Group Theory and Molecular Orbital (MO) Diagrams Lecture Notes

A set of vocabulary flashcards covering basic and complex molecular orbital diagrams using group theory, specifically focusing on symmetry matching and polyatomic case studies like water, ammonia, and methane.

Symmetry Matching

The core principle stating that for any two orbitals or SALCs to interact and form a bond, they must share the exact same symmetry label (irreducible representation); otherwise, the Hamiltonian integral is zero.

LCAO

Linear Combination of Atomic Orbitals. In polyatomic molecules, peripheral orbitals are combined into SALCs first.

SALCs

Symmetry Adapted Linear Combinations; these are the building blocks of peripheral atoms that possess specific symmetry labels.

Non-Bonding Orbitals

Orbitals on a central atom with a symmetry that finds no match in the SALCs and remain as lone pairs.

Water ($H_2O$) Symmetry Point Group

$C_{2v}$.

Hydrogen SALCs in Water ($H_2O$)

$(s_1 + s_2)$ which transforms as $a_1$, and $(s_1 - s_2)$ which transforms as $b_2$.

Oxygen $2p_x$ ($b_1$) in $H_2O$

An orbital that has no partner and remains a non-bonding lone pair because its symmetry finds no match in the Hydrogen SALCs.

Ammonia ($NH_3$) Symmetry Point Group

$C_{3v}$.

E symmetry

A representation signifying degeneracy, where two orbitals share the same energy, introduced in systems like ammonia.

Nitrogen Orbitals in $NH_3$

The $2s$ ($a_1$), $2p_z$ ($a_1$), and $(2p_x, 2p_y)$ as a pair ($e$).

Ammonia $a_1$ SALC

$s_1 + s_2 + s_3$.

Ammonia $e$ SALC pair

$(2s_1 - s_2 - s_3)$ and $(s_2 - s_3)$, which are confirmed as eigenvalues in pairs through electronic structure calculations.

Methane ($CH_4$) Symmetry Point Group

$T_d$.

Methane Carbon Orbitals

Includes the $2s$ ($a_1$) and $(2p_x, 2p_y, 2p_z)$ as a triplet ($t_2$).

Photoelectron Spectroscopy (PES) of Methane

Shows two peaks in a $3:1$ ratio, proving valence orbitals reside in two energy levels ($a_1$ and $t_2$) despite physically equivalent C-H bonds.

Chemical Intuition (MO Construction)

The principle that only valence orbitals close in energy interact strongly, while core orbitals (such as the $1s$ on Carbon) are ignored.

Peripheral p-orbital Phase Flip

Occurs when p-orbital phases on peripheral atoms in systems like $NO_2$ flip by a factor of $-1$ during symmetry operations.