Bonding Bonding: H2+ and H2 molecules ( h h h ) 2 2 2 + = - - PowerPoint PPT Presentation

Bonding

Bonding: H2+ and H2 molecules

+

• +

R HA HB rA rB r e-

+

• +

R HA HB r1A r1B r1 e-

• e-

r2B r2A r2

µ (

)

+ = −

∇ − ∇ − ∇ − − + h h h

2 2 2 2 2 2 2 2 2 2

2 2 2

A B e A B e A B

H H m m m e e e Q Q Q r r R

µ (

) = −

∇ − ∇ − ∇ − ∇ − − − + + h h h h

2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 1 1 2 2 2 2 12

2 2 2 2

A B A B e e e e A B A B

H H m m m m e e e e Q Q Q Q r r r r e e Q Q r R

Born – Oppenheimer Approximation Nuclei are STATIONARY with respect to electrons ignore

µ (

)

+ = −

∇ − ∇ − ∇ − − + h h h

2 2 2 2 2 2 2 2 2 2

2 2 2

A B e A B e A B

e e e H H Q Q Q m m m r r R

µ (

)

+ = −

∇ − − + h

2 2 2 2 2 2

2

e e A B

e e e H H Q Q Q m r r R

µ (

)

+ = −

∇ − ∇ − ∇ − − + h h h

2 2 2 2 2 2 2 2 2 2

2 2 2

A B e A B e A B

e e e H H Q Q Q m m m r r R

Born – Oppenheimer Approximation ignore

µ (

) = −

∇ − ∇ − ∇ − ∇ − − − + + h h h h

2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 1 1 2 2 12

2 2 2 2

A B e e A B e e A B A B

H H m m m m e e e e e e Q Q Q Q Q Q r r r r r R

µ (

) = −

∇ − ∇ − − − + + h h

2 2 2 2 2 2 2 2 2 2 2 1 2 1 1 2 2 12

2 2

e e e e A B A B

e e e e e e H H Q Q Q Q Q Q m m r r r r r R

Bonding: H2+ Molecule Difficult; but can be solved using elliptical polar co-ordinates

µ (

)

+ = −

∇ − − + h

2 2 2 2 2 2

2

e e A B

e e e H H Q Q Q m r r R

µ (

) ψ ψ

+ ×

= ×

2

( , ) ( ) ( , ) H H r R E R r R

Bonding: H2 molecule CANNOT be Solved For all the molecules except the simplest molecule H2+ the Schrodinger equation cannot be solved. We have approximate solutions

µ (

) ψ

ψ × = ×

2

( , ) ( ) ( , ) H H r R E R r R

µ (

) = −

∇ − ∇ − − − + + h h

2 2 2 2 2 2 2 2 2 2 2 1 2 1 1 2 2 12

2 2

e e e e A B A B

e e e e e e H H Q Q Q Q Q Q m m r r r r r R

Bonding For all the molecules except the simplest molecule H2+ the Schrodinger equation cannot be solved. We have only approximate solutions Valance-Bond Theory & Molecular Orbital Theory are two different models that solve the Schrodinger equation in different methods

Valance Bond Theory R=∞ R= Re Resonance H−−H ←→ H+−−H− ←→ H− −−H+ Inclusion of Ionic terms ψ ψ Ψ = ×

(1) (2) A B

ψ ψ ψ ψ Ψ = × + ×

(1) (2) (2) (1) A B A B

ψ

(1) A

ψ

(2) B

( ) ( )

ψ ψ ψ ψ λ ψ ψ λ ψ ψ λ λ

+ − − +

Ψ = × + × + × + × Ψ = Ψ + Ψ + Ψ

(1) (2) (2) (1) (1) (2) (1) (2) cov A B A B A A B B H H H H

Valance Bond Theory R= Re λ λ

+ − − +

Ψ = Ψ + Ψ + Ψ

cov H H H H

+

• +

R HA HB rA rB r e- A molecular orbital is analogous concept to atomic orbital but spreads throughout the molecule It’s a polycentric one-electron wavefunction (Orbital!) It can be produced by Linear Combination of Atomic Orbitals LCAO-MO Molecular Orbital Theory of H2+ ψ ψ   − ∇ − − + = ×  ÷   h

2 2 2 2 2

2

e e A B

e e e Q Q Q E m r r R

+

• +

R HA HB rA rB r e- LCAO-MO Molecular Orbital Theory of H2+ ψ ψ   − ∇ − − + = ×  ÷   h

2 2 2 2 2

2

e e A B

e e e Q Q Q E m r r R ψ φ φ = +

1 1 2 1

A B

MO s s

C C

( )

ψ φ φ φ φ = + +

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

2

A B A B

MO s s s s

C C C C = ⇒ = ±

2 2 1 2 1 2

Symmetry requirement C C C C

+

• +

R HA HB rA rB r e- + + +

• Molecular Orbital Theory of H2+

= ⇒ = ±

2 2 1 2 1 2

Symmetry requirement C C C C ( )

( )

ψ φ φ = = = + = +

1 2 1 1 1

1 1

A B

a a s s a A B

C C C C C s s ( )

( )

ψ φ φ = − = = − = −

1 2 2 1 1

1 1

A B

b b s s b A B

C C C C C s s

( )

ψ = + 1 1

Bonding a A B

C s s

( )

ψ

−

= − 1 1

Anti bonding b A B

C s s

Molecular Orbital Theory of H2+

( )

ψ = + 1 1

Bonding a A B

C s s

( )

ψ

−

= − 1 1

Anti bonding b A B

C s s

Bracket Notation

µ µ

φ φ τ φ φ δ φ φ τ φ φ

∗ ∗

= = = =

∫ ∫

i i

j i j ij allspace j i j ij allspace

d A d A A

= = = ≠ 1 (for ) 0 (for ) i j i j

Normalization 1 S is called Overlap-Integral ( ) ( )

[ ] [ ] [ ]

ψ ψ φ φ φ φ φ φ φ φ φ φ φ φ = = + +   = + + +   = + = + = −

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

1 1 1 2 2 1 2 2 Similarly 1 2 2

A B A B A A B B A B B A

a s s s s a s s s s s s s s a a b

C C C S C S C S φ φ φ φ φ φ φ φ = = = =

1 1 1 1 1 1 1 1

1

A A B B A B B A

s s s s s s s s

S

Overlap Integral Overlap-Integral S can be positive or negative or zero

Molecular Orbital Theory of H2+

[ ]

( )

[ ]

( ) ψ φ φ ψ φ φ = + + = − −

1 1 1 2 1 1

1 2 2 1 2 2

A B A B

s s s s

S S

µ µ

ψ ψ ψ ψ = =

1 1 1 1 2 2

E H E H

Molecular Orbital Theory of H2+

µ

[ ]

( ) µ

[ ]

( )

[ ] (

) µ ( )

[ ]

µ µ µ µ

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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

A B A B A B A B A A B B A B B A

s s s s s s s s s s s s s s s s

E H E H S S E H S E H H H H S

Molecular Orbital Theory of H2+

µ

[ ]

( ) µ

[ ]

( )

[ ] (

) µ ( )

[ ]

µ µ µ µ

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

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

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

A B A B A B A B A A B B A B B A

s s s s s s s s s s s s s s s s

E H E H S S E H S E H H H H S

Molecular Orbital Theory of H2+

[ ]

( )

[ ]

( ) ψ φ φ ψ φ φ = + + = − −

1 1 1 2 1 1

1 2 2 1 2 2

A B A B

s s s s

S S

µ µ

ψ ψ ψ ψ = =

1 1 1 1 2 2

E H E H

[ ]

µ µ µ µ

φ φ φ φ φ φ φ φ = + − − −

2 1 1 1 1 1 1 1 1

1 2 2

A A B B A B B A

s s s s s s s s

E H H H H S

[ ]

µ µ µ µ

φ φ φ φ φ φ φ φ = + + + +

1 1 1 1 1 1 1 1 1

1 2 2

A A B B A B B A

s s s s s s s s

E H H H H S

Molecular Orbital Theory of H2+ Ĥ is Hermitian

[ ]

µ µ µ µ

φ φ φ φ φ φ φ φ = + + + +

1 1 1 1 1 1 1 1 1

1 2 2

A A B B A B B A

s s s s s s s s

E H H H H S

µ µ µ µ

φ φ φ φ φ φ φ φ φ φ φ φ = = = = = = = = =

1 1 1 1 1 1 1 1 1 1 1 1

i i j j i j j i i j j i

s s ii jj s s s s ij ji s s s s ij ji s s

H H H H H H H H S S

[ ]

µ µ µ µ

φ φ φ φ φ φ φ φ = + − − −

2 1 1 1 1 1 1 1 1

1 2 2

A A B B A B B A

s s s s s s s s

E H H H H S + + = =     + +     − − = =     − −    

1 2

2 2 2 2 1 2 2 2 2 1

ii ij ii ij ij ij ii ij ii ij ij ij

H H H H E S S H H H H E S S

Molecular Orbital Theory of H2+ + + = =     + +     − − = =     − −    

1 2

2 2 2 2 1 2 2 2 2 1

ii ij ii ij ij ij ii ij ii ij ij ij

H H H H E S S H H H H E S S

Molecular Orbital Theory of H2+

µ µ µ µ

= − ∇ − − +   = − ∇ − − +  ÷   = − + h h

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

2 2

e e A B e e A B e B

e e e H Q Q Q m r r R e e e H Q Q Q m r r R e e H H Q Q r R

µ µ

φ φ φ φ φ φ φ φ = = = + −

1 1 2 2 1 1 1 1 1 1 1

(or ) 1 1

i i i i i i i i

ii AA BB s s e s s s s s s B

H H H H H Qe Qe R r

Molecular Orbital Theory of H2+ Constant at Fixed Nuclear Distance J ⇒ Coulomb Integral

µ µ µ

φ φ φ φ φ φ φ φ φ φ φ φ φ φ = = = + − = + − = + − ×

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

(or ) 1 1 1

i i i i i i i i i i i i i i

ii AA BB s s e ii s s s s s s B e ii s s s s s s B ii s

H H H H H H Qe Qe R r Qe H H Qe R r Qe H E Qe J R φ φ φ φ = =

1 1 1 1

1 1

i i i i

s s s s B

J r

Molecular Orbital Theory of H2+ K ⇒ Exchange Integral Resonance Integral Constant K is purely a quantum mechanical

• concept. There is no classical

counterpart

µ µ µ

φ φ φ φ φ φ φ φ φ φ φ φ φ φ = = = + − = + − = + − ×

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

(or ) 1 1 1

i j i j i j i j i j i j i j

ij AB BA s s e ij s s s s s s B e ij s s s s s s B ij s

H H H H H H Qe Qe R r Qe H H Qe R r Qe H E S S Qe K R φ φ φ φ = =

1 1 1 1

1

i j i j

s s s s B

S K r

Molecular Orbital Theory of H2+

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

+       = = + − + + −  ÷  ÷   +   +           = + + + − +   +   + = + − + −       = = + − − − −  ÷  ÷   −   −         = − −

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

ii ij s s ij s s ii ij s s ij s

H H S E E Qe J E S Qe K S R R S Qe E E S S Qe J K S R Qe J K Qe E E R S H H S E E Qe J E S Qe K S R R S E E S S

[ ] [ ] [ ] [ ]

  + − − −     − = + − −

2 2 2 2 2 1

1 1

s

Qe S Qe J K R Qe J K Qe E E R S

Molecular Orbital Theory of H2+

[ ]

( )

[ ] [ ] [ ]

( )

[ ] [ ]

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

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

1 2 2 1 1 2 2 1

A B A B

s s s s s s

S Qe J K Qe E E R S S Qe J K Qe E E R S

Molecular Orbital Theory of H2+ Destabilization of Anti-bonding orbital is more than Stabilization of Bonding orbital J - Coulomb integral - interaction of electron in 1s orbital around A with a nucleus at B K - Exchange integral – exchange (resonance) of electron between the two nuclei.

[ ] [ ] [ ] [ ]

+ = + − + − = + − − ≤ ≤ < <

2 2 1 1 2 2 2 1

1 1 1 ; 0 &

s s

Qe J K Qe E E R S Qe J K Qe E E R S S J K

Molecular Orbital Theory of H2+

E1

1

E2

Sigma Bonding with 1s Orbitals

Bonding with 2p Orbitals Note the signs, symmetries and nodes