[PPT] - x 2 x 1 x x 2 1 PowerPoint Presentation

SLIDE 1

Bonding in Diatomic Molecules such as H Bonding in Diatomic Molecules such as H2

2

r

r1

1

r r2

2

H Nucleus A H Nucleus A H Nucleus B H Nucleus B ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(B) = 1/(

(B) = 1/(π π π π π π π π) )1/2

1/2(1/a

(1/a0

0)

)3/2

3/2 exp

exp[ [-

r

r2

2/a

/a0

0], 1s orbital for atom B

], 1s orbital for atom B ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A) = 1/(

(A) = 1/(π π π π π π π π) )1/2

1/2(1/a

(1/a0

0)

)3/2

3/2 exp

exp[ [-

r

r1

1/a

/a0

0], 1s orbital for atom A

], 1s orbital for atom A Note the two Note the two orbitals

rbitals are centered at different points in space.

are centered at different points in space. The simplest molecular orbital or The simplest molecular orbital or wavefunction wavefunction is obtained is obtained by adding or subtracting the atomic by adding or subtracting the atomic wavefunctions wavefunctions for A and B. for A and B. z z1

1

x x1

1

z z2

2

x x2

2

y y1

1

y y2

2

Bonding Bonding Axis Axis

SLIDE 2

σ σ σ σ σ σ σ σ1s

1s* = C

* = C2

2[

[ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A)

(A) -

ψ

ψ ψ ψ ψ ψ ψ ψ1s

1s(B)], Sigma 1s

(B)], Sigma 1s Anti Anti-

bonding Molecular

bonding Molecular

rbital. C
rbital. C2

2 is a constant.

is a constant. σ σ σ σ σ σ σ σ1s

1s = C

= C1

1[

[ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A) +

(A) + ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(B)], Sigma 1s

(B)], Sigma 1s Bonding Molecular Bonding Molecular

rbital. C
rbital. C1

1 is a constant.

is a constant. Note that probabilities for finding electron at some position in Note that probabilities for finding electron at some position in space scale like [ space scale like [σ σ σ σ σ σ σ σ1s

1s]

]2

2 and [

and [σ σ σ σ σ σ σ σ1s

1s*]

*]2

2:

: [ [σ σ σ σ σ σ σ σ1s

1s]

]2

2 = {C

= {C1

1[

[ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A) +

(A) + ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(B)]}

(B)]}2

2 =

= (C (C1

1)

)2

2{[

{[ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A)]

(A)]2

2 + [

+ [ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(B)]

(B)]2

2 + 2[

+ 2[ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A)][

(A)][ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(B)]

(B)]} } [ [σ σ σ σ σ σ σ σ1s

1s*]

*]2

2 = {C

= {C2

2[

[ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A)

(A) -

ψ

ψ ψ ψ ψ ψ ψ ψ1s

1s(B)]}

(B)]}2

2 =

= (C (C2

2)

)2

2{[

{[ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A)]

(A)]2

2 + [

+ [ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(B)]

(B)]2

2 -

2[

2[ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A)][

(A)][ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(B)]

(B)]} } Extra Term Extra Term (note difference (note difference in sign!). in sign!). [Zero for large [Zero for large A, B separation.] A, B separation.] “Non “Non-

interacting” part of

interacting” part of [ [σ σ σ σ σ σ σ σ1s

1s]

]2

2

“Non “Non-

interacting” part of

interacting” part of [ [σ σ σ σ σ σ σ σ1s

1s*]

*]2

2

“Non “Non-

interacting” part is result for large

interacting” part is result for large separation between nucleus A and B separation between nucleus A and B

SLIDE 3

ANTIBONDING BONDING N ON-INTERACTING

Wave Func t i

ns

Wave Func t i

ns

El e c t ron Den s i t i e s El e c t ron Den s i t i e s

[ψ ψ ψ ψ − − − − ψ ψ ψ ψ ]

1s 2 1s 1s

σ σ σ σ * = C

A B 1s 1s 1s

[ψ + ψ ψ + ψ ψ + ψ ψ + ψ ]

1

σ σ σ σ

= C

A B

A B A B B A + + + + +

A

A B B

1s

[ σ [ σ [ σ [ σ

* ]

] ] ] 2

1s

[ σ [ σ [ σ [ σ ]

] ] ] 2

1s 1s 1s 1s

ψ ψ ψ ψ ψ ψ ψ ψ 2

2(n.i.)

(n.i.) ~

~

[(ψ ) + (ψ ) ] [(ψ ) + (ψ ) ] [(ψ ) + (ψ ) ] [(ψ ) + (ψ ) ] [(ψ ) + (ψ ) ] [(ψ ) + (ψ ) ] [(ψ ) + (ψ ) ] [(ψ ) + (ψ ) ]

2 2 2 2 A A B B

H 2

+

ψ

ψ ψ ψ ψ ψ ψ ψ1s

1s(B)

(B) ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A)

(A)

2[ψ

ψ ψ ψ1s(A)][ψ ψ ψ ψ1s(B)] + 2[ψ ψ ψ ψ1s(A)][ψ ψ ψ ψ1s(B)] Pushes e Pushes e-

away from

away from region between region between nuclei A and B nuclei A and B

Pushes e Pushes e-

between

between nuclei A and B nuclei A and B

ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(A)

(A) ψ ψ ψ ψ ψ ψ ψ ψ1s

1s(B)

(B)

A B

SLIDE 4

Potential Energy of H2

+

V(R) R R 1.07 Å 1.07 Å ∆E ∆Ed

d=

= 255 kJ 255 kJ mol mol-

1

1

H + H H + H+

+

H R R H Single electron Single electron holds H holds H2

2 + + together

together Separated H, H+

σ σ σ σ1s σ σ σ σ*

*1s

SLIDE 5

E

σ σ σ σ σ σ σ σ1

1 1 1 1 1 1 1s s*

* H H2

2 Molecular

Molecular Orbitals Orbitals

σ σ σ σ σ σ σ σ1

1 1 1 1 1 1 1s s

1s (H Atom A)

Atomic Orbital

1s (H Atom B)

Atomic Orbital CORRELATION DIAGRAM CORRELATION DIAGRAM

H H2

2

Bond order =(1/2)[# electrons in bonding Bond order =(1/2)[# electrons in bonding orbitals

rbitals -
#electrons in

#electrons in antibonding orbitals antibonding orbitals] ]

For H For H2

2 (

(σ σ σ σ σ σ σ σ1s

1s)

)2

2, Bond order =

, Bond order = (1/2)(2 (1/2)(2-

0) = 1

0) = 1 H-H Bond Length = 0.74Å; Bond Energy = 431 kJ/mole. Strong bond! Note: is about 1/2 bond Energy of H2

+ (255 kJ/mole)

Energy Ordering:

σ σ σ σ σ σ σ σ1

1 1 1 1 1 1 1s s < 1s < σ

σ σ σ σ σ σ σ1

1 1 1 1 1 1 1s s*

*

SLIDE 6

E σ σ σ σ σ σ σ σ1

1 1 1 1 1 1 1s s*

* He He2

2 Molecular

Molecular Orbitals Orbitals

σ σ σ σ σ σ σ σ1

1 1 1 1 1 1 1s s

1s (He Atom A)

Atomic Orbital

1s (He Atom B)

Atomic Orbital CORRELATION DIAGRAM CORRELATION DIAGRAM

He He2

2

He He2

2 barely exists under the most extreme

barely exists under the most extreme conditions: He conditions: He-

He bond length 52

He bond length 52± ± ± ± ± ± ± ±4Å!! Bond 4Å!! Bond Energy ~ 0.008 Joules/mole!!!! Energy ~ 0.008 Joules/mole!!!! (Phys. Rev. (Phys. Rev. Letts Letts. . 85 85, 2284 , 2284-

87(2000))

87(2000))

For He For He2

2 (

(σ σ σ σ σ σ σ σ1s

1s)

)2

2 (

(σ σ σ σ σ σ σ σ1s

1s*)

*)2

2 Bond

Bond

rder = (1/2)(2
rder = (1/2)(2-
2) = 0

2) = 0 Z for He =2

SLIDE 7

E

σ σ σ σ σ σ σ σ2

2 2 2 2 2 2 2s s*

* Li Li2

2 Molecular

Molecular Orbitals Orbitals

σ σ σ σ σ σ σ σ2

2 2 2 2 2 2 2s s

2s (Li Atom A)

Atomic Orbital

2s (Li Atom B)

Atomic Orbital CORRELATION DIAGRAM CORRELATION DIAGRAM

Li Li2

2

For Li For Li2

2 (

(σ σ σ σ σ σ σ σ2s

2s)

)2

2, Bond order =

, Bond order = (1/2)(2 (1/2)(2-

0) = 1

0) = 1 Bonding for Second Row Bonding for Second Row Diatomics Diatomics Involves the n=2 Atomic Shell Involves the n=2 Atomic Shell Lithium atomic configuration is 1s22s1 (Only the 2s electron is a valence electron.) Li2 dimer has the configuration: [( [(σ σ σ σ σ σ σ σ1s

1s)

)2

2(

(σ σ σ σ σ σ σ σ1s

1s*)

*)2

2] (

] (σ σ σ σ σ σ σ σ2s

2s)

)2

2 = [KK](

(σ σ σ σ σ σ σ σ2s

2s)

)2

2

SLIDE 8

Bonding for Second Row Diatomics: the Role of 2p Orbitals Once the σ σ σ σ2s, σ σ σ σ2s* molecular orbitals formed from the 2s atomic orbitals on each atom are filled (4 electrons, Be2), we must consider the role of the 2p electrons (B2 is first diatomic using 2p electrons). There are 3 different sets of p orbitals (2px, 2py, and 2pz), all mutually perpendicular. If we choose the molecular diatomic axis to be the z axis (this is arbitrary), we have a picture like this:

SLIDE 9

Nucleus A

Nucleus B x1 y1 x2 y2 z z1

1

z z2

2

Bonding Axis + 2p 2pz

z obital on

atom 1 and atom 2 +

2p

2pz

z orbitals

rbitals

point point at at each other. each other.

Nucleus A

Nucleus B x1 y1 x2 y2 z z1

1

z z2

2

Bonding Axis + +

2p

2px

x orbital on

rbital on

atom 1 and atom 2 atom 1 and atom 2 2px

rbitals

are parallel to each

ther.

SLIDE 10

B A

+

σ

σ σ σ (Bonding) (Bonding)

2pz

Atom ic Atom ic Orbitals Orbitals

f Atom s A,B
f Atom s A,B

z z A B 2p 2p

+

+

Molecular Molecular Orbitals Orbitals

A

+

+

B

σ σ σ σ * ( (Antibonding Antibonding) )

2pz Nodal Plane

2pz

2pz
+

+

A B

2p 2pz

z(A) + 2p

(A) + 2pz

z(B)

(B) 2p 2pz

z(A)

(A) -

2p

2pz

z(B)

(B) z Axis is taken to be line through A, B z Axis is taken to be line through A, B z axis z axis Bonding with 2p Bonding with 2p Orbitals Orbitals

SLIDE 11

Atom ic Atom ic Orbitals Orbitals

f Atom s A,B
f Atom s A,B

Molecular Molecular Orbitals Orbitals

A B

+ +

2px
2px

A B

+ +

2p x

2p x 2p

π π π π

(bonding)

A B

+

A

B

+ +

(antibonding)

π π π π*

2p

Nodal Plane

2p 2px

x(A)

(A) -

2p

2px

x(B)

(B) 2p 2px

x(A) + 2p

(A) + 2px

x(B)

(B) z x x z Note: orbital is Note: orbital is ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ to bond (z) axis to bond (z) axis A,B Nodal Plane Bonding with 2p Bonding with 2p Orbitals Orbitals ( (cont cont) )

SLIDE 12

CORRELATION DIAGRAM CORRELATION DIAGRAM

for second period diatomic molecules

E

σ σ σ σ*

2pz

π π π π*2py π π π π*2px π π π π2px π π π π2py σ σ σ σ 2pz σ σ σ σ*

2pz

π π π π*2px π π π π*2py σ σ σ σ 2pz π π π π2px π π π π2py

2p 2p 2s 2s

σ σ σ σ*

2s

σ σ σ σ 2s

2p 2p 2s 2s

σ σ σ σ*

2s

σ σ σ σ 2s

(6 Valence (6 Valence electrons electrons for each for each atom) atom) (3 Valence (3 Valence electrons electrons for each for each atom) atom) B B2

2

Z ≤ 7 Z ≤ 7 O O2

2

Z ≥ 8 Z ≥ 8 Molecular Molecular Orbitals Orbitals Molecular Molecular Orbitals Orbitals Note unpaired

Note unpaired electrons electrons

Note

rder of

filling σ σ σ σ2pz Degenerate Degenerate π π π π π π π π orbitals

rbitals

SLIDE 13

Molecular Orbitals of Hom onuclear Diatom ic Molecules

(Å)

He He2

2

Li Li2

2

Be Be2

2

B B2

2

C C2

2

N N2

2

O O2

2

F F2

2

Ne Ne2

2

2 4 2 4 6 8 10 12 14 16 1 1 1 2 3 3 2 1 0.74 0.74 52 52 2.67 2.67 2.45 2.45 1.59 1.59 1.24 1.24 1.10 1.10 1.21 1.21 1.41 1.41

431

.000008 105 9 289 599 942 942 494 154

Valence Electron Configuration

# Valence # Valence Electrons Electrons Bond Length Bond Order Bond Bond Energy Energy (kJ/mole) (kJ/mole) Mole Mole-

cule

cule

H H2

2

( (σ σ σ σ σ σ σ σ2s

2s)

)2

2(

(σ σ σ σ σ σ σ σ2s

2s*)

*)2

2(

(σ σ σ σ σ σ σ σ2p

2pz z)

)2

2(

(π π π π π π π π2p

2p)

)4

4(

(π π π π π π π π2p

2p*)

*)4

4

( (σ σ σ σ σ σ σ σ2s

2s)

)2

2(

(σ σ σ σ σ σ σ σ2s

2s*)

*)2

2(

(σ σ σ σ σ σ σ σ2p

2pz z)

)2

2(

(π π π π π π π π2p

2p)

)4

4(

(π π π π π π π π2p

2p*)

*)2

2

( (σ σ σ σ σ σ σ σ2s

2s)

)2

2(

(σ σ σ σ σ σ σ σ2s

2s*)

*)2

2(

(π π π π π π π π2p

2p)

)4

4 (

(σ σ σ σ σ σ σ σ2p

2pz z)

)2

2

( (σ σ σ σ σ σ σ σ2s

2s)

)2

2(

(σ σ σ σ σ σ σ σ2s

2s*)

*)2

2(

(π π π π π π π π2p

2p)

)4

4

( (σ σ σ σ σ σ σ σ2s

2s)

)2

2(

(σ σ σ σ σ σ σ σ2s

2s*)

*)2

2(

(π π π π π π π π2p

2p)

)2

2

( (σ σ σ σ σ σ σ σ2s

2s)

)2

2(

(σ σ σ σ σ σ σ σ2s

2s*)

*)2

2

( (σ σ σ σ σ σ σ σ2s

2s)

)2

2

( (σ σ σ σ σ σ σ σ1s

1s)

)2

2(

(σ σ σ σ σ σ σ σ1s

1s*)

*)2

2

( (σ σ σ σ σ σ σ σ1s

1s)

)2

2

( (σ σ σ σ σ σ σ σ2s

2s)

)2

2(

(σ σ σ σ σ σ σ σ2s

2s*)

*)2

2(

(σ σ σ σ σ σ σ σ2p

2pz z)

)2

2(

(π π π π π π π π2p

2p)

)4

4(

(π π π π π π π π2p

2p*)

*)4

4(

(σ σ σ σ σ σ σ σ2p

2pz z*)

*)2

2