Quantum Chemistry

For each pair of indistinguishable electrons, the system is stabilized by the amount of exchange energy \( \Pi_e \).

• AC
• Intended Crossing, Non-Crossing, Anticrossing

Two eigenvalues of an observable that depends on \( k \) continuous real parameters cannot cross each other, except on a manifold of dimension \( k-2 \).

Two degenerate state, when combined by interacting one with one another, yield one higher and one lower energy state. Therefore, in chemistry, resonance structure (especially in benzene) is more stable.

When \( \sigma_s \) and \( \sigma_p \) has the same symmetry, they can interact, Lowering the energy of one, and raising the energy of the other.

In \( \rm O_2 \), the mixing between 2s and 2p is weak, because the energy gap is relatively large. However, for \( \rm N_2, C_2, B_2 \), the energy gap is sufficiently small for the mixing to occur. For them, the energy of \( \sigma_{2p_z} \) is raised and the energy of \( \sigma_{2s} \) is lowered, making \( \sigma_{2p_z} \) higher than \( \pi_{2p} \) in terms of energy. For the oxygen the \( \pi_{2p} \) is higher, giving the molecule its magnetic property.

The interaction is due to the non-crossing rule (avoided crossing).

The absolute hardness \( \eta \) is defined by: \[ \eta := \frac{I - A}{2} \] where \( I \) is the ionization energy, and \( A \) is the electron affinity.

Notice that it is half the gap between HOMO and LUMO, with the center being the absolute electronegativity \( \chi \): \[ \chi := \frac{I + A}{2}. \]

If \( \eta \) is high, it is hard for the electron in the HOMO to "squish", that is, to be in a superposition of excited states and ground state.

• SALC

The s-orbital of core electron of heavy elements shrinks due to relativity, the spin-orbit coupling completely mixes spin and angular momentum and only the total angular momentum \(J\) is conserved.

• Molecular orbital diagram - Wikipedia
• Avoided crossing - Wikipedia
• 삼중 결합의 상대론적 붕괴 관측! 100년 간 당연한 일이 깨졌다! 화학 결합에 대한 최신 발견에 대해 - YouTube