LCAO, MO diagrams, and BDE Flashcards
What is a wave function? How can it be used?
An equation, called a molecular orbital, which describes an electron
The square of the wave function gives the probability distribution of an electron at a point in space
What does it mean to be polycentric? How is energy associated to an orbital?
Polycentric: electron density is centred around more than 1 nucleus
Each orbital has a specific energy, which is defined and quantised
What are the two main approximations used in the linear combination of atomic orbitals?
Born-Oppenheimer approximation: the nuclear and electron positions can be treated independently
Nuclei can be assumed to be fixed but electrons can move (lighter)
Orbital Approximation: Wavefunction of N electrons can be written as the product of N one electron wavefunctions
i.e ψ (1,2,3,4,…)=ψ(1) x ψ (2) x ψ (3) …
What is the equation for calculating LCAO?
ψ = ∑ᵢ cᵢϕᵢ
where c is a weighting factor of the contribution of the wavefunction for the bonding/antibonding orbital
How can the equation for LCAO be applied to H₂ + ?
Let one of the orbitals be 1a and the other 1b
As below, weighting factor x wave function, and squared for born interpretation
(Xa1a + Xb1b)²
= (Xa1a)² + (Xb1b)² + 2XaXb1a1b
Close to A, resemble the first term, close to B the second, and the internuclear region is the final term
How does LCAO give rise to bonding and antibonding MOs, using H₂+ as an example?
The contribution to the wavefunction must be equal as we are using two H wave functions
Using the born interpretation:
(Ca)² = (Cb)²
Ca= Cb or Ca=-Cb
Same sign=bonding, Opposite, out of phase= anti-bonding
What does the electron density along the bonding H₂+ axis look like relative to H alone?
H alone: high electron density at the nucleus alone
H₂+: Electron density at the nuclei reduced but increased in the internuclear region, stabilising
What does the electron density along the antibonding H₂+ axis look like relative to H alone?
H alone: high electron density at the nucleus alone
H₂+: Electron density at the nuclei increased and steep decline in internuclear region, falling to 0 as there exists a nodal plane
What are the balance of forces which determine the bond length in a molecule?
Coulombic nuclear repulsion decreasing with increasing radius
Nuclear-electronic attraction decreasing with increasing radius
There is a minimum of the sum of these forces at the bond length, where the energy is minimised
There exists no minimum for the antibonding orbitals as increasing radius will stabilise both factors
where B= repulsion A= attraction C= sum
What is the variation principle? How can it be applied to H₂+ ?
The variation principle states that an estimated wavefunction will always be at the same or higher energy than the true wavefunction
This means lower energy estimates are more reflective of the actual wavefunction
Contracted orbitals give lower energies so a closer representation of the wavefunction
How do you name the MOs based on the orbitals which form?
S orbitals will always form σ bonds
P orbitals can form either σ bonds or π bonds
D orbitals can form σ, π or δ bonds
If a molecule consists of the same species i.e O2, then the MOs can be classified with u or g to distinguish bonding / antibonding
σ g= bonding σ u = antibonding
π u = bonding π g= antibonding
From g meaning the same on inversion
How does the electron density change from the H atoms alone to the bonding MO of H₂ ?
Decreased electron density at the nuclei and increased in the internuclear region
Why is the bond strength of H₂ not twice that of H₂ + ?
Whilst there is twice the amount of stabilising binding energy with the additional electron - nuclear attraction, there is also additional inter electron repulsion weakening the bond
The net effect is strengthening of the bond compared to with 1 electron, but not twice as strong
What does the electron density along the nuclear axis of an antibonding molecule look like for H₂ ?
Higher electron density at the nucleus
Falls to 0 at the centre of the axis, and changes sign
Opposite phase peak at the other nucleus
How do you calculate bond order?
Bond Order= 1/2 ( n of bonding electrons - n of antibonding electrons)
Compare the bond strength for H₂ and He₂ and ions? Why is this the case?
H₂ > H₂ + > He₂ + > He₂
H₂ : 1σg 2 1
H₂ + : 1σg 1 1/2
He₂ +: 1σg 2 1σu 1 1/2
He ₂ : 1σg 2 1σu 2 0
Bond strength lies with the increasing bond order, as there is greater nuclear-electron attraction, stabilising the bond and increasing strength
When comparing the ions, they have the same bond order, but there is an electron in the antibonding for He but not in H
As antibonding orbitals are more destabilising than bonding orbitals are stabilising, He would be at a higher energy, and so weaker bond
What does the MO diagram for H₂ look like?
Larger gap between atoms and antibonding than atoms and bonding
What does the MO diagrams for Li₂ look like? How does the bond strength of Li₂ compare to H₂ ?
Still the same diagram with1σg 2 configuration
This is because the 1s orbitals are too contracted to be involved with bonding to a large extent, as there is a greater nuclear charge
However, Li₂ is a weaker bond, lower bond dissociation energy, and so the distance from bonding/antibonding is smaller
This is because electrons are removed from a larger atom, possessing larger bond lengths
There is a reduced nuclear attraction between the bonding electrons and nuclei, so a weaker bond
How can the p orbitals overlap in bonding? How do the energies of these compare?
The Pz orbitals can overlap head on, forming a σ bond
The Px and Py can only overlap sideways, forming π bonds
These form bonding and antibonding MOs
σg< πu < πg < σu
What characteristics favours MO formation?
Energies of the atomic orbitals are similar
Orbitals should overlap well
Atomic orbitals should have the same same symmetry relative to the molecular axis
What does the MO diagram for O₂ look like?
How do the bond disassociation energy for O₂ and its first ions compare? Why?
O₂ : 2σg2 2πu4 2πg2 2
O₂ + : 2σg2 2πu4 2πg1 5/2
O₂ 2+ : 2σg2 2πu4 3
O₂ - : 2σg2 2πu4 2πg3 3/2
As before, increasing bond order reflects a greater nuclear-electron attraction and so stronger bond, larger bond dissociation energy
This arises from removing antibonding electrons which would otherwise destabilise the bond
Adding electrons increases the number of antibonding electrons, destabilising the bond
If further electrons were to be removed, they would be bonding in nature, thus reducing the nuclear-electron attraction, shown in the decrease in bond order, so weaker bond
How do bond dissociation enthalpies change across a period for the same bond order?
Same bond order, then the BDE increases
Increased Zeff and so electronegativity across the period, resulting in greater electron-nuclear attraction, and smaller bond lengths, so stronger bonds
What is sp mixing and its implications?
For G3-G5, the s and the p orbitals are close enough in energy that they mix, effectively forming sp hybrid orbitals
This occurs with the p z orbital only as they have the same molecular axis symmetry
This results in the s σ bonding and antibonding decreasing in energy, whilst the p σ increasing in energy
As a result, the p σ (2σg) is higher in energy than the p π (1πg)
So π bonds filled before σ
Why does sp mixing only occur before g6?
The energy gap between the 2s and 2p orbital increases across period
This results in the difference in energy being too large to enable effective orbital mixing, preventing sp mixing
Across the period, the Zeff increases, resulting in greater nuclear attraction experienced by the electrons
As the 2s is more effective at penetrating the core region than 2p, it will experience a greater more stabilisation, and thus the energy difference between the two increases
What does the MO diagram for N2 look like? How do the diagrams and BDE differ down G5 and why?
When drawing the MO, for both the sp mixed bonding and antibonding, show sp hybrids, but opposite sides overlapping
Same MO style down G5, but relative, energies would be higher
The size of the atoms increases, meaning the bonding electrons are further away from the nuclei
This reduces the electrostatic nuclear-electron attraction, decreasing bond strength / BDE and increased bond length
What happens to BDE down the halogens and why?
Fluorine<Chlorine>Bromine
Cl>Br explained as before, larger atom size</Chlorine>
F<Cl is anomalous
This is because the antibonding 1πg is filled
F is a very small atom, arising from the large Zeff
Therefore, the inter-electron repulsion is greater than for the other halogens, weakening the bond
What are the general steps for drawing an MO diagram for 2 different species forming a bond?
Consider the relative energies of the AOs and put on the diagram
See if sp mixing occurs, if so the π will go below σ
Generally the S will still mix, as will pz as a σ and the other ps as a π
Fill in electrons
Special cases:
NF: sp mixing does not occur
Bonding with H: will result in non-bonding with the p x/y and likely s
What does the MO diagram for HF look like?
1s of H higher is energy than 2s of F due to Zeff
What does the MO diagram for CO look like?
sp mixing still occurs
What does the MO diagram for NF look like
sp mixing does not occur enough to enable the σ to switch with the π
How does the difference in energy between AOs affect bonding?
The closer in energy between the orbitals, the smaller the HOMO-LUMO gap in general, the larger the stabilisation of bond formation
What does the MO diagram for HB look like?
Sp mixing occurs:
The lone sp orbital is effectively non bonding, as as the px and py
This is because there is no similar energy p in H to enable bonding, so non-bonding
The antibonding is higher in energy than the non-bonding
How do MO diagrams explain the existence of polar bonds?
There is an unequal contribution to the bonding MO when considering heteroatomic bonds
The species with a lower energy AO, and so greater electronegativity, has a greater orbital coefficient for the bonding MO
Therefore, the electron density has a greater probability to be found closer to this more electronegative species, resulting in a polar
What does it mean to be paramagnetic and diamagnetic? What gives molecules these characteristics?
Paramagnetic: attracted into a magnetic field
Diamagnetic: (slightly) repelled by a magnetic field
Paramagnetism is caused by a species containing unpaired electrons
Why does the BDE of N2 and CO differ despite being isoelectronic?
The isoelectronic species contain 5 electrons pairs
There are 3 total electron pairs in C/O compared to 2 for N/N
There is a greater reduction in inter-electron repulsion forming N/N stabilising this species, and acting as a driving force to break the bond, resulting in the N2 bond being weaker than CO
Alternatively, N/N contain more electrons with parallel spins, so contains greater exchange energy
When forming the molecules, this exchange energy must be overcome, thus reducing the net energy released from bond formation as offset, so a weaker bond
What does the MO diagram for the terminal B-H-B in borane look like? How would you describe the bond?
3-centre, 2-electron bond
What does the MO diagram for the HF2- look like? How would you describe the bond?
3-centre, 2 electron bond
Non-bonding electrons do not contribute to the bond type
What does the MO diagram XeF2 look like, assuming no d interaction? How would you describe the bond?
3 centre, 2 electron bond
Non-bonding electrons do not contribute to the bond type
How can the dz2 orbital interact in XeF2? Why is this a weak interaction?
There is some overlap between the dz2 and non-bonding orbitals, resulting in weak stabilisation
This is weak because the 5dz2 is far higher in energy than the 2p of the fluorines
The large energy gaps results in very poor orbital overlaps, so weak stabilisation
Why does the 1s bond with the 2px in HF? Why does nothing else occur?
1s is similar in energy to the 2px
The 2s of F is too low in energy for any significant overlap
And as the 2p of H is far too high in energy for overlap, they do not form pi bonds, and instead non-bonding
How would you approach an MO diagram for elements with the same period? And different periods?
Same period: there will be normal orbital overlap s-s, p-p, potential sp mixing
Different period: i.e with H, overlap the orbitals with the same symmetry and similar in energy
It could mean non-bonding, if no orbitals are of similar energy, or different bondings i.e s to p, with or without sp mixing
How does sp mixing vary down a group?
The effect of sp mixing decreases
This is because the energy gap between the np and ns increases with n
This is because as the s orbital is more penetrating, the electrons can penetrate the core region and experience the full, greater nuclear charge more than p
As nuclear charge increases down the group, the s is stabilised more than p, resulting in larger s-p gaps, reducing the impact of sp mixing
How is exchange energy calculated?
For each subshell
= k (n spin up ( n spin up -1))/2 + same for spin down
Why is the BDE for SiS, P2, and AlCl different despite having the same electron configuration?
First consider exchange energies
The exchange energy of P is greater than for Si/S=Al/Cl
Therefore, more energy is required to overcome this exchange energy in the process of pairing, resulting in a weaker bond, and so a driving force for the bond breaking, so SiS stronger
The orbital overlap between Si and S is stronger Al/Cl as there is a smaller difference in orbital energy
This arises from Zeff increasing across the period
