5.03, Inorganic Chemistry Prof. Daniel G. Nocera Lecture 25 Apr 11: Angular Overlap Method and Electronic Structure of Metal Complexes

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1 5.0, Inorganic Chmistry Prof. Danil G. Nocra Lctur 5 Apr 11: Angular Ovrlap Mthod and Elctronic Structur of Mtal Complxs Th intraction nrgy rsulting from th intraction of th mtal and ligands is dfind on th following nrgy lvl diagram, whr E M is th nrgy of th mtal orbital, E L is th nrgy of th ligand orbital, E E M E L, S is th nrgy of ovrlap (i.., th ovrlap intgral) btwn th ligand and mtal orbital and th intraction nrgis ar, ε σ M E S ΔE ε σ * L E S ΔE Th nrgis ε σ and ε σ* ar th molcular orbital (MO) nrgis th nrgis arising from th mtal and ligand orbitals. Not that E M, E L and E in th abov xprssions ar constants, hnc, th nrgy of intraction btwn mtal and ligand scals dirctly with th ovrlap intgral, S M E S ε σ β S ΔE EL S β S * ΔE ε σ whr β and β ar constants. Thus by dtrmining th ovrlap intgral, S, th nrgis of th MOs may b ascrtaind rlativ to th mtal and ligand atomic orbitals. 1

2 Th Angular Ovrlap Mthod (AOM), provids a masur of S and hnc MO nrgy lvls. In AOM, th ovrlap intgral is also factord into a radial and angular product, S S(r) F(θ,φ) Analyzing S(r) as a function of th M L intrnuclar distanc, Undr th condition of a fixd M L distanc, S(r) is invariant, and thrfor th ovrlap intgral, S, will dpnd only on th angular dpndnc, i.., on F(θ,φ). Bcaus th σ orbital is symmtric, th angular dpndnc, F(θ,φ), of th ovrlap intgral mirrors th angular dpndnc of th cntral orbital. p-orbital is dfind angularly by a cos θ function. Hnc, th angular dpndnc of a σ orbital as it angularly rotats about a p-orbital rflcts th cos θ angular dpndnc of th p-orbital. F( )

3 Similarly, th othr orbitals tak on th angular dpndnc of th cntral mtal orbital. Hnc, for a d yz -orbital d z -orbital Diatomic Complxs To bgin, lt s dtrmin th nrgy of th d-orbitals for a M L diatomic dfind by th following coordinat systm, Thr ar thr typs of ovrlap intractions basd on σ, π and δ ligand orbital symmtris. For a σ orbital, th intraction is dfind as, ( σ ) β Fσ ( θ, φ ) β 1 E d S z

4 Th nrgy for maximum ovrlap, at θ 0 (s abov) is st qual to 1. This nrgy is dfind as. Th mtal orbital bars th antibonding intraction, hnc d z is dstablizd by (th corrsponding L orbital is stabilizd by (β ) 1 ). For orbitals of π and δ symmtry, th sam holds maximum ovrlap is st qual to 1, and th nrgis ar π and δ, rspctivly. E ( d ) E ( d ) S ( π ) π E ( d ) E d S ( δ ) δ yz xz xy x y As with th σ intraction, th (M Lπ)* intraction for th d-orbitals is dstabilizing and th mtal-basd orbital is dstablizd by π, whras th Lπ ligands ar stabilizd by π. Th sam cas occurs for a ligand possssing a δ orbital, with th only diffrnc bing an nrgy of stabilization of δ for th Lδ orbital and th nrgy of d-stabilization of δ for th δ mtal-basd orbitals. S (δ) is small compard to S (π) or S (σ). Morovr, thr ar fw ligands with δ orbital symmtry (if thy xist, th δ symmtry ariss from th pπ-systms of organic ligands). For ths rasons, th S (δ) ovrlap intgral and associatd nrgy is not includd in most AOM tratmnts. Rturning to th problm at hand, th ovrall nrgy lvl diagrams for a M L diatomic molcul for th thr ligand classs ar:

5 6 Octahdral Complxs Of cours, thr is mor than on ligand in a typical coordination compound. Th powr of AOM is that th and π (and δ), nrgis ar additiv. Thus, th MO nrgy lvls of coordination compounds ar dtrmind by simply summing and π for ach M(d) L intraction. For th ligands in an octahdral complx, Th nrgy for ach o fth ligands is collctd in th following tabl. Th total nrgy for a ligand is obtaind by summing th total nrgy of ach d-orbital, 5

6 L1 L L L L5 L6 E TOTAL E(d z ) + δ E(d yz ) π δ π δ π π π E(d xz ) π π δ π δ π π E(d xy ) δ π π π π δ π E(d x y) δ σ δ δ + δ Rmmbr, δ << or π thus δ may b ignord. Th O h nrgy lvl diagram is: Not th d-orbital splitting is th sam rsult obtaind from th crystal fild thory (CFT) modl taught in frshman chmistry. In fact th nrgy paramtrization scals dirctly btwn CFT and AOM, whr Δ o is th octahdral crystal fild splitting, Δ o π 6

7 Th AOM angular scaling factors (S ) for ligands in th various positions shown by labls L1-L1 is summarizd in th tabl blow. L z x y xz yz xy σ π σ 1/ / π σ 1/ / π σ 1/ / π σ 1/ / π σ π σ 1/ / /16 π 0 / 1/ / 1/ σ 1/ / /16 π 0 / 1/ / 1/ σ 0 0 1/ 1/ 1/ π / / / / / σ 0 0 1/ 1/ 1/ π / / / / / σ 0 0 1/ 1/ 1/ π / / / / / σ 0 0 1/ 1/ 1/ π / / / / / 7

8 You can choos any combination of ligands to construct th ligand filds of common mtal complxs. Structur Ligands Linar 1 and 6 Trigonal planar,7 and 8 Squar planar -5 Ttrahdron -1 Trigonal bipyramid 1,,7,8 and 6 Squar pyramid 1-5 Octahdron 1-6 Lt s us th and π paramtrs to dtrmin th crystal fild splitting nrgis for a ttrahdral molcul. 8

9 Calculating th ttrahdral crystal fild splitting, π σ π π σ t + Δ But raliz that Δ t is / that of Δ o, ( ) 0 t Δ Δ π σ This is th rsult from and 5.11, 0 t Δ Δ i.. that th crystal fild splitting for a ttrahdral complx is ( /) that of an octahdral complx. Not that AOM dos not giv th sign but w not that th and t lvls of th ttrahdral ligand fild ar invrtd from that of th octahdral crystal fild, hnc th ( ) sign.