Molecules and binding
Linus Pauling
Nobel 1954
The Nature of the Chemical Bond
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Max Born
Nobel 1954
Robert Oppenheimer
The Born-Oppenheimer Approximation
Molecules and binding
Stability
Variety
Complexity
due to chemical bonds
electromagnetic forces +
subtle quantum phenomena
lowering of energy if bond is formed
H2+
classical and static model:
+
+
eR/2
R/2
+ e2
e2
e2
−2
= −3
VCoulomb =
4πε 0 R
4πε 0 (R / 2 )
4πε 0 R
centered electron keeps
positively charged nuclei together
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Born-Oppenheimer Approximation
Nuclei fixed in a frame: use
Schrödinger equation:
A
k=
1
4πε 0
⎧⎪ h 2 2
ke 2
ke 2
ke 2 ⎫⎪
∇r − r
− r
+
⎬Ψe (r ) = ε e Ψe (r )
⎨−
m
r
R
r
R
R
2
/
2
/
2
+
−
⎪⎭
⎪⎩
e
T
Va
VR
Vb
B
Solutions, for the case of
only one potential:
φ1As (r − R / 2)
φ1Bs (r + R / 2)
φ1s
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
not exact
approximations
Symmetric and anti-symmetric wave functions
(
1 A
φ1s + φ1Bs
Define: Ψ+ (r ) =
2
)
and
Ψ− (r ) =
Two hydrogenic wave functions for 1s orbital:
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
(
1 A
φ1s − φ1Bs
2
)
φ ∝ e − a (r ± R / 2 )
Linear combinations of single 1s orbital solutions
(
)
(
(
)
)
1 A
Ψ+ (r ) =
φ1s + φ1Bs
2
Insert in SE:
ε = ε± =
(T + Va )φ A = ε1sφ A
2. Define:
(
)
)
(
)
)
(
Coulomb pot with B
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
)
(
)
2
2
φ
V
+
V
dV
=
φ
A
R
B
∫ b
∫ Va + VR dV = G
electron at A
1 A
φ1s − φ1Bs
2
(
(T + Vb )φB = ε1sφB
)
(
Ψ− (r ) =
1
A
B
A
B
φ
±
φ
[
T
+
V
+
V
+
V
]
φ
±
φ
dV =
a
b
R
∫
2∫
1
φ A ± φ B [(T + Va )φ A ± (T + Vb )φ B ]dV
∫
2
1
1
+ ∫ φ A2 Vb + VR dV + ∫ φ B2 Va + VR dV
2
2
1
1
± ∫ φ A Va + VR φ B dV ± ∫ φ B Vb + VR φ AdV
2
2
(
(
1.
and
)
solutions with hydrogen wave
functions
direct Coulomb energy
Evaluation of energies in H2+ system
(
)
1
A
B
[(T + Va )φ A ± (T + Vb )φB ]dV
φ
±
φ
∫
2
1
1
+ ∫ φ A2 Vb + VR dV + ∫ φ B2 Va + VR dV
2
2
1
1
± ∫ φ A Va + VR φ B dV ± ∫ φ B Vb + VR φ AdV
2
2
ε =
(
(
Define:
S
)
)
(
)
(
)
1
1
= ε1s + ε1s
2
2
1
1
= G+ G
2
2
1
1
=± S± S
2
2
∫ φ A (Va + VR )φB dV = ∫ φB (Vb + VR )φ AdV = S
- is a quantum interference, or an overlap integral
- it contributes if electron has density at A and B
- integral contributes if electron is near z=0
Energies in H2+:
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
ε = ε1s + G ± S
Bonding in a molecule as a result of S integral
Energies in H2+:
ε = ε1s + G ± S
Integral calculations show that:
S >G
One of the two gives binding
+S
interference integral is larger
or
−S
(can be calculated that S is negative, but)
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Ψ+ (r )
must be the bonding state
Ψ− (r )
is then higher in energy
by 2S
Bonding and anti-bonding in H2+
ε + = ε1s + G − S
σ* anti-bonding
ε − = ε1s + G + S
σ bonding
The Nature of the Chemical bond:
Wave function overlap of identical orbitals
give rise to chemical stability
Linus Pauling
Nobel 1954
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Molecular “Potential Energy Curves”
S integral is a function of R
Curves for electronic energies
Bound states in a potential well
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
From H2+ to H2, the hydrogen molecule
Problem is similar, but two electrons occupying the orbitals
ε = 2ε1s + G '± S '
triplet
singlet
sym
(φ1As − φ1Bs )χ spin
anti
(φ1As + φ1Bs )χ spin
Two electrons in a bonding orbital Æ chemical bond is stronger
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
Bonding in molecules
H2+
(σ1s)
H2
(σ1s)2
He2+
(σ1s)2(σ1s*)
one effective bond
He2
(σ1s)2(σ1s*)2
no effective bond
B2
(σ1s)2(σ1s*)2(σ2s)2(σ2s*)2(σ2p)2
C2
(σ1s)2(σ1s*)2(σ2s)2(σ2s*)2(σ2p)4
N2
(σ1s)2(σ1s*)2(σ2s)2(σ2s*)2(σ2p)6
O2
(σ1s)2(σ1s*)2(σ2s)2(σ2s*)2(σ2p)6(σ2p*)2
F2
(σ1s)2(σ1s*)2(σ2s)2(σ2s*)2(σ2p)6(σ2p*)4
Ne2
(σ1s)2(σ1s*)2(σ2s)2(σ2s*)2(σ2p)6(σ2p*)6
Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs
6 effective bonds:
The most strongly
bound molecule