@ Behnam Farid, I can elaborate the question as calculated value of g factor from resonance experiments is always different from the theoretical value and can be less than or greater than the value for free electron but it is never described that how much difference can be a good consideration. Also when it differs than how spin orbit coupling affects the quantity.
“g” values are dependent on L,S, J and Spin Orbit Coupling* [SOC](λ complex ). SOC it is an important factor which can make the value of “g” to VARY FROM THEIR THEORETICAL VALUES.
First about THEORETICAL VALUE OF “g” .
Theoretically:
g =1+ [(J(J+1)+S(S+1) –L(L+1) ]/ [2J(J+1)] -------(I)
We can divide the discussion on the calculation of theoretical “g” values in three parts.
[A] In case of free electron ,L=0.
{ J values lie in between [J=L-S], [L-S+1]---------[L+S]( Modulas)]
So J=S=I/2.
Putting these values of J and S in the above relation;
g =2.0.
[B] In almost all the ORGANIC FREE RADICALS, L=0 .So”g” value should come out to almost equal to 2. But, it is ,generally, taken to be 2.0023; the 0.0023 being the relativistic correction (Discussion of this correction omitted intentionally).
[C]Now ESR of the paramagnetic complexes. Here the unpaired electron is spread over whole of the paramagnetic complexes( transition , lanthanides and actinide ions) with the central ion genally possessing non zero value of L. So “g” can be MORE as well as LESS THAN 2.In order to prove the point, I intentionally take a case where “g” is in fractions as:
Ce^3 ,with outermost configuration 4f^1, has maximum L value is =3 .With one unpaired electron, its max S=1/2. So its J= 3-1/2=5/2.
Put these L, S and J values in (I) to obtain g=6/7.
[D] SOC and “ g “ relation:
We know the the electron has two types of motions- the “spin motion” and the “orbital motion”, Both these motions cobtribute to the magnetism. Of course, the contribution from “spin motion” is much more than the “orbital motion”. The SOC(λ complex) may RESTRICT the “orbital motion” either COMPLETELY or PARTIALLY or even MAY NOT RESTRICT AT ALL. The terms used are COMPLETE QUENCHING, PARTIAL QUENCHING or NO QUENCHING** of the orbital motion respectively. This, in simple sense, means that the value of L may differ thom the theoretically calculated values of L.So the relation(I) does not hold good for the complexes.
Depending upon the ground term [A or E or T ] of the transtion metal ion , the following relations can be given to show as how the “g” and SOC are mutally related :
[1] When transition metal ion has “A” ground term:
g= 2.0(1-4 λ complex/ 10Dq) -----------(II)
[1I] When transition metal ion has “E” ground term:
g= 2.0(1-2 λ complex/ 10Dq) ------------(III)
[III] Different relations are reported for “T” terms if the terms differ in multiplicity.
[E] Lastly:
One very important point which needs to be taken into consideration is that:
λcomplex is positive when the “d” sub shell is less than half full( d^1-d^4) but is negative if the “d” sub shell is more than half full( d^6-d^9). Accordingly, the values of “g” will be less in complexes with metal ions with configuration-d^1-d^4.The reverse will be observed in the metal ion complexes with d^6-d^9 configurations.
*λcomplex have different values for different complexes of even the same metal ion.
** There are certain rules which need separate discussion.
I am also not sure, even after your second post, what answer exactly would help you the most.
If the question is what may yield a measured g factor in a crystal or a molecule deviate from that of the free electron (ge=2.0023...), then a simple (simplified) answer is the following.
There are two basic cases.
1. If the spin-orbit (LS) coupling is large enough (larger than crystal fields), and the ion of interest has both orbital and spin momenta in the ground state, one has to use Lande's formula to calculate the g factor, which may be very different from ge.
2. If crystal field is stronger than LS coupling, your ion is likely to be in a "quenched" orbital momentum state, so your g factor will be close to ge. However, LS coupling partly "de-quenches" the orbital moment, and orbital moment will give some contribution to the Zeeman splitting, i.e. your measured g factor will deviate somewhat from ge. A rough estimate is:
gmeasured=2*(1-lambda/Delta)
where lambda is LS coupling and Delta is the crystal field splitting. This topic is covered in C. P. Slichter: Principles of Magnetic Resonance (ISBN: 978-3-642-08069-2), Sect. 11, and in fact the above formula is a rough copy of its eq. (11.21). You can get more insight from this chapter, but note that it contains only a one electron treatment, and it is rather a toy model than a serious calculation. I would say that it is good mostly just to demonstrate how a g factor shift and anisotropy arise.
As Behnam already remarked, in any non-trivial (i.e. many-electron) system, a correct and precise theoretical treatment is extremely complicated and there exists no general solution to calculate g factors from/to LS coupling to date.
Theoretically, just like that has been mentioned above spin orbit coupling is a critical factor affect the g factor of a system, so it should be ask how the SOC affect the g factor. However, in practice, the SOC for a system, especially with the Rashba SOC , also can be expressed as a function containning variable g factor. Proximately, they are propotional.
In calculation from theory one makes certain suppostions about the various parameters such as S.O. coupling. However experimentally if one obtains a different g value then one coud attribute it to the S.O. coupling being quenched to some extent . One always uses alpha which is never = 1.