Chemistry · Lesson

Molecular orbitals and approximate electronic structure

Treat molecular orbitals as approximate one-electron functions with explicit overlap and occupancy. Separate basis flexibility, electron-interaction treatment and convergence from a measured molecular excitation.

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By the end, you can

Normalize linear combinations using their overlap.
Solve a stipulated two-basis generalized eigenproblem.
Distinguish orbital, many-electron, basis and method descriptions.

A basis expansion is a mathematical representation

An orbital expansion writes φ = Σ cμ χμ using selected basis functions χμ. Atomic-centred functions are common, but they need not be orthogonal. Their overlap matrix has Sμν = ⟨χμ|χν⟩, so normalization is c†Sc = 1. A squared basis coefficient is not generally an independent spatial probability.

For two real normalized functions φA and φB with overlap S = 0.200, normalized symmetric and antisymmetric combinations are (φA ± φB)/√[2(1 ± S)]. The cross term changes the required coefficients. Orbital occupancy in an approximate many-electron description supplies additional information beyond that one-electron shape.

Use the same Hamiltonian and overlap in a variational calculation

The stationary finite-basis problem is Hc = ESc. H contains Hamiltonian matrix elements and S the overlaps. For a normalized admissible wavefunction and the same bounded-below Hamiltonian, the variational expectation cannot lie below its exact ground energy. Comparing different Hamiltonians would not use that guarantee.

In an invented one-electron matrix model, H_AA = H_BB = −10.0 eV, H_AB = H_BA = −3.00 eV and S = 0.200. The symmetric eigenvalue is (−10 − 3)/(1 + 0.2) ≈ −10.8333 eV and the antisymmetric one is (−10 + 3)/(1 − 0.2) = −8.75 eV. This is a finite model, not a real hydrogen-molecule calculation or an observed excitation gap.

Overlap changes both normalization and the matrix eigenvalues

Combination	Normalization coefficient	Eigenvalue relation	Model energy
Symmetric: φA + φB	1/√[2(1+S)] ≈ 0.64550	(H_AA + H_AB)/(1+S)	−10.8333 eV
Antisymmetric: φA − φB	1/√[2(1−S)] ≈ 0.79057	(H_AA − H_AB)/(1−S)	−8.75 eV
Both normalized states	c†Sc = 1	Hc = ESc	Splitting ≈ 2.08333 eV

Original two-function one-electron model with normalized real basis functions, overlap S=0.200, diagonal Hamiltonian elements −10.0 eV and off-diagonal elements −3.00 eV. Coefficients are amplitudes in a nonorthogonal basis. These finite-model energies are not a real molecule’s total energies or an experimentally established transition gap.

The many-electron approximation matters separately

Hartree–Fock optimizes a single-determinant description and includes Coulomb and exchange effects. It does not discard all electron–electron repulsion. The conventional correlation-energy comparison uses the exact nonrelativistic energy and the complete-basis Hartree–Fock limit for the same Hamiltonian, with the state and other assumptions held consistent.

Practical density-functional calculations approximate their functional. Kohn–Sham orbitals are auxiliary one-electron quantities, not the interacting many-electron wavefunction. Approximate density-functional energies do not automatically inherit every wavefunction variational upper-bound guarantee. The electron density integrates to the electron number, rather than necessarily to one.

Convergence and observables need their own checks

Increasing basis flexibility addresses basis incompleteness; changing the electronic method addresses other approximations. A Gaussian basis function facilitates integrals but one Gaussian does not reproduce the exact nuclear cusp and long-range shape of a general atomic orbital. More functions do not by themselves validate every predicted observable.

Self-consistent-field convergence means the chosen iterative equations have met their numerical criterion. It does not prove the globally lowest allowed state or a complete basis. Orbital-energy differences, including a HOMO–LUMO gap, are method-dependent quantities and are not generally identical to measured molecular excitation energies. An excitation calculation and a comparison with relevant data require further decisions.

Worked examples

Include the cross term in a symmetric combination

Two real basis functions are individually normalized and have overlap S = 0.200. Find C for φ = C(φA + φB).

The squared norm is C²(1 + 1 + 2S).
Set C²2(1 + S) = 1.
C = 1/√2.400 ≈ 0.645497.
Using 1/√2 would instead give squared norm 1.200 for this overlapping basis.

C ≈ 0.6455 under the supplied overlap. The cross term prevents interpreting the two coefficient squares as probabilities that simply sum to one.

Solve the specified symmetric matrix model

Use the supplied H diagonal −10.0 eV, off-diagonal −3.00 eV and real overlap S = 0.200. Find both symmetry-adapted eigenvalues.

The symmetric numerator is −10.0 − 3.00 = −13.0 eV and denominator is 1.200.
E_symmetric ≈ −10.8333 eV.
The antisymmetric numerator is −10.0 + 3.00 = −7.0 eV and denominator is 0.800.
E_antisymmetric = −8.75 eV.

Approximately −10.8 and −8.8 eV at the supplied model-input precision. These are one-electron model eigenvalues, with no real spectral intensity claimed.

Try it yourself

For S = 0 and two individually normalized real functions, what coefficient normalizes φA + φB?
Show the answer and reasoning
1/√2.
The two cross terms vanish in the orthogonal case.
Which normalization replaces c†c = 1 in a nonorthogonal basis?
Show the answer and reasoning
c†Sc = 1.
S is the overlap matrix.
Does Hartree–Fock omit every electron–electron interaction?
Show the answer and reasoning
No.
Its mean-field determinant treatment includes Coulomb and exchange contributions.
Is a computed HOMO–LUMO difference generally a measured excitation energy without further analysis?
Show the answer and reasoning
No.
The orbital quantity, excitation treatment and observed transition require separate checks.

What to carry forward

Normalize overlapping basis functions with S.
Apply a variational bound only to its stated Hamiltonian and admissible states.
Keep basis and electronic-method errors distinct.
Convergence and orbital gaps do not replace observable validation.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.

Massachusetts Institute of Technology3 sources