The following table reports the energy of the triplet and singlet states of CH2, in atomic units, derived from Hartree-Fock calculations with different basis sets.

 

  Basis set STO-3G 6-31G 6-31G(d) 6-31G(d,p)  
  Triplet -38.4342 -38.9112 -38.9208 -38.9249  
  Singlet -38.3363 -38.8353 -38.8493 -38.8538  
a)  According to these results, which is the ground state of CH2? Explain your
  answer.       [2]

 

  1. Explain what is meant by the following terms, and identify one or more entries from the table above with each term:
i) minimal basis set; [2]
ii) split-valence basis set; [2]
iii) polarization function. [2]

 

  1. Evaluate the energy difference (in kJ mol±1) between singlet and triplet states of CH2 with each basis set. Comment on how your values vary with basis set and state, with reasons, which should be the most reliable value.

 

[6]

 

  1. Explain the terms static correlation and dynamic correlation and hence suggest why the Hartree-Fock prediction of the singlet-triplet energy gap in

 

CH2 differs markedly from the experimental value of 38 kJ mol±1. [6]

1

  1. Answer ALL parts a) ± f).

 

VO2 undergoes a metallic to insulator phase transition as a function of temperature. Below 340 K it is insulating. Above 340 K, it has the rutile structure, which is characterised by edge-sharing octahedra. TiO2 can also adopt the rutile structure and is semiconducting.

 

a) Sketch the density of states (DOS) of TiO2 in the rutile structure, and indicate  
  the positions of d-orbitals and Fermi level [2]
b) The metallic state can be attributed to a chain of V atoms. Sketch a possible  
  orbital interaction in a chain of V atoms that explains this behaviour. [2]
c) Draw a band that results from your answer to part b), denoting the spacing  
  between V atoms as r. [4]
d) Draw the DOS that results from this band. Highlight the key difference  
  between this and that for TiO2, and explain the origin of this difference. [4]
e) Comment on the properties of this system as a conductor. [2]

 

  1. Explain why this system would distort, and what would be expected in terms of structure and electronic properties. Use a sketch of the DOS to illustrate

2

  1. Answer ALL parts a) ± c).

 

  1. A general formula for the molecular partition function is given in the Appendix. With reference to this formula, describe and explain the relative magnitudes of the translational, rotational and vibrational partition functions for a diatomic molecule. As part of your answer explain the number of degrees of freedom
for each contribution and explain how they are combined to obtain an overall  
partition function.                 [6]
b)  The translational partition function is given by the formula:  
  § · 3          
  2          
q3     ¨ 2S m ¸   V = V  
    /3    
t ¨ h2 E ¸          
© ¹            
             
By comparing these two formulae obtain an expression for the thermal  
wavelength of a particle, ȁ, and calculate its value for O2 at 300 K. [6]

 

  1. The change in entropy when a system moves from state R to state P at constant temperature can related to the canonical partition function, Q, via:

 

§ Q  ·

S                                       k ln ¨¨                P  ¸¸

©QR ¹

 

  1. qN N!

 

  1. Using the above formulae, calculate the change in entropy when one mole of a monoatomic ideal gas undergoes an isothermal expansion in which the volume is doubled. Note that only the translational partition function need be

 

considered for such a gas. [3]
ii) Calculate the change in entropy when one mole of a monoatomic ideal gas  
is heated at constant pressure from a temperature T to 2T. [5]

3

Appendix 1: General Data Sheet

 

Physical constants and units conversions

  Name   Common Value Units    
      Symbol          
  Speed of light   c = 2.9979 × 108 m s-1    
  Elementary charge e = 1.6022 × 10-19 C    
  Planck constant   h = 6.6261 × 10-34 J s    
     

 

  K

 

1.0546 × 10-34 J s    
  Standard acceleration of free g =   9.8067 m s-2    
  fall              
  Boltzmann constant k = 1.3806 × 10-23 J K-1    
  Avogadro constant NA =   6.0221 × 1023 mol-1    
  Gas constant   R = kNA =   8.3145 J K-1 mol-1  
  Electron mass   me = 9.1094 × 10-31 kg    
  Proton mass   mp = 1.6726 × 10-27 kg    
  Neutron mass   mn = 1.6749 × 10-27 kg    
  atomic mass constant mu = 1.6605 × 10-27 kg    
  Bohr radius   a0 = 5.2918 × 10-11 m    
  Rydberg constant   =   1.0974 × 107 m-1    
  Hartree   Eh =   27.2114 eV    
      Eh = 4.3597 × 10-18 J    
      EhNA =   2625.50 kJ mol-1    
  Electron Volt   eV = 1.6022 × 10-19 J    
  Faraday constant   F =   96485 C mol±1    
  Debye-Hückel   A =   0.509 mol±1/2 kg1/2  
  Limiting Law parameter            
  ( aqueous solutions at 25 °C )            
  Nuclear spin quantum numbers for selected nuclei    
  Nucleus I Nucleus I Nucleus I  
  1H 1/2 15N 1/2 29Si   1/2  
  2H 1 17O 5/2 31P   1/2  
  13C 1/2 19F 1/2        
  14N 1 27Al 5/2        

 

 

 

 

 

 

 

 

 

 

 

 

 

4

Appendix 2

 

The equations set out below may be useful in answering the questions for this paper. In each case the symbols used have their usual meanings.

 

Counting microstates

 

The number of microstates available to a system of N particles which can occupy M states can be calculated using:

 Ǩ

Ǩ  Ǩ ଶǨǥǤெǨ

 

6WLUOLQJ¶V

IRUPXOD lnN!         N ln N N

 

 

Relative populations in mixtures            
  f (i)   g(i) exp
 E
E E
j              
                 
  f ( j)   g( j)   i              
                     
Partition function general definition       f    
                           
                        q   ¦ gi e EHi  
General expression for the molecular partition function:    
i  0    
                           
    E   1       E 1        
with kT  for energy per molecule or RT  for molar energies since R = k NA.  
       

 

Partition functions for molecular degrees of freedom

 

qt3 V    
/3 with  
Translational:  

 

qR2

 

Rotational diatomic molecule:

 

 

1

  • §¨ h2E ·¸2¨© 2S m ¸¹

1

EhcBV ,

 

Rotational constant for a diatomic molecule:   with moment of inertia:  
 
Vibrational: qv 1   ଼గమ ூ      
  1  eE hQ : Note here frequency is in s-1 units.      
                   

Canonical partition function and entropy

The canonical partition function is related to the molecular partition function via:

 

  1. qN N!

The canonical partition function is linked to the entropy via:

S UU(0)  k ln Q
 
  T

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