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o  Hydrogen molecule ion (H2+)

o  Overlap and exchange integrals

o  Bonding/Anti-bonding orbitals

o  Molecular orbitals

o  Chapter 24 of Haken & Wolf Chapter 11 of Atkins

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o  Simplest example of a chemical bond is the hydrogen molecule ion (H2

+).

o  Consists of two protons and a single electron.

o  If nuclei are distant, electron is localised on one nucleus. Wavefunctions are then those of atomic hydrogen.

rab

rb ra

+ +

-

a b

!

"!2

2m#2 "

e2

4$%0ra

&

' (

)

* + ,a (ra ) = Ea

0(ra ),a (ra )

o  !a is the hydrogen atom wavefunction of electron belonging to nucleus a. Must therefore satisfy

and correspondingly for other wavefunction, !b. The energies are therefore Ea

0 = Eb

0 = E0

(1)

Ha

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o  If atoms are brought into close proximity, electron localised on b will now experience an attractive Coulomb force of nucleus a.

o  Must therefore modify Schrödinger equation to include Coulomb potentials of both nuclei:

where

o  To find the coefficients ca and cb, substitute into Eqn. 2:

!

"!2

2m#2 "

e2

4$%0ra"

e2

4$%0rb

&

' (

)

* + , = E,

!

" = ca#a + cb#b

(2)

!

"!2

2m#2 "

e2

4$%0ra"

e2

4$%0rb

&

' (

)

* + ca,a +

"!2

2m#2 "

e2

4$%0rb"

e2

4$%0ra

&

' (

)

* + cb,b = E(ca,a + cb,b )

Ha

Hb

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o  Can simplify last equation using Eqn. 1 and the corresponding equation for Ha and Hb.

o  By writing Ea0 !a in place of Ha !a gives

o  Rearranging,

o  As Ea0 = Eb

0 by symmetry, we can set Ea0 - E = Eb

0 - E = -!E,

!

Ha "e2

4#$0rb

%

& '

(

) * ca+a + Hb "

e2

4#$0ra

%

& '

(

) * cb+b = E(ca+a + cb+b )

!

Ea0 "

e2

4#$0rb

%

& '

(

) * ca+a + Eb

0 "e2

4#$0ra

%

& '

(

) * cb+b = E(ca+a + cb+b )

!

Ea0 " E " e2

4#$0rb

%

& '

(

) * ca+a + Eb

0 " E " e2

4#$0ra

%

& '

(

) * cb+b = 0

!

"#E " e2

4$%0rb

&

' (

)

* + ca,a + "#E " e2

4$%0ra

&

' (

)

* + cb,b = 0 (3)

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o  !a and !b depend on position, while ca and cb do not. Now we know that for orthogonal wavefunctions

but !a and !b are not orthoganal, so

o  If we now multiply Eqn. 3 by !a* and integrate the results, we obtain

o  As -e|!a|2 is the charge density of the electron => Eqn. 4 is the Coulomb interaction energy between electron charge density and nuclear charge e of nucleus b.

o  The -e!a !b in Eqn. 5 means that electron is partly in state a and partly in state b => an exchange between states occurs. Eqn. 5 therefore called an exchange integral.

!

"a*# "adV = "b

*# "bdV =1 Normalisation integral

!

"a*# "bdV = S(rab ) Overlap integral

!

"a*# (ra ) $

e2

4%&0rb

'

( )

*

+ , "a(ra )dV = C(rab )

"a*# (ra ) $

e2

4%&0ra

'

( )

*

+ , "b(rb )dV = D(rab )

(4)

(5)

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o  Eqn. 4 can be visualised via figure at right.

o  Represents the Coulomb interaction energy of an electron density cloud in the Coulomb field on the nucleus.

o  Eqn. 5 can be visualised via figure at right.

o  Non-vanishing contributions are only possible when the wavefunctions overlap.

o  See Chapter 24 of Haken & Wolf for further details.

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o  If we mutiply Eqn 3 by !a* and integrate we get

o  Collecting terms gives,

o  Similarly,

o  Eqns. 6 and 7 can be solved for c1 and c2 via the matrix equation:

o  Non-trivial solutions exist when determinant vanishes:

!

"a* #$E # e2

4%&0rb

'

( )

*

+ , ca"a + #$E # e2

4%&0rb

'

( )

*

+ , cb"b

-

. /

0

1 2 3 dV

!

"b* #$E # e2

4%&0rb

'

( )

*

+ , ca"a + #$E # e2

4%&0rb

'

( )

*

+ , cb"b

-

. /

0

1 2 3 dV

!

("#E + C)ca + ("#E $ S + D)cb = 0

!

=> ("#E $ S + D)ca + ("#E + C)cb = 0

(6)

(7)

!

"#E + C "#E $ S + D"#E $ S + D "#E + C%

& '

(

) * cacb

%

& '

(

) * = 0

!

"#E + C "#E $ S + D"#E $ S + D "#E + C

= 0

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o  Therefore (-"E + C)2 - (-"E·S + D)2 = 0

=> C - "E = ± (D - "E ·S)

o  This gives two values for "E: and

o  As C and D are negative, "Eb < "Ea

o  "Eb correspond to bonding molecular

orbital energies. "Ea to anti-bonding MOs.

o  H2+ atomic and molecular orbital energies

are shown at right.

!

"Ea =C #D1# S

!

"Eb =C + D1+ S

Energy

(eV)

-13.6

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o  Substituting for "Ea into Eqn. 6 => cb = -ca = -c.The total wavefunction is thus

o  Similarly for "Eb => ca = cb = c, which gives

o  For symmetric case (top right), occupation probability for # is positive. Not the case for a asymmetric wavefunctions (bottom right).

o  Energy splits depending on whether dealing with a bonding or an anti-bonding wavefunction.

!

"# = c($a #$b )

!

"+ = c(#a + #b )

Anti-bonding orbital

Bonding orbital

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o  The energy E of the hydrogen molecular ion can finally be written

o  ‘+’ correspond to anti-bonding orbital energies, ‘-’ to bonding.

o  Energy curves below are plotted to show their dependence on rab !

E = E 0 + "E = E 0 +C ± D1± S

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o  Energies of bonding and anti-bonding molecular orbitals for first row diatomic molecules.

o  Two electrons in H2 occupy bonding molecular orbital, with anti-parallel spins. If irradiated by UV light, molecule may absorb energy and promote one electron into its anti-bonding orbital.

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