Physical Chemistry V Exercise V 1 May 2020

1 Molecular oxygen from a group theoretical point of view

In the �rst question of Exercise Set 4 we already looked at the electronic structure of O2. In this

exercise, we will use molecular group theory to obtain the allowed states for the two energetically

lowest-lying electronic con�gurations and their corresponding term symbols. Please note that

you can �nd character tables and direct product tables at http://www.webqc.org or in most

textbooks on spectroscopy.

(a) List all symmetry elements for O2 and determine the corresponding point group.*

(b) Derive the transformation properties of the 2p orbitals of O2. To do so, draw vectors

(x, y, z) on each oxygen atom. Set up matrices which represent the symmetry operations

of the group. Determine the character χ(Ai) of each symmetry operation Ai.��

(c) Through inspection of the character table of the point group, express the reducible rep-

resentation determined in part (b) as a combination of irreducible representations.

(d) With the obtained information construct the orbital diagram for p orbitals (compare with

Exercise 4), �ll in the electrons corresponding to the most stable con�guration and deduce

the term symbols. To obtain the term symbols of the states you have to multiply the

characters of the irreducible representations that represent the partially �lled molecular

orbitals. Then use inspection to reduce this reducible representation into irreducible

representation, which gives you the required term symbols.

(e) Using equation (4.36) from your notes, show why the 3Σ−u ← 3Σ−g transition is electroni-

cally allowed and the 1∆g← 3Σ−g transition is electronically forbidden with light polarized

parallel to the molecular axis (i.e. you only have to look at the parallel component of the

dipole moment).

2 Vibrations of BrF5

Group theory of molecules gives an easy access to the concept of normal modes and a descriptive

picture of molecular vibrations. In this task the normal modes of BrF5 in square pyramidal

geometry will be analysed.

*We use �symmetry element� as used e.g. in Bishop, Group Theory and Chemistry and Bunker, Jensen,Fundamentals of Molecular Symmetry. To quote Bishop: �The symmetry operation is an action, the symmetryelement is a geometrical quantity (a point, a line or a plane) about which an action takes place. A symmetryelement is e.g the C3 rotation axis in NCl3 which has two symmetry operations: clockwise rotation by 2π/3and clockwise rotation by 4π/3. The symmetry operations form the group, but the symmetry elements alreadydetermine the group.

�The character is given by the trace of the matrix, which is the sum of its diagonal elements.�The character for the rotations along the C∞ symmetry element (which you hopefully found) is a function

of the rotation angle ϕ. All members of a class in a character table have the same character for a givenrepresentation. Note that a class is composed of conjugate elements of the group (check what that means). Tofully introduce the confusing terminology, note that the elements of the group are the symmetry operations butthe classes of the group are equivalent to the symmetry elements of the molecule.

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Physical Chemistry V Exercise V 1 May 2020

(a) List the seven symmetry elements of BrF5.

(b) What is the point group of BrF5?

(c) How many nuclear degrees of freedom does BrF5 have in total?

(d) Derive the characters of the reducible representation for all degrees of freedom, Γtot

(e) What are the irreducible representations corresponding to rotation and translation?

(f) Derive the representation of all vibrations Γvib. Using the reduction formula below, derive

the irreducible representation (i.e. the symmetry) of all individual vibrational modes,

al =1

h

∑c

g(c)χ(l)(c)∗χ(c), (1)

where al is the multiplicity of irrep l, c is the class index, h is the group order, χ(l)(c)∗ is

the character of class c in irrep l, χ(c) is the character of the reducible representation of

class c and g(c) is the number of symmetry operations in a class c.

(g) Consider the symmetries of vibrational modes you obtained. Which of them are IR active,

and which are Raman active?

3 Hückel MO treatment of Cyclobutadiene

In this task, we will determine the energy levels of cyclobutadiene using the Hückel MO method.

The HMO model represents a simple semiempirical method to calculate the electronic energy

level structure of molecules that exhibit conjugated π molecular orbitals such as polyenes and

aromatic molecules. The model is useful to gain a semi-quantitative description of the π

molecular orbitals and their relative energies and is widely used in physical-organic chemistry.

Instead of diagonalising the Hückel Hamiltonian He� directly, we will build from the relevant

atomic orbitals a set of symmetry-adapted molecular orbitals. This will allow us to build the

molecular orbital diagram of cyclobutadiene and determine the energy levels associated with

the ground state con�gurations along with their symmetry. We consider the π-system which

spans all carbon atoms.

Figure 1: Structure of cyclobutadiene ([4]annulene).

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Physical Chemistry V Exercise V 1 May 2020

(a) List all symmetry elements of cyclobutadiene in a square-shaped nuclear con�guration.

Which point group does it belong to?

(b) In a Hückel-MO picture one assumes that the valence orbitals are given by the π-bonding

and -antibonding orbitals. Which orbitals contribute to the π-system of the molecule?

(c) These orbitals form a basis {φi} for the Hückel Hamiltonian

He� =

α β 0 β

β α β 0

0 β α β

β 0 β α

Name and explain the approximations that were used to set up this Hamiltonian.

(d) Determine the reducible representation of the π molecular orbitals (Γπ) by applying the

symmetry operations of the associated point group to this basis set. Decompose the re-

ducible representation into irreducible representations by applying the reduction formula

(Eq. (1)).

(e) Now we construct a new, symmetry-adapted orthonormal basis set {ψr} by applying the

projection operator

P (l) =dlh

∑R

χ(l)(R)∗R (2)

to the appropriate basis functions of the original set of basis functions. Here, h is the

group order, dl is the dimension of the irreducible representation l and χ(l)(R) is the

character of operation R for the irreducible representation l.

Hint: For the derivation of the new set of basis functions you can ignore the prefactors

dl/h since you normalize the functions afterwards.

(f) In the previous task you should have obtained the new orbitals ψj in terms of the old

ones φj . For this task, use the following solution:

ψ1 =1

2(φ1 + φ2 + φ3 + φ4)

ψ2 =1

2(φ1 − φ2 + φ3 − φ4)

ψ3 =1√2

(φ1 − φ3)

ψ4 =1√2

(φ2 − φ4).

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Physical Chemistry V Exercise V 1 May 2020

Set up the Hückel Hamiltonian in the new symmetry adapted basis set. To obtain the

elements in the new basis, calculate H ′jk = 〈φj |He�|φk〉. Use the approximation 〈φj |φk〉 =

δjk.�

(g) Calculate the Hückel energies of the orbitals and draw an energy level diagram. For this

purpose, set up the Hückel determinant det(H ′ − εI), where I is a 4× 4 unit matrix, set

it equal to zero, an determine the possible values of ε.

(h) Fill in the electrons and derive the molecular term symbol for the ground state.

(i) Bonus: As one can see, the ground state of the cyclobutadiene molecule is degenerate.

In accordance with the Jahn-Teller theorem, such states are unstable and any non-totally

symmetric distortion that lowers the overall energy of the system (e.g. geometrical dis-

tortion due to vibrational movement of the nuclei) removes the degeneracy. Consider the

rectangular structure of cyclobutadiene. How does decreasing the symmetry a�ect the

basis functions? Compare with the result you obtained for the square conformation of

the molecule. Draw the molecular orbital energy diagram. Derive the molecular term

symbol for

� the most stable electronic con�guration

� the excited con�guration where one electron is transferred to the next higher lying

orbital

Hint: While constructing energy level diagrams, use the fact that orbitals describing

double bonds have a lower energy than orbitals describing single bonds.

�We could include the overlap Sjk = 〈φj |φk〉, which would lead to a generalized eigenvalue equation. Thisis no practical problem for e.g. a computer implementation but it complicates the discussion.

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