Idea Transcript
11.2 Quantum Numbers of Multielectron Atoms ▪ Table11.2:Microstate table for p2 (according to their ML & MS)
▪ Example on p.413) microstate for an s1p1 s: ml = 0, 0 ms = ±1/2 p: ml = +1, 0, -1, ms = ±1/2
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11.2 Quantum Numbers of Multielectron Atoms ▪ the quantum # that describe states of multielectron atoms L = total orbital angular momentum quantum # S = total spin angular momentum quantum # J = total angular momentum quantum # determine by vector sums of the individual quantum # L, S : describe collections of microstates : largest possible values of ML & MS related to ml
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related to ms
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11.2 Quantum Numbers of Multielectron Atoms ▪ ml: describes the component of the quantum # l in the direction of a magnetic field for an eML: describes the component of L in the direction of a magnetic field for an atomic state ms: describes the components of an e- spin in a reference direction MS: describes the components of S in a reference direction for an atomic state L → atomic states S, P, D, F L = 0 S state L = 1 P state L = 2 D state L = 3 F state S → spin multiplicity (2S + 1), left superscript ex)
1: singlet 2: doublet 3: triplet
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11.2 Quantum Numbers of Multielectron Atoms ▪ atomic states characterized by S & L called free-ion term ( = Russell-Saunders terms) labels: terms symbols
describe individual atoms & ions, free of ligands
(2S+1)
▪ term symbol: ex) 3D (2S+1)
5F
L=3
related to value of L =2
- free-ion terms: important for the spectra of coordination compound - how to determine L, ML, S, MS for a given term how to prepare microstate tables from them Inorganic Chemistry 2
2011 Fall
Using the next example!! T.-S.You
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11.2 Quantum Numbers of Multielectron Atoms ▪ Example on p.415) 1) 1S (singlet S) L = 0, ML = 0 2S + 1 = 1, S = 0, MS = 0 only one state microstate 2) 2P (doublet P) L = 1, ML = +1, 0 -1 2S + 1 = 2, S = 1/2, MS = +1/2, -1/2 six micro states (3 rows X 2 columns) if one e- case,, * spin multiplicity = # column in the microstate table
Inorganic Chemistry 2
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11.2 Quantum Numbers of Multielectron Atoms ▪ Reducing the p2 microstate table to its constituent atomic states (terms) designate each microstate → x need to know # microstates but not necessary to write out microstate but, need to find the rectangular arrays
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11.2 Quantum Numbers of Multielectron Atoms ▪ Table 11.4: for each term,, spin multiplicity = # column of microstates ex)
1D
(singlet term) → single column
3P
(triplet term) → three column
p2 e- configuration → three free-ion terms 3P, 1D, 1S
w/ diff. E
three states w/ diff. degree of e-—e- interactions
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11.2 Quantum Numbers of Multielectron Atoms ▪ final step: determining which term has the lowest E : following Hund’s rules 1) highest spin multiplicity (= Hund’s rule of max. multiplicity) for p2 case: it is 3P
2) if same max. spin multiplicity,, highest value of L ex) 4P and 4F terms,, identical spin multiplicity but, 4F, L = 3 4P,
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L=1 2011 Fall
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11.2.1 Spin Spin--Orbit Coupling ▪ spin-orbit coupling: the spin & orbital angular momenta couple each other J = L + S, L + S - 1, L + S - 2,,,, IL - SI ▪ Example on p.418) possible J for C three term symbols: 1D, 1S, 3P J = 1 + 1, 1 + 1 -1, 1 + 1 -2 = 2, 1, 0 J=2+0=2
J=0+0=0
▪ spin-orbit coupling: split free-ion terms into states w/ diff. E ▪ total energy energy-level level diagram for C:
five E states for C atom Inorganic Chemistry 2
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11.2.1 Spin Spin--Orbit Coupling ▪ the lowest E state (spin-orbit coupling): predicted from Hund’s third rule 3) if subshell is less than half-filled → the lowest J if subshell of more than half-filled → the highest J ▪ spin-orbit coupling → significant effects on the electronic spectra of compd.
11.3 Electronic Spectra of Coordination Compounds ▪ Now we know how to determine the microstates & free-ion terms for electron configurations ex) d2 config. → five free-ion terms: 3F, 3P, 1G, 1D, 1S lowest E ▪ absorption spectra → involve d orbitals of M ∴ it is important to know the free-ion terms for the possible d config.
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11.3. Electronic Spectra of Coordination Compounds ▪ However, determining microstates & free-ion terms for config. of 3 or more e- is,,, tedious process reference Table 11.5
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11.3. Electronic Spectra of Coordination Compounds ▪ quick & simple way to determine the lowest–E term,, ex) d3 in Oh 1) sketch E levels, showing the d e2) spin multiplicity of lowest-E state
spin multiplicity = 3 + 1 = 4
= # unpaired e- + 1 3) max. possible value of ML (= ∑ml)
Max. possible value of ml for three e- as shown:
→ determine type of free-ion (S, P, D)
2+1+0=3
4) combine results of steps 2 & 3 to get
therefore, F term
the ground term
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4F
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11.3. Electronic Spectra of Coordination Compounds ▪ Example on p.420) d4 (low spin) 1)
2) spin multiplicity = 2 + 1 = 3
3) max. ML = 2 + 2 +1 + 0 = 5 → H term 4) ∴ 3H
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11.3.1 Selection Rules ▪ Relative intensities of absorption bands governed by a series of selection rules 1) Laporte selection rule: transition b/w states of the same parity (center of inversion ) are forbidden ex)) g → g transition: ii b/ b/w d orbitals bi l → forbidden f bidd (∵ d orbitals: symmetric to inversion) : transition b/w d & p are allowed ex) g → u transition: p orbitals are antisymmetric 2) spin selection rule: transition b/w states of diff. spin multiplicities are forbidden. ex) 4A2 → 4T1: “spin-allowed” spin-allowed 4A 2
→ 2A2: “spin-forbidden”
▪ seems like most of the e- transition → ruled out by selection rules However,,, still many complexes are vividly colored!! ∵ these rules can be relaxed!! Inorganic Chemistry 2
2011 Fall
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11.3.1 Selection Rules ▪ The selection can be relaxed!! 1) bands are not rigid!! → vibrations changes the symm. ex) Oh → vibration coupling center of inversion is temporarily lost relax the 1st selection rule ∴ d-d transition (molar absorptivities of 5-50 L/mol·cm) responsible for the bright color of many of these complexes 2) tetrahedral complex: stronger absorption than octahedral complexes : σ-bonding sp3 & sd3 hybridization : mixing p-orbital w/ d-orbital character relax the 1st selection rule
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2011 Fall
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11.3.1 Selection Rules ▪ The selection can be relaxed!! 3) spin-orbit coupling → relax the 2nd rule transition b/w states w/ diff. spin multiplicity is possible!! p bands for 1st TM veryy weak absorption for 2nd & 3rd TM more important !!
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11.3.2 Correlation Diagram : to relate the electronic spectra of TM complexes to the ligand field splitting ∆o ▪ For d2 config. ([V(H2O)6]3+) → there are two extremes 1) free ion (no ligand field): five terms, 3F, 3P, 1G, 1D, 1S lowest : locate on the far left 2) strong ligand field: three possible config. for 2 d e- in octahedral ligand field
: extremely strong ligand field → far right
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11.3.2 Correlation Diagram ▪ In actual coordination compds.,, → intermediate b/w these extremes @ 0 field → five atomic states w/ diff. E: 3F, 3P, 1G, 1D, 1S @ very high ligand field strength → t2g2, t2geg, eg2 - the correlation diagram shows the full range of in-between cases free-ion terms: important : to be reduced to their constituent irreducible representations in the Oh
strong field limit config. (t2g2, t2geg, eg2): irreducible representation is possible !! irreducible representation for the two limit situation must match !! Inorganic Chemistry 2
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11.3.2 Correlation Diagram ▪ Characteristic of the correlation diagram 1) free-ion states (from LS coupling) → far left 2) extremely strong field → far right 3) irreducible rep. rep of both extremes → must match transition b/w same spin multiplicity Heavy lines in Fig. 11.3
more likely
transition b/w diff. spin multiplicity
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2011 Fall
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11.3.3 Tanabe Tanabe--Sugano Diagram ▪ Special correlation diagram useful in the interpretation of electronic spectra of coordination compound ▪ the lowest-E state: as a horizontal axis vertical distances → indicates a measure of E of the excited state this is the most useful characteristic easy to determine E/B !! ▪ ex) d2 config. lowest-E state: 3T1g state (from 3F free-ion term) line to 3T1g state (strong field t2g2) horizontal line excited states: w/ the same spin multiplicity as ground state 3T , 3T (p), 3A 2g 1g 2g
Fig.11.6 Inorganic Chemistry 2
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11.3.3 Tanabe Tanabe--Sugano Diagram ▪ quantities in a Tanabe-Sugano diagrams 1) horizontal axis: ∆o/B → ∆o: octahedral ligand field splitting B → Racah parameters indicating repulsion b/w terms of the same multiplicity ex) d2: b/w 3F & 3P → 15 B
2) vertical axis: E/B → E: E of excited states above the ground state
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11.3.3 Tanabe Tanabe--Sugano Diagram ▪ Example on p.425) [V(H2O)6]3+ (d2) - ground state: 3T1g(F) → three excited states: 3T2g, 3T1g(p), 3A2g three bands are expected !! but, actually only two bands are observed @ 17,800 & 25,700 cm-1 Fig.11.5
3rd band @ 38,000cm-1 → obscured in aqueous solution by charge-transfer bands nearby!! → observed in solid-state
Fig.11.6 Inorganic Chemistry 2
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11.3.3 Tanabe Tanabe--Sugano Diagram ▪ Other Electron Configurations - Fig. 11.7: d1 → d8 : for d4, d5, d6, d7 → have discontinuities (vertical lines) → low/high-spin both possible - ex) d4: 1) high-spin (weak–field) → 4 unpaired espin i multiplicity: l i li i 5 - S = 4(1/2) = 2 - spin multiplicity = 2S + 1 = 5 : 2) low-spin (strong-field) → 2 unpaired espin multiplicity: 3 - S = 2(1/2) =1 - spin multiplicity = 2S + 1 = 3 - in the Tanabe-Sugano diagram: @ weak-field → 5Eg ground state @ strong field → 3T1g ground state vertical line divides: weak-field & strong field case high-spin complexes
Fig.11.7
low-spin complexes
: changes ground state - 5Eg → 3T1g : changes spin multiplicity – 5 → 3 Inorganic Chemistry 2
2011 Fall
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11.3.3 Tanabe Tanabe--Sugano Diagram ▪ Fig.11.8: absorption spectra of 1st TM complexes of [M(H2O)6]n+ H2O: weak-field ligand ∴ [M(H2O)6]n+ → high-spin complexes represented by left side of the T-S diagram : vertical scale – molar absorptivities 2 - similar i il (1 (1-20 20 L/mol∙cm) L/ l ) except [Mn(H [M (H2O)6]2+
- pale-pink (weakly colored than others) Why ??
@ d5 T-S diagram: ground state of weak-field d5 → 6A1g no excited it d states t t w/ / 6 spin i multiplicity lti li it ∴ no spin-allowed absorption!!
Pale pink color is due to a very weak forbidden transition b/w states w/ diff. spin
Fig.11.8 Inorganic Chemistry 2
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