Diatomic Molecules
Chem 2430
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Diatomic Molecules: Nuclear and Electronic Interactions, Lecture notes of Chemistry
The interactions between nuclei and electrons in diatomic molecules using the Born-Oppenheimer approximation. Topics include potential energy curves, rigid rotor and harmonic oscillator approximations, centrifugal distortion, and vibrational-rotational coupling. The document also discusses the use of a minimal basis set in a LCAO approach.
What you will learn
- How does a minimal basis set LCAO approach help in understanding the ground state of diatomic molecules?
- What is vibrational-rotational coupling and how does it arise in diatomic molecules?
- What is the Born-Oppenheimer approximation and how is it used to study diatomic molecules?
- What is centrifugal distortion and how does it affect the rotational terms in diatomic molecules?
- What are the rigid rotor and harmonic oscillator approximations in the context of diatomic molecules?
Typology: Lecture notes
2021/2022
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Diatomic Molecules
Chem 2430
2
2
2
2
2
2
ˆ^2
i^2 i^
i^
j^ i
e^
B^
i^
ij
Z Z e^
Z e^
e
H^
m^
m^
r^
r^
r
^
^
^
^
^ ^
^ ^
^
^
^
^
^
^
,^ :i j^
,^ : ^ nuclei
electrons
Born-Oppenheimer approximation
2
2
2
2
ˆ^1 e^2
i i
i^
j^ i
e^
i^
ij
Z e
h^
e
H^
m^
r^
r
^ ^
^
^
Assumes that we can separate the electronic andnuclear degrees of freedom, since electrons movemuch faster than nuclei
For fixed nuclear positions
^
^
1 1
1 1
ˆ^
i^
i^
i e^ e^
e^ e
H^
E
v = 0^
De
U
2
ˆ^
(^ )
N^2
H^
U R
^ m ^
^
Full^ non
‐relativistic
H^ for
molecules
( )^
( ) 1
(^ )^
(^ )^
(^ )
i^
i e
NN
U R^
E^ R
V^
R
^
^
Here^ R
is^ used
to^ represent
the
collection
of^ all^
nuclear
positions
v = 1
D^0 R
Potential
energy
curve
of
a^ diatomic
molecule
V^ representsNN^
the^ nuclear repulsion
^ ^
^
^ ^
^
^
^
^
^
2
2
2
2 2 e 2
2
3
R
e
e
e^
e
J^ J^
R^ R
J^ J^
J^ J
U^ R^
U^
k^ R^
R
R^
R^
R
^
^
^
^
^
^
^
^
^
^
If we keep only the first three terms we obtain
^ ^
1 ^ v^
1 2 e^
e
E^ U^
R^
J^ J^
B ^
^ ^
^
^
^ ^
i.e., the rigid rotor plus harmonic oscillator
^
^ 1 ^202 e
J^ J e
dU^ k
R^ R
dR^
R
^
^
^
(^2)
min^
e
J^ J e
R^
R^
R^
k
^
^
However, due to the last term in the Eq. for
U(R),
the potential energy minimum is shifted from R
.e
To find the new minimum calculate dU/dRSubstituting back into the rotational terms gives
^ ^
(^22 2) (^42) 2
3 1
1 2
2 e^
e J^ J^
J^ J R^
R^ k
^
^
^
Centrifugal distortion
^
(^1) v
e
J^ J
^ ^ R
^
^
The vibrational/rotational couplingcan be interpreted as a v-dependentrotational constant, B
v^ e^
e
B^ B
^ v
^
^
^
If we keep the next term (quadratic in R-R
) in the series expansion of the rotational,e
this leads to a modified force constant for the vibration, and leads to vibrational-rotational coupling:^ Actually
the^ coefficient
of^ the
(v^ +^ ½)J(J+1)
is supposed to be negativeThere is another contribution whicharises from the cubic term in the potential
1 |^
AA^
A^
A
H^
s^ H^
s ^
1 |^
BB^
B^
B
H^
s^ H^
s
1 |^
AB^
A^
B
H^
s^ H^
s
^
1 |1AB A^
B
S^
s^ s
For H^2
+^ AA
H^ H^ BB
(if we were considering
(^2)
,^
AA^
BB
HeH^
H^
H
^
kR
S^ e^ AB
kR^
k R
^
^
^
^
Let
3
krA,
ks A
e
3
krB
ks B
e
^
2
2
1
kR
H^ AA
k^ k^
e^ k
R
R
^
^ ^
^
^
kR
AB^
AB H^
k S^
k^ k
kR e
^
^
For^ derivation
see http://www2.chem.umd.edu/groups/alexander/chem691/Chap3.pdf
^
^
^
^
2
2
2
2 2
kR^
kR
kR
k^ k^
kR e^
k^ k^
kR e
R^ R
E^
k^
k R
e^
^ kR
^
^ ^
^
^
^
^
^
^
^
As^
R^
E^
k^ k^
k
^
^
12 k^ k ^2
Note if
1,^
k^
E
which is just the energy of anH atom
U^ E
R
For^
k^ goes from
1 at^ R =
∞^ to^2
at^ R = 0
at^ R,^ (~ 2.0 Bohrs)e
k = 1.
E
U^ and^ E
for^ theel
ground
state^ of
+ H 2
(from^ Levine)
k^ =^2 in
the^ R^ ‐
^0 limit from^ the
above eq.
R^ ^0 R
H^
H
He
U^
E^ ^
U^ 2 E^
E at minimum = -1.1033auU at minimum= -0.6026au
(D
= 2.79eV)e
1 ^ g 1 ^ u^11 bondinganti-bonding ^ g ^ u
1 1 A^
B
s^ s 1 1 A^
B
s^ s
in the separated atom limit
^1 He s^ ^ ^2 He p^