Orbital, Spin, Total Angular Momentum

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Orbital, Spin, Total Angular Momentum ?Term
Symbol Pauli Exclusion Principal Multi-electron
Atom Spin Operations
F. Grieman
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• Pauli Exclusion Principle Chem 1 style
• In an atom, no 2 electrons can have the same 4
  quantum s
• Application
• He ground state L 0, as shown before e-s
  have same n, l, ml q. s
• ms(1) ½ so ms(2) must -½
• So, Ms ½ (-½) 0 only, so ST 0.
• ? 1s(1)1s(2)a(1)ß(2) (??) (not really
  correct)
• b) He excited state 1s(1) 2s(2)
• (1) n1, l0, ml0, ms ½ or ½
• (2) n2, l0, ml0, ms ½ or ½
• Possibilities
• ms(1) ms(2) Ms
• ½ ½ 1
• ½ -½ 0

Ms 0, 1 ? ST 1 gS 2S1 3
1 orbital configuration 4 spin states
Ms 0 ? ST 0 gS 2S1 1
Need to keep track of these states Term Symbol
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• Term Symbol
• Orbital Angular Momentum L 0, 1, 2, 3,
  4,
• 
  __ __ __ __ __
• Spin Angular Momentum - Degeneracy 2S1
  superscript
• Term Symbol _______
• He examples
• He ground state ? 1s(1)1s(2)a(1)ß(2) only one
  quantum state
• L 0, Ms 0 so S 0 ? 2S1 1 ?
  ____
• For ground state,
  use an X
• b) He excited state 1s(1) 2s(2) L 0
• ms(1) ms(2) Ms
• a a 1
• a ß 0
• ß ß -1
• ß a 0

Ms 0, 1 ? S 1 2S1 3 ? _____
Ms 0 ? S 0 2S1 1 ? _____
But theres more!
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_____ Angular Momentum J (from spin-orbit
interaction) 1) L S interact magnetically
(they couple) to give J J L S J2?
J(J1)? J ______________________ 2)
Hamiltonian term Hso ?(r) ____ (change in
energy) 3) Complete Term Symbol
_______ Examples 1) H atom ground state
1s(1)a(1) L0, S½ J__ ? ____
1s(1)____ 2) He
atom ground state L0 S0 J0 ?
_____ 3) He excited state 1s(1)2s(2) L0
S1 J1 ? _____
L0 S0 J0 ?
_____ 4) He higher excited state
1s(1)2p(2) a(1)a(2)
a(1)ß(2) L1 S1 J____ ? b
__________
ß(1)ß(2)
ß(1)a(2) L0 S0 J0 ? B ____
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Na (grd. state) Ne 3s a or ß
L0 S½ J½ ? _____
(excited state) Ne 3p a or ß
L1 S½ J3/2,1/2 ? A
_________(different energies)
LS __ L-S __ The
_______________ are finally solved!!! E(2P3/2) gt
E(2P1/2) _________ 2P3/2
_________ 2P1/2
16973.379 cm-1
?E 17.2
cm-1
16956.183 cm-1
______ X 2S½ Similar lines for H atom
2P3/2 ? X 2S½ 82,259.272 cm-1

2P1/2 ? X 2S½ 82,258.917 cm-1


-------------------------------
--


0.355 cm-1 (small!!!)
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Pauli Exclusion Principle (not chem 1 version) ?
__________ product function, space
spin coordinates Electrons probabilities not
exact trajectories they are
____________________ How must our wave functions
indicate this? Permutation operator _______
________________________ Now (i,j) ?? must
__________ 2 possibilities (i,j) ? ___
or (i,j) ? ___
____________ _____________
___________
____________
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Apply to He atom (first ground state X 1So)
(1,2) 1s(1) 1s(2) a(1) a(2)
__________________ (1,2) 1s(1) 1s(2) a(1)
ß(2) __________________ What to do??? Use
________________________ ? ___________________
_________________ (1,2) ?
____ (1,2) ?- ____ _________________
__ 1sa 1sß 1
1sa(1)1sß(2) -
1sß(1)1sa(2) 2
_____________________________
_____________
_________
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He excited states _________________
?space ___________ _______________________
_________ ? (2)½/2 _______________________
_ ?spin a(1)a(2) ß(1)ß(2)
ß(1)a(2) a(1)ß(2) ?total
?space ?spin Allowed Wavefunctions
______________________________
__________
__
(2)½/2 ____________________
__________
____________ ____________
(2)½/2 _________________ (2)½/2
_________________
__
sym sym sym anti anti sym anti anti
__
__
__
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Example of Spin Operations on He excited state
1s(1)2s(2) configuration ? (2)½/2
1s(1)2s(2) 2s(1)1s(2) ?(1So) ?__ (2)½/2
_____________ __ __ ______
________
__ ?(3S1) ?- (2)½/2 ________________
_______ ________
__ ______________ on 1So
Sz (2)½/2 ______________
____ (2)½/2 ?(1)?(2) - ?(1)?(2)
S2 (2)½/2 _______________
____ (2)½/2 ?(1)?(2) - ?(1)?(2)
S ___ Ms ___
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Spin Operations on 3S1
?(1)?(2)
___ ________________
Sz (2)½/2 ?(1)?(2) ?(1)?(2) ___
________________ ?(1)?(2)
___ ________________
?(1)?(2)
___ ?(1)?(2) S2
(2)½/2 ?(1)?(2) ?(1)?(2) ___
(2)½/2 ?(1)?(2) ?(1)?(2)
?(1)?(2) ___
?(1)?(2)
S __ S(S1) __ Ms _______
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Lets do one in detail Sz ?(1)?(2) _____
?(1)?(2) _____ ?(1)?(2) _______________
_________________ ?(2) (___) ?(1) ?(1)
(___) ?(2) (__________) ?(1)?(2)
________________
___________ ?(1)?(2)