20_01fig_PChem.jpg

1. 20_01fig_PChem.jpg Hydrogen Atom Potential Energy Kinetic Energy m r C + M R

2. Hydrogen Atom Radial Angular Coulombic 20_01fig_PChem.jpg

3. 20_01fig_PChem.jpg Hydrogen Atom will be an eigenfunction of Separable

4. 20_01fig_PChem.jpg Hydrogen Atom Recall Bohr Radius

5. 20_01fig_PChem.jpg Hydrogen Atom It is a ground state as it has no nodes Assume Let’s try

6. 20_01fig_PChem.jpg Hydrogen Atom The ground state as it has no nodes n=1, and since l=0 and m = 0, the wavefunction will have no angular dependence

7. 20_01fig_PChem.jpg Hydrogen Atom In general: 1S- 0 nodes Laguerre Polynomials 2S- 1 node 3S-2 nodes

8. Energies of the Hydrogen Atom In general: kJ/mol Hartrees

9. Wave functions of the Hydrogen Atom In general: Z=1, n = 1, l = 0, and m = 0:

10. Wave functions of the Hydrogen Atom Z=1, n = 2, l = 0, and m = 0:

11. Hydrogen Atom Z=1, n = 2, l = 1 m = 0: m = +1/-1: - + + - + _ + - - - + - + +

12. 20_06fig_PChem.jpg Radial Distribution Functions For radial distribution functions we integrate over all angles only Prob. density as a function of r.

13. Radial Distribution Functions 20_09fig_PChem.jpg

14. Probability Distributions Z Y X 20_08fig_PChem.jpg

15. Atomic Units Set: Hartrees a.u. Much simpler forms. 20_12fig_PChem.jpg

16. Atoms Potential Energy me Kinetic Energy C M =r12 me

17. Helium Atom Hydrogen like 1 e’ Hamiltonian me C M =r12 Cannot be separated!!! i.e. r12 cannot be expressed as a function of just r1 or just r2 What kind of approximations can be made? me

18. Ground State Energy of Helium Atom Ionization Energy of He I2 = 54.416 ev I1 = 24.587 ev EFree E2 E2 Eo=- 24.587 - 54.416 ev =- 79.003 ev E1 E1 =- 2.9033 Hartrees Eo Eo Perturbation Theory

19. Ground State Energy of Helium Atom Not even close. Off by 1.1 H, or 3000 kJ/mol H Therefore e’-e’ correlation, Vee, is very significant

20. Ground State Energy of Helium Atom

21. Ground State Energy of Helium Atom Closer but still far off!!! Perturbation is too large for PT to be accurate, much higher corrections would be required

22. Variational Method The wavefunction can be optimized to the system to make it more suitable Consider a trail wavefunction and Is the true wavefunction, where: is a complete set Then The exact energy is a lower bound Assume the trial function can be expressed in terms of the exact functions We need to show that

23. Variational Method Variational Energy Evar(a) Since min E0

24. Variational Method For He Atom Let’s optimize the value of Z, since the presence of a second electrons shields the nucleus, effectively lowering its charge.

25. Variational Method For He Atom

26. Variational Method For He Atom

27. Variational Method For He Atom

28. Variational Method For He Atom Similarly Recall from PT

29. Variational Method For He Atom Much closer to -2.9033 H (D E= 0.055 H =144.4 kJ/mol error)

30. Variational Method For He Atom Optimized wavefunction

31. Variational Method For He Atom Optimized wavefunction Other Trail Functions Optimizes both nuclear charges simultaneously (D E= 0.027 H =71.1 kJ/mol error)

32. Variational Method For He Atom Other Trail Functions Z’, b are optimized. Accounts for dependence on r12. (D E= 0.011 H =29.7 kJ/mol error) In M.O. calculations the wavefunction used are designed to give the most accurate energies for the least computational effort required. The more accurate the energy the more parameters that must be optimized the more demanding the calculation.

33. Variational Method For He Atom Experimental -79.003 ev -2.9003 H -2.862879 H -2.862871 H -2.84885 H In M.O. calculations the wavefunction used are designed to give the most accurate energies for the least computational effort required. The more accurate the energy the more parameters that must be optimized the more demanding the calculation.