Molecular Term Symbols

A molecular configuration is a specification of the occupied molecular orbitalsin a molecule. For example,

N2 : 12g1

2u1pi

4u2

2g

A given configuration may have several different states depending on how theelectrons are arranged in the valence orbital(s); a molecular term symbol labelsthese states and specifies the total spin and orbital angular momentum of themolecule, along with various other symmetries:

2S+1 ||(+/)(g/u)The various symbols here have the following meanings:

S is the total spin angular momentum quantum number, formed from theindividual electrons spin quantum numbers, si =

12 , from the Clebsch-

Gordan series, as for atomic term symbols. For one unpaired electron,S = s1 =

12 (a doublet state with 2S + 1 = 2); for two unpaired electrons

the possible values are S = s1 + s2 = 1, S = s1 + s2 1 = 0, giving triplet(2S + 1 = 3) and singlet (2S + 1 = 1) states respectively. A closed-shellconfiguration is necessarily a singlet state;

is the quantum number for the total orbital angular momentum of theelectrons about the internuclear axis. Unlike in atoms, the cylindricalsymmetry created by the strong electric field of the nuclei in a linearmolecule destroys the relationship [H, L2] = 0, and so L ceases to bea good quantum number. Instead, we can use the quantum number ,corresponding to the component of L along the internuclear (z-) axis:

=i

i

where i is equal to 0 for a electron, and 1 for a pi electron. || =0, 1, 2, 3, . . . is written ,,,, . . .. Note that in this context, it is betterto think of the degenerate pi orbitals as the pair (pi+, pi), where the +and subscripts represent clockwise and anticlockwise rotation (Lzpi =hpi), rather than the functions (pix, piy). The latter (real) functions arelinear combinations of the former with the same energy but they are noteigenfunctions of the Lz operator, so they do not contain the necessaryinformation about ;

the g/u subscript applies only to molecules with a centre of symmetryand labels the symmetry of the electronic wavefunction with respect toinversion through this centre;

the +/ superscript applies only to states, and labels the symmetry ofthe electronic wavefunction with respect to reflection in a plane containingthe nuclei.

For the ground state configurations of first row diatomic molecules, term sym-bols are deduced as follows:

Closed-shell configurations

This is the easy case; all electrons are paired off in closed orbitals, e.g.

F2 : 12g1

2u2

2g1pi

4u1pi

4g

so there is no net orbital or spin electronic angular momentum: S = 0, = 0and the term symbol is

1+g

One unpaired -electron

S = 12 , and = 0, so we have a2+ term, e.g.

N+2 : 12g1

2u1pi

4u2

1g

has the molecular term symbol:2+g

One or three pi-electrons

The one unpaired electron (S = 12 ) is in a pi-orbital, so || = 1, and the termsymbol is 2. For example,

B+2 : 12g1

2u1pi

1u

gives rise to the term symbol:2u

For three pi electrons (e.g. O2 ), the configuration can be thought of as eitherpi2+pi

1 (giving = +1 + 1 1 = +1) or pi2pi1+ (giving = 1 1 + 1 = 1).

In both cases, || = 1 and S = 12 , and the term symbol is the same as for a pi1configuration.

Two (identical) pi-electrons

For example, O2, B2. The easy way is to note that the states arising from a pi2

configuration will have symmetries given by the direct product of the irrepsof the Dh molecular point group:

= + []

Here, [ ] indicates the antisymmetrized product. The total electronic wavefunc-tion, = spatialspin must be antisymmetric (electrons are fermions), so the (spatial) part is associated with the triplet (symmetric) spin part and the+ and parts are associated with singlet (antisymmetric) spin parts; the termsymbols are therefore:

3, 1+, 1

A longer way around is to construct suitably-symmetrized wavefunctions for thestates derived from the electronic configuration. For example, the ground stateconfiguration of molecular oxygen is:

O2 : 12g1

2u2

2g1pi

4u1pi

2g

Let the degenerate valence pi MOs be called pi+ and pi corresponding to oppositerotations of the electron around the internuclear axis. Consider the state inwhich both electrons occupy pi+, and hence = +2:

(a)spatial = pi+(1)pi+(2)

This wavefunction is symmetric with respect to permutation of the (identical)electrons, and so must be associated with the antisymmetric spin wavefunction.This is the singlet state (S = 0):

1spin =12

[(1)(2) (1)(2)]

The term symbol corresponding to the total wavefunction spatialspin, is there-fore:

1g : pi+(1)pi+(2)12

[(1)(2) (1)(2)]

(of course, spatial = pi(1)pi(2), for which = 2, gives rise to the same termsymbol: the 1g state is doubly degenerate).If we place one electron in each of pi+ and pi, giving = +1 1 = 0 and a term, we must be careful to construct appropriately symmetrized spatial wave-functions: pi+(1)pi(2) wont do because it does not have a definite symmetrywith respect to permutation of the identical electrons. We can have:

(b)spatial =

12

[pi+(1)pi(2) + pi(1)pi+(2)]

which is symmetric, and hence goes with the singlet spin wavefunction:

1g :12

[pi+(1)pi(2) + pi(1)pi+(2)]12

[(1)(2) (1)(2)]

Or we can have:

(c)spatial =

12

[pi+(1)pi(2) pi(1)pi+(2)]

which is antisymmetric, and hence can go with one of the three symmetric tripletspin wavefunctions:

3g :12

[pi+(1)pi(2) pi(1)pi+(2)]

(1)(2)12

[(1)(2) + (1)(2)]

(1)(2)

Now, the +/ superscript on these states labels the symmetry of the wave-function with respect to reflection in any plane containing the nuclei (the op-eration ). Such a reflection sends a clockwise rotation to an anti-clockwise

rotation and vice versa, and so interchanges pi+ and pi. For example,

(c)spatial =

12

[pi+(1)pi(2) pi(1)pi+(2)]

=12

[pi(1)pi+(2) pi+(1)pi(2)]

= (c)spatialSimilarly,

(b)spatial =

(b)spatial

Thus, the term symbols are:

3g ,1+g ,

1g

Note that the 1g state does not have a +/ label: the reflection inter-changes the degenerate pi+(1)pi+(2) and pi(1)pi(2) functions, so degenerateparity-conserved spatial wavefunctions, 1

2[pi+(1)pi+(2) pi(1)pi(2)], can be

constructed. This means that the 1g state consists of one + and one - state,and could be written 1g , but in practice the is usually omitted.

Spin-orbit coupling

Terms with electronic orbital angular momentum (|| > 0) and spin angularmomentum (S > 0) can undergo spin-orbit coupling; the component of the totalangular momentum along the internuclear axis, , is given as a subscript to theterm symbol. For example, the NO molecule has a 1222321pi42pi1 electronicconfiguration and so a 2 term symbol (S = 12 and = 1). The spin angularmomentum can make a projection Sz = 12 onto the internuclear axis, andthe quantum numbers and add to give = + :

= +1 + 12 =32

= +1 12 = 12 = 1 + 12 = 12 = 1 12 = 32

Therefore, the spin-orbit levels are = 32 and = 12 (each is doubly de-generate, since the total electronic angular momentum can be clockwise or an-ticlockwise about the internuclear axis); these are denoted 2 3

2and 2 1

2. The

splitting is about 120 cm1, somewhat smaller than kBT at room temperature(207 cm1).

Energy ordering

Hunds rules are a useful guide to the energy ordering of terms arising from theground state electron configuration:

1. The term with the highest spin multiplicity, 2S + 1, is lowestin energy. This is often explained in terms of electron spin correlation:electrons with parallel spins have a tendency to spend more time furtherapart, on average, than those with paired spins.

2. For terms of the same multiplicity, the term with the largestorbital angular momentum, given by , is lowest in energy. Aclassical explanation of this says that if two electrons orbit the internuclearaxis in the same sense (say, clockwise) they will encounter each other lessoften than if they orbit in opposite senses.

For example, the terms arising from the ground state of O2 lie in the order3g ,

1g,1+g .

Selection Rules

Electric dipole allowed transitions between electronic states of linear moleculesare subject to the following selection rules:

= 0,1 S = 0, in the absence of spin-orbit coupling = 0,1 + +,

g u