term symbol

TERM SYMBOLS

Dr. Cyriac Mathew

Atomic Term Symbols

In many-electron atoms the electron configuration is rather ambiguous. For example,

consider the ground state electron configuration of a carbon atom, 1s22s

22p

2. The two 2p

electrons could be in any of the three 2p orbitals and have any spin consistent with Pauli

Exclusion Principle. This may give rise to several atomic energy states because of the effects of

inter-electron repulsions. Therefore, we need a more detailed designation of the electronic states

of the atom. Such detailed designation of the electronic states of the atom is called term. The

corresponding energy levels are represented using the term symbol. A term symbol tells us

something about the angular momentum of the electrons in the atom.

Spin-orbit coupling

An electron is a charged particle, and hence its orbital angular momentum gives rise to a

magnetic field just as an electric current in a

loop gives rise to magnetic field in an

electromagnet. Thus an electron with an orbital

angular momentum possesses magnetic

moment. The magnetic moment due to orbital

motion is given as l el , e is the

magnetogyric ratio of the electron and l is the

orbital angular momentum. An electron also has

a spin angular momentum. This intrinsic

spinning motion also gives rise to magnetic

moment. The spin magnetic moment is given by

2s es .Where s is the spin angular

momentum. The spin magnetic moment is twice

the value of the spin angular momentum. For

electrons the magnetic moment is opposite to

the angular momentum; see figure 1.

Thus there are two sources for magnetic

moment for the electrons. These two magnetic

moments can interact and give rise to shifts in

energies of the states of the atom. The

interaction of these two magnetic moments is

called spin-orbit coupling. The strength of the

coupling, and its effect on the energy levels of

the atom depend on the relative orientations of the spin and orbital magnetic moments, and hence

on the relative orientations of the spin and orbital angular momenta. The total angular

NN

SS

N

N

S

S

High J

Low J

S

N

l

l

s

s

l

l

s

s

l : Orbital angular momentum

l

: Orbital magnetic moment

s : spin angular momentum

s spin magnetic moment:

J: Total angular momentum

Fig. 1

2 Term Symbols Lecture notes

momentum of the electron is the vector sum of its spin and orbital momenta. Thus when the spin

and orbital angular momenta are nearly parallel the total angular momentum is high; when the

two angular momenta are opposed, the total angular momentum is low. When the total angular

momentum (J) is high, the total energy also is high (unfavourable orientation of magnetic

moments), and when the total angular momentum (J) is low, the total energy also is low

(favourable orientation of magnetic moments), as shown in figure 1.

The probable levels that may arise as a result of vector addition of the two angular momenta

from a d1 electron configuration is shown in fig. 2

When l = 0 the electron is having no orbital

angular momentum and the total angular

momentum is same as that of the spin angular

momentum.

Total electronic angular momentum

When several electrons are present in an

atom there are, generally, two ways in which the

orbital and spin angular momenta add together

or oppose each other.

(i) Each spin may couple to its own orbital

angular momentum as in a one-electron case.

This type of coupling is called jj coupling.

(ii) All the individual orbital angular momenta

(il

) couple to produce a total orbital angular

momentum ( L

). It is written as vector sums:

i

i

L l

. Here the summation is over the

electrons in the atom. Similarly, the individual

spin angular momenta ( is

) couple together to produce a total spin angular momentum ( S

).

i

i

S s

. Here also the summation is over the electrons in the atom.

Now, the orbital and spin angular momenta couple to produce the total angular momentum ( J

).

This type of coupling is called the Russell-Saunders or L-S coupling. This coupling scheme is

used when the spin-orbit coupling is weak, and generally used for atoms of low atomic number

(z ≤ 30).

(i) Total electronic orbital angular momentum

When several electrons are present in an atom the total orbital angular momentum is

obtained by the vector addition of individual orbital angular momenta.ie. in order to find out the

value of total orbital angular momentum we need an operator, L . It can be shown that L

commutes with the atomic Hamiltonian, H when spin-orbit coupling of individual electrons are

j = 3/2

j = 5/2

l = 2

s = 1/2

s = 1/2

l = 2

Fig. 2 The coupling of the spin and

orbital angular momenta of a d electron

(l =2) and s = ½ gives two possible

values of j depending on the relative

orientations of the spin and orbital

angular momenta of the electron.


neglected. Therefore we can characterize an atomic state by a quantum number L, so that the

magnitude of the total orbital angular momentum is given as ( 1)L L .

The total orbital angular momentum (a non-negative integer) is obtained by coupling the

individual orbital angular momenta, empirically, using the Clebsch-Gordan series. If we have

two electrons with orbital angular momenta l1 and l2, then the total angular momentum is given

as;

L = l1 + l2, l1 + l2 – 1, ………1 2l l

The maximum value of l1 + l2 is obtained when the two orbital angular momenta are in the same

direction. The lowest value1 2l l , is obtained when they are in opposite directions. The

intermediate values represent possible intermediate relative orientations of the two momenta.

The total electronic orbital angular momentum quantum number L of an atom is specified by a

code letter as shown below:

L 0 1 2 3 4 5 6

letter S P D F G H I

The total orbital angular momentum is designated by a capital letter, while the individual

electronic orbital angular momentum is represented by lower case letter.

Example 1: Let us find out the possible values of the quantum number L, for the states of the

carbon atom that arise from the electron configuration of 1s22s

22p

13d

1. The s-electrons have zero

orbital angular momentum and contribute nothing to the total orbital angular momentum. Now

the p-electron has l = 1and designated as l1 and the d-electron has l = 2 and is designated as l2.

From the Clebsch-Gordan series the total orbital angular momentum quantum number ranges

from,

L = l1 + l2 ………1 2l l = 3 …….. 1 2 = 3……..1. Therefore the possible values of L are 3, 2,

1. The electron configuration, 1s22s

22p

13d

1, thus gives rise to P, D and F states.

If more than two electrons are to be considered in a configuration then we use two series in

succession. First we couple two electrons and then we couple the third to each combined state

and so on. For eg. Let us consider the configuration 2p13p

14p

1. Three electrons l1 = 1, l2 = 1and

l3 = 1. First we couple l1 and l2 and to each of the combined angular momentum we couple l3.

First coupling:

L’ = l1 + l2 ………1 2l l

L’ = 1 + 1 ………1 – 1 = 2 ….. 0, hence

L’ = 2, 1, 0.

Now we couple l3 with each value of L’

l3 = 1, L’ = 2

L = 1 + 2, ……….., 1 – 2 = 3, ……. 1

L = 3, 2 ,1.

l3 = 1, L’ = 1


L = 1 + 1, …. 1 – 1 = 2,……., 0 = 2, 1, 0

l3 = 1, L’ = 0

L = 1 + 0 ,……………….1- 0

L = 1

The total L terms are :3,2,1,2,1,0,1. The corresponding states are:

F, D, P, D, P, S, P

(ii) Total electronic spin angular momentum

When there are several electrons to be considered we must assess their total spin angular

momentum quantum number S (a non-negative integer or half-integer). Using the Clebsch-

Gordan series we can decide the values of S;

S = s1 + s2 , s1 + s2 – 1, .…..1 2s s

For example let us consider two electrons, each of them with spin s = ½.

s1 = ½, s2 = ½

S = ½ + ½ , …….. 1 12 2 =1, ……0 = 1,0

If there are three electrons the total spin angular momentum is obtained by coupling the third

spin with each value of S.

The spin multiplicity of a term is the value of 2S + 1. When S = 0 (as for a closed shell) the

electrons are all paired and there is no net spin. Hence the spin multiplicity is 2×0 + 1 = 1 and

the state is called a singlet state. If S = ½ , 2 × ½ + 1 = 2, a doublet state. When S =1 the state is

a triplet state, 2×1 + 1 = 3, and so on. A singlet S term is written as, 1S (read as singlet S), a

doublet term as 2S (read as doublet S).

(iii) Total electronic angular momentum

If there are several electrons outside a closed shell we have to consider the coupling of all

the spin angular momenta and all the orbital angular momenta separately. For atoms of low

atomic number the spin-orbit coupling is weak, and we are following the Russell-Saunders

coupling scheme. We imagine that all the orbital angular momenta of the electrons couple to give

the total orbital angular momentum L, and that all the spin angular momenta are similarly

coupled to give total S. Now we imagine that the two kinds of angular momenta couple through

the spin-orbit interaction to give a total angular momentum J. J is the total angular momentum

quantum number. The permitted values of J are given by the Clebsh-Gordan series.

J = L + S, L + S – 1, ……… J S . For example if L = 2 and S = 1, then

J = 2 + 1, ………… 2 1 = 3, …….., 1 = 3,2,1(each value differ by 1)

It is to be noticed that L, S and J are zero for completely filled sub-shells because, for

every electron with a negative value of li, there is another electron with a corresponding positive

value to cancel it; the same case exists for spin angular momentum quantum number si also. Thus

we can ignore the electrons in completely filled sub-shells in finding the terms.


L is the total orbital angular momentum and its z component Lz can have 2L+1 values

ranging from L to – L. These are represented by, ML = L, L – 1,…..,0,…. – L. Similarly MS can

take 2S+1 values and MJ can take 2J +1 values.

MS = S, S – 1, ………– S. and MJ = J, J – 1, … – J.

Now L = (max)LM and S =

(max)SM

Atomic Term symbol

The atomic terms symbol is represented in the following way;

2 1SJL

L is the total orbital angular momentum, (2S + 1) is the spin multiplicity and J is the total

electronic angular momentum quantum number.eg 3P2; (read as triplet P two).

Two electrons in different sub-shells are called nonequivalent electrons. Nonequivalent

electrons have different values for n or l or both, and we need not worry about any restrictions

imposed by Pauli Exclusion Principle when we derive the terms. For example, the excited state

of helium 1s1 2s

1, or configurations like 2s

12p

1, 3p

13d

1etc.

Two electrons in the same sub-shell are called equivalent electrons. Equivalent electrons

have the same value of n and l. Example ground state carbon 1s22s

22p

2, or configurations like

2p3, 3d

2 etc. The situation is complicated by the necessity to avoid giving two electrons the same

four quantum numbers. Hence not all the terms derived for nonequivalent electrons are possible

in the case of equivalent electrons.

Term symbols for nonequivalent electrons

As an example let us consider the excited state electron configuration 1s12s

1, of the He

atom. Let us set up a table showing the possible ML and MS values, in the following manner.

ML 0 0 0 0

MS

1 0 0 −1

Col. 1 2 3 4

There are four microstates in the table because there are two possible spins, ±½, for the electron

in the 1s orbital and also for the electron in the 2s orbital. Since both the electrons are on

different orbitals the Pauli Exclusion Principle need not be considered. ML is equal to zero for all

the microstates in the table because the electrons are s-electrons, and they correspond to L = 0.

The largest value of MS is 1 and therefore, S = (max)SM = 1. All the values of MS = 1, 0, −1

correspond to L = 0 and S = 1. When L = 0 the term is S and the spin multiplicity is 2S+1 =

2×1+1 = 3. Therefore the term is 3S. Thus the microstates of columns 1, 2 (or 3), and 4 give rise

to 3S term. Now column 3 corresponds to ML =0 and MS = 0, and since there is only one entry,

L = 0 and S = 0. It represents a 1S term. For the

3S term L = 0 and S = 1. Hence,

−½

+½


J = L+S, …… L S = 1+0, … 1 – 0 = 1

The term symbol is 3S1. For the 1S term Both L and S are zero. Hence J =0 and the term

symbol is 1S0.

A more convenient and easier way to derive term symbols for nonequivalent electrons is

to use the Clebsch-Gordan series. For the excited state electron configuration of He, 1s12s

1, l1 = 0

and l2 = 0, and s1= s2 = ½. Therefore,

L = l1 + l2, ……. 1 2l l = 0

S = s1 + s2, .…..1 2s s = ½ + ½ , …….. 1 1

2 2 =1, ……0 = 1,0. Thus S has two values.

Taking L = 0 and S = 1, the term is 3S. Now the value of J is,

J = L+S, …… L S = 1+0, … 1 – 0 = 1

Therefore the term symbol is 3S1

Taking L = 0 and S = 0 the term is 1S. Since L and S are zero J also is zero and the term

symbol is 1S0. Thus the excited state of He can be represented by two states 3S1 and 1S0.

Example 2: Find the term symbol for the excited state of carbon atom represented by the

electron configuration 1s22s

22p

13s

1.

Since 1s and 2s are completely filled we need consider only 2p13s

1.

l1 = 1, l2 = 0 and s1 = s2 = ½

L = l1 + l2, ……. 1 2l l = 1 + 0 …… 1 – 0 = 1

S = s1 + s2, .…..1 2s s = ½ + ½ , …….. 1 1

2 2 =1, ……0 = 1,0.

Taking L =1 and S =1, the term is 3P

J = L+S, …… L S = 1+1, …….1 – 1 =2, ….,0 = 2, 1, 0. The term symbols are: 3P2,

3P1, and

3P0.

Taking L =1 and S = 0 the term is 1P

J = L+S, …… L S = 1 + 0, …, 1- 0 = 1 (only one value since first and last are the same).

Therefore the term symbol is 1P1. The different levels for this electron configuration are: 3P2,

3P1,

3P0 and

1P1.

Example 3: Derive the term symbols for the electron configuration np1nd

1

For the p electron l = 1 and for the d electro l = 2

L = l1 + l2, ……. 1 2l l = 1 + 2, ……… 1 2 = 3, ……, 1= 3, 2, 1

S = s1 + s2, .….. 1 2s s = ½ + ½ , …….. 1 12 2 =1, ……0 = 1,0.

For each value of L there can be two values of S, 1 and 0.

Taking L = 3 and S = 1the term is 3F

J = L+S, …… L S = 3+1, ….., 3-1 = 4,…, 2 = 4, 3, 2. Hence the term symbols are: 3F4,

3F3,

3F2

Taking L = 3 and S = 0 we have 1F term


J = L+S, …… L S = 3+0, … 3- 0 = 3. The term symbol is 1F3

Taking L =2 and S =1we have 3D term. The corresponding term symbols are

J = L+S, …… L S = 2+1, …2-1 = 3,..,1 = 3,2,1. 3D3,

3D2,

3D1

Taking L = 1and S = 0 the term is 1D and the corresponding term symbol is

1D1

For L = 1 and S =1 we have a 3P term. The corresponding term symbols are,

3P2,

3P1,

3P0

When L = 1 and S = 0 we have a 1P term and the corresponding term symbol is

1P1

Thus the total terms for this electron configuration are: 3F,

1F,

3D,

1D,

3P and

1P.

Example 4: Write down the term symbol for the ground state configuration of F and Na.

a. Fluorine

The ground state configuration of fluorine is [He]2s22p

5 or [Ne]2p−1. We treat this as a p

1

configuration. For a p electron l =1and since there is only one electron, L = 1. For a single

electron s = ½ and hence S = ½. Therefore 2S+1 = 2. J values are:

J = 1 + ½, …… 1 - ½ = 32 , …, ½ = 3

2 ,½. Hence the term symbols are,

3 122

2 2 and P P

b. Sodium

The ground state electron configuration of Na is [Ne]3s1. For an s electron l = 0 and hence L = 0.

For a single electron s =½ and hence S = ½. Therefore, 2S+1 = 2. Since L =0 and S = ½, J = ½.

The term symbol is 2S½

Exercise 1

(a) Write down term symbols for the configurations (i) 3d10

4s2 (ii) 3d

14s

2 (iii) 1s

22s

1

(b) What values of J may occur in the following terms?

(i) 1S (ii)

2P (iii)

3D (iv)

4F

Configurations with equivalent electrons

Electrons in the same sub-shell are equivalent electrons. They have the same n and l

values. As an example we shall consider the ground state carbon atom. The electron

configuration is 1s22s

22p

2. Since we need not consider completely filled sub-shells we focus on

the electron configuration 2p2 or in general np

2. Since the two 2p electrons do not differ in their n

or l values, only the terms that are consistent with Pauli Exclusion Principle need be considered.

We are going to assign two electrons to two of the six possible spin-orbitals (2pxα, 2pxβ, 2pyα,

2pyβ, 2pzα, 2pzβ).The number of distinct ways (microstates) of assigning N electrons to G

spin-orbitals belonging to the same sub-shell (equivalent orbitals) is given by;!

!( )!

G

N G N .

For the 2p2 configuration N = 2 and G = 6. (For p orbitals G = 6, for d orbitals G = 10 etc.). The

number of distinct ways are: 6!

152!(6 2)!

. We write down all these spin combinations in the

form of a table.


Sl.

No

ml

ML

MS

(½)

(-½)

L

((max)LM )

S

((max)SM

)

Term +1 0 −1

1 0 0 0 0 1S

2 2 0

2 0 1D

3 1 0

4 0 0

5 −1 0

6 −2 0

7 1 1

1 1 3P

8 1 0

9 1 −1

10 0 1

11 0 0

12 0 −1

13 −1 1

14 −1 0

15 −1 −1

Table 1

For the spin combination given in 1st row ML and MS are zero and hence M and S also are

zero. Therefore this arrangement corresponds to 1S term. For the spin combinations given in

rows 2 to 6 the maximum value of ML is 2. Hence all these five combinations correspond to a D

term. Also, for all the five combinations S = 0, and hence represents 1D term.

For the spin combinations given in rows 7 to 15 the maximum value of ML is 1.Hence it

corresponds to a P term. Now there are three values for MS and the maximum is 1. Therefore S =

1. Hence all these 9 combinations correspond to 3P term.

Therefore, the corresponding term symbols are 1S0,

1D2,

3P0,

3P1 and

3P2.If we compare

this with an arrangement of two non-equivalent electrons, such as 2p13p

1 where Pauli Exclusion

Principle need not be considered, the possible terms will be 3D,

3P,

3S,

1D,

1P and

1S instead of

1S,

1D and

3P.

The number of terms that arise from an np4 configuration also will be the same as that of np

2

configuration. The following table gives the terms that would arise from different configurations.

(Any sub-shell that contains n electrons will give exactly the same term symbols as the same

sub-shell when it is n electrons short of being full). Table 2 shows the terms that may arise from

equivalent and non-equivalent electron configurations.


Electron

Configuration

(Equivalent

electrons)

Terms

s1

2S

p1; p

5 2P

p2; p

4

1S,

1D,

3P

p3

2P,

2D,

4S

d1; d

9

2D

d2; d

8

1S,

1D,

1G,

3P,

3F

d3; d

7

2P,

2D (two),

2F,

2G,

2H,

4P,

4F

d4; d

6

1S (two),

1D (two),

1F,

1G (two),

2I,

3P (two),

3D,

3F

(two), 3G,

3H,

5D.

d5

2S,

2P,

2D (three),

2F (two),

2G (two),

2H,

2I,

4P,

4D,

4F,

4G,

6S

(non-equivalent electrons)

s1s

1

1S,

3S

s1p

1

1P,

3P

s1d

1

1D,

3D

p1p

1

3D,

1D,

3P,

1P,

3S,

1S

Table 2

Hund’s Rule

Even though we could calculate the energy associated with each state, we can predict which one

of the terms arising from a given electron configuration is lowest in energy using the empirical

Hund’s Rule:

1. The term with the largest value of S is most stable (has the lowest energy).

2. For terms with same value of S, the term with largest value of L is most stable.

3. If the terms have the same value of L and S, then, for a sub-shell that is less than half-filled,

the term with smallest value of J is most stable; for the sub-shell that is more than half-filled,

the term with largest value of J is most stable. For example the ground state term symbol for

oxygen is 3P2 and for carbon it is

3P0. (No rule is needed to find the lowest level of the lowest

term of a half filled sub-shell configuration because for such sub-shells the term with largest S

value will be an S term. For an S term J will be zero. From Table 2, for the p3 configuration

the lowest term is 4S and for d

5 the lowest term is

6S. For an f

7 it will be the

8S term).

Hund’s rule work very well for the ground-state configuration, but occasionally fails for an

excited configuration. Hund’s rule gives only the lowest energy term of a configuration, and

should not be used to decide the order of the remaining terms.


The classical explanation of Hund’s rule is that electrons with the same spin tend to keep

out of each other’s way, thereby minimizing the Coulombic repulsion between them. The term

that has the greatest number of parallel spins (highest value of S) will therefore be lowest in

energy. But this traditional explanation turns out to be wrong in most cases. For example

consider the 3S (1s

12s

1) term and the

1S (1s

12s

1) term of Helium. The

3S term is found to be

lower in energy than the 1S term. But calculations using accurate wave functions have shown that

the average distance between the two electrons is slightly less for 3S term than

1S term. Hence

the traditional explanation could not be applied here. The reason for the 3S term lies below in

energy than the 1S term is because of a substantially greater electron-nucleus attraction in

3S term

as compared with 1S term.

The following explanation is given for this: the ‘repulsion’ between electrons of like spins makes

the average angle between the radius vectors of two electrons large for the 3S term than for the

1S

term. This reduces the screening of the nucleus and allows the electrons to get closer to the

nucleus in the 3S term, making electron nucleus attraction grater for

3S term which results in

energy lowering compared to 1S term.

Terms, Levels and States

The electrostatic interaction between electrons

give rise to different terms while the spin-orbit

(magnetic) interaction gives rise to different levels. The

interaction with external magnetic field gives rise to

different states. Different states are represented by MJ

values. A summary of the different types of interaction

that are responsible for the various kinds of splitting of

energy levels in atoms is shown in Fig 3. Only some

terms, levels and states are shown as an example. This

kind splitting is applicable in lighter atoms only. In

heavy atoms magnetic interaction may dominate

electrostatic (charge-charge) interaction.

For the ground state electron configuration of

carbon the term symbols are 3P0,

3P1,

3P2,

1D2 and

1S0.

Fig 4 shows the different terms, levels and states that

are possible for this arrangement. Each level is 2J+1

degenerate and the splitting of levels (MJ values) occur

in the presence of an external magnetic field only.

Configuration

Electrostatic interaction

SPDFG

Spin correlation

1S3S1P

Magnetic interaction (spin-orbit)

1S03P0

3P1

Interaction with external magnetic field

TERMS

LEVELS

STATES(+J, ........, -J)

Fig 3:


3P

1S1S0

3P0

3P1

3P2

0

0

10

-1

210 -1-2

Terms Levels States MJ

1D

0

Observed atomic energy levels in ground state carbon

10194 cm-1

11454 cm-1

(The separation of the levels for 3P term is very small

and is exaggerated for clarity)

1D2

Fig 4

Example 5

Using Hund’s rule deduce the lowest energy level of an excited state of beryllium atom whose

electron configuration is 1s22s

13s

1

The term symbols for the configuration are, 3S1 and

1S0. From Hund’s first rule

3S1 is the lowest

energy level.

Example 6

Using Hund’d rule select the ground state term from the following

(a) 3P,

1P,

3F,

1G (b)

4P,

4G,

6S,

2I

Ans

(a) 3F (b)

6S

Example 7

Explain the fine structure of sodium D-line.

The atomic Hamiltonian does not include electron spin. But in reality the existence of

spin introduces an additional term (usually small), to the Hamiltonian. This term is called spin-

orbit interaction. This spin-orbit interaction splits the atomic terms into levels. When we include

the spin-orbit interaction, the energy of an atomic level depends on its total angular momentum J.

Thus each atomic term is split into levels, each level having a different value of J. For example

the 1s22s

22p

63p

1 configuration of sodium has the single term

2P, which is composed of the two


levels 2P3/2 and

2P1/2 .These two levels give rise to the observed fine structure of the sodium D-

line. The ground state configuration 1s22s

22p

63s

1 has only one electron in the 3s orbital and

hence the term symbol is 2S1/2. Fig 5 explains the fine structure of sodium D line.

2P3/2

2P1/2

1s22s22p63p1

1s22s22p63s1

589.6 nm589.0 nm

2S1/2

Fig.5 fine structure of sodium D line Example 8: Predict the lowest energy level for the d

5 electron configuration

The d5 configuration is a half-filled configuration. ie., one electron in each orbital.

Therefore all the orbital angular momenta will get cancelled ( for a +2 electron there will be a –2

electron and so on) so total orbital angular momentum L will be zero. Hence an S term. The

lowest energy term has the maximum multiplicity and hence all the five electrons are considered

to be parallel with ∝-spin. Hence the total spin is 5/2. Therefore the term is 6S. Since L = 0 and S

= 5/2, J has only one value, 5/2. The term symbol is 6S5/2

MS 5/2

ML +2 +1 0 −1 −2 0

Atomic Spectroscopy – selection rules

For a hydrogen-like atom the electron transition between two energy levels is restricted

by the selection rules; ∆n = any integer, ∆l = ±1.That is an electron in the 1s orbital can undergo

transition to any p state, 1s → np, n ≥ 2. Similarly an electron in the p-state can go to any s-state

or d-state. For poly electronic atom the situation is more complicated, but the concept of term

symbols makes it simpler. It provides us the number of energy states of the atom as a whole. The

transitions between all these energy states are not allowed by the selection rule. For poly-electron

atoms the selection rules are:

∆n = ± any value

∆L = 0, ±1 (excluding L = 0 to L = 0)

∆S = 0

∆J = 0, ±1 (excluding J = 0 to J = 0)

Figure 6 shows the transitions responsible for the formation of the spectrum of hydrogen. Here

we consider the transitions between terms arising from configurations with n = 3, 2, and 1. The

ground level is 1s1

(2S1/2). The configuration 3s

1gives rise to another

2S1/2 with a single level. The


configuration 3p1 gives rise to

2P3/2 and

2P1/2. Similarly with other excited states. Consider

transitions (1) and (2). For these transitions;

∆L = +1 (S to P)

∆S = 0 (2S to

2P)

∆J = 0 for (1) and +1 for (2)

1s1 2S1/2( )

2s1 2S1/2( )

3s1 2S1/2( )

3p1 (2P3/2)

3p1(2P1/2)

2p1 (2P3/2)

2p1 (2P1/2)

3d1(2D5/2)

3d1(2D3/2)

Fig. 6 The energy levels of a hydrogen atom showing

possible transitions which is responsible for the

fine structure of the spectrum

(1)

(2)

The Zeeman Effect

In 1896, Zeeman observed that application of an external magnetic field caused a

splitting of atomic spectral lines. Electrons possess magnetic moments due to orbital and spin

angular momenta. In the presence of an external magnetic field the magnetic moment of the

electrons interact with the external field resulting shifts in energy level which causes the apparent

split in spectral lines.


For a poly-electron atom the magnetic moment associated with the orbital motion of the

electrons is given by 2

L

e

eL

m and that due to spin motion is given by

2S

e

eS

m ; where

e is the charge of the electron, me is the mass of the electron considered as a point mass, L is the

total orbital angular momentum and S is the total spin. The magnetic moment due to total

angular momentum J is given by2

e

e

egJ gJ

m ; where g is Landé g-factor and βe is the

Bohr magneton (which has a value of 9.274 ×10-24

JT-1

). The value of g is given by

( 1) ( 1) ( 1)1

2 ( 1)

J J S S L Lg

J J

. It depends on the state of the electron in the atom and in

general, g lies between 0 and 2.

When an external magnetic field is applied to the atom, say in the z direction, the

component of the magnetic moment along z-direction, μz, will interact with the applied field. μz

is given by −βegMJ. Now the interaction energy E is given by

E = − μzB = βegMJB

where B is the strength of the applied magnetic field, MJ is the components of the total angular

momentum in the z-direction. MJ has (2J + 1) values ranging from + J to – J. Therefore the

interaction energy also will have (2J + 1) different values. These energy states are degenerate in

the absence of the external field.

Thus in the presence of an external magnetic field a particular energy level splits into

(2J +1) different energy states. The splitting of the MJ energy levels in the presence of an

external magnetic field is called the Zeeman Effect. The splitting is proportional to the strength

of the applied field and is very small in magnitude. For

example in an applied field of one tesla (SI unit of magnetic

field strength, 1T = 10,000 gauss) the splitting is of the order

of 0.5 cm−1.

Normal Zeeman Effect

For singlet states the total spin magnetic moment (S)

is zero. The magnetic moment of the electron system is then

due to the orbital motion alone. Hence there will not be any

coupling between spin and orbit motion and hence MJ is equal

to ML and g become 1. The interaction energy is now, E =

βeMLB. This type of splitting of energy

levels due to applied field when S = 0 is called Normal

Zeeman Effect. As an example let us consider the spectral

lines that may arise due to the transition 1P →

1S, in the

presence of an external magnetic field. A 1S term has neither

orbital nor spin angular momentum, so it is unaffected by the external magnetic field.

B = 0 B > 0

ML

+1

0

-1

0

1P

1S

Normal Zeeman Effect

Fig. 6


The 1P level splits into three levels which give rise three lines in the spectrum. The selection rule

is ∆ML = 0, ±1. In the absence of magnetic field there will be only a single line. The normal

Zeeman Effect is observed wherever spin is not present (singlet to singlet transitions).

The anomalous Zeeman Effect

Anomalous Zeeman Effect is observed wherever

spin is not zero. It is more common than the normal

Zeeman Effect. In anomalous Zeeman Effect a more

complex pattern of lines is observed. For example consider

the transition 2D3/2 →

2P1/2, in the presence of an external

magnetic field. When S≠0, the value of g depends on the

values of L and S and so different terms split to different

extents. The selection rule ∆MJ = 0, ± continues to limit the

transitions. Anomalous Zeeman Effect provides an

experimental evidence for spin-orbit coupling.

When a strong field is applied, the coupling

between L and S may be broken and they tend to couple

directly with the applied magnetic field. As a result the

anomalous Zeeman Effect disappears and normal Zeeman

Effect is observed. This switch from the anomalous effect

to the normal effect is called Paschan-Back effect.

Molecular Term Symbols

We have seen that the electronic states of atoms are designated by term symbols.

Similarly, the electronic states of molecules are also designated by term symbols. As in poly-

electron atoms, we consider the coupling

between the motions of the electrons. For

diatomic molecules the most suitable

coupling scheme is that which is analogous

to the Russell-Saunders coupling employed

in atoms. The orbital angular momenta of

all the electrons in the molecule are

coupled to give a resultant L and all the

electron spin momenta are coupled to give

a resultant S. However, the coupling

between L and S is usually weak, and

LS

Nuclei

Electron

Figure 8

2D3/2

2P1/2

MJ

+3/2

+1/2

-1/2

-3/2

+1/2

-1/2

The anomalous Zeeeman EffectFig: 7


instead of being coupled to each other, they couple to the electrostatic field produced by the two

nuclear charges of the diatomic molecule. This situation is shown in Figure 7 and is referred to

as the Hund’s case.

The direction of the electrostatic field of the two nuclei is taken as the inter-nuclear axis

(taken as the z-axis). The vector L is strongly coupled to the electrostatic field so that it precess

about the inter-nuclear axis. As a result, the magnitude of L is not defined. Only the component

of L along the inter-nuclear axis is defined, which is l Lm M . (ml is for individual

electrons and to calculate ML we simply add algebraically the lm ’s of individual electrons).

Here is a quantum number taking values, 0, 1, 2, 3 …… All electronic states with > 0 are

doubly degenerate. Classically, the degeneracy is ascribed to the electrons being orbiting,

clockwise or anticlockwise around the inter-nuclear axis represented by ±ml. The value of is

similar to the value of L in the case of atoms. The electronic terms corresponding to different

values of are , , , …..

0 1 2 3 …

Symbol …

The different types of molecular orbitals are σ, π, δ … We use the symbol λ to represent the

angular momentum of an electron in a molecule; lm = λ. Depending upon the type of MO

the value of λ varies as shown in Table 3.

MOs ml λ (lm )

AOs from which

MOs are formed

σ 0 0 s, pz, 2zd

π ±1 1 px, py, dxz, dyz

δ ±2 2 dxy, 2 2x yd

Table 3

The coupling of S to inter-nuclear axis is caused by magnetic field along the axis due to

the orbital motion of the electrons, and the electrostatic field has no effect on S. The component

of S along z-axis can be taken as and the quantum number (beta) is analogues to Ms in

atoms. is the component of S along z-axis (the symbol is , but to avoid confusion is used).

can have values, S, S – 1, ….. – S. It can be computed from the Clebsch-Gordan series. For

states > 0, there are (2S + 1) components corresponding to the values that can take. ie the

multiplicity of the level is (2S + 1).

The component of the total (orbital + electron spin) angular momentum along the inter-

nuclear axis is given by , where (omega) is the absolute value of (.


. is analogous to the quantum number J in atoms. It is actually the quantum number

for the z-component of the total electronic angular momentum and therefore can take on negative

values. It is the value of ( and not the value of is written as the subscript on the term

symbol. For eg., consider the 3 term. Since the term is , = 1. Now (2S + 1) = 3 or S = 1.

Hence the values of are S, S – 1, …. S = 1, 0, 1.

= S, (S), …..+ (S) = 1+1, 1+11, 11 = 2, 1, 0. Therefore the term symbols

are, 32,

31,

30.

A 4 term has four levels,

21

4

21

4

23

4

25

4 ,,,

. This arises in the following way:

Spin multiplicity (2S + 1) = 4, therefore S = 32

. Hence the values of B are: 3 31 12 2 2 2, , , .

= S, (S), …..+ (S) = 3 3 5 31 1 1 12 2 2 2 2 2 2 2

1 ,1 ,1 ( ),1 ( ) , , ,

For a state, there is no orbital angular momentum ( = 0). Hence, a state has only one

component irrespective of the multiplicity. The quantum numbers and are not defined for

state. A filled molecular orbital has

both and equal to zero and

gives rise to only a 1 term. It

always corresponds to a single non-

degenerate energy level. Example

is the ground state electronic

configuration of hydrogen

molecule.

For atoms, the electronic

energy states may be classified entirely by the use of L, S, and J. In diatomic molecules, the

corresponding quantum numbers , , and are not quite sufficient. We must also use the

symmetry properties of the electronic wave function. For homopolar diatomic molecules, the

states are labeled as g or u, which indicates the wavefunction is symmetric or anti-symmetric

respectively to inversion through a centre

of symmetry. The other symmetry

property concerns the symmetry of

electron wave function with respect to

reflection across any plane (V)

containing the inert-nuclear axis. If the

wave function is unchanged by this

reflection (ie. symmetric), the state is

labeled +, and if it changes in sign by this reflection (ie. Anti-symmetric), the state is labeled – ,

as in 3g

+ or

2

g. This symbolism is normally used for states only. Similar to atomic term

symbol molecular term symbols also are also represented by writing spin multiplicity as left

superscript and total angular momentum as right subscript to the code letter for .ie.

u g

g u

Parity - behaviour under inversion.

Figure 9

v(-) (+)

anti-symmetric

with respect to

plane

symmetric with

respect to

plane

v

Figure 10


2 1 /

( / )

S

g u

Generally the total angular momentum is not shown, and the parity sign (+/−) is shown for

terms only. g/u subscript is used only for homonuclear diatomic molecules.

For simple homonuclear diatomic molecules the following guidelines can be used for assigning

parity. (a detailed way of assigning +/− is discussed in the case of O2 molecule)

If all MOs are filled, + applies

If all partially filled MOs have σ symmetry, + applies

For partially filled MOs of π symmetry (for example B2 and O2), if Σ terms arise,

the triplet state is associated with −, and the singlet state is associated with +.

For diatomic molecules of closed-shell configuration all the electrons are paired. Hence

the quantum number MS, which is the algebraic sum of individual ms values, must be zero.

Therefore S = 0 for configurations containing only filled molecular shells. A filled σ shell has

two electrons with ml = 0, so ML is zero. A filled π-shell has two electrons with ml = +1 and two

electrons with ml = −1, so ML is zero. The same situation hold for filled δ, ϕ, … shells. Thus a

closed-shell molecular configuration has both S and Λ equal to zero and give rise to only a 1Σ term.

For example consider the ground state configuration 1σg2, of H2 molecule. The term is

1Σ. Since

the electrons are in a gerade orbital, and the σ-orbital being symmetric the term symbol is 1

g

.

Electrons in the same molecular shell are called equivalent. For equivalent electrons, there are

fewer terms than for the corresponding non-equivalent electron configuration, because of the

Pauli Principle. Table 4 gives a list of terms for equivalent and non-equivalent electron

configurations of diatomic molecules.

Configuration Terms

Non-equivalent

δ1σ1 1Σ+,3Σ+

σ1π1; σ1π3 1Π, 3Π

π1 π1; π1 π3 1Σ+,3Σ+,1Σ−, 3Σ−, 1∆, 3∆

π1δ1, π3δ1, π1δ3 1Π, 3Π,1Φ, 3Φ

Equivalent

σ1 2Σ+

σ2; π4; δ4 1Σ+

π1; π3 2Π

π2 1Σ+,3Σ−,1∆

δ1; δ3 2∆

Δ2 1Σ+, 3Σ−, 1Γ

Table 4


Examples

(i) H2+

The electron configuration is 1σg1. Since there is only one electron, S = s = ½ . Spin multiplicity

is 2. For σ electron λ = 0 and hence, Λ = 0. The term symbol is 2Σg

+.

(ii) B2+

The electron configuration is, 1σg21σu

22σg

22σu

21πu

1. Only one electron need to be considered.

For that electron λ = Λ = 1 (π-electron) and S = s = ½ . Hence the term is Π. The spin

multiplicity is 2. Therefore the term symbol is 2Π (or 2Πu).

(iii)N2+

The electron configuration is 2σg22σu

21πu

43σg

1. All orbitals except 3σg are filled. Only one

electron is present in the 3σg orbital. Hence, Λ = 0 and S = s = ½. Therefore the term symbol is 2Σg

+.

(iv) O2 (or B2)

The electron configuration for O2 is 2σg22σu

23σg

21πu

41πg

2. All orbitals up to 1πg are filled. There

are two electrons in the πg orbital. For π-electrons λ = 1 (and ml = ±1). The two electrons in

the1πg orbital can be arranged in the following way (Table 5). π+ and π− are the two different

angular momenta of the electron. Ψ1, Ψ2, … are the spin-orbit wave functions

Ψ1 Ψ2 Ψ3 Ψ4 Ψ5 Ψ6

π+ ⇵ ↓ ↑ ↑ ↓

π− ⇵ ↑ ↑ ↓ ↓

ML +2 −2 0 0 0 0

MS 0 0 0 +1 0 −1

term 1∆ 1

g

3g

Table 5

In the table, ML values +2 and −2 represent a term with Λ = 2. MS values for both the values

are zero and hence S = 0. Therefore this arrangement represents a 1Δ term. When ML and

MS are zero it represents a 1Σg term. For the remaining three arrangements ML = 0, but the

values MS are +1, 0, −1, and it corresponds to S = 1. ie. Λ = 0 and S = 1. Hence a 3Σg term.

Now we have to assign +/− to Σ terms.

The O2 molecule has a π2 configuration, and anti-symmetric wave functions can be

formed either by combining symmetric spatial functions with anti-symmetric spin functions or

vice versa. All the six combinations are shown below in table 6. The spatial part of Ψ1 and Ψ2

are symmetric and this being a ∆ term we do not assign +/− to it. For Ψ3 the spatial

function is symmetric and hence we assign + sign. Ψ4 to Ψ6 are anti-symmetric so it

combine with symmetric spin function. We assign ‘–‘ for these three terms.


Spatial (orbital)

function Spin function

Symmetry of

spin function

Symmetry of orbital

function

Ψ1 π+ π+ α(1)β(2) - β(1) α(2) Anti-symmetric symmetric 1Δ

Ψ2 π− π− α(1)β(2) - β(1) α(2) Anti-symmetric symmetric

Ψ3 (π+ π− )+ (π− π+) α(1)β(2) - β(1) α(2) Anti-symmetric symmetric 1Σ+

Ψ4 (π+ π− ) − (π− π+) α(1) α (2) symmetric Anti-

symmetric

3Σ− Ψ5 (π+ π− ) − (π− π+) α(1)β(2) + β(1) α(2) symmetric Anti-

symmetric

Ψ6 (π+ π− ) − (π− π+) β(1)β(2) symmetric Anti-

symmetric

Table 6

Thus the term symbols for the ground state of O2 are: 1Δ, 1

g

and 3

g

.

Exercise:

Write the term symbol for (i) Li2+ with the electron configuration 2σg

2, (ii) F2 and (iii) H2

Electronic spectra of diatomic molecules.

Let us take the simplest molecule H2 as example. For H2 molecule the configuration is 1σg2,

hence the term symbol is 1Σg

+. We can also imagine a large number of singlet excited states and

let us consider some lower energy levels in which only one electron has been raised from the

ground state into some higher molecular orbitals (ie. singly excited states). We can ignore any

promotion into any anti-bonding states since this would lead to the formation of an unstable

molecule leading to immediate dissociation of the molecule. Thus we may consider, for example,

three states; 1sσg12sσg

1, 1sσg

12pσg

1, 1sσg

12pπu

1

(a) 1sσg12sσg

1

Here both the electrons are σ electrons. Hence Λ = λ1 + λ2 = 0. We are considering only

singlet states, so S = 0. Since both the constituent orbitals are gerade and symmetrical, the

over all state will be the same. Therefore we have 1

g

state.

(b) 1sσg12pσg

1

Since both the electrons are σ- electrons we have a 1Σ state. But the

overall state is now odd. This can be understood in the following

way. Imagine that one of the electrons is coming from a

hydrogen atom in the gerade 1s state, and the other electron

from a hydrogen atom in the ungerade 2p state. Combination of

g

1s gerade

u

2p ungerade


an ungerade and gerade states lead to an overall ungerade state. Hence the term symbol is 1Σu

+.

(c) 1sσg1 2pπu

1

Since one electron is σ and the other is π, Λ = λ1 + λ2 = 0 + 1 = 1. Therefore we have a 1Π term. As in the above case, since one electron comes from a 2p orbital we can consider the overall state as u. Hence 1Πu

Thus the three excited state configurations give rise to 1

g

,

1

u

1and u.

Selection Rules

The transition between different electronic state can occur according to the following selection

rules:

1. ΔΛ = 0, ±1

For eg. Σ ↔Σ, Π ↔Π, Π↔Σ are allowed while Δ ↮ Σ or Φ ↮ Π are not allowed.

2. ΔS = 0

That is only transitions like singlet – singlet, triplet – triplet are allowed. But this

rule breaks down with increase of nuclear charge. For example triplet – singlet

transitions are strictly forbidden in H2 but in CO, 3Π -

1Σ

+ transition is observed

although very weekly.

3. ΔΩ = 0, ±1

4. There are also rules based on symmetry: ↔ indicates allowed transition, and ↮ indicates forbidden transition + ↔ + and − ↔ − are allowed + ↮ − is forbidden g ↔ u allowed

g ↮ g and u ↮ u are forbidden

Figure 11 shows the molecular orbital energy level diagram for H2 molecule. For hydrogen 2s

and 2p are having same energy.

Heteronuclear diatomic molecules

For heteronuclear diatomics the gerade ungerade designation is not used. For example let us find

the term for the NO molecule.

The electron configuration for NO is 32, 4

2, 5

2 1

4, 2

1. Only one electron in the -orbital,

so = 1 and S = ½. Hence the term is 2. Now S has only two values +½ and ½. Hence =

+ S = 1 + ½, 1 ½. Therefore the term symbols are; 23/2 and

21/2

Exercise: Find the term symbols for the ground state of CO


1s 1s

2s, 2p 2s, 2p

2sg

1sg

2su

1su

2pg

u

2pu

2p g

2p

Fig: 11

1sg

2s

3s

4s

2p

2p

3p

4p

3p

3p

(ground state)

(a) MO energy level diagram for hydrogen

molecule(b) singlet-singlet one-electron transitions in

hydrogen molecule

Energy

1

g

+

1

+

uu

1

g

+


References

1. Molecular quantum mechanics, 4th edn., Atkins and Friedman

2. Quantum Chemistry, 5th edn Ira N. Levine

3. Physical chemistry a molecular approach, Donald A. McQuarrie and John D. Simon

4. Atkins’ Physical chemistry, Peter Atkins and Julio de Paula

5. Electronic Absorption Spectroscopy and related techniques, D. N. Sathyanarayana

term symbol

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