chem 373- lecture 28: heteronuclear diatomic molecules
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Lecture 28: Heteronuclear Diatomic Molecules The material in this lecture covers the following in Atkins.
14.7 Heteronuclear diatomic molecules (a) Polar bonds (b)Electronegativity (c) The variation principle (d) Two simple cases
Lecture on-line Heteronuclear diatomic molecules (PowerPoint) Heteronuclear diatomic molecules (PDF)
Handout for this lecture
Heteronuclear diatomic molecules
A heteronuclear diatomic molecule is a diatomic molecule formed from atoms of two different elements CO HCl
The electron distribution in a covalentbond between the atoms is not evenly shared
Because it is energetically favorable for the electron pair to be closer to one atom than the other
The imbalance results in a polar covalent bond in which the electron pair is hared unequally by the atoms
The bond in HF, for instance, is polar , with the electron pair closer to F.
+ −−
δ δH F
H F
Heteronuclear diatomic molecules Polar BondIn homonuclear diatomics the atomic orbitals from the two nuclei contribute equally to the molecularorbitals :
1 of Hg 2σ : 11 1
2 1σg
A Bs sS
= ++( )
σg 2 of F :12 2
2 1σ σ σ
gA Bp p
S=
++( )
πu of N2 : 12 2
2 1π π π
uA Bp p
S=
++( )
In heteronuclear diatomics the atomic orbitals from the two nuclei do not contribute equally to the molecular orbitals :
1 of HAuσ :=+
+C s C s
SH H Au Au1 6
2 1( )
σ of FI : = ++
C p C pS
F I2 52 1σ σ( )
π of NP : =++
C p C p
SN N P P2 3
2 1π π
( )
The ability of an element to attract electrons is a measure of its : Electronegativity
Electronegativity according to Pauling :
χA = Electronegativity of A χB = Electronegativity of B
|
{
χ χA =
0.102 D(A -B) -12
D(A - A) -12
D(B -B)
−
×
B
AB bond energy AA bond energy
BB bond energy
Heteronuclear diatomic molecules
Electronegativity
Heteronuclear diatomic molecules
Electronegativity
Electronegativity according to Mulliken :
χA = Electronegativity of A
χA I EA ea= +12
( )
Ionization potential of A
Electron affinity of A
L earg value indicatesthat electron is difficult to remove poor electron donor
L earg value indicatesthat element is a good electron acceptor
Heteronuclear diatomic molecules Electronegativity
We have for the total energy of the trial wavefunction
= C + C A Bψ ψ ψA B
E =H dv
dv
ψ ψψ ψ
*
*∫∫
This energy is optimal if E
C and
EC
A B
δδ
δδ
= =0 0
We now express theenergy in terms of thecoefficients C and CA B
ψ ψ
ψ ψ ψ ψ
*
( )( )
dv
C C C C dvA A B B A A B B
=∫
+∫ +
= +∫ +
= + ∫∫
+ ∫
( )
(
C C C C dv
C dv C dv
C C dv
A A B B A B A B
A A B B
A B A B
2 2 2 2
2 2 2 2
2
2
ψ ψ ψ ψ
ψ ψ
ψ ψ
= + +C C C C SA B A B2 2 2
1 1 Overlap S
Linear variationHeteronuclear diatomic molecules
Heteronuclear diatomic moleculesψ ψ
ψ ψ ψ ψ
*
( ) ( )
H dv
C C H C C dvA A B B A A B B
=∫
+∫ +
= + ∫∫
+ ∫
C H dv C H dv
C C H dv
A A A B B B
A B A B
2 2
2
ψ ψ ψ ψ
ψ ψ )
Introducing
H dvA A Aα ψ ψ= ∫ α ψ ψB B BH dv= ∫ b β ψ ψ= ∫ A BH dv
Linear variation theory
ψ ψ α α β*H C C C CA A B B A B∫ = + +2 2 2
Heteronuclear diatomic molecules Linear variation theory
ψ ψ α α β*H C C C CA A B B A B∫ = + +2 2 2 E =
H dv
dv
ψ ψψ ψ
*
*∫∫
ψ ψ*∫ = + +C C C C SA B A B2 2 2
EnergyKinetic
of electron in orbital energy + attraction energy
from both nuclei + repulsion from all other electrons
Aψ : Always negative stabilizing
EnergyKinetic
of electron in orbital energy + attraction energy
from both nuclei+ repulsion fromall other electrons
Bψ : Always negative stabilizing
Heteronuclear diatomic molecules Linear variation theory
ψ ψ α α β*H C C C CA A B B A B∫ = + +2 2 2 E =
H dv
dv
ψ ψψ ψ
*
*∫∫
ψ ψ*∫ = + +C C C C SA B A B2 2 2
Overlap between and A Bψ ψ
Resonance integral represents energy of overlap density S
β
Heteronuclear diatomic molecules Linear variation theory
ψ ψ α α β*H C C C CA A B B A B∫ = + +2 2 2 E CA(
*
*,C ) =H dv
dvB
ψ ψψ ψ
∫∫
ψ ψ*∫ = + +C C C C SA B A B2 2 2
hus
E C C C C C C S
C C C
A B A B A B
A A B B A B
( , )[ ]2 2
2 2
2
2
+ + =
+ +α α β
This energy is optimal if E
C and
EC
A B
δδ
δδ
= =0 0
Heteronuclear diatomic molecules Linear variation theory
We shall now differentiate respect to Cand make use of that for the optimal C
AA
dE C C dC oA B A( , ) / =hus
dE C CdC
C C C C S E C C S
C C
A B
AA B A B A B
A A B
( , )[ ] [ ]2 2 2 2
2 2
+ + + +
+
2
α β
Thus
E C C C C C C S
C C C CA B A B A B
A A B B A B
( , )[ ]2 2
2 22
2
+ + =+ +α α β
o
rE C ES CA A B
:( ) ( )α β− + − = 0
Heteronuclear diatomic molecules Linear variation theory
We shall now differentiate respect to Cand make use of that for the optimal C
BB
dE C C dC oA B B( , ) / =hus
dE C CdC
C C C C S E C C S
C C
A B
BA B A B B A
B B A
( , )[ ] [ ]2 2 2 2
2 2
+ + + +
+
2
α β
hus
E C C C C C C S
C C C
A B A B A B
A A B B A B
( , )[ ]2 2
2 2
2
2
+ + =
+ +α α β
o
rE C ES CB B A
:( ) ( )α β− + − = 0
Heteronuclear diatomic molecules Linear variation theory We have derived :( ) ( )( ) ( )α ββ α
A A BA B B
E C ES CES C E C− + − =
− + − =00
This is a homogeneous linear equation in the unknown C and CA B
Homogeneousx A y
A x A y
eq :A11 + =
+ =1222
021 0
Has only solutions ifthe secular determinants :
AA
1121
0 A A
1222
=
Our secular equationreads
α ββ α
AB
-E -ES-ES -E
= 0
This gives us aquadratic equation inE from which we can determine the two roots E and E+ -
Here E corresponds tothe orbital of lower energyand E to the orbital of higher energy
+
-
Heteronuclear diatomic molecules Linear variation theory
A substitution of E = E intothe secular equation
+
( ) ( )( ) ( )α ββ α
A A BA B B
E C ES CES C E C− + − =
− + − =00
affords the coefficients C and CA
+B+
We thus obtain the orbital
energy E
+
+
ψ ψ ψ= ++ +C C
with
A A B B
A substitution of E = E into the secular equation
-
( ) ( )( ) ( )α ββ α
A A BA B B
E C ES CES C E C− + − =
− + − =00
affords the coefficients C and CA
-B-
We thus obtain the orbital
energy E
-
-
ψ ψ ψ= +− −C C
with
A A B B
Heteronuclear diatomic molecules Linear variation theory
αA
αB
E+
E−
ψ ψ ψ+ = ++ +C CA A B B
ψ ψ ψ- = +− −C CA A B B
This is an in - phasebonding orbital withthe largest contributionfrom of lowest energyAψ
This is an out - of-phase anti - bonding orbital withthe largest contribution from of highest energy
Bψ
Heteronuclear diatomic molecules Linear variation theory
ES
A+ = +
+α β1
αB
αA
αB −αA
Ε+
Ε−
ES
B− = −
−α β1
The solutions
1. or << 1A BA Bα α α α
β≈ −| |
| |
ψ ψ ψ+ = +12
12A B
ψ ψ ψ- = +12
12A B
αB
αA
αB −αA
Ε+
Ε−
Heteronuclear diatomic molecules Linear variation theory
ES
AA
A BA
A B+ = + −
−≈ +
−α β α
α αα β
α α( )2 2
The solutions
2. or << 1A BA B
α α βα α
<<−
| || |
ψ ψ β αα α
ψ+ = + −−A
A
A BB
S( )
ES
BB
A BB
A B− = − −
−≈ −
−α β α
α αα β
α α( )2 2
ψ ψ β αα α
ψ− = + −−B
B
A BA
S( )
αB
αA
αB −αA
Ε+
Ε−
Heteronuclear diatomic molecules Linear variation theory
E B+ = +α β ζcot
The general solution
ψ ψ ζ ψ ζ+ = +A Bcos sin
E A− = −α ζcot
ψ ψ ζ ψ ζ− = − +B Asin cos
ζ βα α
=−
12
2arctan
| |
B A
Heteronuclear diatomic molecules Linear variation theory
Molecular orbitals og HF
Molecular Orbital Theory Diatomics Term symbols
HD+
Molecule Configuration Term symbol
( )1 1σ 2Σ+
2 1ST +
SYM Lz( )
Spin multiplicity
Σ Π ∆
LTz : 0 1 2Reflection
Molecular Orbital Theory Diatomics Term symbols
HD
Molecule Configuration Term symbol
( )1 2σ Σ+
2 1ST +
SYM Lz( )
Spin multiplicity
Σ Π ∆
LTz : 0 1 2Reflection
HD− ( ) ( * )1 22 1σ σ 2Σ+
2 3He He ( ) ( *)1 22 2σ σ Σ+
LiNa ( ) ( *) ( )1 2 32 2 2σ σ σ Σ+ 2σ *
BeMg ( ) ( *) ( ) ( *)1 2 3 42 2 2 2σ σ σ σ Σ+σ1s s+
1s s−
3σ2 3s s+
4σ *2 3s s−
Molecular Orbital Theory Diatomics Term symbols
2 1ST +
SYM Lz( )
Spin multiplicity
Σ Π ∆
LTz : 0 1 2Reflection
Molecule Configuration Term symbol
BAl ( )1 2π
CSi ( )1 4π
3 Σ− 1∆+ 1Σ+
NP+ ( ) ( )5 11 4σ π
NP ( ) ( )5 12 4σ π 1Σ+
1Σ+
2 Σ+
1π
5σ
Molecular Orbital Theory Diatomics Term symbols
2 1ST +
SYM Lz( )
Spin multiplicity
Σ Π ∆
LTz : 0 1 2Reflection
Molecule Configuration Term symbol
5σ
π
6σ *
2πNO ( ) ( ) ( *)5 1 22 4 1σ π π 2Π− 2Π+
SO ( ) ( ) ( )5 1 22 4 2σ π π3Σ− ∆+ Σ−
FCl ( ) ( ) ( )5 1 22 4 4σ π π Σ−
Heteronuclear diatomic molecules
Metal - main - groupdiatomics
1σ1πx 1πy
δ1 δ2
2πx 2πy
2σ
Main groupMetal
Heteronuclear diatomic molecules Metal - Metal
diatomics
1σg
1πxu1πyu
1δg1 1δg2
1δu1 1δu2
1πxg1πyg
1σu
What you should learn from this lecture
1. You should be able to construct in qualitative terms the compositionsand energies of molecular orbitals for heteronuclear diatomicmolecules
2.You
C C C C
changes
A A B B A A B B
should be able to account for how the coefficients changes in and + +
as A becomes more electronegative than B
ψ ψ ψ ψ ψ ψ= + = ++ + + +
3. You should know qualitatively howE E behaves when A and B havethe same electronegativity and whenA is much more electronegative than B
+ - and
What you should learn from this lecture 4. You should be able to figure out the electron configuration for maingroupdiatomics as well as the term symbol(in simple cases)
5. It would be really cool if you also could account for the bonding in diatomics containing transition metals. But it is not required
6
00
.
( ) ( )( ) ( )
You are required to be able to derive the equations
you will not be askedto solve for E or and
general case
α ββ α
A A BA B B
A B
E C ES CES C E C
HoweverC C in
the
− + − =− + − =
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