Quantum Numbers

• Electronic state, S
  • e.g. X1Sigma+, b3Pi
  • For non-singlet states, this will also contain the omega quantum number, e.g. b3Pi_1, b3Pi_2, b3Pi_3
  • For Pi states, this will also contain the parity; e.g. +/- or e/f (ask your tutor)
• Vibrational quantum number, v
  • v=0,1,2,...
• Rotational quantum number, J, (strictly, this is total angular momentum)
  • J is either integer or half integer
  • Some sources may use N instead of J; talk to your supervisor about which quantum number you will be using

MARVEL Input Format (suggested)

1. Wavenumber (cm-1)
2. Uncertainty
3. S' (Upper electronic state)
4. v' (Upper vibrational quantum number)
5. J' (Upper rotational quantum number)
6. S" (Lower electronic state)
7. v'' (Lower vibrational quantum number
8. J" (Lower rotational quantum number)
9. ID (unique for each transition)

Interpreting Spectroscopic Data

For Rotational Levels:

• P branch means J' = J" -1
• Q branch means J' = J''
• R branch means J' = J''+1

Quantum Numbers and Division of Energy: More Details

Beginning

• At a basic level, the energy of a diatomic molecule can be divided into
  • Electronic energy (energy of the electrons)
  • Vibrational energy (energy of the nuclei)
  • Rotational energy (energy of the whole molecule rotating)
• Each type of energy has an associated quantum number
• Electronic quantum numbers are complicated - e.g. X2Sigma+
• Vibrational quantum numbers are n=0,1,2,...
• Rotational quantum numbers are usually J=0,1,2,3,.... or J=0.5,1.5,2.5,..
• Electronic quantum number has a big effect on the energy of the state, vibrational quantum number has a small effect on the energy of the state and rotational quantum number has a small effect on the energy of the state

Intermediate

• The electronic quantum number contains a number of components
• A superscripted number, e.g. 1, 2, 3, 4 --> this refers to the number of unpaired electrons + 1 -- i.e. its spin
  • 1 : singlet (no unpaired electrons)
  • 2 : doublet (1 unpaired electron)
  • 3 : triplet (2 unpaired electrons)
• A greek capital letter, e.g. Sigma+, Sigma-, Pi, Delta, Phi, Gamma, etc.
  • This is the diatomic extension of s, p, d, f, g atomic orbitals
  • This refers to the symmetry or shape of the orbital
• A letter at the beginning - this is usually either capital (X,A,B,C,...) or lower-case (a,b,c,d,...)
  • There are ideal rules for what you call different states, but there are significant deviations from these rules due to historical reasons (people didn't know which order the electronic states actually were in)
  • Ideally, the X state is the lowest energy state. The A, B, C, D, etc states are the 2nd, 3rd, 4th and 5th lowest electronic state of the same spin (e.g. triplet, singlet etc).
  • Ideally, the a,b,c,d, etc states are the 1st, 2nd, 3rd, 4th lowest energy states of the other spin (e.g. singlet if the ground state is triplet)

Advanced

• What we are colloquially calling the rotational quantum number J is actually the total angular momentum of the system. J is actually the rotational angular momentum (N) plus the angular momentum associated with the electron spin (Sigma) and the electronic orbital angular momentum (Lambda)
  • e.g. a doublet has a spin 1/2 and spin -1/2 component; the spin 1/2 component will increase the total angular momentum by 1/2, while the spin -1/2 component will decrease the total angular momentum by 1/2.
  • Sigma have Lambda = 0, Pi have Lambda =1, Delta have Lambda=2 etc
  • For doublet, quartet states, there is non-integer J
  • For singlet, triplets etc states, there is integer J
• The electronic ground state of a molecule is normally Singlet Sigma; in this case, there is no additional angular momentum to worry about.
• However, for non-singlet Sigma states, we often have to worry about a few other quantum numbers:
  • Pi, Delta, Phi etc state (i.e. Lambda > 0) have two component states/levels which have different parity--> think of this like px and py orbitals. Usually, the energies of these two components are the same, but sometimes (particularly for Pi states and/or for very high resolution calculations) they are split. If they are split, we need to use another quantum number. Usually, we use e/f (rotational-less parity) or +/- (total parity). In terms of molecular constants, the effect of splitting due to parity is normally included in the lambda-doubling constants, o+p+q and p+2q.
• For non-singlet states, the different spin components often have very different energy levels, particularly for non-Sigma states. This can be treated in two ways
• 1. Produce a unified set of molecular constants associated with the electronic state, then add a spin-orbit coupling constant (usually A)
• 2. Produce a set of molecule constant for each spin-electronic state
• We will use whatever is in the original document at this stage (feel free to put both representations if it is in the original paper).