1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

2. Homework Phys 452 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 AtomNumber of electrons Number of protons (Z) Number of neutrons (isotopes) Atomic mass A Hydrogen H 1 e-Z=1 1 H (0 neutron): 99.985% 2 H (1 neutron): 00.015% A~ 1.01 (in nature) Helium He 2e-Z=2 3 He (1 neutron): 00.0001% 4 H (2 neutrons): 99.9999% A~ 4.0 Lithium Li 3e-Z=3 6 Li (3 neutrons): 7.5% 7 Li (4 neutrons): 92.5% A~ 6.94 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. H atom Hydrogen molecule ion H 2 + Phys 452 H atom electron

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

6. Hydrogen molecule ion H 2 + Phys 452 electron 3-bodies system… Hard to Solve! Assumption: the two nuclei are at rest the distance R is fixed R is going to be our parameter in the variational principle

7. Hydrogen molecule ion H 2 + Phys 452 electron 3-bodies system… Hamiltonian?

8. Hydrogen molecule ion H 2 + Phys 452 electron Trial wave function? LCAO Technique (linear combination of atomic orbitals)

9. electron Hydrogen molecule ion H 2 + Phys 452 Trial wave function? Normalization Overlap integral

10. Hydrogen molecule ion H 2 + Phys 452 Overlap integral Great Overlap Small Overlap The normalization coefficient A will depend on the distance R!

11. Hydrogen molecule ion H 2 + Phys 452 Overlap integral calculation

12. Hydrogen molecule ion H 2 + Phys 452 Trial wave function: Normalization coefficient where

13. Hydrogen molecule ion H 2 + Phys 452 Use the trial wave function: To calculate Hydrogen Hamiltonian with first nucleus Hydrogen Hamiltonian with second nucleus Cross terms

14. Hydrogen molecule ion H 2 + Phys 452 Direct integral Dexchange integral X Pb. 7.8

15. Hydrogen molecule ion H 2 + Phys 452 Finally…! where