Since we're still dealing with diatomics, the energy of the contributing atomic orbitals of each atom is the same. The 2s orbitals combine to form a bonding 1σg and an antibonding 1σu, as well as a bonding 2σg, although that MO is mainly 2p in character. The 2p orbitals combine as follows: the lowest energy orbitals are two degenerate π orbitals, denoted 1πu. The next lowest is a single 2σg orbital (which has some 2s character). Next in energy are the antibonding orbitals, 1πg (doubly degenerate), and 2σu. Remember: to order the molecular orbitals, (Lowest energy) σ, σ, π, σ, π, σ (highest energy) For simplicity's sake, this discussion is restricted to H-(something that bonds through 2p and 2s orbitals). Because we are now working with two different atoms, we need to take into account the relative energies of the atomic orbitals. In a molecule that is hydrogen bonded to some other atom, the H1s orbital will always be higher in energy than the highest energy orbital of the other atom (this is not a universal truth, but for molecules that actually form, it is a good rule of thumb). The 4 atomic orbitals of the X atom mix with the 1s of H to give 5 molecular orbitals. From lowest E to the highest, the order of the orbitals is 1σ, 2σ, 1π (doubly degenerate), and 3σ. The antibonding 3σ is mostly H in character, while the other 4 are mainly X in character. E: The identity operator. It's always there.
Cn: Rotation by 360o/n. If n=2, there is only one rotation operation involved. If n=3, there are two operations associated with it, C3 and C3'. There will always be n-1 rotational operations involved, only considering that axis. The principal axis is the highest order rotational axis, and it defines the z-axis of the molecule.
σ: Mirror plane. A mirror plane is a vertical mirror plane, σv, if it contains the z-axis, as defined by rotational symmetry. The horizontal mirror plane, σh, is in the plane of the molecule, or alternatively, perpendicular to σv. There can also be a dihedral mirror plane, σd, which bisects two C2 axes.
I: Center of Inversion. The inversion operation has each atom of the molecule projected through a single point, and out the same distance on the other side, succeeding to interchange diametrically opposite pairs. Only Oh, D ∞h, , D4h, D2h, and Ci point groups have the inversion operator.
Sn: Improper rotation/Screw axis. This operation consists of a rotation through a certain angle followed by a reflection in a mirror plane perpendicular to the rotation.
There is a character table corresponding to every point group. The table contains all of the symmetry species (Γ) and all of the symmetry operators. In the table are values corresponding to how they transform under the operations.
This is the character table for the C4v point group. The numerical elements in the body of the table are the characters, χ, and these are what tell us how the symmetry species transform under the operation. A value of 1 means it is unchanged, a value of -1 means it changes sign, 0 means it undergoes a more complicated change (higher math, and such). The letters in the column directly to the right of the main body are translations, rotations, and components of dipole moments (s and p orbitals) which are relevant to IR activity. In the column to the right of those are the quadratic functions (d orbitals) which are of relevance to Raman activity.
The letter A in the symmetry species column means that it is symmetric with respect to rotation about the principal axis. B means that it is inverted with respect to rotation about the principal axis. If it is labeled A1, then it is symmetric with respect to all symmetry operators. The subscript one means that it is symmetric with respect to reflection in the principal vertical plane, the subscript two means it is inverted with respect to reflection in the principal vertical plane. If the letter E is used for a symmetry species, it is a degenerate species, and the number of degenerate orbitals is given under the identity operator.
For a molecule of N atoms, there are 3N degrees of freedom. 3 are translational, 3 are rotational, so 3N-6 vibrational modes (if non-linear). If it is linear, there is no rotation about the axis of the molecule, so there are 3N-5 vibrational modes in a linear molecule.
The symmetry species of the vibration must be the same as that of x, y, or z in the character table for the vibration to be IR active, and the same as that of a quadratic function, such as xy or x2, for it to be Raman active.