Download 5. Quantum mechanics of chemical binding

5.
Quantum mechanics of chemical binding
According to quantum mechanics:
ĤΨ = EΨ
where:
• Ĥ is the Hamiltonian, includes the interactions:
Ĥ = T̂el (r) + V̂e−n (r, R) + V̂e−e (r) + V̂n−n (R) + T̂nucl (R)
|
{z
Ĥe (r,R)
}
|
{z
}
T̂n (R)
Ĥ(r, R) = Ĥe (r, R) + T̂n (R)
where
– r stands for the coordinates of the electrons
– R stands for the coordinates of the nuclei
– see also earlier notations.
The Born-Oppenheimer approximation
the electrons are much lighter than the nuclei ( mMel ≈ 1836)
⇓ equipartition principle
electrons are much faster
⇓
electrons follow the nuclei instantaneously (adiabatic approximation)
⇓
from the point of view of the electrons the nuclei are not moving
⇓
equation for the electronic part: Ĥe (r; R)Φ(r; R) = E(R)Φ(r; R)
equation for nuclei: (T̂n (R) + E(R)) χ(R) = ET OT χ(R)
Notes:
• in Born-Oppenheimer (BO) approximation the equations for nuclei and electrons
have separated;
• nuclei are not still (not motionless);
• the potential the nuclei feel is E(R), the electronic energy calculated at different
bond distances;
• E(R) potential energy surface is a consequence of the Born-Oppenheimer approximation;
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• Usually very good approximation, but fails if the potential energy surfaces cross
each other (photochemical reactions).
From now on, we deal with the electronic part:
Ĥe (r; R)Φ(r; R) = E(R)Φ(r; R)
How can we deal with this? The same technique as for atoms:
• use the Independent Particle Approximation: obtain orbitals (φi ) and orbital energies (εi );
• use the Aufbau principle to occupy the orbitals and get the configurations;
• obtain the states.
5.1.
Simplest example: H2 +
This is a single electron problem, like hydrogen atom, and can be solved analytically
(within the Born-Oppenheimer approximation): we get the wave function of the system
(orbital) and the corresponding energies (orbital energies).
The two lowest energy orbitals:
The energy of these orbitals depend on the H-H distance:
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Why one decreases, the other increases? Look at an other representation of the orbitals
(along the bond)
u1 is a binding orbital, u2 is anti-bonding (meaning it eases the bond).
Prinzipal question: what is chemical bond according to quantum mechanics?
Answer from the above figures:
• according to the energy curves: energy decreases if the two atoms get closer;
• form of the orbitals and the density: there is an increase in electron density between
the two atoms.
Quantum mechanics can not say more, but this is mathematically a perfect explanation. For chemists there are of course other explanations based on approximate models.
How can we approximate the orbitals? Simplest is to start with atoms and approximate
the molecular orbital by linear combination:
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By equations:
u1 = N1 · (1sA + 1sb )
u2 = N2 · (1sA − 1sb )
where 1sA and 1sB are the 1s orbital of the two atoms.
The energy of one of the combinations (plus) increases, that of the other one (minus)
decreases with respect to the energy of the atomic 1s orbital:
The single electron of the H2+ should be put on the orbital with lower energy.
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5.2.
The simplest molecule: H2
The same MO-s can be constructed than in case of H2+ . Now we have to put two electrons
to the orbitals. One with α spin (ms =1/2), one with β spin (ms =-1/2):
5.3.
Other diatomic molecules
We just follow the same route: obtain MO-s from the atomic 1s, 2s, 2p orbitals.
He2 :
He atomic configuration: 1s2
Both binding and anti-binding orbitals are double occupied: no energy gain with
respect to the atoms.
Li2 :
Li atomic configuration: 1s2 2s1
We create MOs also from 2s orbitals:
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One bond, energy gain.
Be2 :
Be atomic configuration : 1s2 2s2
No bond!!
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N2 :
N atomic configuration 1s2 2s2 2p3
New MOs from the 2p orbitals
The orbital diagram:
Triple bond, since three bonding orbitals are double occupied.
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O2
O atomic configuration 1s2 2s2 2p4
Az oxigénmolekula elektronszerkezete
2p
2p
2s
2s
1s
1s
O
O2
O
Double bond, but open shell!
(Remember C atom: we got three states there.)
1
1 −
Here also three states can be constructed: 3 Σ+
g , Σg and ∆g .
Lowest energy is the 3 Σ+
g : triplet, therefore paramagnetic property.
F2
F atomic configuration 1s2 2s2 2p5
Three bonding and two anti-bonding orbitals → single bond.
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