symmetry operations and elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · symmetry...

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Symmetry Operations and Elements The goal for this section of the course is to understand how symmetry arguments can be applied to solve physical problems of chemical interest. To achieve this goal we must identify and catalogue the complete symmetry of a system and subsequently employ the mathematics of groups to simplify and solve the physical problem in question. A symmetry element is an imaginary geometrical construct about which a symmetry operation is performed. A symmetry operation is a movement of an object about a symmetry element such that the object's orientation and position before and after the operation are indistinguishable. A symmetry operation carries every point in the object into an equivalent point or the identical point.

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Page 1: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

Symmetry Operations and Elements

• The goal for this section of the course is to understand how symmetry arguments can be

applied to solve physical problems of chemical interest.

• To achieve this goal we must identify and catalogue the complete symmetry of a system and

subsequently employ the mathematics of groups to simplify and solve the physical problem

in question.

• A symmetry element is an imaginary geometrical construct about which a symmetry

operation is performed.

• A symmetry operation is a movement of an object about a symmetry element such that the

object's orientation and position before and after the operation are indistinguishable.

• A symmetry operation carries every point in the object into an equivalent point or the

identical point.

Page 2: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

The Electronic Structure of Ferrocene

• The two cyclopentadienyl (Cp) rings of ferrocene may be orientated in the two

extremes of either an eclipsed (D5h) or staggered (D5d) conformation.

• The energy of rotation about the Fe-Cp axis is very small (~ 4 kJmol-1) and ground

state structures of ferrocene may show either of these conformations.

• There is also very little difference in electronic states between the D5h and D5d

symmetries however the D5d point group irreducible representations are used here

in the description of the electronic structure of ferrocene as they simplify the

symmetry matching of ligand molecular orbitals (SALCs) and metal atomic orbitals.

• The primary orbital interactions that form the metal-ligand bonds in ferrocene

occur between the Fe d orbitals and the π-orbitals of the Cp ligand.

• If D5d symmetry is assumed, so that there is a centre of symmetry in the ferrocene

molecule through the Fe atom there will be centro-symmetric (g) and anti-

symmetric (u) combinations.

Page 3: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

ππππ MOs of Cyclopentadienyl, C5H5-

• C5H5− , has three pairs of electrons delocalized in a π system extending around the

pentagonal ring.

• The five 2p orbitals perpendicular to the ring on the five carbon atoms combine to

form three bonding (π1, π2, π3) and three antibonding (π4*, π5*, π6*) MOs.

• The symmetries and forms of these MOs can be deduced by applying the operations of

the point group D5h to a set of five vectors perpendicular to the ring, one at each

carbon, to generate a reducible representation Γπ .

Page 4: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

ππππ MOs of Cyclopentadienyl, C5H5-

Page 5: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

ππππ MOs of Cyclopentadienyl, C5H5-

Page 6: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

• The five p-orbitals on the planar Cp ring (D5h symmetry) can be combined to

produce five molecular orbitals according to the reducible representation:

• One combination has the full symmetry of the ring (a2’’)

• There are two doubly degenerate combinations (e1’’ and e2

’’) having one and two

planar nodes at right angles to the plane of the ring.

• The relative energies of these orbitals increase as the number of nodes increases.

• The a2’’ and e1

’’ orbitals are both fully occupied in the electronic configuration of

the Cp- anion whereas the e2’’ orbitals are net anti-bonding and are unfilled.

Page 7: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

The ππππ-molecular orbitals of the cyclopentadienyl ring (D5h)

A2''

(0 nodes)

bonding

E1''

(2 nodes)

weakly bonding

E2''

(4 nodes)

anti-bonding

Energ

y

• The five p-orbitals on the planar Cp ring (D5h symmetry) can be combined to

produce five molecular orbitals.

Page 8: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

• For a bis-cyclopentadienyl metal complex (η5-Cp)2M , such as ferrocene, the π-

orbitals of the two Cp ligands are combined pairwise to form the symmetry-

adapted linear combination of molecular orbitals (SALC’s) which are described by

the irreducible representations of the D5d point group.

Page 9: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

• For a bis-cyclopentadienyl metal complex (η5-Cp)2M , such as ferrocene, the π-

orbitals of the two Cp ligands are combined pairwise to form the symmetry-

adapted linear combination of molecular orbitals (SALC’s) which are described by

the irreducible representations of the D5d point group.

Page 10: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

• The Γπ SALCs of the (Cp)2 framework are also defined by the sum and difference

of SALCs from the two contributing Cp ligands whose results must correspond to

the irreducible components of Γπ (Cp)2 , i.e.

(ψ1+ψ1), (ψ1-ψ1); (ψ2+ψ2), (ψ2-ψ2) ...etc.

where, for example, 'ψ1+ψ1' gives rise to a molecular orbital of A1g symmetry.

• This gives rise to three sets of ligand molecular orbitals of gerade (g) and

ungerade (u) symmetry with respect to the centre of inversion;

� a low lying filled bonding pair of A1g and A2u symmetry

� a filled weakly bonding pair of E1g and E1u symmetry

� an unfilled anti-bonding pair of E2g and E2u symmetry.

Page 11: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

Z

Y

X

a1g a2u

e1ue1g

e2ue2g

SALCs for a (ηηηη5-Cp)2M complex

Page 12: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

• Using the reducible representation of SALCs the corresponding metal AOs are

found

• Thus, in D5d the bonding metal orbitals transform as

A1g : (s , dz2)

E1g : (dyz , dxz)

E2g : (dx2 - dy

2, dxy)

A2u : (pz)

E1u : (px, py)

• By considering the AO and SALC symmetries and how overlap can be affected the

MO bonding picture of ferrocene can be constructed.

• Each combination of AOs and SALCs leads to a bonding molecular orbital

[(ψligand molecular orbital)+(ψmetal atomic orbital)] and a corresponding anti-bonding molecular

orbital [(ψligand molecular orbital)-(ψmetal atomic orbital)] providing that the energies of the

two component sets are sufficiently close for overlap.

Page 13: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

Symmetry matching of the SALC’s with the metal atomic orbitals

Z

Y

X

px

pzs dz2

dx2-dy2dxy

dxzdyz

no metal

orbital of

appropriate

symmetry

available

a1g a1u

e1ue1g

py

e2ue2g

no metal

orbital of

appropriate

symmetry

available

Page 14: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

FeII

Fe

SALC's

Page 15: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

a1g

FeII

Fe

SALC's

Page 16: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

a1g

a1g*

FeII

Fe

SALC's

Page 17: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

a1g

a1g*

FeII

Fe

SALC's

Page 18: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a2u a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

a1g

a1g*

a2u*

FeII

Fe

SALC's

Page 19: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a2u

e1g

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

e1g*

a1g

a1g*

a2u*

FeII

Fe

SALC's

Page 20: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a2u

e1g

e1u

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

e1g*

a1g

a1g*

a2u*

e1u*

FeII

Fe

SALC's

Page 21: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a2u

e1g

e1u

e2g*

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

e2g

e1g*

a1g

a1g*

a2u*

e1u*

FeII

Fe

SALC's

Page 22: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a2u

e1g

e1u

e2g*

e2u

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

e2g

e1g*

a1g

a1g*

a2u*

e1u*

FeII

Fe

SALC's

Page 23: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a2u

e1g

e1u

e2g*

e2u

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

e2g

e1g*

a1g

a1g*

a2u*

e1u*

FeII

Fe

SALC's

Page 24: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

A qualitative molecular orbital diagram for ferrocene (D5d)

dxzdyz

e1g

a2u

dz2

a1g'

p z

a1u

a1g

a2u

e1g

e1u

e2g*

e2u

a1g , a2u

e1g , e1u

e2g , e2u

a1g , e1g , e2g

a1g

a2u , e1u

e2g

e1g*

a1g

a1g*

a2u*

e1u*

FeII

Fe

SALC's

HOMO

LUMO

Page 25: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

• Due to a difference in energies the lowest energy a1g molecular orbital is mainly

ligand based with a slight admixture of the Fe 4s and 3dz2 orbitals.

• Similarly the a2u level has little if any metal character due to higher lying Fe 4pz

orbital with which it is formally able to combine.

• The e1g molecular orbital arises from the bonding combination of the ligand e1g

orbitals with the Fe 3dxz and 3dyz orbitals. This is the only symmetry combination

of orbitals in the two Cp rings that has appreciable overlap with the metal 3d

orbitals to act as an efficient donor and it is thus this interaction which is mainly

responsible for the stability of the complex.

• The corresponding anti-bonding orbitals, e1g*, are unfilled in the ground state of

ferrocene but they are involved in excited state transitions.

• The e1u bonding molecular orbitals are again mainly ligand based but with a small

contribution from the higher energy Fe 3 px , py orbitals.

• The a1g HOMO mostly consists of the Fe 3dz2 orbital as the a1g SALC and the metal

dz2 orbital result in little or no overlap.

• The e2g (dx2-y2, dxy) metal orbitals are considered weakly-bonding due to poor

overlap with the e2g SALC orbitals.

Page 26: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

• Since the occupied orbitals are of either a1g , e1g or e2g type symmetry no intrinsic

barrier to internal rotation is predicted as each of these molecular orbitals are

symmetric about the axis of rotation.

• The very low values observed for this rotation (~ 4 kJmol-1) may be attributed to

van der Waals forces between the two Cp rings.

• The attachment of additional groups or ligands destroys the D5d/D5h symmetry of

ferrocene thus significantly altering the MO diagram.

1. S. Barlow, H. E. Bunting, C. Ringham, J. C. Green, G. U. Bublitz, S. G. Boxer, J. W. Perry, S. R. Marder, J. Amer. Chem. Soc. 1999,

121, 3715-3723.

2. J. C. Calabrese, L. T. Cheng, J. C. Green, S. R. Marder, W. Tam, J. Amer. Chem. Soc. 1991, 113, 7227-7232.

3. D. R. Kanis, M. A. Ratner, T. J. Marks, Chem. Rev. 1994, 94, 195-242.

Page 27: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

Example: Constructing a MO for

Platinum Tetrachloride, [PtCl4]2-

PtCl Cl

point group = D4h

Cl

Cl2-

Pt bonding AOs

A1g : 6s , 5dz2

Eu : (6px , 6py)

B1g : 5dx2-y2

Pt non-bonding AOs

A2u : 6pz

B2g : 5dxy

Eg : (5dxz, 5dyz)

D4h E 2C4 C2 2C2' 2C2'' i 2S4 h 2 v 2 d /h

4 0 0 2 0 0 0 4 2 0

A1g 4 0 0 4 0 0 0 4 4 0 16 1

A2g 4 0 0 -4 0 0 0 4 -4 0 0 0

B1g 4 0 0 4 0 0 0 4 4 0 16 1

B2g 4 0 0 -4 0 0 0 4 -4 0 0 0

Eg 8 0 0 0 0 0 0 -8 0 0 0 0

A1u 4 0 0 4 0 0 0 -4 -4 0 0 0

A2u 4 0 0 -4 0 0 0 -4 4 0 0 0

B1u 4 0 0 4 0 0 0 -4 -4 0 0 0

B2u 4 0 0 -4 0 0 0 -4 4 0 0 0

Eu 8 0 0 0 0 0 0 8 0 0 16 1

h = 16

Page 28: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the
Page 29: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the
Page 30: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

px py

p z

Page 31: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

PtCl ClCl

Cl 2-

px py

p z

Page 32: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

PtCl ClCl

Cl 2-

a1g*px py

p z

Page 33: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

a1g*

PtCl ClCl

Cl 2-

a1g*px py

p z

Page 34: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

a1g*

PtCl ClCl

Cl 2-

a1g*px py

p z

Page 35: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

a1g*

PtCl ClCl

Cl 2-

eu

eu*

a1g*px py

p z

Page 36: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

a1g*

PtCl ClCl

Cl 2-

b1g

eu

b1g*

eu*

a1g*px py

p z

Page 37: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

eg

a1g*

PtCl ClCl

Cl 2-

b1g

eu

b2g

b1g*

eu*

a1g*px py

p z

Page 38: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

eg

a1g*

PtCl ClCl

Cl 2-

b1g

eu

b2g

b1g*

eu*

a2u

a1g*px py

p z

Page 39: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

eg

a1g*

PtCl ClCl

Cl 2-

b1g

eu

b2g

b1g*

eu*

a2u

a1g*px py

p z

Page 40: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

6s (a1g)

6p (a2u , eu)

a1g

Pt2+ 4Cl-

5d (a1g , b1g , b2g , eg)

b1g

eu

a1g

eg

a1g*

PtCl ClCl

Cl 2-

b1g

eu

b2g

b1g*

eu*

a2u

a1g*px py

p z

HOMO

LUMO

∆∆∆∆

Page 41: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

Example: Constructing a MO for

Tetrakis(triphenylphosphine)Palladium, Pd(PPh3)4

Pd bonding AOs

A1 : 5s

T2 : (5px , 5py , 5pz)

(4dxy, 4dxz, 4dyz)

Pd non-bonding AOs

E : (4dx2-y2 , 4dz2 )

Page 42: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

• Consider first the A1 SALC. It must have the same symmetry of the s orbital on

the central metal atom. This requires that it be everywhere positive and

unchanged by all symmetry operations

A1 → σ1 + σ2 + σ3 + σ4

Construction of SALCs for σσσσ bonding in Td complexes

σ1

σ2

σ3

σ4

Page 43: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

Construction of SALCs for σσσσ bonding in Td complexes

• The T2 SALC’s must match the symmetries of the (px , py , pz ) and (dxy, dxz, dyz)

orbitals, e.g. must have positive amplitude where the p orbital is positive and

negative amplitude where the p orbitals are negative.

σ1 - σ2 + σ3 - σ4

T2 σ1 - σ2 - σ3 + σ4

σ1 + σ2 - σ3 - σ4

σ1

σ2

σ3

σ4

Page 44: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

Construction of SALCs for σσσσ bonding in Td complexes

Page 45: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

SALCs

AOs

dxy dxz

T2(1 node)

dyz

T2(1 node)

MOs

dz2 dx2-dy2

Page 46: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

dx2-y2dz2

Page 47: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

5s (a1)

5p (t2)

a1

Pd0 4 PPh3

4d (t2 , e)

t2

PPh3

PdPPh3Ph3P PPh3

dx2-y2dz2

Page 48: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

5s (a1)

5p (t2)

a1

Pd0 4 PPh3

4d (t2 , e)

t2

PPh3

PdPPh3Ph3P PPh3

dx2-y2dz2

Page 49: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

5s (a1)

5p (t2)

a1

Pd0 4 PPh3

4d (t2 , e)

t2

a1

a1*

PPh3

PdPPh3Ph3P PPh3

dx2-y2dz2

Page 50: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

dx2-y2dz2

Page 51: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

dx2-y2dz2

Page 52: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

dx2-y2dz2

Page 53: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

dx2-y2dz2

Page 54: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

5s (a1)

5p (t2)

a1

Pd0 4 PPh3

4d (t2 , e)

t2

a1

t2

e

t2

t2*

a1*

PPh3

PdPPh3Ph3P PPh3

dx2-y2dz2

The tetrahedral geometry is electronically

favored by d4 or d10 metal complexes

where the non-bonding orbitals are either

1/2 or entirely filled, respectively.

Page 55: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

5s (a1)

5p (t2)

a1

Pd0 4 PPh3

4d (t2 , e)

t2

a1

t2

e

t2

t2*

a1*

PPh3

PdPPh3Ph3P PPh3

dx2-y2dz2

HOMO

LUMO

∆∆∆∆t

The tetrahedral geometry is electronically

favored by d4 or d10 metal complexes

where the non-bonding orbitals are either

1/2 or entirely filled, respectively.

Page 56: Symmetry Operations and Elementsalpha.chem.umb.edu/chemistry/ch612/documents/week12.pdf · Symmetry Operations and Elements ... • A symmetry operation carries every point in the

The Electronic Structure of bisbenzenechromium

Cr(ηηηη6-C6H6)2

• Similar to ferrocene, primary orbital interactions that form the metal-ligand bonds

in occur between the Cr d orbitals and the π-orbitals of the benzene ligand.

• The two benzene rings of Cr(η6-C6H6)2 are ideally orientated in an eclipsed (D6h)

conformation.

Homework – generate an MO diagram for the Cr(ηηηη6-C6H6)2 molecule