1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Phys 452 Homework Tuesday Feb14: assignment # 9 Pb 7.6, 7.7, 7.15, 7.16 Thursday Feb 16: assignment # 10 Pb 7.8, 7.9, 7.13, 7.17

3. Phys 452 Review of atomic properties Ionizing energy: to extract or add 1 electron at a time Binding energy: to extract all the electrons from atom (equal in norm to energy of ground state)

4. Phys 452 electron H atom H atom Hydrogen molecule ion H2+

5. Phys 452 Quiz 14 When looking for the ground state of hydrogen molecule ion using the variational principle, what is the most relevant parameter to optimize? • The apparent Z due to screening effect • The reduced mass of the system • The distance between the electron and first nucleus • The distance between the electron and first nucleus • The distance between the two nuclei

6. Phys 452 Hydrogen molecule ion H2+ electron the distance R is fixed R is going to be our parameter in the variational principle 3-bodies system… Hard to Solve! Assumption: the two nuclei are at rest

7. Phys 452 Hydrogen molecule ion H2+ electron 3-bodies system… Hamiltonian?

8. Phys 452 Hydrogen molecule ion H2+ LCAO Technique (linear combination of atomic orbitals) Trial wave function? electron

9. Phys 452 Hydrogen molecule ion H2+ electron Normalization Overlap integral Trial wave function?

10. Phys 452 Small Overlap Great Overlap Hydrogen molecule ion H2+ Overlap integral The normalization coefficient A will depend on the distance R!

11. Phys 452 Hydrogen molecule ion H2+ calculation Overlap integral

12. Phys 452 Normalization coefficient Hydrogen molecule ion H2+ Trial wave function: where

13. Phys 452 Hydrogen molecule ion H2+ Hydrogen Hamiltonian with first nucleus Hydrogen Hamiltonian with second nucleus Cross terms Use the trial wave function: To calculate

14. Hydrogen molecule ion H2+ Phys 452 Direct integral D exchange integral X Pb. 7.8

15. Hydrogen molecule ion H2+ Phys 452 Finally…! where