Homework Class

4.1 Carter 4.1 Determine the sets of specific atomic orbitals that can be combined to form hybrid orbitals with the following geometries: (a) trigonal planar, (b) square planar,(c) trigonal bipyramidal,(d) octahedral

 How do you determine  point group Symmetry?

B

Group table are attached.

What are the point groups?

A B B trigonal planar

(a) D3h

4.1a-- Determine the sets of specific atomic orbitals that can be combined

B

to form hybrid orbitals

Hybrid orbital construction, Carter, page 100

(a) trigonal planar -- D3h

A B B trigonal planar

(x axis coincident with C'2 axis)

“To construct the reducible representation for any set of hybrid orbitals, count the number of vectors in the basis set that remain nonshifted by a respresentative operation of each class in the point group of the system. The number of unshifted vectors is the the character for the class in each case”

What is the reducible representation for the hybrid orbital vectors? Count # unshifted vectors = A’1 + E’ Which orbitals?

t

0

3

1

3

0

1

t

A1': s, d

2

E': (p , p ) , (d , d

)

A B B trigonal planar

(a) trigonal planar -- D 3h A1': s, dz2

4.1b --Determine the sets of specific atomic orbitals that can be combined to form hybrid orbitals for a square planar geometry-- XeF 4

B

4.1a

E': (px, py), (dxy, dx2-y2)

What are the hybrid orbitals? Use 3 atomic orbitals to make 3 hybrid orbitals

E'

s

(px, py)

dz2

can have

Notation

A1'

s



(dxy, dx2-y2) (px, py)

dz2 (dxy, dx2-y2)

What is the point group?

sp2 sd2 p2d

D4h

dp2

d3

What is the reducible representation for the hybrid orbital vectors?

4.1b What is the reducible representation for the hybrid orbital vectors for a square planar geometry-- XeF 4?

What are the irreducible representations?? By inspection or……

D4h

USE

(x axis coincident with C'2 axis)

ni

1 =

h

 gc i r c

ni = number of times irreducible representation i occurs in the reducible representation h = order of the group = number of elements in the group c = class of operations gc = number of operations in the class

Reducible representation

= character of the irreducible representation for the operations of the class r = character for the reducible representation for the operations of the class i

t

4

0

0 2

0

0

0 4

What are the irreducible representations??

2

0

4.1b Irreducible representations for square planar hybrids

t

 /16

4

0

0

2

0

0

0

4

2

0



4 4

0 0

0 0

4 -4

0 0

0 0

0 0

4 4

4 -4

0 0

16 0

1 0

4

0

0

4

0

0

0

4

4

0

16

1

4 8

0 0

0 0

-4 0

0 0

0 0

0 4 0 -8

-4 0

0 0

0 0

0 0

4 4

0 0

0 0

4 -4

0 0

0 0

0 -4 0 -4

-4 4

0 0

0 0

0 0

4

0

0

4

0

0

0

-4

-4

0

0

0

4 8

0 0

0 0

-4 0

0 0

0 0

0 -4 0 8

4 0

0 0

0 16

0 1

t = A1g + B1g + Eu

4.1b -- Which orbitals can be used? t = A1g + B1g + Eu

A1g: s , dz2

B1g: dx2-y2 A1g

What are the hybrid orbitals?

Which orbitals?

Eu: (px, py) B1g

Notation

s

dx2-y2 (px, py)

dsp2

dz2

dx2-y2 (px, py)

d2p2

4.1c --Determine the sets of specific atomic orbitals that can be combined to form hybrid orbitals for trigonal bipyramidal

-- D3h

Eu

4.1c (a) trigonal bipyramidal -- D 3h

(x axis coincident with C'2 axis)

2A1': s, dz2

A2": pz

E': (px, py), (dxy, dx2-y2)

What are the hybrid orbitals? Use 5 atomic orbitals to make 5 hybrid orbitals

What is the reducible representation for the hybrid orbital vectors? Count # unshifted vectors Determine Irreducible reps

2 t 5 = 2A 1 + A 2 + E Which orbitals? A1': s , dz2 A2": pz

1

3

0

3

t

E':(px, py) ,(dxy, dx2-y2)

2A1'

A2"

E'

Notation

s, dz2

pz

(px, py)

dsp3

s, dz2

pz

(dxy, dx2-y2)

d3sp

4.1d Determine the sets of specific atomic orbitals to form hybrid orbitals for octahedral geometry

Which orbitals do we use? t=

A1g + Eg + T1u

A1g: s

Eg: (dz2, dx2-y2)

T1u: (px, py, pz)

Determine reducible representation t 6 Irreducible rep’s? – A 5 1g t = A1g + Eg + T1u  – Eg 3

0 0 -1 -1 0 -1

2 1 1

2 1 -1

0 -1 -3

0 0 -1 -1 -1 0

4 3 1

2 1 1

Only possibility is d2sp3

4.3 Carter-- Borane Borane, BH3, is an unstable compund produced by thermal decomposition of H 3B•PF3. Although it has not been isolated and structurally characterized, it probably is trigonal pl anar.

MO’s for BH3

For simple molecule such as MXn, match symmetries of the atomic orbitals of M with symmetry adapted linear combinations (SALC’s) of atomic orbitals of the X ligands.

(a) Develop a general MO scheme for BH3. Assume that only the boron 2s and 2p orbitals interact with the hydrogen 1s orbitals (I.e., the boron 1s orbital is nonbonding). (b) The photoelectron spectrum of BH3 has not been observed. Nonetheless, if it could be taken, what would you expect it to look like, based on your MO scheme? (c) Compare and contrast the general MO description of BH3 with a valence bond (VB) model an its related loacalized MO model.

mo

=a

AO(M)

SALC(nX)

=

±b

ic i i

nX). SALC(nX) where

i

are atomic orbitals of the X ligands

BH3 Molecular Orbitals

Molecular Orbital Generation 1.

2.

Use the directional properties of potentially bonding atomic orbitals on the outer atoms (depict as vectors) as a basis for a reducible representation of  the SALC’s in the point group of the molecule.

What is the symmetry of BH3? Draw Lewis dot structure…there are 3 boron valence electrons and 3 hydrogen valence electrons. The Lewis octet rule cannot be satisfied.

Generate the reducible representation characters for these vectors. +1 if not shifted; -1 if shifted into the negative of itself (head becomes tail); and 0 if taken into another vector.

BH3

3. Decompose the SALC reducible representation into its irreducible components. # of SALC’s must equal the number of starting basis AO’s

H

4. Determine symmetries of potentially bonding AO’s from central atom by inspecting the group table unit vector and direct product transformations listed in the table.

1–1 = 0 for H

General MO scheme

B

A B B trigonal planar

(x axis coincident with C'2 axis)

3 – 3 = 0 for B

Expect D3h Symmetry -- no lone pair

6. Central atom AO’s and pendant atom SALC’s form non-bonding MO’s if no symmetry match

(a) trigonal planar -- D3h

Formal Charge

xo

o   B   o x       x H H

5. Central atom AO’s and pendant atom SALC’s of the same symmetry will form both bonding and antibonding LCAO’s (molecular orbitals)

4.3-- Use the Hybrid orbital approach

3x1 + 3 = 6 valence electrons

(a) Develop a general MO scheme for BH3. Assume that only the boron 2s and 2p orbitals interact with the hydrogen 1s orbitals (i.e., the boron 1s orbital is nonbonding).

Hydrogen atoms

Boron

A’1: s

What are the reducible representations for the H(1s) SALC’s? Count # unshifted vectors SALC

SALC

= A’1 + E’

Which Boron orbitals?

- A’1

3 2

0 -1

A’ : s

1 0

3 2

E’ : p , p

0 1 -1 0 A 2: p

E’ : px, py

A 2: pz

SALC

= A’1 + E’

MO’s for BH3

1. Bonding MOs always lie lower in energy than the anti-bonding MOs formed from the same AOs. 2. Nonbonding MOs tend to have energies between those of bonding and antibonding MOs formed from similar AOs. 3.

interactions tend to have less effective overlap than sigma

interactions. -bonding MOs tend to have higher energies than -bonding MOs  formed from similar AOs. * MOs tend to be less anti-bonding and have lower energies than  * MOs formed from similar AOs

pz

4. MO energies tend to rise as the number of nodes increases.  MOs with no nodes tend to lie lowest, and those with the greatest number of nodes tend to lie highest in energy. 5. Among s-bonding MOs, those belonging to the totally symmetric representation tend to lie lowest.

4.3 b

The photoelectron spectrum of BH3 has not been observed. Nonetheless, if it could be taken, what would you expect it to look like, based on your MO scheme?

Two bands, both with vibrational fine structure. The lowerenergy band, from the doubly degenerate (e') level, should be roughly twice as big as the higher-energy band, from the nondegenerate (a1') level.

4.3c Compare and contrast the general MO description of BH3 with a valence bond (VB) model an its related localized MO model.

Similar overall electron distributions. VB model ------ sp 2 hybridized boron AOs forming 2c-2e bonds with hydrogen 1s orbitals. Localized MO description----- all three bonding pairs in localized sigma-bonding MO’s.

VB suggestion of energetic equivalence of all three pairs disagrees with the symmetry restrictions of the MO’s.

Octahedral Coordination- Transition Metals

What about



bonding?

Only difference is when we derive the reducible representation, have to include x and y vectors that are perpendicular to the bond vector. Example: CO2 Carter, Page 119

Vector basis for a representation of oxygen SALCs of CO 2

Filled ligand p orbitals

Empty ligand p orbitals

Use Descent in Symmetry (subgroup) approach ---Carter page 73

Octahedral ML6 molecular orbitals where L is an arbitrary donor ligand

Projection Operator How do we determine the SALC combinations in practice? SALC(nX) =

ic i

i

where

Use Projection operator:

i are atomic orbitals of the X ligands

Molecular Orbital Orthonormal Properties Orthogonality for wave functions that are solutions to the wave equation i.e. can be defined by symmetry

   i jd   ij =

Slater overlap integral for wave functions on two atoms A and B

 ij 

=

 ij 

=

Sij

1

=

0

i=j ij

    A  Bd 

Projection Operator for BH 3

To get SALC’s for the H atoms---Apply Projection Operator

Hc B Hb Ha

=

Do same for E’ irreducible representation D3h

E

C13(z)

C2a

C23(z)

C2b C2c

h(xy)

S 13

Hb S2

Hc B

Ha

va

vb

vc

3

R  j   

  a 

  b 

  c 

  a 

+2 

-1

-1

0 

  c 

  b 

0 

0 

  a 

  b 

  c 

  a 

  c 

  b 

2 

-1

-1

0 

0 

0 

a 

Apply symmetry operations on all orbitals. If class, include all operations in class. “Expanded” table may be useful. For our problem....

D3h

C13(z)

E

C23(z)

C2a

C2b C2c

h(xy)

S 13

S2

va

vb

E’ vc

Get 4

a 

R  j   

  a 

  b 

  c 

  a 

  c 

  b 

  a 

  b 

  c 

  a 

  c 

  b 

A’1 +1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

a 

Multiply, add, and normalize to get hydrogen A1 SALC

1(A1) =

1

3

(

a + b + c

)

 Midterm Monday, November 7  Class Notes

 2 (E)

- 2 

1

3 =

6

b 

- 2 

c 

Normalization gives:

(2 a   b   c )

See page 143 Carter --- need to get other MO for this irreducible representation. This function must be orthogonal to 1(A1) and  2 (E) Result:

( )

3 E

1

=

2

( b   c )

6.1 Carter Determine the number of frequencies, their symmetries , and the infrared and Raman activities of the normal modes for the following molecules. Indicate the number of polarized Raman bands and the number of frequencies that should be coincident between the two spectra. Representations of the normal modes for these structures can be found in Appendix C. (a) NH 3 (b) FeCl63–  (c) H2CO (d) PF5 (e) C2H6 (staggered configuration), (f) H2O2

Homework Carter, Chapter 3, pages 66-73 Note: Systematic Reduction of  Reducible Representations!! Carter, Chapter 4 (Symmetry and Chemical Bonding) Carter Chapter 6 (Vibrational Spectroscopy) Huheey Chapter 3, pp 71-74 Huheey Chapter 5 (Bonding Models in Inorganic Chemistry: The Covalent Bond) Huheey Chapter 6, pages 203-218 (The Structure of Molecules)

Derive the reducible representation for all 3N degrees of freedom of the molecule.

Then, categorize symmetries of all of the 3n degrees of freedom by point group irreducible representations……

Can simplify this greatly! (Carter, page 174)

 Atomic contributions, by symmetry operation, to the reducible representation for the 3N degrees of freedom for a molecule

The character, I, that contributes to the reducible representation for a given operation is universal for all point groups. I=

character of the 3 x 3 block matrix of which the operation is composed

Operation E C2 C3 C4 C6

Thus, to find R, the character for the overall operation, count the number of atoms that remain nonshifted by the operation , No , and multiply by the contribution per unshifted atom, i R

= Ni

i

The contribution per nonshifted atom for a particular operation is the same regardless of the orientation of its associated symmetry element.

C3v

R = Ni i Determine reducible representations for 3N = 12 degrees of freedom for NH 3 all atoms unchanged

1 + 1 + 1+ 1 = +4

3 -1 0 1 2 1 -3 -2 -1 0

i S3 S4 S6

Moreover, the value of the contribution per nonshifted atom for a particular operation is the same in any point group in which the operation is found.

6.1 a NH 3

Contribution per atoma,

I

a

Cn = 1 +2cos(2  /n) Sn = -1 + 2cos(2  /n) Where do these come from?

6.1 Carter Determine the number of frequencies, their symmetries , and the infrared and Raman activities of the normal modes for the following molecules. Indicate the number of polarized Raman bands and the number of frequencies that should be coincident between the two spectra. (a) NH3

Ni From the group table below,

trans

-(trans+

+ E

trans+ rot

=

6

0

ni

C3v 2

3

0

1

12

0

2

0

to give

+2

1

A1 A2 E

E +1 +1 +2

0

The remaining irreducible rep’s can be obtained from this reducible rep by using

2 atoms unshifted

4

6

A2 + E

+1

so that

rot)

=

rot

1 atom unchanged

= A1

2C3 (z) 3 v +1 +1 +1 -1 -1 0

z Rz

(x, y) (Rx, Ry)

1 =

h

 g c i  r

3n-6

c

= 2A1 + 2E

x2+y2, z2 y(3x2-y2) (x2-y2, xy) (xz, yz)

2

3n

Ammonia -- C3v

= 3A1 + A2 + 4E

trans = A1 + E = A2 + E

Summary:

rot

3n-6

= 2A1 + 2E

pp 183-184 Carter How many independent Raman and IR frequencies is this? 4

6.1 Carter Determine the number of frequencies, their symmetries , and the infrared and Raman activities of the normal modes for the following molecules. Indicate the number of polarized Raman bands and the number of frequencies that should be coincident between the two spectra. (a) NH 3 (b) FeCl63– (c) H2CO (d) PF5 (e) C2H6 (staggered configuration), (f) H 2O2

4 (2A1+2E) R

4 (2A1+2E)

mode active in both Raman and IR (Carter, page 183)--exclusion rule?

4 (2A1+2E) 0

C3v

tot

E +1 +1 +2

2C3 (z) 3 v +1 +1 +1 -1 -1 0

4

1

normal modes (irreducible rep of 3n-6) without x,y,z or quadratic symmetry? z Rz

(x, y) (Rx, Ry)

E C2 C3 C4 C6 i S3 S4 S6

7

1

1

3

3

1

1

1

5

3

3

0

-1

1

-1

-3

-1

0

1

1

21

0

-1

3

-3

-3

-1

0

5

3

x2+y2, z2 y(3x2-y2) (x2-y2, xy) (xz, yz)

2

What are the irreducible representations?? By inspection or……

 Atomic contributions, by symmetry operation, to the reducible representation for the 3N degrees of freedom for a molecule

Operation

i

# totally symmetric normal modes of 3n-6

2(2A1)

A1 A2 E

= Ni

Contribution per atoma, 3 -1 0 1 2 1 -3 -2 -1 0

USE I

ni

1 =

h

 gc i r c

ni = number of times irreducible representation i occurs in the reducible representation h = order of the group = number of elements in the group a

Cn = 1 +2cos(2  /n) Sn = -1 + 2cos(2  /n)

c = class of operations gc = number of operations in the class = character of the irreducible representation for the operations of the class r = character for the reducible representation for the operations of the class i

Where do these come from?

ni

1

 gc  i  r

rot

= T1u = T1g

3n

= A1g + Eg + T1g + T2g + 3T1u + T2u

trans

3n-6

=

h

c

= A1g + Eg + T2g + 2T1u + T2u

2(2T1u) 3(A1g+Eg+T2g) 1(A1g) 0

1(T2u)