Symmetry Adapted Linear Combinations and Molecular Orbitals

The names and symbols used to label atomic orbitals (s, p,d etc.) only apply to the orbitals as they appear in a spherically symmetric system ie., an atom. The labels, and their symmetry properties, can be found grouped together in a character table for the system symmetry, which is referred to as the point group. Point groups contain the characters of irreducible respresentations. Hence whilst it is not appropriate to label an orbital with atomic labels on anything other than an atom, when we consider the orbitals on multi-atomic species, we label them according to their behaviour under the symmetry of the species' point group. By convention, vibrations are written in lower case, as are orbitals.

Orbital label assignment

The first step to naming a molecular orbital is to determine the molecular point group, as it is from the point-group's character table that we obtain the set of names - the labels used to denote 'irreducible representations.' Referring to the particular point group's symmetry operations, we consider how the atomic orbitals behave. Then, symmetry allowing, combining the various atomic orbitals, we produce a set of molecular orbitals, labelled according to their symmetry behaviour.

As an example we shall consider the water molecule, H₂O. We can, using the algorithm for symmetry determination or by inspection, deduce that the H₂O molecule belongs to the point group C_2v point-group. Hence the symmetries of all the constituent orbitals conform to one of the four symmetry labels found in the point group below.

`bb"C"_2v`	E	`C_2(z)`	`sigma_v(xz)`	`sigma_v(yz)`	linear, rotations	quadratic
a1	+1	+1	+1	+1	z	x2, y2, z2
a2	+1	+1	-1	-1	Rz	xy
b1	+1	-1	+1	-1	x, Ry	xz
b2	+1	-1	-1	+1	y, Rx	yz

The functions on the right can be used as 'bases' for the representations,
ie., x is the basis for B₁ representation.

The process of orbital label assignment involves applying the operators (listed along the top row of the point group) to each of the orbitals and establishing whether the symmetry transformations leave the orbital unchanged or inverted: if the orbital remains the same, it is assigned a character of +1, and if inverted, -1. Once we have applied each operation individually, we compare the characters obtained to those in the character table, then identifiying that which matches exactly a row, we use the label found in the left hand column.

Whilst this approach works for the oxygen atom, for which there is only one, the caveat is we cannot apply such a direct procedure to the hydrogens. Since the two hydrogen atoms are 'symmetrically equivalent', we have to consider them as a pair, as we shall see shortly. First, a worked example using an oxygen-p orbital.

Singular atomic orbitals

For this example we shall consider the 2p_x orbital. Oxygen is at the centre of the molecular, and all symmetry elements pass through it. We sketch the orbital, and systematically apply each of the symmetry operations.

Element	Effect of Operation	Character
`E`		+1
`C_n`		-1
`sigma`		+1
`sigma'`		-1

Collecting the results in the form of a representation (row), we can read off the symmetry label for the orbital based on the way it responds to these various symmetry operations,

	E	`C_n`	`sigma_v`	`sigma_v'`	label
O 2px	+1	-1	+1	-1	b1

Evidently, the irreducible representation from the character table which matches exactly the behaviour of the O 2p_x orbital is the species b₁. The remaining oxygen orbitals can be considered similarly and give rise to the following orbital labels,

Oxygen atomic orbital	C2v orbital label
2s	a1
2px	a1
2py	b1
2pz	b2

Degenerate atomic orbitals: symmetry adapted combinations

The two H atoms are symmetrical equivalent and different to oxygen. Since the H atoms are equivalent, they must be considered together, as a pair. The representation for these two H atoms must be obtained, and to do this, we apply a similar procedure. This time rather than assigning a value of +1 for unchanged and -1 for inverted, we count the number of atoms which remain in the same place, and hence, the character, `chi` will either be 0 (both atoms moved), or 2 (both atoms in the same place).

Effect of Symmetry Operation	Character
	Both orbitals unchanged, `chi=2`
	Both orbitals changed, `chi=0`
	`chi=0`
	`chi=2`

To determine the symbol, we now construct the representation, and look to see if there an irreducible representation corresponding to it from the molecular point group character table,

C2v	E	C2	`sigma_v(xz)`	`sigma_v'(yz)`
`Gamma{1s,1s}`	2	0	0	2

Evidently this is not an irreducible representation (there is no symbol in the character table that has any symbol `|chi|>1`, hence the combination, as we may expect, is reducible. This means that the symmetry adapted orbital combination is composed of a combination of those from the irreducible set. By inspection we have,

C2v	E	C2	`sigma_v(xz)`	`sigma_v'(yz)`
a1	1	1	1	1
b2	1	-1	-1	1
`Gamma{1s,1s}`	2	0	0	2

Hence, `Gamma{1s,1s}=a_1+b_2`. The hydrogen orbitals do not have the symmetries individually, but their combinations do. As there are only two H1s orbitals, the combinations can only be the in-phase and out-of-phase combinations. One of these is the 'a₁' Symmetry-Adapted Linear-Combination (SALC), and the other, 'b₂'.

Comparing the above to the character table, it is clear the left hand side is a₁ and the right hand side is b₂. With orbital fragments identified, we can now go about finding the various linear combination of orbitals which constitute the molecular orbital framework, and then from there, a qualitative molecular orbital diagram can be constructed.

Construction of Molecular Orbitals

Since orbital wavefunctions extend to infinite distance, all orbitals overlap with every other - however, only certain combinations result in a net change in energy, either by construction or destruction. For example, if we overlap a 1s orbital with a p_z orbital, laterally, then whilst one lobe will result in construction, the other lobe with result in destruction - the net result, nothing. The overlap is neither stabilising nor destabilising, and we refer for this as a non-bonding orbital combination. In specific, the criteria for 'effective orbital overlap' is that the symmetry species (label) of each constituent atomic orbital (or symmetry adapted combination) must match. In fact, all orbitals of the same label interact with each other simultaneously, in various in-phase and out-of-phase combinations. When the symmetry and the phase of the constituent orbitals match, a bonding-orbital results which lowers the system energy, and when the symmetry matches but the phases are opposite, an anti-bond results, which were the orbital to be occupied, destabilises the system, raising its energy. When adding orbitals together, it is worth remembering that 'orbital quantity is conserved' and this means, that if we start with two orbitals and add them together, then what results is a new set of two orbitals.

So continuing the explanation with water as an example, having assigned the symmetry labels to each of the constituent orbitals, we consider their various combinations. To do this, we work our way down the character table (in no particular order), identifying each component orbital that matches the symmetry species: a₁, a₂, b₁ and b₂.

1. a₂ - there are no orbitals, neither on oxygen nor from the set of symmetry adapted hydrogen orbitals with this symmetry.

2. b₁ - the only oxygen orbital of that symmetry is p_x. As there is no symmetry adapted combination of hydrogen atomic orbitals with this symmetry label, then this O P_x has nothing to effectively overlap with (neither constructive nor destructive), as such it is unchanged and is non-bonding,

ie.,

3. b₂ - there are two fragment orbitals of b₂ symmetry - the oxygen p_y orbital and the out of phase H1s orbital combination. These SALCs can then combine. Noting there are two orbitals to begin with, we obtain two different combinations.

4. a₁ - There are three fragment orbitals of a₁ symmetry: oxygen 2s and 2p_z, and the in-phase H1s SALC, these give rise to three molecular orbitals:

The strongly bonding 1a₁ ({H1s}+O2s+O2p_z),

The middle energy molecular orbital results from the in-phase combination of in-phase H1s SALC and oxygen 2p_z and out-of-phase oxygen 2s ({H1s}-O2s+O2p_z),

The highest energy molecular orbital results from the fully out-of-phase combination ({H1s}-O2s-O2p_z),

Qualitative Molecular Orbitals

Now we have considered each of the possible orbital combinations, we can tabulate a list of qualitative orbitals. When it comes to determining the order of orbitals, from most bonding (stabilising) to anti-bonding (destabilising), in general it is easiest to consider them in terms of constructive density - the greater the density - or venn diagram-esque overlap of in-phase components - the more stabilising will be the resulting orbital. Additionally, the fewer the number of nodes (when an orbital changes phase), the more strongly bonding it will be. For example, from the list below, the 1a₁ orbital has only one node, whereas the 2b₂ orbital has three nodes. Also we must consider the spacial positions of these nodes - in terms of molecular bonding, if a node occurs in between nuclei, then occupation of that orbital will act to

The opposite to a node, or change in orbital phase, is an anti-node: these represent the region of maximal construction. When an anti-node exists directly between two nuclei, occupation of that orbital will act to draw the nuclei together - consider, the nuclei are positively charged and will naturally be repelled from one another, but if there exists a high density of electrons between the two nuclei, then an attract force exists between each nucleus and the 'bonding' electrons, thereby in turn drawing the nuclei themselves together, ie., we have a bonding orbital. In contrast, when an orbital node exists between two nuclei there is a reduction in electron density between the nuclei, and so the nuclei repel one-another, ie., we have an anti-bonding. Thus, molecular orbitals with nodes between nuclei are always higher in energy than those with anti-nodes between them.

We introduce a new element of nomenclature, an index to the orbitals for cases where there are several orbital combinations of that said symmetry, for example, there are three constitute orbitals of a₁ symmetry, and these give rise to three new molecular, symmetry-adapted linear combination of orbitals, all with the symmetry of a₁ - to distinguish them from one another, we number them, 1a₁, 2a₁, 3a₁, with the lowest in energy given the index of 1.

Energy			Label	Comments
High			2b2	most antibonding
			3a1
			1b1	is non-bonding - orbital does not overlap with hydrogen orbitals at all
			2a1
			1b2
Low			1a1	most bonding

 

Molecular Orbital Diagram for Water

Now that we have established which orbitals will overlap constructively, destructively and in a non-bonding capacity, produced a qualitative representation of the molecular orbitals, and determined their (approximate) relative energies, we can construct a qualitative molecular orbital energy level diagram. Our diagram contains three colums - one for the orbitals localised on oxygen, those that belong to the hydrogen-pair SALCs, and a third for the molecular orbitals.

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