Matrix Elements: The matrix elements are constructed using an uncoupled basis comprising of spin elements (the magnetic spin quantum number, M_s, value) for each magnetic centre. If a magnetic centre has N unpaired electron, then the magnetic spin quantum number values for that particular magnetic centre range from N/2 to –N/2; the consecutive values differing from each other by unity (total (N+1) values).

Hamiltonian: Only the two major contributors – the isotropic exchange term and the Zeeman term – to the Hamiltonian have been taken into consideration.

Isotropic Exchange term:[Boca1]

where J_AB is the coupling constant between the magnetic centres A and B, I and J represent the particular M_s terms in the basis elements that Ŝ_A.Ŝ_B operate on and

where Ŝ_A,+ and Ŝ_B,- are increment and decrement operators that are defined as follows:

Zeeman term:[Boca1]

Here B⃗ represents the applied magnetic field, g_A is the gyromagnetic tensor and μ_B is the unit Bohr Magneton.

where δ_AB is the kroenecker delta operator which is equal to 1 if A=B. Otherwise its value is 0. B_i and g_Ai represent the value of the magnetic field and the gyromagnetic tensor along a given direction. The effective Hamiltonian is thus given as

The Hamiltonian matrix can thus be represented as

where A_I and A_J represent the elements of the basis and N is the total number of elements in the basis.

The eigenvalues ε_i and hence the energy levels of the various magnetic states are obtained by the diagonalisation of the Hamiltonian matrix. Once the eigenvalues are obtained, it can be used to plot the magnetic susceptibility with respect to temperature. To calculate the magnetic susceptibility, the Van Vleck equation is used which is:[Kahn]

where ϵ_i⁽¹⁾ and ϵ_i⁽²⁾ are the first and second derivative of energy with respect to the magnetic induction, μ₀ is the permeability of free space, N_A is the Avogadro’s number, k is the Boltzmann constant and T is the temperature.

By Taylor expansion, the eigenvalues can be represented as:[Boca]

with

being the coefficients to be determined. To obtain these coefficients, 5 sets of eigenvalues, ε_i,m are generated using 5 different magnetic fields B_m , where

where N = 0 and 1, and δ has been chosen to be equal B₀/10.
Therefore,

for

In matrix form,

or

The matrix of coefficients can be calculated, provided the inverse of B exists, as follows:

By assembling the rows together, we obtain,

i.e.,

Thus all the coefficients can be determined by the following equation:

These coefficients can then be used to calculate the magnetic susceptibility using the Van Vleck equation which can now be given as: