Electron orbitals of Molecules. See Molecular physics, and this book, and these notes

A common approach to approximate molecular orbitals, is to construct them as a Linear Combination of Atomic Orbitals (LCAO). Atomic orbital

Diatomics

For homonuclear diatomic molecules, it is easy to see that the Hamiltonian has a symmetry corresponding to inversion of the electron coordinate through the center of mass of the molecule. This means that the Energy eigenstates can be made to be Eigenfunctions of this reflection operator, which can have eigenvalue +1 or -1. The corresponding classes of molecular orbitals are known as gerade or g, and ungerade, or u, respectively.

• The gerude orbital is lower in energy, and can is responsible for bonding two atoms. It is thus known as a bonding orbital
• The ungerude orbital is higher in energy, and makes atom not bond. It is thus known as a antibonding orbital.

hydrogen mol

To determine whether a bond forms, we define the bond order, as

$\text{bond order} = \frac{\text{no. of electrons in bonding orbitals} - \text{no. electrons in antibonding orbitals}}{2}$

Some of the ideas from homonuclear diatomics, naturally extend to heteronuclear diatomics, and polyatomic molecules, as they capture the essentials of Chemical bonding.

Molecular orbitals

See Molecular physics to see derivation of the electron orbitals.

Principal quantum number

$n$. The extension of the atomic Principal quantum number

Angular momentum

$l$, as for Atomic structure

Magnetic quantum number

Electronic state in multi-electron orbitals

As for multi-electron atoms, individual $l_z$ don't commute with the Hamiltonian, but the $z$ component of the total angular momentum, $L_z$ does, and this becomes the good quantum number, $\Lambda$.

We also have the total spin $\mathbf{S}$

There is finally a binary quantum number, $+$/$-$, corresponding to reflection along the $y$ plane.