Discussion  Fr 22.05.2009 at 12:15,  PK 11.2

Home work 6            H₂⁺-ion
 
Exercise 1:  wave function of H₂⁺

The wave function of  the H₂⁺ molecular ion can be written in the LCAO form: Y = N(j_A ± j_B)_, where j_A and j_B are ground state hydrogen atom wave functions: j_A= ¹/_p1/2 exp(-r_A),  j_B = ¹/_p1/2 exp(-r_B)  (in atomic units).

1. The wavefunctions j_A and  j_B are solutions of the Schroedinger equation for the hydrogen atom, where r measures the distance from the nuclei. However, in general j_A and j_B are not orthogonal to each other. Why?
2. Considering normalization condition for the wave  functions Y. Obtain the normalization factor N which was given at the lecture.
3. Show explicitly which of the wave functions Y is symmetric (g) and which is anti-symmetric (u) under inversion (I) of the electron coordinates in the middle of the internuclear distance R.

          Hint: draw the molecule in the frame centered in the middle of the internuclear distance and prove the equality: I j_A = j_{B .}

       d.   Does this type of symmetry exist for the wavefunctions of the HeH⁺² molecular ion? Why?

Exercise 2:  Integrals, bond length and dissociation energy of the H₂⁺-ion

1. The overlap integral S for the ground state of H₂⁺ can be calculated as  S = e^–R(1 + R+ ¹/₃ R²). Analyze the limiting cases R=0 and R->infinity.  Draw this function S from R=0 till R=10 a.u.
2. Let E_±(R) be the energies of the bound and unbound electronic states. What is the physical meaning of the asymptotic values of  E_± for R->infinity?
3. Draw schematically the potential curve of the H₂⁺ ground state. Show in this graph the bond length and the dissociation energy.

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