \frac{\phi_A\pm\phi_B}{\sqrt{2(1\pm S)}}

Reading. Two atomic orbitals combine constructively into a lower-energy bonding molecular orbital, delocalised over both nuclei, and destructively into a higher-energy antibonding orbital carrying a node between them.

Scope. The minimal two-orbital treatment above applies directly to a diatomic with matching-symmetry valence orbitals; polyatomic molecules require a larger basis and many more coefficients (basis-sets), and mismatched-symmetry orbital pairs contribute no bonding interaction at all by the Hypotheses' third assumption.

Corollaries & converses

huckel-theory is exactly this LCAO machinery restricted to a conjugated molecule's pi system, with simplified, standardised values assumed for the Coulomb and resonance integrals.
variational-principle guarantees that any LCAO-computed energy is an upper bound on the true ground-state energy, and that enlarging the basis set can only lower (never raise) the computed energy.
Bond order can be read directly from how many electrons occupy bonding versus antibonding combinations: equal occupation of both gives zero net bond order, which is why \(\text{He}_2\) (four electrons, filling both \(\psi_+\) and \(\psi_-\)) does not exist as a stable molecule, while \(\text{He}_2^+\) (three electrons, net bond order \(0.5\)) is weakly bound.

Fails without

Combine two atomic orbitals of very different energy (large \(|H_{AA}-H_{BB}|\)) as if they mixed as strongly as two comparable-energy orbitals: the secular-equation solution shows mixing is heavily suppressed by a large energy gap even when overlap \(S\) is nonzero, so treating such a pair with the equal-energy formulas of Step 3 badly overestimates the resulting bonding interaction.
Use only a single, minimal atomic orbital per atom where genuine polarisation or diffuse character is needed (Hypotheses' basis assumption): a minimal-basis LCAO calculation systematically misrepresents bond angles and energies for such cases; only enlarging the basis set (basis-sets) toward completeness recovers the missing flexibility.

Common errors

Treating LCAO molecular orbitals as exact solutions rather than a basis-dependent approximation whose accuracy improves only as the basis is enlarged.
Forgetting the overlap integral \(S\) in the normalisation and energy denominators, implicitly treating \(\phi_A\) and \(\phi_B\) as orthogonal when they are not.
Expecting significant bonding interaction between atomic orbitals of mismatched symmetry, when the relevant overlap integral vanishes by symmetry and no interaction results.
Filling only bonding orbitals with electrons irrespective of electron count, rather than respecting the Pauli exclusion principle and populating antibonding orbitals once the bonding orbitals are full.

Discussion

The LCAO approach to molecular orbital theory took shape through the work of Friedrich Hund, Robert Mulliken, and John Lennard-Jones in the late 1920s and 1930s, developing alongside (and initially in some rivalry with) valence bond theory as a competing description of the chemical bond. Mulliken received the 1966 Nobel Prize in Chemistry substantially for this body of work.

Beyond the minimal two-orbital treatment given here, real molecular orbital calculations expand the basis with many more functions per atom (basis-sets), and the variational principle guarantees systematic, if computationally increasingly expensive, improvement as the basis grows — the direct conceptual bridge to the fuller electronic-structure methods (hartree-fock, density-functional-theory) developed later in the computational chemistry unit.

Common misconception: that a bonding orbital being lower in energy automatically means a molecule is stabilised. Stabilisation depends on how many electrons actually occupy the bonding versus antibonding combinations (Corollaries); a bonding orbital that exists but goes unoccupied contributes nothing to bond formation.

Worked examples

\text{H}_2^+: \ \text{one electron in } \psi_+

The single electron occupies the lower-energy bonding combination, giving a net stabilisation relative to a separated proton and hydrogen atom; this is the simplest possible chemical bond, with bond order \(1/2\) by the usual (bonding\(-\)antibonding)/2 electron count. A

\text{He}_2: \ \psi_+^2\psi_-^2 \ \Rightarrow\ \text{bond order}=\tfrac{2-2}{2}=0; \qquad \text{He}_2^+: \ \psi_+^2\psi_-^1 \ \Rightarrow\ \tfrac{2-1}{2}=0.5

Filling both bonding and antibonding orbitals equally in \(\text{He}_2\) cancels any net stabilisation, consistent with helium's chemical inertness; removing one antibonding electron to form \(\text{He}_2^+\) leaves a net bonding interaction, and this weakly bound cation is indeed observed spectroscopically. A

\text{Bond order} = \tfrac{1}{2}(n_{\text{bonding}}-n_{\text{antibonding}})

Reading. Simply counting how electrons distribute between the bonding and antibonding combinations derived in the Result predicts, correctly, which diatomics bond and which do not.

Scope. The same electron-counting logic extends directly to heavier homonuclear diatomics once their full set of molecular orbitals (from \(s\) and \(p\) atomic orbitals) is built up by the identical LCAO procedure.

Problems

Using the bond-order formula, compute the bond order of \(\text{Li}_2\) (two valence electrons, both in the lowest bonding combination of the \(2s\) orbitals).
Solution
Both valence electrons occupy the bonding \(\sigma_{2s}\) combination with none in the corresponding antibonding orbital: bond order \(=\tfrac12(2-0)=1\), consistent with \(\text{Li}_2\) existing as a genuine, if weakly bound, diatomic molecule.
Explain, using symmetry, why an \(s\) orbital on one atom and a \(p\) orbital oriented perpendicular to the internuclear axis on the neighbouring atom form no LCAO bonding interaction.
Solution
The overlap integral \(S=\int\phi_A\phi_B\,d\tau\) integrates the product of the two orbitals over all space; for an \(s\) orbital (spherically symmetric, everywhere positive) combined with a \(p\) orbital oriented perpendicular to the bond axis (equal positive and negative lobes symmetric about that axis), the positive-overlap contribution from one lobe exactly cancels the negative-overlap contribution from the other, giving \(S=0\) identically — no net bonding or antibonding interaction results (Hypotheses, third assumption).
A student claims that adding more atomic orbitals to an LCAO basis could, in principle, make the computed ground-state energy come out lower than the true, exact ground-state energy. Explain why this cannot happen.
Solution
This is precisely what variational-principle forbids: any trial wavefunction built as a linear combination of basis functions, however large the basis, gives a computed energy that is a strict upper bound on the true ground-state energy. Enlarging the basis can only lower (improve, approach the true value from above) the computed energy, never push it below the true value.