Contents:

1. Spectra of Helium (He) Atom

2. Spectra of two valence electron systems

The students will be able to lean spectra of Helium and alkaline metals

and in general spectra of two electron system including their energy

levels and the fine structure

1. Spectra of Helium (He) Atom:

The Helium (He) atom has electronic configuration of 1s2. These two

electrons in the configuration are responsible for the valency of two and

hence in deciding the general characteristics of the optical spectrum. The

ground state Helium has two 1s electrons. Both the electrons have l = 0

and as n, l and ml are same; therefore, according to the Pauli Exclusion

Principle their spins are antiparallel.

Thus for 1s2, S and L are zero and hence J = 0 and the ground state is 1S0.

Now assuming that one of the two electrons is excited to the 2 p orbits

yielding a configuration 1s2p due to n =1 for the s electron and n = 2 for

the p electron. Pauli Exclusion Principle does not apply, as the two

electrons are non-equivalent. For the s and p electrons, l = 0 and 1,

respectively and thus L = 1.The state is represented by the symbol P. The

spins are either parallel or antiparallel giving S =1 and 0, respectively. As

the multiplicity is 2S+1, one gets triplet and singlet and for S = 0 the).

The J values for triplet are 2, 1, 0 and for singlet, it is 1. The possible

terms are therefore, 3P210 and 1P1.

In case, one of the two electrons is promoted to various excited

configuration, then the resulting states are singlet and triplet (The ground

state configuration leads only to singlet) and according to Hund’s rule,

the triplet states lie lower than the corresponding singlet states.

Normal

configuration

Excited

configuration

n Term

1s2 -- -- 1S0

1s2 1sns 2,3,4,…. 1S0 ,3S1

1s2 1snp 2,3,4,…. 1P1,3P210

1s2 1snd 3,4,5,…. 1D2,3D321

1s2 1snf 4,5,6… 1F3,3F432

The energy level diagram of Helium is shown in the figure. It depicts the

energies of the ground state and first few excited states of helium atom obtained

4P 4S

3S

2P

4P 4F

1S

Singlets

2S

4D

3P 3D

Triplets

4S

3S

3P

4D

3D

4F

2P

2S

from an analysis of the optical line spectra. The energy of the term increases in

the order of nS, nP, nD, nF,…….., for the same n. The lowest triplet state is

2 3S, and the lowest singlet state is 11S. The state 13S is not allowed because of

Pauli’s Exclusion Principle. The energy difference between the 11S and 23S is

quite large ~19.72 eV reflecting the tight binding closed shell electrons.

Helium atoms in singlet states (antiparallel spins) constitute parahelium and

those in triplet states (parallel spins) constitute orthohelium. The lowest state of

orthohelium, 23S does not combine with the ground state 11S. Those terms, that

cannot go to a lower state with the emission of radiation and correspondingly

cannot be reached from a lower state by absorption, are called metastable states.

The 21S state is also metastable, as the selection rule ∆l = ± 1 does not allow

any transition to 11S. The metastability of the 23S state is however, stronger than

that of 21S, as the transition 23S →11S would contradict the prohibition, of an

ortho-para transition as well as ∆l = ± 1. However, 23S → 11S transitions can be

caused by collision with other atoms or with the wall of a containing vessel that

distorts the electronic structure of the atom, and allow a change of spin to occur.

The transition from one energy state to another is governed by the selection

rule ∆L = ± 1, ∆S = 0, ∆J =0, ±1. Allowed transitions between different energy

levels gives rise to a large number of lines that can be put into two distinct

system of lines: a system of closely spaced triplets and a system of singlets. In

the emission spectrum of helium atom, lines of singlet and triplet system may be

grouped into four series each:

(i) Principle series: Two principle series for singlet are observed one is

from higher P state to normal 11S0 state and member of this series lie

in far ultraviolet region. This series also exists in absorption. Besides

this there exists another principle series whose spectral lines arise

from transitions from higher 1P states to the 21S state. This principle

series lie from infrared to near ultraviolet region. The members of the

triplet principle series are due to transitions from higher 3P state to

23S state. The triplet principal series lie from infrared to ultraviolet

region.

(ii) Sharp series: The sharp series arise due to transitions from S states to

the lowest P state. The lines of sharp series of singlet and triplet

system lie in visible and ultraviolet region.

(iii) Diffuse series: The diffuse series arise from transition between higher

D states to the lowest P state. The lines of this series for both singlet

and triplet system lie in visible and ultraviolet region.

(iv) Fundamental series: This series arise from transitions between higher

F states to the lowest D state. The lines of this series for singlet and

triplet system lie in infrared region.

Under high resolution, the lines of parahelium system are single and those of

orhtohelium show closely spaced triplet structure, that is very difficult to

resolve. The triple fine structure is due to spin-orbit interaction. The figure

shows 3P→3S transition.

3S1

3P2

3P1

3P0

3D3

3P2

3P1

3P0

3D1

3D2

Splitting of 3P210→3S1 and 3D321→3P210

state

3S state has one component 3S1 while 3P has three components 3P2, 3P1

and3P0. Three lines are observed according to selection rules. ∆L = ± 1,∆J =

0, ±1 and ∆S = 0, between these states.

The adjacent figure shows the fine structure of triplet diffuse series.

For triplet D we have S = 1, L = 2 and hence J = 3, 2, 1. According to

allowed selection rules, six transitions are possible. The complete spectrum

should consist of six lines. Normally, the very close spacing is not resolved

and only three lines are observed. For this reasons the spectrum is referred to

as compound triplet.

For He, the component separations 3P0-3P1 = 0.996 cm-1 and 3P1-

3P2 = 0.078

cm1. The terms of 23P of He are inverted with 3P0 component lying highest in

energy and 3P2 the lowest. Further there is breakdown of Lande interval rule.

This is due spin-spin interaction.

The order is inverted in both P and D state for He

2. Spectra of two valence electron systems

3S1

3P2

3P1

3P0

3D3

3P2

3P1

3P0

3D1

3D2

Splitting of 3P210→3S1 and 3D321→3P210

state

The electronic configuration of neutral atoms of group II elements (alkaline

elements) are given as

Elements Atomic

Number,Z

Configuration

Be 4 1s22s2

Mg 12 1s22s22p63s2

Ca 20 1s22s22p63s23p64s2

Zn 30 1s22s22p63s23p64s23d104s2

Sr 38 1s22s22p63s23p64s23d104s24p65s2

Cd 48 1s22s22p63s23p64s23d104s24p64d105s2

Ba 56 1s22s22p63s23p64s23d104s24p64d105s25p66s2

Hg 80 1s22s22p63s23p64s23d104s24p64d105s25p65d106s2

The last two equivalent electrons in the group II elements are responsible for the

general characteristics of the optical spectra. These elements in their ground

state have two equivalent s electrons in the nth (n = 2, 3, 4, 5 and 6) orbit and

therefore the ground state is n1S0. On exciting one of these two electrons to

higher s, p, d or f orbit one obtains a system of two non-equivalent electrons.

These systems gives singlet and triplet S, P, D and F state as described in the

case of helium.

The ground state configuration leads only to singlet states whereas each excited

configuration arising from the promotion if an electron gives rise to singlet and

triplet states with later lying lower than the corresponding singlet state.The

figure depicts the energies of the ground states and first few excited states of Sr

atom obtained from the analysis of the optical line spectra. This figure also

shows some of the transitions allowed by selection rules. (The fine structure

splitting is not shown).

7S 7S

6S

6S

6P

4D 4D

6P

7P 4F

5D

5S

5D

4F

5P

5P

Singlet Triplet

Allowed transitions between different energy levels give rise to a large number

of lines that can arise from transitions

(a) between closely spaced triplets and

(b) between a systems of singlet.

In the spectra of alkaline earth metals, the lines of tripled system and of

singlet system can be further classified into four chief series. These are:

(i) Principal series: This series arise from transition from higher P states

to the lowest S state.

(ii) Sharp series: This series arise from transition from higher S states to

the lowest P state.

(iii) Diffuse series: This series arise from transition from excited D states

to the lowest P state.

(iv) Fundamental series: This series arise because of transitions from

excited F states to lowest D state.

The regions in which the lines of these four series of singlet and triplet system

lie for the Sr element

Element Principle

series

Sharp series Diffuse series Fundamental

series

Sr (Singlet) VIS,UV IR,VIS IR,VIS VIS

Sr (Triplet) IR,VIS VIS,UV IR,UV VIS,UV

Each series converge towards shorter wavelength to fit the wavenumber of the

spectral line to hydrogen like formula

�̅� = 𝑅𝑍2 (1

𝑛𝑒" 2 −

1

𝑛𝑒′ 2)

are successful only if non integral quantum number are used. For each spectral

series the term T’’ = RZ2/𝑛𝑒′′2 is constant and is the wavenumber of series limit.

ne = n - µ, is the effective quantum number where n is an integer and µ is a

fraction called quantum defect that is zero for non-penetrating orbits. (As in

case of alkali atoms, the penetration of various possible electron orbits into the

atom core is measured by the deviation of the term values from those of

hydrogen). Two methods are used to measure the penetration

(i) by modifying the nuclear charger in the term value expression for hydrogen,

(ii) modifying the principal quantum number.

These leads to term values for (i) as

T = 𝑅(𝑍−𝜎)2

𝑛2

and for (ii) as

T = 𝑅𝑍0

2

(𝑛− µ)2

Where 𝜎 is a screening constant, Z0 is the effective nuclear charge when the

electron is well outside the core. From the spectrum, ignoring fine structure, it is

observed that:

(i) Triplet sharp and diffuse series have one common limit and singlet

sharp and diffuse another.

(ii) The frequency difference between common limit of triplet sharp and

diffuse series and triplet principal series is equal to the frequency of

the first member of the principal series (Rydberg-Schuster law).

(iii) The frequency difference between the common limit of the triplet

sharp and diffuse series and the triplet fundamental series is equal to

the frequency of the first member of the triplet diffuse series (Runge

law).

(iv) The same is true even if fine structure is taken into account.

(v) (i), (ii), and (iii) holds for singlet series also.

Fine structure

The examination of alkaline spectra reveals that lines of various triplets’

series have a structure known as fine structure. It is observed that:

(i) All members of the principal series are composed of three lines with

decreasing separation with increasing frequency, and approach a

single limit.

(ii) All member of sharp series are composed of three lines with the same

separation, and approach a triple limit.

(iii) All members of diffuse and fundamental series contain six lines, three

strong and three satellites and approach a triple limit.

The fine structure in triplet series is due to splitting of various energy levels

by spin-orbit interaction as described previously in case of alkali atoms. In case

of alkaline metals, in triplet levels we have S = 1 and L can take values 0, 1, 2

and 3 for S, P, D and F states, respectively. For S state, S = 1 L = 0, therefore, J

= 1. For 3S1 there is no splitting of the level. For P state, L = S =1, therefore J =

2, 1, 0 and P state is split into three components with different J values.

Similarly, D and F states, represented by 3D321 and 3F432, respectively splits into

three components each.

In the triplet principal series transitions take place from 3P210 to lowest 3S1 level

as shown in the figure and according to selection rules

∆S = 0, ∆L = ±1, ∆J = 0,±1

three transitions take place.

The spacing between the transitions depends on the splitting of P level which

decreases with increasing n as per the equation.

Δ�̅� =𝑅𝐴𝛼2𝑍4

𝑛3𝑙(𝑙 + 1)

For large value of n, the splitting of the P level is negligible and as a result, one

observes a single transition in series limit.

Fig. shows the fine structure transition for sharp series that take place from

different excited triplet S states to lowest triplet P state.

3S1

3P2

3P1

3P0

Figure shows the Fine structure transitions for sharp series

As a result of allowed transitions, three lines are observed. The spacing between

the lines of triplet is governed by the splitting of the lowest triplet P state. As

the sharp series triplet separation is produced by the splitting of the lower 3P210

level on which all the transitions terminates, therefore it remains constant.

Figure shows the fine structure transitions for the triplet diffuse series that take

place from different excited 3D levels to lowest 3P level. Each of 3P and 3D level

splits into three levels. As a result of allowed selection rules, six transitions are

possible. The spacing between the transitions depends on the splitting of 3P and

3D levels. For large value of n, the spacing between 3D321 levels diminishes and

3P2

3P1

3P0

3D1

3D2

3D3

3S1

3P2

3P1 3P0

therefore 3D321 levels are not resolved. As a result of this only three transitions

are observed whose spacing depends on the splitting of 3P210 levels. Same

situation exists for the fundamental series. However, now for large value of n,

the spacing between the transitions depends on the splitting of 3D321 levels,

which is constant.

As for sharp and diffuse series the transitions terminate at lowest 3P level

therefore, for large value of n, the spacing between transitions is same for both

sharp and diffuse series.

Qualitatively, the relative intensity can be obtained by rules given for alkali

atom spectra. Let us apply those rules for the members of diffuse series.

For transitions (a) ∆J = + 1 with J = 3,

(b) ∆J = 0, J = 2

(c) ∆J = + 1, J = 2

(d) ∆J = -1, J = 1

(e) ∆J = 0, J = 1 and

(f) ∆J = +1, J = 1.

For all these transitions ∆ L has the same value. Thus transition ‘a’ is the

strongest and ‘d’ is the weakest. The intensities of other transitions are

intermediate between the intensities of ‘a’ and ‘d’.

The transitions in order of decreasing intensities are a >c >f >b >e >d.

Assignment 1.

Calculate the Rydberg denomionators for the first term value of the Principle

series of sodium. The wavenumber of the transition is 16973.7 cm-1. The R for

Na is 109734 cm-1

Write down the spectral term

T= R/ne2

So ne2=109734/16973.7=254

Assignment 2.

The first ionization potential of sodium is two fifth that of hydrogen. Calculate

the effective nuclear charge of the sodium atom as far as the 3s electron is

concerned.

For hydrogen

Z = 1, n = 1.

Therefore using En=-RZ2/n2 E1 = -R cm-1 = - 13.6 eV. The ionization energy

of hydrogen is 13.6eV.

Now 𝐼𝐻

𝐼𝑁𝑎=

(𝑅𝑍𝑒𝑓𝑓2 𝑛2⁄

[𝑅𝑍𝑒𝑓𝑓2 𝑛2]𝑁𝑎⁄

For Na, n = 3 and for hydrogen Zeff = 1. Putting (INa/IH) = 2/5 in the above

equation

Z2 eff =1.897

Assignment 3.

The principal and sharp series for the Li atom converge to continua at 43487

and 28583 cm-1 respectively. Calculate the quantum defect for the common term

in each of the series (RLi = 109729 cm-1).

The common term for sharp and principal series is 22P term. The wavenumber

position of this term is 28535 cm-1 and we know T= R/ne2

Therefore ne2=R/T= 109729/28583 =3.83896

So, ne= 1.9593

Μ= n- ne= 2-1.9593=.0407

Assignment 4

The principal and sharp series for the sodium atom converge to continua at

41450 and 24477 cm-1, respectively. Calculate the ionization potential of Na.

The wavenumber corresponding to series limit of principal series give the

position of normal ground state 32S of Na. Hence ionization energy corresponds

to 41450cm-1 as 1 eV= 8065 cm-1. Thus 41450 cm-1 is equal (41450/8065) eV =

5.139eV.

Summary:

A two-electron atom or helium-like ion is a quantum mechanical system

consisting of one nucleus with a charge of Ze and two electrons. This is

the first case of many-electron systems where the Pauli Exclusion

Principle plays a central role. It is considered as a three body problem.

The important two-electron atoms are:

Hydrogen anion (Z=1) Helium atom (Z=1) Lithium ion (Z=1)

Beryllium ion(Z=1) Boron ion (Z=1)

The optical spectrum of the two electron atom has two systems of lines:

A para system of single lines, and an ortho system of triplets (closely

spaced group of three lines).

The energy levels in the atom for the single lines are indicated

by 1S0 1P1

1D2 1F3 etc., and for the triplets, some energy levels are

splitted: 3S1 3P2

3P1 3P0

3D3 3D2

3D1 3F4

3F3 3F2

Alkaline earths and Mercury also have spectra with similar features, due to the

two outer valence electron.