, Molecular Orbital Theory
The molecular hydrogen ion
* simplest ,
stable molecular system
↳ 2 protons and I e
* to solve , ser up Hamiltonian + love
↳ Use diff Co-ords than for H arom >
-
elliptical nor spherical
Born-Oppenheimer approximation
A nuclear repulsion term that depends on invernuclear separation , R
↳ assume that nuclei more much more slowly than e and fix R
↳ then solve electronic Schrodinger eq for . each D
↑ =
- -un
Calculating the energy
* Integrals involving the Hamiltonian
solve this for each
val .
of R + fir
curve
through
=Sindr Yi *
-energy
all vals .
Ei funcionis energy
LCAO approximation
A Add aromic wavefunctions to
vogether make molecular
Y+ =
N + (c , 1st +
C1Sp)
Y .
= N - (C :1Sa ,
-
G'1sp)
Normalisation
* wavefunctious much
comply wI Born Interpretation of warefanc .
↳ ones that comply = normalised
denom in energy func.
Sinc Yi(r)
>
-
*
when i
Yj(rldr = 1 =
j
Normalisation constant
* If integral #1 but = another constant insread find scaling factor (normalization constant
Sirs Ya (r)
*
YB(r)dr =
C = 1/N
Y
N2 Sin3 YA(r)
*
UB(r)dr = 7 =
Sir > NUa(r)
*
NYp(r]dr
Overlap integral
A aromic 4 on diff aromic centres are NOT
ourhogonal
↳ leads to overlap S- R 1
invegral ,
depends on distance apart the aroms are
S =
Sir Ya (r)
*
Yp(r)drc1
, Using LCAO method
* Assume U made 2 4
is up of hydrogenIs
Find normalisation conse
*
by solving denom .
Y+ N + (15a
=
=
+ 154)
it (1Sa
7)
+
- exchange integral
41mb +
Y (1Sa 1sp)
.
= N -
Ksa-1se)
-
U
=
* Wavefunction + conjugate changed in
exchange inverval /resonance integral
* plor values of 2 invegrals as function or distance
want-re E
>
- >
- -
j K
E+ -
Exs = DE = +
(1 + S) (1 5) +
superposition or aromic
↑formmoleculaa
T
* Coulomb integral =
min .
at infinite separation i. e. . I separated aroms
superposition + exchange
* Exchange invegual :
may have min. .
I than 2 aroms -> can stabilise molecule invegral
chem
=
bond
source of
I ... e . i .
=
source of Chem . bond
* in u amplitude bown 2 aroms (ae density) is a feature of chem bond
(bonding orbital M145/
I
* indicates antibonding orginal
,
molecule is decrabilised 11 IVA
Diatomic Molecules
s-orbital overlap
* overlapping any pair of s-orbitals =
Sigma m . .
o
* add e is 1st
, energy
A anribonding -destabilise molecule
Homonuclear diatomics & parity
* homonuclear diatomics 168168 and 178 160
charge
: molecules we drome of same e
.
g .
↳ label u loubinals) in terms of their
parity
*
parily :
effect on U on
inverting axis >
- either invers
Sign of Y (4) or leaves in same
(g)
Overlapping wavefunctions
* bond order = 112 (no · of bonding e--no . Of anribonding ef
↳ if bo .
= 0 >
-
molecule only held by weak
long-range vow's forces
* symmetric U = ↓ in energy for 2e mo
Other sigma bonds
constructive
* 6 bonds =
volationally symmetric about bond axis
A form them s , p , d etc . Orbitals that lie bond axis
wi along destructive
A orbitals wI'Z eg P2 d2
generally ..
,
The molecular hydrogen ion
* simplest ,
stable molecular system
↳ 2 protons and I e
* to solve , ser up Hamiltonian + love
↳ Use diff Co-ords than for H arom >
-
elliptical nor spherical
Born-Oppenheimer approximation
A nuclear repulsion term that depends on invernuclear separation , R
↳ assume that nuclei more much more slowly than e and fix R
↳ then solve electronic Schrodinger eq for . each D
↑ =
- -un
Calculating the energy
* Integrals involving the Hamiltonian
solve this for each
val .
of R + fir
curve
through
=Sindr Yi *
-energy
all vals .
Ei funcionis energy
LCAO approximation
A Add aromic wavefunctions to
vogether make molecular
Y+ =
N + (c , 1st +
C1Sp)
Y .
= N - (C :1Sa ,
-
G'1sp)
Normalisation
* wavefunctious much
comply wI Born Interpretation of warefanc .
↳ ones that comply = normalised
denom in energy func.
Sinc Yi(r)
>
-
*
when i
Yj(rldr = 1 =
j
Normalisation constant
* If integral #1 but = another constant insread find scaling factor (normalization constant
Sirs Ya (r)
*
YB(r)dr =
C = 1/N
Y
N2 Sin3 YA(r)
*
UB(r)dr = 7 =
Sir > NUa(r)
*
NYp(r]dr
Overlap integral
A aromic 4 on diff aromic centres are NOT
ourhogonal
↳ leads to overlap S- R 1
invegral ,
depends on distance apart the aroms are
S =
Sir Ya (r)
*
Yp(r)drc1
, Using LCAO method
* Assume U made 2 4
is up of hydrogenIs
Find normalisation conse
*
by solving denom .
Y+ N + (15a
=
=
+ 154)
it (1Sa
7)
+
- exchange integral
41mb +
Y (1Sa 1sp)
.
= N -
Ksa-1se)
-
U
=
* Wavefunction + conjugate changed in
exchange inverval /resonance integral
* plor values of 2 invegrals as function or distance
want-re E
>
- >
- -
j K
E+ -
Exs = DE = +
(1 + S) (1 5) +
superposition or aromic
↑formmoleculaa
T
* Coulomb integral =
min .
at infinite separation i. e. . I separated aroms
superposition + exchange
* Exchange invegual :
may have min. .
I than 2 aroms -> can stabilise molecule invegral
chem
=
bond
source of
I ... e . i .
=
source of Chem . bond
* in u amplitude bown 2 aroms (ae density) is a feature of chem bond
(bonding orbital M145/
I
* indicates antibonding orginal
,
molecule is decrabilised 11 IVA
Diatomic Molecules
s-orbital overlap
* overlapping any pair of s-orbitals =
Sigma m . .
o
* add e is 1st
, energy
A anribonding -destabilise molecule
Homonuclear diatomics & parity
* homonuclear diatomics 168168 and 178 160
charge
: molecules we drome of same e
.
g .
↳ label u loubinals) in terms of their
parity
*
parily :
effect on U on
inverting axis >
- either invers
Sign of Y (4) or leaves in same
(g)
Overlapping wavefunctions
* bond order = 112 (no · of bonding e--no . Of anribonding ef
↳ if bo .
= 0 >
-
molecule only held by weak
long-range vow's forces
* symmetric U = ↓ in energy for 2e mo
Other sigma bonds
constructive
* 6 bonds =
volationally symmetric about bond axis
A form them s , p , d etc . Orbitals that lie bond axis
wi along destructive
A orbitals wI'Z eg P2 d2
generally ..
,
