## Part 3

Part 1 (Introduction. Crystal Field splitting of the d-type Atomic Orbitals. Taylor's notation

Part 2 (Energy Plane. Energy Surfaces. g-Tensor Surfaces

### Discussion

#### Sign choice of g-values

The g-value is a measure of energy splitting of electronic states in the external magnetic field. Both in experiment  and in theory only the energy gap (the absolute value) can be determined. In other words, the g-values are defined only up to the sign. Below are the g-values vs. at V* = 0 (D4 symmetry):

The above | gz | plot is a cross-section of the  3D | gz | surface along the  - axis (V* = 0):

The gz dependence can be made smooth if we change the sign of one of its branches, for example like this:

This is Oosterhuis-Lang sign choice. The above picture is the cross-section along  V* = 0 of the three g-value surfaces in the previous section if the sign of gz is changed to the opposite.

 gz gy gx

For this sign choice in the pure octahedral field ( = 0, V* = 0) gx = gy = 2 and gz = -2.

There are two other popular choices.

Taylor's sign choice: the gx surface is taken with the sign opposite to the images in Part 2. At ( = 0, V* = 0) gx = gy = 2 and gx = -2.

McGarvey's sign choice: all g-value surfaces are taken with the signs opposite to the images in Part 2. For the pure octahedral field gx = gygz = -2. For this choice all three g-values tend to 2 (free electron value) when the upper level is well separated from the others.

The g-value signs can only be determined  with the use of additional information (either theoretical or experimental).

Note that the three-level model cannot be continuously reduced to the case of the free electron because it is already assumed that the crystal field is strong enough so that the eg doublet is shifted high upward.

#### Sign-independent properties

There are two sign-independent quantities: absolute value and zero.

Absolute value. For each g-tensor component 0 | g | 4. At the origin of the energy plane  | gx | =  | gy | =  | gz | = 2. The | g | levels divide the energy plane into three narrow (1, 2 and 3) and three wide (4, 5 and 6) regions:

| g |-value regions

The table below summarizes the qualitative properties of the regions. The asymptotic values along the six semi-axes are presented in the last column.

 Region MO position |g|-pattern Semi-axis asymptote 1 (narrow) dxy is the lowest |gz| > 2 > |gx|, |gy| |gz| = 4,  |gx|, |gy| = 0 2 (narrow) dyz is the lowest |gx| > 2 > |gy|, |gz| |gx| = 4,  |gy|, |gz| = 0 3 (narrow) dxz is the lowest |gy| > 2 > |gx|, |gz| |gy| = 4,  |gx|, |gz| = 0 4 (wide) dxz is the highest |gy| < 2 < |gx|, |gz| |gy|, |gx|, |gz| = 2 5 (wide) dxy is the highest |gz| < 2 < |gx|, |gy| |gz|, |gx|, |gy| = 2 6 (wide) dyz is the highest |gx| < 2 < |gy|, |gz| |gx|, |gy|, |gz| = 2

The mechanical angular momentum is the sum of the spin ( S ) and orbit ( L ) momentums: . If J = 0 then the system displays no rotation. However, it can still interact with the external magnetic field because in the Hamiltonian they are combined with different weights ( g 2 ).

It can also be that for non-zero S, L and J the combination ( L + ge S ) is equal to zero. This is the case for the third level:

 gx = 0 Jx = 0 gy = 0 Jy = 0 gz = 0 Jz = 0

The above results were obtained under the assumption that ge = 2 and the orbital reduction factor k = 1 (no reduction due to the presence of a covalent bond). For non-zero orbital reduction ( k < 1 ) g g (1 + 2 k )/3.

Part 4 (Calculations. Programs. References)

Part 5 (Experimental data processing)