**3. Russel-Saunders or L-S coupling scheme**

An orbiting electronic charge produces magnetic field perpendicular to the plane of the orbit. Hence the orbital angular momentum and spin angular momentum have corresponding magnetic vectors. As a result, both of these momenta couple magnetically to give rise to total orbital angular momentum. There are two schemes of coupling: Russel-Saunders or L-S coupling and j-j coupling.

a. The individual spin angular momenta of the electrons, si, each of which has a value of ± ½, combine to give a resultant spin angular momentum (individual spin angular momentum is represented by a lower case symbol whereas the total resultant value is given by a upper case symbol).

<sup>i</sup> s =S

Two spins of each ± ½ could give a resultant value of S =1 or S= 0; similarly a resultant of three electrons is 1 ½ or ½ .The resultant is expressed in units of h/2 π . The spin multiplicity is given by (2S+1). Hence, If n is the number of unpaired electrons, spin multiplicity is given by n + 1.

b. The individual orbital angular momenta of electrons, li, each of which may be 0, 1 ,2, 3 , 4 ….. in units of h/2π for s, p, d, f, g, …..orbitals respectively, combine to give a resultant orbital angular momentum, L in units of h/2π . ∑ li =L

The resultant L may be once again 0, 1, 2, 3, 4…. which are referred to as S, P, D, F G,… respectively in units of h/2π.The orbital multiplicity is given by (2L+1).


c. Now the resultant S and L couple to give a total angular momentum, J. Hence, it is not surprising that J is also quantized in units of h/2π.The possible values of J quantum number are given as

$$\text{I} = \begin{pmatrix} \text{L} + \text{S} \\ \end{pmatrix}, \begin{pmatrix} \text{L} + \text{S} \cdot \text{I} \\ \end{pmatrix}, \begin{pmatrix} \text{L} + \text{S} \cdot \text{I} \\ \end{pmatrix}, \begin{pmatrix} \text{L} + \text{S} \cdot \text{I} \\ \end{pmatrix}, \begin{pmatrix} \text{L} + \text{S} \cdot \text{J} \\ \end{pmatrix}, \begin{pmatrix} \text{L} + \text{S} \cdot \text{J} \\ \end{pmatrix}, \begin{pmatrix} \text{L} - \text{S} \\ \end{pmatrix}, \begin{pmatrix} \text{L} - \text{S} \\ \end{pmatrix}$$

The symbol | | indicates that the absolute value (L – S) is employed, i.e., no regard is paid to ± sign. Thus for L = 2 and S = 1, the possible J states are 3, 2 and 1 in units of h/2π.

The individual spin angular momentum, si and the individual orbital angular momentum, li, couple to give total individual angular momentum, ji. This scheme of coupling is known as spin-orbit coupling or j -j coupling.

## **4. Term symbols**

4 Advanced Aspects of Spectroscopy

above.

infrared, visible and U.V. region.

**3. Russel-Saunders or L-S coupling scheme** 

Saunders or L-S coupling and j-j coupling.

given by a upper case symbol).

Electronic absorption spectrum is of two types. d-d spectrum and charge transfer spectrum. d-d spectrum deals with the electronic transitions within the d-orbitals. In the charge –

transfer spectrum, electronic transitions occur from metal to ligand or vice-versa.

Electronic absorption spectroscopy requires consideration of the following principles:

a. *Franck-Condon Principle:* Electronic transitions occur in a very short time (about 10-15 sec.) and hence the atoms in a molecule do not have time to change position appreciably during electronic transition .So the molecule will find itself with the same molecular configuration and hence the vibrational kinetic energy in the exited state remains the

b. *Electronic transitions between vibrational states:* Frequently, transitions occur from the ground vibrational level of the ground electronic state to many different vibrational levels of particular excited electronic states. Such transitions may give rise to vibrational fine structure in the main peak of the electronic transition. Since all the molecules are present in the ground vibrational level, nearly all transitions that give rise to a peak in the absorption spectrum will arise from the ground electronic state. If the different excited vibrational levels are represented as υ1, υ2, etc., and the ground state as υ0, the fine structure in the main peak of the spectrum is assigned to υ0 → υ0 , υ0 → υ1, υ<sup>0</sup> → υ2 etc., vibrational

states. The υ0 → υ0 transition is the lowest energy (longest wave length) transition. c. *Symmetry requirement:* This requirement is to be satisfied for the transitions discussed

Electronic transitions occur between split 'd' levels of the central atom giving rise to so called d-d or ligand field spectra. The spectral region where these occur spans the near

An orbiting electronic charge produces magnetic field perpendicular to the plane of the orbit. Hence the orbital angular momentum and spin angular momentum have corresponding magnetic vectors. As a result, both of these momenta couple magnetically to give rise to total orbital angular momentum. There are two schemes of coupling: Russel-

a. The individual spin angular momenta of the electrons, si, each of which has a value of ± ½, combine to give a resultant spin angular momentum (individual spin angular momentum is represented by a lower case symbol whereas the total resultant value is

50,000 - 26300 26300 -12800 12800 -5000 cm 200 - 380 380 -780 780 - 2000 nm

Ultraviolet UV Visible Vis Near infrared NIR


**2. Electronic spectra of transitions metal complexes** 

same as it had in the ground state at the moment of absorption.

#### **4.1. Spectroscopic terms for free ion ground states**

The rules governing the term symbol for the ground state according to L-S coupling scheme are given below:


The term symbol is given by 2S+1 LJ. The left-hand superscript of the term is the spin multiplicity, given by 2S+1 and the right- hand subscript is given by J. It should be noted that S is used to represent two things- (a) total spin angular momentum and (b) and total angular momentum when L = 0. The above rules are illustrated with examples.

For d4 configuration:


Hence, L = 3 -1 = 2 i.e., D; S = 2; 2S+1 = 5; and J = L- S = 0; Term symbol = 5D0

For d9 configuration:


Hence, L = +2+1+0-1 = 2 i.e., D ; S = 1 /2 ; 2S+1 = 2 ; and J = L+ S = 3/2 ; Term symbol = 2D5/2

Spin multiplicity indicates the number of orientations in the external field. If the spin multiplicity is three, there will be three orientations in the magnetic field.- parallel, perpendicular and opposed. There are similar orientations in the angular momentum in an external field.

The spectroscopic term symbols for dn configurations are given in the Table-1. The terms are read as follows: The left-hand superscript of the term symbol is read as singlet, doublet, triplet, quartet, quintet, sextet, septet, octet, etc., for spin multiplicity values of 1, 2, 3, 4, 5, 6, 7, 8, etc., respectively.1S0 (singlet S nought); 2S1/2 (doublet S one–half); 3P2 (triplet P two ); 5I8 (quintet I eight). It is seen from the Table-1 that dn and d10-n have same term symbols, if we ignore J values. Here n stands for the number of electrons in dn configuration.


**Table 1.** Term symbols

It is also found that empty sub -shell configurations such as p0, d0, f0, etc., and full filled subshell configurations such as p6, d10, f14, etc., have always the term symbol 1S0 since the resultant spin and angular momenta are equal to zero. All the inert gases have term symbols for their ground state 1S0 .Similarly all alkali metals reduce to one electron problems since closed shell core contributes nothing to L , S and J; their ground state term symbol is given by 2S1/2. Hence d electrons are only of importance in deciding term symbols of transition metals.

#### **5. Total degeneracy**

We have seen that the degeneracy with regard to spin is its multiplicity which is given by (2S+1). The total spin multiplicity is denoted by Ms running from S to -S. Similarly orbital degeneracy, ML, is given by (2L+1) running from L to -L. For example, L= 2 for D state and so the orbital degeneracy is (2x2+1) =5 fold. Similarly, for F state, the orbital degeneracy is seven fold. Since there are (2L+1) values of ML, and (2S+1) values of Ms in each term, the total degeneracy of the term is given by: 2(L+1)(2S+1).

Each value of ML occurs (2S+1) times and each value of Ms occurs (2L+1) times in the term. For 3F state, the total degeneracy is 3x7 =21 fold and for the terms 3P, 1G, 1D, 1S, the total degeneracy is 9,9,5,1 fold respectively. Each fold of degeneracy represents one microstate.
