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Quantum chemistry / INTRODUCTORY EXPLORER

Orbital mixing.

Bring two orbital energies together and explore how coupling changes their energies and composition.

From basis orbitals to mixed states

Orbital mixingEnable JavaScript to display the interactive diagram.

Explore the relationship.

The relationship

The model and its limits.

H=(0VVΔ)H=\begin{pmatrix}0&V\\V&\Delta\end{pmatrix}
E±=Δ±Δ2+4V22E_\pm=\frac{\Delta\pm\sqrt{\Delta^2+4V^2}}{2}

A and B are orthonormal basis orbitals with uncoupled energies 0 and Δ. The coupling V mixes them. Squared eigenvector coefficients give the composition percentages. At equal basis energies, any nonzero coupling produces equal mixing.

This is a two-state Hamiltonian, not a computed molecule. It neglects basis overlap, electron–electron interactions, orbital occupancy, and geometry changes. A lower eigenvalue alone does not establish molecular stability or bond order. Changing the sign of coupling changes relative eigenvector signs but not these energies or composition percentages.

Sources

Andrei Tokmakoff, Two-Level System develops the coupled Hamiltonian and eigenstates.

Inspired by the extra-material topics in UT Dallas CHEM 3322.