Colors of Co(III) solutions. Electronic-Vibrational Coupling. Vibronic Coupling

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1 Colors of Co(III) solutions Electronic-Vibrational Coupling Vibronic Coupling Because they have g g character, the d-d transitions of complees of the transition metals are forbidden (LaPorte forbidden). Complees with noncentrosymmetric coordination geometries (e.g., tetrahedral) have more intense d-d spectra. Spectra in centrosymmetric (e.g., octahedral) complees acquire intensity via vibronic coupling. The total molecular wavefunction can usually be approimated as a product of electronic, vibrational, and rotational parts: H Ψ It is good first approimation to assume that electronic, vibrational, and rotational motion can be separated: H H elec. + H vib. + H rot. In this approimation, the wavefunction is a product: Ψ = ψ elec. ψ vib. ψ rot. ( H elec. ψ elec. ) + ( ) + ψ elec. ψ vib. ( H rot. ψ rot. ) H Ψ = ψ vib. ψ rot. +ψ elec. ψ rot. H vib. ψ vib. H Ψ = ( E elec. + E vib. + E rot. )Ψ However, the "separability" is not eact. More accurately, H H elec. + H vib. + H rot. + H elec-vib

2 log ε Including the effects of coupling demands a modified wavefunction. In the simplest approimation, ψ elec. and ψ vib. not separable. Consider the product, ψ elec. ψ vib., for eamining selection rules: ψ gnd ψ gnd elec. ˆdψ e e vib. elec. ψ vib. dτ =? ψ gnd vib. generally belongs to totally symmetric rep. (otherwise, "hot bands" are involved). consider, ψ gnd elec. ˆdψ e e elec. ψ vib. dτ =? Vibronic Coupling in [trans-co(en) Cl ] + Spectra: Fig in Cotton - (solution spectrum is nearly indistinguishable from [trans-co(nh 3 ) 4 Cl ] + ) Virtual D 4h symmetry, d 6, low-spin In approimate O h symmetry, ground state configuration is (t g ) 6, 1 state. Dipole allowed transitions? Lowest energy singlets? See Tanabe-Sugano diagram. O h to D 4h correlations? Vibrations of the [trans-cocl N 4 ] grouping Dichroism of [trans-cocl (en) ] D 4h E C 4 C (C 4 ) C C i S 4 σ h σ v σ d y, z A g R z B 1g y B g y E g (R ) (z, yz) u A u z B 1u B u E u (, y) y-polarization z-polarization 1,, 3, 4, Fig in Cotton

3 Qualitative MO Diagram z Configurations and States y L 6 L 4 (replace L 5 and L 6 with X 1 and X ) X 1 L 4 L 1 L M L 3 L 1 L M L 3 b 1g - y L 5 X a 1g z e g (z,yz) b g y E g B g E g A g X 1 and X lower in the spectrochemical series and are weaker σ donors. Antibonding z orbital pushed up less by antibonding interaction with the X ligands. e g a 1g = b g a 1g = e g b 1g = b g b 1g = E g B g E g A g A g Tanabe-Sugano Diagram & State Symmetry Correlation In O h symmetry, two spin-allowed transitions are 1 to 1 T 1g and 1 T g. When symmetry is lowered to D 4h, the ecited states are split as: 1 T 1g : 1 A g + 1 E g 1 T g : 1 B g + 1 E g Splitting of T 1g in O h on lowering symmetry to D 4h O E 8C 3 3C (= C 4 ) 6C 4 6C y + z A E 1 (z y, y ) T (R, R z ); (, y, z) T (y,z, yz) D 4 E C 4 C (C 4 ) C C y, z A z, R z B y B y E (, y), (R ) (z, yz) T 1 (O h )

4 Electronic Symmetries ψ gnd e elec. zψ elec. ψ gnd e elec. (,y)ψ elec. Electronic Transitions B g E g D 4h E C 4 C (C 4 ) C C i S 4 σ h σ v σ d y, z A g R z B 1g y B g y E g (R ) (z, yz) u A u z B 1u B u E u (, y) Γ all Γ t+r 6 Γ vib Vibrations of the trans-[cocl N 4 ] group _ On p. 93 of Cotton s tet, he gives, B 1g, B g, E g, A u, B 1u, 3E u Allowedness w/ Vibronic Coupling Electronic Polarization Transition z (,y) A g forbidden allowed This should be B g forbidden allowed, B 1g, B g, E g, A u, B u, 3E u E g allowed allowed Use info. with Qualitative Energy Diagram to assign spectrum

5 Graphical Summary Because of the O h D 4h symmetry correlations, the specific configurations shown correspond to only the states shown - even in O h. The dashed transitions are dipole and vibronically forbidden in z-polarization. The,y-polarized transition at ~3, cm 1 is difficult to assign. The 1 B g state should be relatively favored by the weaker ligand field of the Cl ligands, but there is less e -e repulsion in the 1 A g state. y -y y z O h 1 T g 1 T 1g 1 O h : E( 1 T g ) E( 1 T 1g ) = 16B 17 cm 1 D 4h 1 E g 1 B g 1 A g 1 E g 1 -y z -y z y y Graphical Tools for getting relative Energies of States Energies of each configuration are given by counting the orbital energies, adding up the repulsions (J ij ) and subtracting the echange stabilizations (K ij ) between like spins. A Graphical Scheme for getting relative Energies of States A Graphical Scheme for getting relative Energies of States E gr = ε a + ε b + J a,a + J b,b + 4J a,b K a,b ( ) K b,c ( ) ( ) E (3) e = ε a + ε b + ε c + J a,a + J a,b + J a,c + J b,c K a,b + K a,c E e A = ε a + ε b + ε c + J a,a + J a,b + J a,c + J b,c K a,b + K a,c E e B = ε a + ε b + ε c + J a,a + J a,b + J a,c + J b,c K a,b + K a,c E e A+ B + E e A B = E e A + E e B E (1) e = E A e + E B (3) e E e A+ ; but E B e = E (3) A e and E B (1) e = E e ( ) + K b,c E (1) e = ε a + ε b + ε c + J a,a + J a,b + J a,c + J b,c K a,b + K a,c E (1) e E (3) e = +K b,c

6 J, Comple d-orbital J and K s A + 4B + 3C J,,, A + 4B + C J,1, 1, 1,1 A B + C J,, J 1,1 1, 1 1, 1 J 1, 1, K 1, 1 K, K,1 = K, 1 K, 1 = K,1 K, = K, K 1, = K 1, B cm 1 A 4B + C A + B + C A + B + C 6B + C C 6B + C C 4B + C B + C C 5 55 cm 1 The Coulomb (J ij ) and echange (K ij ) integrals shown here can often be used to calculate state energy differences. Slater-Condon and Racah Parameters The "Slater-Condon parameters" are defined by k r 1 F k e k r r 1 r < k +1 r R (r ) R nl 1 nl (r ) k k +1 dr > dr ; r < 1 k +1 r = r if r > r 1 k > r k +1 r if r > r 1 1 and (in the d-shell): F F, F = F 49, F = F The "Racah Parameters" are related to the Slater-Condon parameters by A = F 49F 4 B = F 5F 4 C = 35F 4 For 1 st row transition metals, Racah parameters B and C have typical ranges shown. (State energy differences don t involve A.) Eample: B parameter for V 3+ 3 F state: 3 F ; 3 1 = E( 3 F) = h d + J,1 K,1 = h d + (A B + C) (6B + C) = h d + A 8B 3 P state: 3 P ; 1 1 = 3 / / E(1 + + ) = h d + J 1, K 1, = h d + (A + B + C) (B + C) = h d + A + B E( ) = h d + J, 1 K, 1 = h d + (A B + C) C = h d + A B E( 3 P) = E(1 + + ) + E( ) E( 3 F) = h d + A + 7B E( 3 P) E( 3 F) = 15B = 1,94 cm 1 B = cm 1 Energies of d and d 3 Terms N Term Symbol Slater Condon Epression Racah Epression 3 F F 8F 9F 4 A 8B 3 P F + 7F 84F 4 A + 7B d 1 G F + 4F + F 4 A + 4B + C 1 D F 3F + 36F 4 A 3B + C 1 S F + 14F 16F 4 A + 14B + 7C 4 F 3F 15F 7F 4 3A 15B 4 P 3F 147F 4 3 A H 3F 6F 1F 4 3A 6B + 3C d 3 P 3F 6F 1F 4 3A 6B + 3C G 3F 11F + 13F 4 3A 11B + 3C F 3F + 9F 87F 4 3A + 9B + 3C D = 3, + = 1, 3F + 5F + 3F 4 ± (193F 165F F F 4 ) 1/ 3A 3B + 5C ± (193B + 8BC + 4C ) 1/

7 Tanabe-Sugano E s and the Scheme Worked out details: singlet states eample for the high-field limit for the d 6 case. 1 : 6ε tg + 3J y,y + 1J y,z 6K y,z 1 T 1g : 5ε tg + ε eg + J y,y + J y,y + 1J y,z 6K y,z + K y,y 1 T g : 5ε tg + ε eg + J y,y + J y,z + 8J y,z + 4J z,z 4K y,z K z,z + K y,z E( 1 T g ) E( 1 T 1g ) = ( J y,z J y,y ) + 4 J J z,z y,z ( ) ( + K K y,z z,z ) ( + K K y,z y,y ) = 8B + 4(4B) + (3B B) + (4B ) = 16B Note: The echange contributions are positive because these are the singlet states - see the diagram - back four slides! Some Coulomb and Echange Integrals Coulomb and Echange Integrals Racah Parameters J y,y z,z yz, yz z,z y, y A + 4B + 3C J z, yz y, yz y,z y, yz y,z A B + C J y,z y,z A 4B + C J yz,z z,z A + B + C J y,y K y, yz = K z, yz = K y,z = K y, yz = K y,z A + 4B + C 3B + C K y,z = K y,z 4B + C K yz,z = K z,z B + C K y,y ϕ z (1)ϕ y ()( 1 r 1 )ϕ yz (1)ϕ y ()dτ 1 dτ 3B B cm 1 C C 5 55 cm 1 The Coulomb (J ij ) and echange (K ij ) integrals shown here can often be used to calculate state energy differences. For 1 st row transition metals, Racah parameters B and C have typical ranges shown. (State energy differences don t involve A.) 849 Cr III d-d spectra [Cr(en) 3 ] [Cr(H O) 6 ] λ (nm) Jahn-Teller Theorem Nonlinear Molecules in orbitally degenerate states are inherently unstable with respect to distortion. We can write the ground state electronic energy, E (Q), as a series epansion in each normal coordinate, Q: E (Q) = E + E 1 (Q)Q + E (Q)Q +... Q - belongs to non-totally symmetric rep.

8 Comments Regarding Stable Structures Movement of the nuclei along a non-totally symmetric normal coordinate represents an instantaneous distortion of the molecule. Normal coordinates are defined in terms of the molecular geometry as it eists in at least a local energy minimum as far as nuclear positions are concerned. At a local minimum, E 1 () = E Q = For stable molecules. What are the conditions under which the first term of the power series epansion is nonzero?in other words, when is the molecule is not stable with respect to distortion? Symmetry Constraints on Structural Stability Ground state energy, E, is given by: E = Ψ H Ψ dτ E Consider Q, where Q does not belong to the totally symmetric representation, Γ A1 : E Q E Q Ψ = Q H Ψ dτ + Ψ H Ψ Q dτ H + Ψ Q Ψ dτ The first two terms vanish, by symmetry. H = Ψ Q Ψ dτ ~ Γ Γ H Q Γ H belongs to totally symmetric representation; H Q belongs to Γ Q E Q + will be nonzero iff Γ Γ contains ΓQ + If the ground state is nondegenerate, Γ Γ is the totally symmetric representation. E Q = if ground state is nondegenerate, (integrand will not belong to Γ A1 ) Summary: J-T active modes If the ground state, Ψ, is orbitally degenerate and transforms as Γ, then then the molecule will distort to remove the degeneracy. Possible symmetries of the distortion are found by taking the symmetric direct product: [Γ Γ ] + (Totally symmetric modes are ignored.)

9 Eamples The best known J-T unstable molecules are Cu II d 9 octahedral cases. What is the epected distortion coordinate? Tetrahedral Cu II complees are common. Are they really tetrahedral? Are octahedral Ni II complees J-T unstable, (i) in their ground states?, (ii) in ecited states of the ground configuration? Is cyclobutadiene J-T unstable, (i) in its ground state?, (ii) in ecited states of the ground configuration? What is the epected structure of the CrN 3 6 ion (in Ca 3 (CrN 3 ))? Would an octahedral V III comple be epected to ehibit a J-T distortion? O E 8C 3 3C (= C 4 ) 6C 4 6C y + z A E 1 (z y, y ) T (R, R z ); (, y, z) T (y,z, yz) D 4 E C 4 C (C 4 ) C C y, z A z, R z B y B y E (, y), (R ) (z, yz) [E E] = A [E E] = B 1 B Tetrahedral Group: Td D3h T d E 8C 3 3C 6S 4 6σ d y + z A E 1 (z y, y ) T (R, R z ) T (, y, z) (y,z, yz) D 3h E C 3 3C σ h S 3 3σ v A y,z A R z E 1 1 (, y) ( y,y) A z E 1 1 (R ) (z, yz)

10 Eamples - Some Answers Cu II d 9 octahedral cases: [E g E g ] + = E g Cu II d 9 tetrahedral cases: [T T ] + = E T CrN 3 6 ion: [E E ] + = E V III comple: Ground State is [T g T g ] = 3 T 1g [T 1g T 1g ] + = E g T g