Discussion  Fr 25.11.2005 at 12:15,  HR 30.1

Home work 4            H₂⁺-ion
 
Exercise 1:  wave function of H₂⁺

The wave function of  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= exp(-r_A)/p^1/2,  j_B= exp(-r_B)/p^1/2 (in atomic units).

1. The wavefunctions j_A and  j_B are two different solutions of the Schroedinger equation, however in general they 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 for the symmetry operation I the equality: 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⁺ molecular ion can be calculated  as S = e^-R(1 + R + 1/3 R²). Analyze limiting cases R=0 and R->infinity.  Draw this function from R=0 till R=10 a.u.
2. Let E_±(R) be the energies of the bonding and anti-bonding electronic state.  What is the physical meaning of the asymptotic values of  E_± for R->infinity?
3. Consider the potential curve of the H₂⁺ ground state. Show in this graph the bond length and the dissociation energy.

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