the angular overlap

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5.04 Principles of Inorganic Chemistry II�

Fall 2008

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5.04, Principles of Inorganic Chemistry IIProf. Daniel G. Nocera

Lecture 14: Angular Overlap Method (AOM) for ML for MLn Ligand Fields

The Wolfsberg-Hemholtz approximation (Lecture 10) provided the LCAO-MO energy

between metal and ligand to be,

EM2SML

2EL

2SML

2

ε σ = ε σ =∆EML

* ∆EML

Note that EM, EL and ΔEML in the above expressions are constants. Hence, the MO

within the Wolfsberg-Hemholtz framework scales directly with the overlap integral,SML

=

EM2SML

2

= β ′ S

2

=

EL2SML

2

= β S

2

ε σ ∆EMLML ε σ * ∆EML

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 atomicorbitals.

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

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

5.04, Principles of Inorganic Chemistry II Lecture 14Prof. Daniel G. Nocera Page 1 of 8



cos sin

9

Because the σ orbital is symmetric, the angular dependence, F(θ,φ), of the overlapintegral mirrors the angular dependence of the central orbital.

p-orbital

…is defined angularly by a cos θ function. Hence, the angular dependence of a σorbital as it angularly rotates about a p-orbital reflects the cos θ angulardependence of the p-orbital.

Similarly, the other orbitals take on the angular dependence of the central metal

orbital. Hence, for a

d yz -orbital d z 2-orbital

0 9045

3cos2 –1

0 0

54.73º

Ll

0

l

0

F (

)

F (

)

L




ML Diatomic Complexes

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

There are three types of overlap interactions based on σ, π and δ ligand orbitalsymmetries. For a σ orbital, the interaction is defined as,

E d 2z

⎛ ⎜⎝ ⎞⎟

⎠ = S ML

2σ

2 )( ) β F (θ ,φ β 1 eσ 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, hencedz2 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 to1, and the energies are eπ and eδ, respectively.

2E (d ) = S (π ) = eπ E (d )xz MLE dyz = xy = E ⎛ ⎜

⎝ dx2 2−y ⎞⎟ ⎠ = S ML

2 ( )δ = eδ




As with the σ interaction, the (M-Lπ)* interaction for the d-orbitals is de-stabilizingand the metal-based orbital is destablized by eπ, whereas the Lπ ligands arestabilized by eπ. The same case occurs for a ligand possessing a δ orbital, with theonly difference being an energy of stabilization of eδ for the Lδ orbital and theenergy of de-stabilization of eδ for the δ metal-based orbitals.

SML(δ) is small compared to SML(π) or SML(σ). Moreover, there are few ligands with δorbital symmetry (if they exist, the δ symmetry arises from the pπ-systems of organicligands). For these reasons, the SML(δ) overlap integral and associated energy is notincluded in most AOM treatments.

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

ML6 Octahedral Complexes

Of course, there is more than 1 ligand in a typical coordination compound. Thepower of AOM is that the eσ and eπ (and eδ), energies are additive. Thus, the MOenergy levels of coordination compounds are determined by simply summing eσand eπ for each M(d)-L interaction.




Consider a ligand positioned arbritrarily about the metal,

We can imagine placing the ligand on the metal z axis (with x and y axes of M and Lalso aligned) and then rotate it on the surface of a sphere (thus maintaining M-Ldistance) to its final coordinate position. Within the reference frame of the ligand,

related by a coordinate transformationSML in complex SML (σ and π) = 1

F (θ,φ)

Note, the coordinate transformation lines up the ligand of interest on the z axis so

that the normalized energies, eσ and eπ (and eδ) may be normalized to 1. Thetransformation matrix for the coordinate transformation is:

z22 y2z2 x2z2 x2y2 x2

2–y22

( )θ 2cos314

1+ 0 θ 2sin

2

3− 0 ( )θ 2cos1

4

3−

θ φ 2sinsin2

3

cosφ cosθ sinφ cos2θ –cosφ sinθ θ φ 2sinsin2

1

−

θ φ 2sincos2

3–sinφ cosθ cosφ cos2θ sinφ sinθ θ φ 2sincos

2

1−

( )θ φ 2cos12sin4

3− cos2φ sinθ θ φ 2sin2sin

2

1cos2φ cosθ ( )θ φ 2cos32sin

4

1+

( )θ φ 2cos12cos4

3− –sin2φ sinθ θ φ 2sin2cos

2

1–sin2φ cosθ ( )θ φ 2cos32cos

4

1+

2z

yz

xz

xy

2 2x –y




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

Ligand 1 2 3 4 5 6θ 0 90 90 90 90 180°

φ 0 0 90 180 270 0

Consider the overlap of Ligand 2 in the transformed coordinate space; thecontribution of the overlap of Ligand 2 with each metal orbital must be considered.This orbital interaction is given by the transformation matrix above. By substitutingthe θ = 90 and φ = 0 for Ligand 2 into the above transformation matrix, one finds,

for d z 2 for L2

d 2 =1 ( 1 + 3cos2θ )d 2 + 0dy z −

3sin2θ dx z + 0dx y +

3 ( 1 − cos2θ )d 2 2z 4 z2 2 2 2 2 2 2 2 4 x2 −y2

1 3= − d + 0d + 0d + 0d + d

2 z2 y2z2 x2z2 x2y2 2 x

2−y

22 2 2

Thus the dz2 orbital in the transformed coordinate, dz22, has a contribution from dz2

and dx2–y2. 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 dz2

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 residingon the z-axis (of energy eσ) is overlapping with dz2. This is the energy for L1. Thenormalized energy for L2 is its overlap with the coordinate transformed dz2

2:

⎞⎟ ⎠

L2

= S σ = β • F θ ,φ = d + d = eσ + eδ ML2 ( ) σ

2 ( ) 1

4 z 2

3

4 x 2−y 2

1

4

3

42 2 22




Note, the dz2 orbital is actually 2z2–x2–y2, which is a linear combination of z2–x2 andz2–y2. Thus in the coordinate transformed system, L2, as compared to L1, is lookingat the x2 contribution of the wavefunction to σ bonding. Since it is ½ the electrondensity 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 the2z2–(x2+y2) orbital functional form. Thus if L2 has an orbital of δ symmetry, then itwill have an energy of ¾ eδ.

The transformation properties of the other d-orbitals, as they pertain to L2 orbitaloverlap, may be ascertained by completing the transformation matrix for θ = 90and φ = 0,

⎡ ⎡⎤ ⎤ 1 3⎡ ⎤ ddz

yz

0 0 0 − 22⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣d

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

z22 2dd 0 0 0 − 1 0

0 0 − 1 0 0

0 1 0 0 0 y2z2

ddxz

xy

=x2z2

dd x2y2

3 1d 0 0 0 2 22 2− −x y x2 y22 2

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

E (d ) e ; E (d ) e ; E (d ) 3 1 = eπ ; ⎛ ⎜

⎝ E d

x2 −y2 ⎞⎟ ⎠ =δ π eσ eδ += =yz xz xy

4 4

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

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

⎡1 0 0⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

0 1 0

0 0 0

0 0 0 1 0

0 0 0 0 1

1

0

0

0

0

⎡ ⎡⎤ ⎤ 1 3 1 30 0 0 0 0 0 0 0 – –⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

2 2 2 2

0 0 –1 0 0 0 0 0 1 0 0 0 L1 : L L 0 0 0 1 0 0 0 1 0 0 : : 1 0 3 4

0 –1 0 0 0 0 1 0 0 0 3 1 3 10 0 0 0 0 0 − –2 2 2 2

⎡ ⎤ 1 0 0 0 3 ⎤ 0 0 0 0 –⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

2 2 ⎥⎥⎥⎥⎥⎥

0 0 1 0 0 –1 0 0 0 L6L 0 0 0 –1 0 0 1 0 0 : : 5

0 –1 0 0 0 0 0 –1 0 −

3 0 0 0 –1

2 20 0 0 1 ⎦




Squaring the coefficients for each of the ligands and then summing the totalenergy of each d-orbital,

L1 L2 L3 L4 L5 L6 ETOTAL

E(dz2)eσ δ σ e4

3e4

1+ δ σ e4

3e4

1+ δ σ e4

3e4

1+ δ σ e4

3e4

1+ eσ = 3eσ + 3eδ

E(dyz) eπ eδ eπ eδ eπ eπ = 4eπ + 2eδ

E(dxz) eπ eπ eδ eπ eδ eπ = 4eπ + 2eδ

E(dxy) eδ eπ eπ eπ eπ eδ = 4eπ + 2eδ

E(dx2–y2) eδ δ σ e4

1e

4

3+ δ σ e

4

1e

4

3+ δ σ e

4

1e

4

3+ δ σ e

4

1e

4

3+ eδ = 3eσ + 3eδ

As mentioned above, eδ << eσ or eπ… thus eδ may be ignored. The Oh energy level

diagram is:

Note the d-orbital splitting is the same result obtained from the crystal field theory(CFT) model taught in freshman chemistry. In fact the energy parameterizationscales directly between CFT and AOM

10 Dq = ∆0 = 3eσ – 4eπ


the angular overlap

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