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

Bonding

Bonding: H2+ and H2 molecules µ ( h h h ) 2 2 2 + = − ∇ − ∇ − ∇ e- 2 2 2 - H H 2 A B e 2 m 2 m 2 m rA rB A B e r 2 2 2 e e e − − + + + Q Q Q R r r R HA HB A B µ ( h h 2 2 ) = − ∇ − ∇ 2 2 H H 2 A B 2 m 2 m A B e- - h h 2 2 r1A r1B − ∇ − ∇ 2 2 e 1 e 2 r1 2 m 2 m e e + + HA HB R 2 2 2 2 e e e e r2A r2 − − − Q Q Q Q r2B r r r r 1 A 1 B 2 A 2 B - e- 2 2 e e + + Q Q r R 12

Born – Oppenheimer Approximation µ ( h h h 2 2 2 2 2 2 + = − ) e e e ∇ − ∇ − ∇ − − + 2 2 2 H H Q Q Q 2 A B e 2 2 2 m m m r r R A B e A B Nuclei are STATIONARY with respect to electrons µ ( h h h 2 2 2 2 2 2 + = − ) e e e ∇ − ∇ − ∇ − − + 2 2 2 H H Q Q Q 2 A B e 2 2 2 m m m r r R A B e A B ignore µ ( h 2 2 2 2 + = − ) e e e ∇ − − + 2 H H Q Q Q 2 e 2 m r r R e A B

Born – Oppenheimer Approximation ignore µ ( h h h h 2 2 2 2 ) = − ∇ − ∇ − ∇ − ∇ 2 2 2 2 H H 2 A B e 1 e 2 2 m 2 m 2 m 2 m A B e e 2 2 2 2 2 2 e e e e e e − − − + + Q Q Q Q Q Q r r r r r R 1 A 1 B 2 A 2 B 12 µ ( h h 2 2 2 2 2 2 2 2 e e e e e e ) = − ∇ − ∇ − − − + + 2 2 H H Q Q Q Q Q Q 2 e 1 e 2 2 m 2 m r r r r r R e e 1 A 1 B 2 A 2 B 12

Bonding: H2+ Molecule µ ( h 2 2 2 2 + = − ) e e e ∇ − − + 2 H H Q Q Q 2 e 2 m r r R e A B µ ( + × ) ψ = × ψ H H ( , r R ) E R ( ) ( , r R ) 2 Difficult; but can be solved using elliptical polar co-ordinates

Bonding: H2 molecule µ ( h h 2 2 2 2 2 2 2 2 e e e e e e ) = − ∇ − ∇ − − − + + 2 2 H H Q Q Q Q Q Q 2 e 1 e 2 2 m 2 m r r r r r R e e 1 A 1 B 2 A 2 B 12 µ ( ) ψ × = × ψ H H ( , r R ) E R ( ) ( , r R ) 2 CANNOT be Solved For all the molecules except the simplest molecule H2+ the Schrodinger equation cannot be solved. We have approximate solutions

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 ψ (1) A Ψ = ψ × ψ + ψ × ψ R= Re R= ∞ Ψ = ψ × ψ (1) (2) (2) (1) A B A B (1) (2) A B ψ Resonance (2) B H −− H ←→ H+ −− H − ←→ H − −− H+ Inclusion of Ionic terms ( ) ( ) Ψ = ψ × ψ + ψ × ψ + λ ψ × ψ + λ ψ × ψ A (1) B (2) A (2) B (1) A (1) A (2) B (1) B (2) Ψ = Ψ + Ψ λ + Ψ λ + − − + cov H H H H

Valance Bond Theory R= Re Ψ = Ψ + Ψ λ + Ψ λ + − − + cov H H H H

Molecular Orbital Theory of H2+ 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 L inear C ombination of A tomic O rbitals LCAO-MO e- - rA rB   h 2 2 2 2 e e e r − ∇ − − + ψ = × ψ 2  ÷ Q Q Q E + + e 2  m r r R  R e A B HA HB

Molecular Orbital Theory of H2+ e- -   h 2 2 2 2 rA e e e rB − ∇ − − + ψ = × ψ 2  ÷ Q Q Q E r e 2  m r r R  e A B + + R HA HB LCAO-MO ψ = φ + φ C C 1 1 2 1 MO s s A B ( ) 2 ψ = φ + φ + φ φ 2 2 2 2 C C 2 C C 1 1 2 1 1 2 1 1 MO s s s s A B A B Symmetry requirement = ⇒ = ± 2 2 C C C C 1 2 1 2

Molecular Orbital Theory of H2+ Symmetry requirement e- - = ⇒ = ± 2 2 C C C C rA rB 1 2 1 2 r + + R HA HB = − = = = C C C C C C 1 2 b 1 2 a ( ) ( ) ( ) ( ) ψ = φ − φ = − ψ = φ + φ = + 1 1 1 1 C C s s C C s s 2 b 1 s 1 s b A B 1 a 1 s 1 s a A B A B A B ( ) ( ) ψ = + ψ = − C 1 s 1 s C 1 s 1 s − Bonding a A B Anti bonding b A B + + + -

Molecular Orbital Theory of H2+ ( ) ( ) ψ = + ψ = − C 1 s 1 s C 1 s 1 s − Bonding a A B Anti bonding b A B

Bracket Notation = = 1 (for i j ) ∫ ∗ φ φ τ = φ φ = δ d = ≠ j i j ij 0 (for i j ) i allspace µ µ ∫ φ ∗ φ τ = φ φ = A d A A j i j ij i allspace

Normalization ( ) ( ) = ψ ψ = φ + φ φ + φ 2 1 C 1 1 a 1 s 1 s 1 s 1 s A B A B   = φ φ + φ φ + φ φ + φ φ 2 1 C   a 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s A A B B A B B A 1 [ ] = + 2 1 C 2 2 S a φ φ = = φ φ 1 1 1 s 1 s 1 s 1 s = A A B B C [ ] a + φ φ = = φ φ 2 2 S S 1 s 1 s 1 s 1 s A B B A Similarly 1 = S is called Overlap-Integral C [ ] b − 2 2 S

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

Molecular Orbital Theory of H2+ ( ) 1 ψ = φ + φ [ ] 1 1 s 1 s + A B 2 2 S ( ) 1 ψ = φ − φ [ ] 2 1 s 1 s − A B 2 2 S µ = ψ ψ E H 1 1 1 µ = ψ ψ E H 1 2 2

Molecular Orbital Theory of H2+ µ = ψ ψ E H 1 1 1 ( ) µ ( ) 1 1 = φ + φ φ + φ E H [ ] [ ] 1 1 s 1 s 1 s 1 s + + A B A B 2 2 S 2 2 S ) µ ( ] ( ) 1 = φ + φ φ + φ E H [ + 1 1 s 1 s 1 s 1 s 2 2 S A B A B 1 µ µ µ µ = φ φ + φ φ + φ φ + φ φ E H H H H [ ] + 1 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 2 2 S A A B B A B B A

Molecular Orbital Theory of H2+ µ = ψ ψ E H 2 2 2 ( ) µ ( ) 1 1 = φ − φ φ − φ E H [ ] [ ] 2 1 s 1 s 1 s 1 s − + A B A B 2 2 S 2 2 S ) µ ( ] ( ) 1 = φ − φ φ − φ E H [ − 2 1 s 1 s 1 s 1 s 2 2 S A B A B 1 µ µ µ µ = φ φ + φ φ − φ φ − φ φ E H H H H [ ] − 2 1 s 1 s 1 s 1 s 1 s 1 s 1 s 1 s 2 2 S A A B B A B B A

Molecular Orbital Theory of H2+ ( ) 1 ψ = φ + φ [ ] 1 1 s 1 s + A B 2 2 S ( ) 1 ψ = φ − φ [ ] 2 1 s 1 s − A B 2 2 S µ µ = ψ ψ = ψ ψ E H E H 1 1 1 1 2 2 1 µ µ µ µ = φ φ + φ φ + φ φ + φ φ E H H H H [ ] + 1 1 1 1 1 1 1 1 1 s s s s s s s s 2 2 S A A B B A B B A 1 µ µ µ µ = φ φ + φ φ − φ φ − φ φ E H H H H [ ] − 2 1 1 1 1 1 1 1 1 s s s s s s s s 2 2 S A A B B A B B A

Molecular Orbital Theory of H2+ 1 µ µ µ µ = φ φ + φ φ + φ φ + φ φ E H H H H [ ] + 1 1 1 1 1 1 1 1 1 s s s s s s s s 2 2 S A A B B A B B A 1 µ µ µ µ = φ φ + φ φ − φ φ − φ φ E H H H H [ ] − 2 1 1 1 1 1 1 1 1 s s s s s s s s 2 2 S A A B B A B B A + + µ µ 2 2 H H H H φ φ = = = φ φ = = H H H H ii ij ii ij E 1 s 1 s ii jj 1 s 1 s     1 + + i i j j 2 2 1 S S     ij ij µ µ φ φ = = = φ φ H H H H 1 s 1 s ij ji 1 s 1 s i j j i − − 2 H 2 H H H φ φ = = = φ φ S S = = ii ij ii ij E 1 s 1 s ij ji 1 s 1 s i j j i     2 − − 2 2 S 1 S     ij ij Ĥ is Hermitian

Molecular Orbital Theory of H2+ + + 2 2 H H H H = = ii ij ii ij E     1 + + 2 2 1 S S     ij ij − − 2 H 2 H H H = = ii ij ii ij E     2 − − 2 2 S 1 S     ij ij

Molecular Orbital Theory of H2+ h 2 2 2 2 e e e µ = − ∇ − − + 2 H Q Q Q e 2 m r r R e A B   h 2 2 2 2 e e e µ = − ∇ − − + 2  ÷ H Q Q Q e 2  m r  r R e A B 2 2 e e µ µ = − + H H Q Q 1 e r R B µ = = φ φ (or ) H H H H ii AA BB 1 s 1 s i i 1 1 µ = φ φ + φ φ − φ φ 2 2 H Qe Qe 1 e 1 s 1 s 1 s 1 s 1 s 1 s R r i i i i i i B

Molecular Orbital Theory of H2+ µ = = φ φ H (or H H ) H ii AA BB 1 s 1 s i i 1 1 µ = φ φ + φ φ − φ φ 2 2 H H Qe Qe 1 e ii 1 s 1 s 1 s 1 s 1 s 1 s R r i i i i i i B Constant 2 Qe 1 µ = φ φ + φ φ − φ φ at Fixed 2 H H Qe 1 e ii 1 s 1 s 1 s 1 s 1 s 1 s R r i i i i i i Nuclear B Distance 2 Qe = + − × 2 H E Qe J φ φ = 1 ii 1 s R 1 s 1 s i i 1 φ φ = J 1 s 1 s r i i B J ⇒ Coulomb Integral