LESSON NOTES · 01

1. Molecular bonding, overlap and symmetry

Position in the course: Lesson 1 of 10. Complete the preceding derivation and use the explained exercises to check understanding.

Prerequisites: Linear algebra and molecular orbital concept

1. Build a minimal orbital model

Molecular orbitals are basis functions for a many-electron approximation. They can also organize qualitative bonding, provided the model assumptions stay visible. Consider two normalized real atomic functions \(a,b\) with overlap \(s=\langle a|b\rangle\). Let \(H_{aa}=H_{bb}=\alpha\) and \(H_{ab}=\beta\). The coefficients solve \(Hc=ESc\), rather than an ordinary eigenproblem, because the basis overlaps.

\[ H=\begin{pmatrix}\alpha&\beta\\\beta&\alpha\end{pmatrix},\quad S=\begin{pmatrix}1&s\\s&1\end{pmatrix},\quad \phi_\pm=\frac{a\pm b}{\sqrt{2(1\pm s)}}. \]

Applying either row of the matrix equation gives \(E_\pm=(\alpha\pm\beta)/(1\pm s)\). The denominator is physical normalization, not an optional correction. Linear independence requires \(|s|<1\). When \(s\) approaches one, the antisymmetric direction becomes poorly represented, a numerical warning that also occurs in large diffuse bases.

2. Interpret interference cautiously

For real orbitals, the plus combination has density \((a^2+b^2+2ab)/[2(1+s)]\). The cross term can increase density between nuclei when the chosen orbital signs agree there. The minus combination introduces a node. Changing the sign convention of \(b\) swaps which algebraic label is called plus without changing physical predictions; bonding should be assigned from energy and spatial character, not from a printed coefficient sign alone.

The energy change is not determined by density accumulation alone. Kinetic energy, electron-nucleus attraction, electron repulsion and nuclear repulsion all contribute. A one-electron model is useful for orbital mixing, but filling its levels is not an exact many-electron calculation.

3. Worked heteronuclear mixing

In an orthogonal teaching basis with unequal diagonal energies, define \(\Delta=\alpha_A-\alpha_B\). Diagonalization gives

\[ E_\pm=\frac{\alpha_A+\alpha_B}{2}\pm\sqrt{(\Delta/2)^2+\beta^2}. \]

If \(\alpha_A<\alpha_B\) and \(|\beta|\ll|\Delta|\), the lower state remains mainly on \(A\) and shifts downward by approximately \(-\beta^2/|\Delta|\). Near equal diagonal energies the mixing becomes strong. This explains why similar-energy orbitals mix efficiently, while symmetry can forbid a coupling even when their energies are close. These are model statements rather than numerical electronegativity predictions.

4. Symmetry organizes allowed couplings

If an operator commutes with a symmetry transformation, its matrix decomposes into symmetry blocks. Functions of incompatible irreducible symmetry do not couple under that operator. For a two-center inversion-symmetric problem, symmetric and antisymmetric combinations have different inversion parity. A distorted geometry or external field can remove the symmetry and allow mixing. Atomic \(s,p,d\) labels alone are not molecular symmetry labels.

5. Occupancy and a bond-order exercise

In a minimal closed-shell picture define bond order \((N_{bonding}-N_{antibonding})/2\). Two electrons in the bonding orbital give one; adding one antibonding electron gives one-half. This count predicts a qualitative weakening, not a guaranteed bond length or dissociation energy. Correlation and ionic configurations become essential when stretching a bond.

Why does a four-electron filling of both minimal orbitals give zero bond order? The bonding and antibonding occupancy cancel in this counting model. It does not prove that all forces vanish or that larger bases cannot describe weak interactions.

Why is an orbital sign not observable? A global sign is a phase convention. Only relative signs within a consistent basis affect interference; changing a basis-function sign must also change its matrix elements and coefficients.

6. Connect the models

The tight-binding pathway generalizes this matrix structure to networks and periodic systems. The next molecular lessons restore the full Hamiltonian and fermionic state before applying Hartree-Fock. For quantitative chemistry, interpret molecular orbitals together with total energies, densities and a documented reference approximation.

Analytical teaching schematic, not simulation data.