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arrangements of electrons in polyatomic atoms

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.

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arrangements of electrons in polyatomic atoms

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  1. 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 mlare 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.

  2. microstates • each valid arrangement of electrons (specified by n, l, ml and ms 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"

  3. consider a d1 configuration • the number of microstates is: • n = 10, e = 1 • N = 10!/[1!(9!)] = 10

  4. pigeonhole diagrams of the 10 d1 microstates

  5. the net orbital angular momentum L • find the value of ML and Ms for each microstate • ML = ∑ml • Ms = ∑ms • group by values of ML and Ms • arrange in a table

  6. like this: one X for each microstate

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

  8. L and S? • L for a given set of microstates is the maximum ML • ML= ∑ ml = L, L -1, L -2, ..., - L • (2 L + 1 values of ML) • S is the maximum Ms • S = ∑ ms and the different values of MS • MS = S, S-1,...,0,..- S • so for the d1 case we could arrange the microstates by ML and MS to obtain these values for different terms.

  9. 10 microstates

  10. 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 = 0 1 2 3 4 5 • notation S P D F G H (then by alphabet, omitting J) • Have a D term with spin multiplicity (2S+1) = 2 (doublet) • 2D term (pronounced doublet-dee)

  11. a p2 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 Ms • a systematic treatment of the microstates is given on the next slide • the notation: ml =1, ms= 1/2 is represented by 1+

  12. EXAMINATION of permutations of ml and ms

  13. summary of microstates

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

  15. Group Work • groups of 3: and write the microstates for a d2 electronic configuration

  16. 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.

  17. 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 (ML (max), and the spin, S (∑ms). J = L + S

  18. What are L and S? • L for a given set of microstates is the maximum ML • ML= ∑ ml = L, L -1, L -2, ..., - L • (2 L + 1 values of ML) • S is the maximum Ms • S = ∑ ms and the different values of MS • MS = S, S-1,...,0,..- S • so for the p2 case we can arrange the microstates by ML and MS to obtain these values for different terms.

  19. Microstates for p2 arranged by Msand ML

  20. See L = 2 and S = 0 term

  21. L = 1, S = 1 term

  22. L = 0, S = 0 term

  23. 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 = 0 1 2 3 4 5 • notation S P D F G H (then by alphabet, omitting J)

  24. 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) 1D term • L = 1, S = 1 (2S +1 = 3) 3P term • L = 0, S = 0 (2S + 1= 1) 1S term

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

  26. Develop the table for the d2 configuration • Make a matrix with rows for MLand MS. • Put an X for each microstate in each box with corresponding MLand MS. • It better be symmetrical!

  27. Now… • Find the largest value of ML and MS and assign the L and S values and the term symbol. • Eliminate those microstates and repeat until all microstates are eliminated.

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