Hybrids: Linear Combination of S and P leads to lowering of energy - - PowerPoint PPT Presentation

MO Energies in Polyatomics: BeH2

( ) ( ) ( ) ( )

* 1 2 2 1 1 2 2 1 1 1 * 3 2 4 1 1 4 2 3 1 1

A B A B A B A B

H H H H g Be g Be s s s s H H H H u Be u Be s s s s

s s pz pz

c c c c c c c c

σ σ σ σ

ϕ ψ ψ ψ ϕ ψ ψ ψ ϕ ψ ψ ψ ϕ ψ ψ ψ = + + = − + = + − = − −

Hybrids: Linear Combination of S and P leads to lowering of energy

• Hybridization is close to VBT. Use of experimental information
• All hybrid orbitals equevalent and are orthonormal to each other

Linus Pauling, ~1930

Linear Environment: s & 1-p mix: sp

Contribution from s=0.5; contribution from p=0.5 Have to normalize each hybridized orbital

S and P orbital of the Same atom! This is not The same as S (overlap) 2 equivalent hybrid orbitals of the same energy and shape (directions different) Linear geometry with Hybridized atom at the center 2S- and 2P- (similar energy) Mixes to form hybrid orbital which forms a MO with H (1S)

1 1 2 2 1 2 sp h sp h

s p s p

c c c c ϕ ψ ψ ϕ ψ ψ = + = −

1 2

1 1 2 2 1 1 2 2

sp h sp h

s p s p

ϕ ψ ψ ϕ ψ ψ = + = −

2 more p-orbital available for bonding

Contours & bonding of sp-hybridization

Trigonal Environment: Mixing s & 2-p

1 1 1 1 2 2 1 3 3

1 2 3

. . . . . .

x y x y x y

s p p hy s p p hy s p p hy

Signs of un normalized hybrid orbitals c Cos Sin c Cos Sin c Cos Sin ϕ ψ θ ψ θ ψ ϕ ψ θ ψ θ ψ ϕ ψ θ ψ θ ψ − ≡ + + ≡ + + ≡ + +

x y

30o 30o

x y

30o 30o

x y

30o 30o

px and py can be combined with s to get three 3 equivalent hybrids at 120o to each other

Θ1 Θ2 Θ3

hy1 hy2 hy3

+y

• y

+x

• x

+x +y

30o 30o

+x +y

30o

+x +y

30o

2 2 2

1 1 2 3 2 4 5 6 2 7 8 9 sp h sp h sp h x y x y x y

s p p s p p s p p

c c c c c c c c c ϕ ψ ψ ψ ϕ ψ ψ ψ ϕ ψ ψ ψ = + + = + + = + +

2 2 2

1 1 2 3 2 4 5 6 3 7 8 9

0.

sp h sp h sp h x y x y x y

s p p s p p s p p

c c c c c c c c c ϕ ψ ψ ψ ϕ ψ ψ ψ ϕ ψ ψ ψ = + + = + − = − −

Signs of AOs for specific Sp2 hybrid orbitals (given an orientation)

h1 h3 h2

2 2 2 1 4 7 1 4 7 2 1 5 8 6 9

1 (Total s-contribution) (s contributes equally) 0 (h along x) ( ) ( ) c c c c c c c c c symmetry c c symmetry + + = = = = = =

2 2 2 1 2 3 2 2 2 4 5 6 2 2 2 7 8 9 2 2 2 2 5 8 2 2 2 3 6 9

is 1 1 1 . 1 1

x y

Each normalized c c c c c c c c c P and P Coeffs c c c c c c ϕ + + = + + = + + = + + = + + =

2 2 2

1 2 3 4 5 6 7 8 9

1 2 3

0.

sp sp sp x y x y x y

h h h

s p p s p p s p p

c c c c c c c c c ϕ ψ ψ ψ ϕ ψ ψ ψ ϕ ψ ψ ψ = + + = + − = − −

1 4 2 5 3 6 4 7 5 8 6 9 7 1 8 2 9 3

, :

i j

• rthogonal

c c c c c c c c c c c c c c c c c c ϕ ϕ + + = + + = + + =

How to obtain coefficients for this specific geometry of Sp2?

+x +y

30o 30o

h1 h2 h3

2 2 2

1 2 3

1 2 0. 3 3 1 1 1 3 2 6 1 1 1 3 2 6

sp sp sp x y x y x y

h h h

s p p s p p s p p

ϕ ψ ψ ψ ϕ ψ ψ ψ ϕ ψ ψ ψ = + + = + − = − −

Square of coefficients  Contribution from s=0.33; from p=0.66

+x +y

30o 30o

h1 h2 h3

Signs and coefficients for these particular Sp2 hybrids

• How do we calculate the coefficients?

Use orthonormality of hybrid-orbitals and symmetry arguments

• There is no unique combination/solution (depends on the geometry!)

Hybridization of s & 3-p:sp3: Tetrahedral

Contributions from s = 25%; p=75%

3 3 3 3

1 2 2 4

1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2

y

sp h sp h p sp h sp h x y z x z x y z x y z

s p p p s p p s p p p s p p p

ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ = + + + = − − + = + − − = − + −

3 3 3 3

1 1 2 3 4 2 1 5 7 8 2 1 9 10 11 4 1 12 13 14

+

y

sp h sp h p sp h sp h x y z x z x y z x y z

s p p p s p p s p p p s p p p

c c c c c c c c c c c c c c c c

ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ = + + + = + + + = + + = + + +

1 2 3 4

1 3 0. 0. 2 2 1 2 1 0. 2 3 2 3 1 1 1 1 2 6 2 2 3 1 1 1 1 2 6 2 2 3

y

h h p h h x y z x z x y z x y z

s p p p s p p s p p p s p p p

ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ ϕ ψ ψ ψ ψ = + + + = + + − = − + − = − − −

What if h1 is oriented along z-axis?