Molecular Orbital (MO) Theory of the H2 molecule:
Molecular Orbital (MO) Theory of the H2 molecule:
Following the MO treatment of H2+, assume the (normalized) ground electronic
state wavefunction is given by:
state wavefunction is given by: Ψ gs = ψ + (1)ψ + (2) [α (1) β (2) − β (2)α (1) ] / 2
with:
ψ + (1)ψ + (2) =
1
G G
[1sA (1) + 1sB (1)][1sA (2) + 1sB (2)] ≡ ψ (r1 , r2 )
2(1 + S )
Evaluate the ground state electronic energy based on this presumed
(approximate) eigenfunction:
(approximate) eigenfunction:
j 2 j '+ 2k ' j + 2k + m + 4l
G G G G
G G
Egs = ∫ dr1dr2ψ (r1 , r2 ) H elψ (r1 , r2 ) = 2 E1s + 0 −
+
R
1+ S
2(1 + S ) 2
with:
ih
j0 ≡
e02
4πε 0
… and j’(R), k’( R), m(R), l(R) defined
explicitly in Atkins
explicitly in Atkins.
Here is the result obtained
For E_gs (R) via Eq. [1]:
Configuration Interaction:
Within the two basis orbital approximation, the following electronic
configurations are possible: 1) Two electrons in Molecular Orbital ψ_+
2) One electron in MO ψ_+ and one electron in MO ψ_‐
3) Two electrons in MO ψ_‐.
σ g 1s or σ 1s
Point of notation:
σ u 1s or σ *1s
Spin‐orbitals of type 1 and 3 have the same symmetry, and therefore
can “mix” (to give improved wavefunctions and energy eigenvalues):
Ψ1 = ψ + (1)ψ + (2) [α (1) β (2) − β (2)α (1) ] / 2
Ψ 2 = ψ + (1)ψ + (2) [α (1) β (2) − β (2)α (1)] / 2
Consider the trial wavefunction:
Ψ tr = c1Ψ1 + c2 Ψ 2
[1]
Inserting this trial function into the Raleigh‐Ritz Variational Principle …
we will determine (c1,c2) that minimize the expectation value of the
Electronic Hamiltonian operator with this trial function. This will provide
us with improved values of the ground state energy (which must be lowered
by this variational calculation) and the ground state energy eigenfunction.
by this variational calculation) and the ground state energy eigenfunction. Note the flexibility of the trial (spatial) wavefunction. Modulo normalization:
c1ψ + (1)ψ + (2) + c2ψ − (1)ψ − (2)
= c1[1s A (1) + 1sB (1)][1s A (2) + 1sB (2)] + c2 [1s A (1) − 1sB (1)][1s A (2) − 1sB (2)]
Note the reductions for specific c1,c2 coefficient choices:
c1 = 1,
1 c2 = −1 ⇒ 1s A (1)1sB (2) + 1sB (1)1s A (2)
c1 = 1, c2 = 1 ⇒ 1s A (1)1s A (2) + 1sB (1)1sB (2)
Covalent bonding
configuration
Ionic bonding
configuration
Thus, the trial function ψ_tr in Eq. [1] confers considerable additional
y
p ,
g
flexibility to the wavefunction shape, hence leading to a more accurate
solution of the Schrodinger Eq.
Performance of various level of electronic structure theory for H2:
[D A McQuarrie Quantum Chemistry]
[D.A. McQuarrie, Quantum Chemistry]
Testing qualitative MO theory prediction of Bond Order with experiment
gq
yp
p
for homonuclear diatomics made from elements in the 1st row of the Periodic
Table (using the “Molecular Orbital Aufbau” principle):
Bond Order ≡ [# bonding e ' s − # antibonding e ' s ] / 2
[D.A. McQuarrie, Quantum Chemistry]
Going to the 2nd row of the Periodic Table … need to build up MO’s from the 2nd Lewis Shell:
from the 2
Lewis Shell:
p_z orbitals can also be “added and subtracted” to form g=gerade and
u=ungerade combinations:
[D.A. McQuarrie, Quantum Chemistry]
p_ and p_y
p_x
p_y orbitals can also be “added and subtracted” to form g=gerade
g g
and
u=ungerade combinations:
Å gerade
ungerade Æ
[D.A. McQuarrie, Quantum Chemistry]
Qualitative MO theory orbital diagram for homonuclear diatomics
composed of 1st or 2nd row elements:
difficult to distinguish the ordering of these MOs:
g
[D.A. McQuarrie, Quantum Chemistry]
Testing qualitative MO theory prediction of Bond Order with experiment
for homonuclear diatomics made from elements in the 2st row of the Periodic
Table:
[D.A. McQuarrie, Quantum Chemistry]
Qualitative MO theory for Heternuclear Diatomics:
Note: Now the AO’s of the two atoms do not line up symmetrically. They
N
N
h AO’ f h
d
li
i ll Th
have to carefully aligned, energetically, and then decisions made as to which
AO’s mix strongly to form MO’s. Example: HF
Note: Bond order =1
[T. Engel, Quantum Chemistry & Spectroscopy]