Home work Physical and Theoretical Chemistry IV

- Molecular Spectroscopy -

Discussion  Fr 24.04.2009 at 12:15,  PK 11.2



 

Home work 2. Atomic structure and atomic spectra
 
 

Exercise 1:  Hydrogen atom

a) As known, the angular parts of the Hydrogen atom wave functions are Spherical Harmonics Ylm(θ,φ) which in general have complex values. However, for calculations quantum chemists often use real wave functions which are the linear combinations of the Spherical Harmonics.  Using the known expressions for the Spherical Harmonics derive the expressions for the three p and five d real wave functions given at the lecture.   

b) Using the expression given at the lecture calculate the energy (in cm-1) and the wavelength (in nm) for the n=2 → n=1  and  n=3 → n=2  radiation transitions in Hydrogen atom, where n is the general quantum number. You have calculated the famous α-Lyman and α-Balmer spectral lines. To which spectral range belongs each of these lines?  Assume that these emission lines are exited in a discharge glass tube filled by hydrogen. Can you see these lines?

 

Exercise 2:  Multielectron atoms

a) The quantum numbers L, ML, S, MS and P (parity) of the electronic terms of a multielectron atom can be calculated using the quantum numbers l, ml, s, ms of the corresponding one-electron terms using the quantum mechanical rule for summation the angular momenta and the Pauli principle. Describe the lowest electronic configuration of the Berillium (Be) atom. Calculate the quantum numbers L, S, J, and P of  the lowest electronic term. Is this state degenerated?

b) Exiting one of the 2s-electrons of  Berillium to the upper-lying 2p-orbital calculate the quantum numbers L, S and P of the electronic excited state terms. You will find that there are several excited states available. Which of these excited states is the lowest?  Can you expect radiation transition between each of these excited states and the ground state?

 

Exercise 3: Spin-orbit interaction

a)  Describe the electronic configuration of Sodium atoms (Na) in the ground state.  Calculate all possible values of the quantum numbers L, ML and S, MS in this state.  Taking into consideration the spin-orbit interaction between the electronic spin  S and the orbital angular momentum L, calculate possible values of the total electronic angular momentum J and its projection MJ.  What is the fine structure of the Sodium ground state?

b)  Excite the 3s-electron in a Na atom to the 3p orbital. You have obtained the electron configuration of the Na first excited state. Repeat the task given in a).  Consider the possibility of  radiation transitions between the first excited state of Sodium and the ground state. 

 

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