24. Atomic Spectra, Term Symbols and Hund s Rules

Transcription

1 Page of 4. Atomc Spectra, Term Symbols and Hund s Rules Date: 5 October 00 Suggested Readng: Chapters 8-8 to 8- of the text. Introducton Electron confguratons, at least n the forms used n general chemstry are ambguous. For example for carbon s s p, where to the two electrons n the p orbtals resde? In whch of the three p orbtals, p -, p 0, p +? What s the spn on the electrons. As we saw n the last lecture, all of the varous quantum numbers affect the energy of the state, not just the prncple quantum number, n mult-electron systems. We need a new way to desgnate a state unambguously. The scheme presented here s based on the total angular momentum, J, formed by addng the total orbtal angular momentum, L, and the total spn angular momentum, S. The symbol used to desgnate a state by ths means s called a term symbol and s of the form S + LJ (4-)

2 Introducton to Quantum Chemstry Lecture # 4 Page of The scheme s called Russell-Saunders couplng and s performed as follows: The total orbtal angular momentum s gven by L j l j (4-) a vector addton. The total spn angular momentum s gven by S j s j (4-3) another vector addton. The z components of L and S are gven by the scalar sums L z l z, m M L (4-4) and S z s z, m s, M S (4-5) Thus there are L + values of M L spannng -L, -L+, -L+,..., L-, L and S + values of M S for S spannng -S, -S+, -S+,..., S-, S.

3 Introducton to Quantum Chemstry Lecture # 4 Page 3 of The leadng S+ superscrpt n the term symbol s called the spn multplcty. Table 4-: Names for the leadng superscrpts of atomc term symbols. S+ Name snglet doublet 3 trplet The value of L s substtuted as n the hydrogen-lke orbtals but wth a captal letter. That s, Table 4-: Letter conversons for atomc term symbols. Value of L Letter 0 S P D 3 F 4 G 5 H The value of J s kept as a number. Thus for example: 3 S - trplet S one D 3 - snglet D three

4 Introducton to Quantum Chemstry Lecture # 4 Page 4 of Examples Hydrogen: H s, The maxmum value of M S s M Smax, m s, -- (4-6) to gve S /. The maxmum value of L s M Lmax, m 0 (4-7) thus L 0 and J L + S /. The term symbol for the ground state of hydrogen s S -- (4-8) The ground state of He, s : M Smax, m s, (4-9)

5 Introducton to Quantum Chemstry Lecture # 4 Page 5 of M Lmax, m (4-0) Therefore, the term symbol s S0 (4-) An excted state of He, s s : M Smax, m s, (4-) therefore M S, 0, - but what value of S? M Lmax, m (4-3) Mcrostates To answer our queston above, there are two methods. One nvolves nspecton of the problem and works quckly but s not relable, the second s

6 Introducton to Quantum Chemstry Lecture # 4 Page 6 of perhaps tedous, but always works. It s ths second methods that s gven n the text and wll be used here. Lets create a table M L M S 0-0 (0 +,0 + ) (0 +,0 - ), (0 -,0 + ) (0 -,0 - ) The notaton (0 +,0 + ) means that (m 0, m s +/; m 0, m s +/) for the two electrons. (0 +,0 - ) and (0 -,0 + ) are not equvalent because the two electrons are n separate orbtals. The four entres are called mcrostates. Lets loot at L 0, S, L 0 -> M L 0 S -> M S, 0, - Ths corresponds to the mcrostates (0 +,0 + ), (0 +,0 - ) and (0 -,0 - ). We have one unaccounted for mcrostate, (0 -,0 + ). Ths mcrostate has S 0, L 0. Thus there are two possble states for He s s : 3 S S0 (4-4) The allowed values of J are J L+S, L+S-, L+S-,..., L-S.

7 Introducton to Quantum Chemstry Lecture # 4 Page 7 of Beryllum s s : M Smax, m s, (4-5) therefore M S 0. M Lmax, m (4-6) therefore the term symbol s S 0 (4-7) Flled subshells have total angular momentum of zero and make no contrbuton to the determnaton of the term symbol. Back to carbon. We have sx possble spn orbtals n whch to place the two electrons p 0 α, p 0 β, p - α, p - β, p α, p β Ths s 6 choose

8 Introducton to Quantum Chemstry Lecture # 4 Page 8 of 6 6! !4! 65 ( ) (4-8) There wll be 5 mcrostates to deal wth. In general, f we have j spn orbtals and need to place k electrons we have j j! k k! ( j k)! (4-9) mcrostates. For example 3d has 0 0! !8! 45 (4-0) mcrostates. How about 3d 8? 0 0! !! 45 (4-) mcrostates. d 0-j has the same number of mcrostates as d j. Back to carbon: The two electrons may go n the m -, 0, orbtals. Thus M L ranges from to -. The spns may be pared or unpared, hence M S ->, 0, -.

9 Introducton to Quantum Chemstry Lecture # 4 Page 9 of M L M S 0 - ( +, - ) (0 +, + ) ( +,0 - ), (0 +, - ) (0 -, - ) 0 ( +,- + ) ( +,- - ), (- +, - ), (0 +,0 - ) ( -,- - ) - (0 +,- + ) (- +,0 - ), (0 +,- - ) (0 -,- - ) - (- +,- - ) Snce the electrons are n the same shell ( +, - ) and ( -, + ) are ndstngushable and hence only one s ncluded. Combnatons that are excluded by the Paul Prncple do not make up the table entres. to gve The largest value of M L s and has M S 0, therefore L, S 0, J D (4-) Ths state has M S 0, M L,, 0, -, - for fve mcrostates. The next largest value of M L remanng s M L. The largest value of M S assocated wth ths s M S. Therefor L, S 3 P (4-3)

10 Introducton to Quantum Chemstry Lecture # 4 Page 0 of wth M L, 0, - and M S, 0, - for nne mcrostates. The possble values of J are J (+), (+-), (-) or J,, 0 hence we have 3 P, 3 P and 3 P 0 wth 5, 3, and mcrostates respectvely. Ths leaves a sngle mcrostate remanng wth M L 0 and M S 0 to gve S0 (4-4) Thus the states for the ground state confguraton of carbon are D, 3 P, 3 P and 3 P 0, S 0 The degeneracy of each state s J+ for 5,, 3, 5, for 5 mcrostates. The term symbols for a gven electron confguraton correspond to that wth the same number of holes. For example p and p 4 or 3d 4 and 3d 6 (see Table 8.4 n the text). Hund s Rules Each of the states desgnated by a term symbol corresponds to a determnantal wave functon that s an egenfuncton of Lˆ and Ŝ. Thus each state corresponds to a gven energy. The spectroscopst Fredrch Hund worked out a set of emprcal rules to order the energes of the states.

11 Introducton to Quantum Chemstry Lecture # 4 Page of. The state wth the largest value of S s the most stable (has the lowest energy) and stablty decreases wth decreasng S.. For states wth the same value of S, the state wth the largest value of L s the most stable. 3. If the states have the same value of L and S, then for a subshell that s less than half flled, the state wth the smallest value of J s the most stable. For a subshell that s more than half flled, the state wth the largest value of J s the most stable. For example for carbon, 3 P 0 s the lowest energy state. [As an exercse work out the state for d usng the mcrostate method.] The term symbols are used to descrbe atomc spectra. Thus far n our development of the Hamltonan operator for atoms we have ncluded knetc energy and electrostatc potental energy terms. That s, Ĥ -- j j j Z r j ---- r j < j (4-5) We need to nclude spn and magnetc terms as well. The most mportant s the spn-orbt nteracton term whch results from the nteracton of the magnetc moment assocated wth the spn of an electron wth the magnetc feld generated by the electrc current produced by the electron s own orbtal moton. Other terms nclude spn-spn and orbt-orbt nteracton (n mult-electron systems) but spn-orbt nteractons are the most mportant.

12 Introducton to Quantum Chemstry Lecture # 4 Page of Ĥ -- j j j Z r j + ξ( r r j )l j s j j < j j (4-6) If the effect of the spn-orbt couplng s small, as t s for lght atoms (approxmately Z < 30) the effect may be treated as a perturbaton. For heaver atoms, the effect s not small and must be dealt wth explctly. Selecton Rules for Atomc Spectra Transtons nvolvng lght are allowed wth atoms for the followng L ± S 0 J 0, ± (4-7) expect that a transton from a state wth J 0 to another state wth J 0 s not allowed (forbdden). Suggested Readng for Next Lecture: Chapter 9 of the text.