2. arrangements of electrons in polyatomic atoms for an atom with several valence electrons, a number of arrangements of these electrons in orbitals of different l and m l are possible. –These arrangements are called microstates. Some of these microstates have the same energy (are degenerate) whereas others have different energy, presenting different energy states.

3. microstates each valid arrangement of electrons (specified by n, l, m l and m s for each electron) is called a microstate. Some have different energies and some the same how many can we have? n, number of electron sites; e, number of electrons; n-e, number of "holes"

4. consider a d 1 configuration the number of microstates is: –n = 10, e = 1 N = 10!/[1!(9!)] = 10

5. pigeonhole diagrams of the 10 d 1 microstates

6. the net orbital angular momentum L find the value of M L and M s for each microstate –M L = ∑m l –M s = ∑m s group by values of M L and M s arrange in a table

7. like this: one X for each microstate

8. These microstates correspond to one spectroscopic term the orbital an-gular momentum L is the same, the projection, M L is different Each microstate corresponds to some orientation of L = 2 and M s = ±1/2

9. L and S? L for a given set of microstates is the maximum M L –M L = ∑ m l = L, L -1, L -2,..., - L – (2 L + 1 values of M L ) S is the maximum M s – S = ∑ m s and the different values of M S –M S = S, S-1,...,0,..- S so for the d 1 case we could arrange the microstates by M L and M S to obtain these values for different terms.

10. 10 microstates

11. assigning terms from microstates have a groups of related microstates #microstates in a term = (2L +1)(2S+1) –L = 2, S = 1/2 (10 microstates) L = 012345 notationSPDFGH (then by alphabet, omitting J) Have a D term with spin multiplicity (2S+1) = 2 (doublet) 2 D term (pronounced doublet-dee)

12. a p 2 system is more complex number of valid (remembering pauli exclusion, etc.) microstates is 15 there are microstates which cannot be described by a single value of L and M s a systematic treatment of the microstates is given on the next slide the notation: m l =1, m s = 1/2 is represented by 1 +

13. EXAMINATION of permutations of m l and m s

14. summary of microstates

15. Today’s DJ question Write all the microstates for the neutral fluorine atom.

16. Group Work groups of 3: and write the microstates for a d 2 electronic configuration

17. energies and angular momentum there are several components to the energy of the atom, excluding the stable core electrons the n values of the valence electrons are pretty much the same the angular momentum of the orbitals added to the net electron spin lead to different energy levels called states.

18. Russell-Saunders (L-S) coupling in L-S coupling, the total angular momentum of the electronic configuration, J, is the sum of the orbital angular momentum, L (M L (max), and the spin, S (∑m s ). J = L + S

19. What are L and S? L for a given set of microstates is the maximum M L –M L = ∑ m l = L, L -1, L -2,..., - L – (2 L + 1 values of M L ) S is the maximum M s – S = ∑ m s and the different values of M S –M S = S, S-1,...,0,..- S so for the p 2 case we can arrange the microstates by M L and M S to obtain these values for different terms.

20. Microstates for p 2 arranged by M s and M L

21. See L = 2 and S = 0 term

22. L = 1, S = 1 term

23. L = 0, S = 0 term

24. assigning terms from microstates have several groups of related microstates #microstates = (2L +1)(2S+1) –L = 2, S = 0 (5 microstates) (D term) –L = 1, S = 1 (9 microstates) (P term) –L = 0, S = 0(1 microstate)(S term) L = 012345 notationSPDFGH (then by alphabet, omitting J)

25. indicating Spin Multiplicity; S the spin is shown by the numerical superscript value = 2S + 1 preceding the letter term symbol –L = 2, S = 0 (2S + 1 = 1) 1 D term –L = 1, S = 1 (2S +1 = 3) 3 P term –L = 0, S = 0 (2S + 1= 1) 1 S term

26. Ground State Term? lowest E term that has highest S: –here the 3 P or 3 F for two terms with same S, that with greater L will be the ground state

27. Develop the table for the d 2 configuration Make a matrix with rows for M L and M S. Put an X for each microstate in each box with corresponding M L and M S. It better be symmetrical!

28. Now… Find the largest value of M L and M S and assign the L and S values and the term symbol. Eliminate those microstates and repeat until all microstates are eliminated.