Angular Overlap Method (AOM) for MLn Ligand Fields in Inorganic Chemistry - Lecture 14, Assignments of Chemistry

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5.04 Principles of Inorganic Chemistry II ��

Fall 2008

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5.04, Principles of Inorganic Chemistry II Prof. Daniel G. Nocera Lecture 14: Angular Overlap Method (AOM) for ML for ML (^) n Ligand Fields

The Wolfsberg-Hemholtz approximation (Lecture 10) provided the LCAO-MO energy between metal and ligand to be,

EM^2 SML^2 EL^2 SML^2

ΔEML^ *^ ΔEML

Note that EM , E (^) L and ΔE (^) ML in the above expressions are constants. Hence, the MO within the Wolfsberg-Hemholtz framework scales directly with the overlap integral, SML

EM^2 SML^2

=β ′^ S 2 =

EL^2 SML^2

εσ =β S 2

ΔEML^ ML

ΔEML^ ML

where β and β´are constants. Thus by determining the overlap integral, SML, the

energies of the MOs may be ascertained relative to the metal and ligand atomic orbitals.

The Angular Overlap Method (AOM), provides a measure of SML and hence MO energy levels. In AOM, the overlap integral is also factored into a radial and angular product,

SML = S(r) F(θ,φ)

Analyzing S(r) as a function of the M–L internuclear distance,

Under the condition of a fixed M-L distance, S(r) is invariant, and therefore the

overlap integral, SML, will depend only on the angular dependence, i.e., on F(θ,φ).

5.04, Principles of Inorganic Chemistry II Lecture 14

ML Diatomic Complexes

To begin, let’s determine the energy of the d-orbitals for a M-L diatomic defined by the following coordinate system,

There are three types of overlap interactions based on σ, π and δ ligand orbital

symmetries. For a σ orbital, the interaction is defined as,

E ⎛⎜d (^) z 2 ⎝

= SML^2 ( σ ) = β • F σ^2 (θ, φ )= β • 1 = e σ

The energy for maximum overlap, at θ = 0 (see above) is set equal to 1. This

energy is defined as eσ. The metal orbital bears the antibonding interaction, hence

dz^2 is destablized by eσ (the corresponding L orbital is stabilized by (β’)^2 • 1 = eσ’).

For orbitals of π and δ symmetry, the same holds…maximum overlap is set equal to

1, and the energies are eπ and eδ, respectively.

E (d ) = S 2 (π ) = e π E (d )

E (dyz ) = xz ML xy = E

dx (^2) − y 2 ⎞⎟ ⎠

= SML^2 ( ) δ = e δ

5.04, Principles of Inorganic Chemistry II Lecture 14

As with the σ interaction, the (M-Lπ)* interaction for the d-orbitals is de-stabilizing

and the metal-based orbital is destablized by eπ, whereas the Lπ ligands are

stabilized by eπ. The same case occurs for a ligand possessing a δ orbital, with the

only difference being an energy of stabilization of eδ for the Lδ orbital and the

energy of de-stabilization of eδ for the δ metal-based orbitals.

SML(δ) is small compared to S ML (π) or SML(σ). Moreover, there are few ligands with δ

orbital symmetry (if they exist, the δ symmetry arises from the pπ-systems of organic

ligands). For these reasons, the SML(δ) overlap integral and associated energy is not

included in most AOM treatments.

Returning to the problem at hand, the overall energy level diagrams for a M-L diatomic molecule for the three ligand classes are:

ML 6 Octahedral Complexes

Of course, there is more than 1 ligand in a typical coordination compound. The

power of AOM is that the eσ and eπ (and eδ), energies are additive. Thus, the MO

energy levels of coordination compounds are determined by simply summing eσ

and eπ for each M(d)-L interaction.

5.04, Principles of Inorganic Chemistry II Lecture 14

For ligands in an octahedral complex, the θ and φ for the six ligands values are,

Ligand 1 2 3 4 5 6

Consider the overlap of Ligand 2 in the transformed coordinate space; the contribution of the overlap of Ligand 2 with each metal orbital must be considered. This orbital interaction is given by the transformation matrix above. By substituting

the θ = 90 and φ = 0 for Ligand 2 into the above transformation matrix, one finds,

for dz^2 for L (^2)

d 2 = 1 ( 1 + 3 cos 2 θ ) d 2 + 0dy z −

sin2 θ dx z + 0dx y +

z (^1 −^ cos 2^ θ^ )^ d^2 4 z^2 2 2 2 2 2 2 2 4 x^2 −y^2 1 3 = − d + 0d + 0d + 0d + d 2 z^

(^2) y 2 z 2 x 2 z 2 x 2 y 2 2 x^ 2 22 −y^22

Thus the dz 2 orbital in the transformed coordinate, dz 22 , has a contribution from dz 2 and dx (^2) –y 2. Recall that energy of the orbital is defined by the square of the overlap integral. Thus the above coefficients are squared to give the energy of the dz 2 orbital as a result of its interaction with Ligand 2 to be,

E dz

Visually, this result is logical. In the coordinate transformation, a σ ligand residing

on the z-axis (of energy eσ) is overlapping with dz^2. This is the energy for L1. The

normalized energy for L2 is its overlap with the coordinate transformed dz 22 :

L = SML^2 ( σ ) = β • F σ^2 ( θ , φ ) = d + d = e σ + e δ

4 z^

2

4 x^

(^2) −y 2

5.04, Principles of Inorganic Chemistry II Lecture 14

Note, the dz 2 orbital is actually 2z^2 –x^2 –y^2 , which is a linear combination of z^2 –x^2 and z^2 –y^2. Thus in the coordinate transformed system, L2, as compared to L1, is looking

at the x^2 contribution of the wavefunction to σ bonding. Since it is ½ the electron

density of that on the z-axis, it is ¼ the energy (i.e., the square of the coefficient)

on the σ-axis, hence ¼ eσ. The δ component of the transformation comes from the

2z^2 –(x^2 +y^2 ) orbital functional form. Thus if L2 has an orbital of δ symmetry, then it

will have an energy of ¾ eδ.

The transformation properties of the other d-orbitals, as they pertain to L2 orbital overlap, may be ascertained by completing the transformation matrix for θ = 90

and φ = 0,

⎡ (^) d ⎤ ⎡^1 3 ⎤^ ⎡ d ⎤ z yz

⎢⎣d

2 2 z^2 d (^0 0 0) − 1 0 d 0 0 − 1 0 0 0 1 0 0 0

y 2 z 2 dxz d xy

= (^) x 2 z 2 d dx 2 y 2 d (^2 23 0 0 0 12 ) x − y (^2 2) x 2 −y 2

The energy contribution from L2 to the d-orbital levels as defined by AOM is,

E (d ) e ; E (d ) e ; E (d )

= e π; ⎛⎜

E dx (^2) −y 2 ⎞⎟ ⎠

yz =^ δ xz =^ π^ xy = e^ σ^ + e^ δ

Until this point, only the L2 ligand has been treated. The overlap of the d-orbitals with the other five ligands also needs to be determined. The elements of the transformation matrices for these ligands are,

0 0 ⎡^ –^1 0 0 0 3 ⎤^ ⎡–^1 0 0 03 ⎤

2 2 2 2 0 0 0 0 – 1^0 0 0 0 0 1 L 1 : 1 0 L 3 : 0 0 0 1 0 L 4 : 0 0 1 0 0 0 – 1 0 0 0 0 1 0 0 0 − 3 0 0 0 –^1 3 0 0 2 2 2 2

L 5 : 0 0 0 – 1 0 L 6 : 0 1 0 0

− 23 0 0 0 –^12 0 0 0 1 ⎦

5.04, Principles of Inorganic Chemistry II Lecture 14