[xshadow]

Hi!!!

I don't understand the meaning of micostate,state and level (energetic level i think) for a molecular configuration.
This argument is very hard (for me) : (


For example, the configuration  np2 has these term symbols:  ¹S , ¹D,³P

In particular ³P "splits"(is correct this term) in:
³P₀
³P₁
³P₂

Where 3 is the Spin Multiplicity (3=triplet) and 0,1,2 the value of the total angular momentum J=L+S

1)BUT i don't understand  the meaning of the  term symbols of triplet...an atom that is a "triplet" means that it is in  three STATE AT THE SAME TIME ?? (each one with a different M_s value, -1,0,+1 )...but how can a system be in more state at the same time?!  OR it means that  an atom is in ONE of these states, for example in the particular state with M_s=+1 (but so is called triplet??)...

2)what is the difference between a state and a microstate??...An atom at a certain istant is an a STATE(or more) or in a MICROSTATE (or more)  Is very important for me to understand this point!

3) for example if an atom X has this term symbol associated: ³P₁
According to this,in which state is the atom??? and in which microstate/s


Please Help me!!!...
My textbook doesn't help me to understand these concepts
Thansk.

[Corribus]

Though not a completely rigorous explanation, you can imagine a state of a system being a way to uniquely describe the system in terms of its arrangement of particles, with an associated energy eigenvalue. However, insofar as there are multiple ways that particles can be arranged in the system that have indistinguishable observables, we call these different ways of organization "microstates". So, the particles that comprise a unique state of the system can be arranged in several basically equivalent ways.

For example, in the ground state of molecular oxygen, there are two half filled orbitals. There are two unpaired electrons, each with an associated spin of ± 1/2. The total spin magnitude is therefore 1, which allows values of 1, 0, or -1, corresponding to  ,  and  , respectively.* In the absence of a magnetic field, these three possible arrangements are degenerate and indisguishable from each other. You may think of it as being impossible for determine which of these three possible states the system is in at any time, if that is helpful. So we label these as microstates and call the overall state a "triplet" - there are three possible spin microstates. Bear in mind that if we introduce a magnetic field, these three configurations are no longer degenerate. In certain experiments we would see a single spectroscopic absorption line split into three separate lines in the presence of a magnetic field. This is called the Zeeman effect, and it is the origin of the term "triplet state".

*It is actually a little more complicated than this, because there is also an associated singlet state. So you actually have four total microstates, comprised of a singlet and a triplet, or four combinations:  ,  ,  ,  . It is customary to describe the two states with opposing spins as linear combinations of each other, as depicted here:

https://en.wikipedia.org/wiki/Triplet_state

The singlet state involves on microstate, and the triplet involves three of them.

3) for example if an atom X has this term symbol associated: ³P₁
According to this,in which state is the atom??? and in which microstate/s

The system is in a triplet state, meaning it is in a superposition of three possible degenerate eigenstates. It is not possible to experimentally distinguish which of these microstates the system is in at a given time, although the number of degenerate microstates does have statistical mechanical manifestations that can be observed experimentally. This is distinguished, say from a state labeled ¹P₁, which has a completely different energy eigenvalue, and is therefore experimentally distinguishable.

Microstates associated with orbital angular momenta (the "P") don't work any differently. The three p orbitals in an isolated atom are degenerate in an isotropic field. If such an atom had a single electron, you could not tell which of the three p orbitals the electron was located in at a given time. In some interpretations of quantum mechanics, it is in all three simultaneously. Only if the degeneracy is broken does this situation change. This is the strongest distinction of "microstate" versus "state". One you can experimentally observe, the other only has a statistical mechanical relevance. Term symbols are a convenient way to keep track of all the possible microstates, and how they are grouped together in experimentally observable states of the system.

[xshadow]

Though not a completely rigorous explanation, you can imagine a state of a system being a way to uniquely describe the system in terms of its arrangement of particles, with an associated energy eigenvalue. However, insofar as there are multiple ways that particles can be arranged in the system that have indistinguishable observables, we call these different ways of organization "microstates". So, the particles that comprise a unique state of the system can be arranged in several basically equivalent ways.

For example, in the ground state of molecular oxygen, there are two half filled orbitals. There are two unpaired electrons, each with an associated spin of ± 1/2. The total spin magnitude is therefore 1, which allows values of 1, 0, or -1, corresponding to  ,  and  , respectively.* In the absence of a magnetic field, these three possible arrangements are degenerate and indisguishable from each other. You may think of it as being impossible for determine which of these three possible states the system is in at any time, if that is helpful. So we label these as microstates and call the overall state a "triplet" - there are three possible spin microstates. Bear in mind that if we introduce a magnetic field, these three configurations are no longer degenerate. In certain experiments we would see a single spectroscopic absorption line split into three separate lines in the presence of a magnetic field. This is called the Zeeman effect, and it is the origin of the term "triplet state".

*It is actually a little more complicated than this, because there is also an associated singlet state. So you actually have four total microstates, comprised of a singlet and a triplet, or four combinations:  ,  ,  ,  . It is customary to describe the two states with opposing spins as linear combinations of each other, as depicted here:

https://en.wikipedia.org/wiki/Triplet_state

The singlet state involves on microstate, and the triplet involves three of them.

3) for example if an atom X has this term symbol associated: ³P₁
According to this,in which state is the atom??? and in which microstate/s

The system is in a triplet state, meaning it is in a superposition of three possible degenerate eigenstates. It is not possible to experimentally distinguish which of these microstates the system is in at a given time, although the number of degenerate microstates does have statistical mechanical manifestations that can be observed experimentally. This is distinguished, say from a state labeled ¹P₁, which has a completely different energy eigenvalue, and is therefore experimentally distinguishable.

Microstates associated with orbital angular momenta (the "P") don't work any differently. The three p orbitals in an isolated atom are degenerate in an isotropic field. If such an atom had a single electron, you could not tell which of the three p orbitals the electron was located in at a given time. In some interpretations of quantum mechanics, it is in all three simultaneously. Only if the degeneracy is broken does this situation change. This is the strongest distinction of "microstate" versus "state". One you can experimentally observe, the other only has a statistical mechanical relevance. Term symbols are a convenient way to keep track of all the possible microstates, and how they are grouped together in experimentally observable states of the system.

Thanks!!
With your explanation i understand much more

But i still have some doubts :
1) in 2p2 configuration i see these term symbols with total angular moment "L"=1=P :

³P₀
³P₁
³P₂

Before, you  said that a ³P is a triplet state and SO  it has THREE possible degenerate eigenstates...
1) so this 3 possible egenstates are three  microstates with the same energy??

But my textbook also say that  the degeneration of these 3 state (each one of these is  a triplet) is:

³P₀ >>> 1  (it means that there is only a microstate? )
³P₁>>> 3   (it means  that there are 3 microstates?)
³P₂ >>> 5   (it means  that there are 5 microstates?)

2)What does it mean this fact???
Every  triplet (ie three "objects" that differ for the "z" component of the total Spin angular momentum M_s(+1,0,-1)  state should have   three   possible degenerate eigenstates (i think) ....so why ,for example  the degeneration of ³P₂ is 5 !!!  A triplet state with L=1="P" should have only three different possibile value of M_s  so why  ³P₂  has 5 degenerations ?? 
This number  "5" what does it mean??



Thanks for your help !!  (and sorry for my bad english)

[Corribus]

Because there can be so many statistical permutation, it's common to break a state down into various types of microstates. All triplet states have 3 spin microstates. Then you have orbital angular moment microstates, designated in this case by the P - which also again implies that each of these states has three possible orbital angular momentum microstates. Finally, you have the j term (the subscript), which arises because the spin and angular momenta also interact with each other. ³P₂ has five total angular momentum microstates because if j = 2, then m_j can equal +2, +1, 0, -1, or -2, representing a situation where the spin and orbital angular momenta are aligned to various degrees of parallel or antiparallel configurations.

If you wanted to know the total number of microstates for spin and orbital microstates, you would take the product of the individual types of microstates. This is effectively the process you do in reverse when you determine a term symbol to begin with - write out the total number of possible electron configurations, place them in a table, and then group them together into various distinct states. Ignoring the total angular momentum term for a moment, for example, a ³P state has 9 microstates involved: 3 from spin times 3 from orbital angular momentum. Ms = +1, 0, -1 and Ml = +1, 0, -1, and you can have any of 9 combinations of the three. All of them together comprise the ³P state. The j term just adds to the complexity because in reality, the spin and orbital momentum values are not actually completely independent.

[xshadow]

Thanks Corribus!!

ONE LAST thing:

 When i do a spectroscopy UV-visible experiment on  gaseous atoms that have a TRIPLET state,for example  ³P , i should see THREE spectral bands  (very close).


 But each one of these spectral bands where are they from?? 

one line  from ³P₀
the second  from ³P₁
and the last from  ³P₂

OR

 ALL these three  lines come from a single Term Symbols of the state ³P
For example only from  ³P₁ ...
I don't think that this possibility is the correct one because we have said that all the microstate of a Term Symbols has the same egeinvalue (so the same energy)...but the three lines fall down to different frequencies ,so at  different energy values.

Which one of these two possibilities is the  correct one?
Thanks!
 

[Corribus]

Ok, well, what I wrote earlier was a bit of a simplification. In principle that would work, yes, but there are other rules that govern the allowedness of spectroscopic transitions. Also, the strength of the magnetic field (particularly compared to the spin-orbit coupling magnitude) also makes a big difference.

As a matter of fact, to a (very good) first approximation, singlet to triplet transitions are strictly forbidden, particularly in atomic transitions. There are conditions that relax this rule, but I wouldn't worry about them too much.

Nevertheless, atomic transitions do split in the presence of a magnetic field because not only degenerate spin but also orbital microstates lose their degeneracy under these conditions. Astronomers use this effect, for example, to determine the strength of magnetic fields surrounding emitting bodies (like stars) because the splitting magnitude correlates to the field strength.

You can find a little more about this effect here:

http://www.chemicalforums.com/index.php?action=post;topic=83124.0;last_msg=301758

Because the allowed (and not-allowed) spectra transitions of atoms can get very complicated, particularly when there are lots of states involved, a common way to depict them graphically is using a Grotrian diagram. The Wikipedia article is not very extensive (https://en.wikipedia.org/wiki/Grotrian_diagram) but most physical chemistry textbooks have a few of them for some common atoms/ions.

Regarding your other questions, if an electron configuration gives rise to multiple term symbols, then each of those term symbols represents a unique, discrete state, with an associated unique, discrete energy value. Assuming transitions are allowed by selection rules, a transition will be observed between each of these states regardless of the presence of an external magnetic field. But only a fraction of transitions are formally allowed. In the presence of an external field there are more transitions apparent, as I've said, because degeneracies are broken. However the (now non-degenerate) states are still usually grouped together into a single term symbol, for book-keeping purposes. E.g., the P_1/2 S_1/2 transition of hydrogen is a single line in the absence of a magnetic field, but splits into four lines in the presence of a magnetic field (each of those states splits into a doublet; see the Zeeman effect article I linked to above). Nevertheless, I believe the term symbol designations are still preserved. As a matter of convenience, we still refer to these as the "P_1/2" and "S_1/2" states, even though they might actually be said to involve multiple uniquely observable states.

[xshadow]

Thank you,Corribus!!